From 5be99242cd3fc5523c81a844400ee3ad5c5a9a7e Mon Sep 17 00:00:00 2001 From: Jose Hales-Garcia Date: Fri, 9 Feb 2024 20:53:34 -0800 Subject: [PATCH] Enhancements to ratio approximation Some of the approximation features were in nascent form in previous releases. In this release, we want to bring them closer to completion. Enhancements include: * Add a Tonal::Ratio::Approximation::Set class to maintain the approximations and on which methods concerned with them can be located. * Add a #by_superparticular strategy for finding approximate ratios. * Rename the #approx method to #approximate, which reads better. Additional changes include: * Enhance the #superparticular method to allow assigning the superior part to the numerator (default) or denominator. This allows approaching target ratios from below as well as above. Also, add a factor argument, allowing finding superparticular ratios approaching any desired target ratio. * Change the method name #normalize to #denominize. Normalize suggests extra undesired meaning. Denominize is a made up term, but its meaning is closer to what the method is actually doing. * Extract the precision argument, located in method signatures, into a constant named PRECISION. * Improve the readability of Tonal::Ratio#efficiency, by limiting its precision to 2 decimal places. * Make available to_cents from instances of Tonal::Interval. * Make ratio an alias to Tonal::Interval#interval. * Add the ability to reduce ratios derived from prime divisions. * Move the gem's version from the gemspec file, into a separate file, that will be changing more often. --- docs/Array.html | 182 +++---- docs/Integer.html | 2 +- docs/Math.html | 6 +- docs/Numeric.html | 2 +- docs/Prime.html | 2 +- docs/Rational.html | 2 +- docs/Tonal.html | 27 +- docs/Tonal/Cents.html | 75 ++- docs/Tonal/Comma.html | 2 +- docs/Tonal/Hertz.html | 2 +- docs/Tonal/Interval.html | 78 +-- docs/Tonal/Log.html | 2 +- docs/Tonal/Log2.html | 2 +- docs/Tonal/Ratio.html | 467 +++++++++-------- ...Approximations.html => Approximation.html} | 474 ++++++++++++++---- docs/Tonal/Ratio/Approximation/Set.html | 423 ++++++++++++++++ docs/Tonal/ReducedRatio.html | 4 +- docs/Tonal/Step.html | 2 +- docs/Vector.html | 6 +- docs/_config.yml | 1 - docs/_index.html | 11 +- docs/class_list.html | 2 +- docs/file.README.html | 2 +- docs/index.html | 2 +- docs/method_list.html | 398 ++++++++------- docs/top-level-namespace.html | 2 +- lib/tonal/approximation.rb | 189 +++++++ lib/tonal/approximations.rb | 148 ------ lib/tonal/attributions.rb | 4 + lib/tonal/cents.rb | 3 +- lib/tonal/extensions.rb | 9 +- lib/tonal/interval.rb | 5 +- lib/tonal/ratio.rb | 58 +-- lib/tonal/tools.rb | 2 +- ...imations_spec.rb => approximation_spec.rb} | 43 +- spec/tonal_tools/extensions_spec.rb | 6 +- spec/tonal_tools/interval_spec.rb | 16 +- spec/tonal_tools/ratio_spec.rb | 10 +- tonal-tools.gemspec | 3 +- 39 files changed, 1746 insertions(+), 928 deletions(-) rename docs/Tonal/Ratio/{Approximations.html => Approximation.html} (65%) create mode 100644 docs/Tonal/Ratio/Approximation/Set.html delete mode 100644 docs/_config.yml create mode 100644 lib/tonal/approximation.rb delete mode 100644 lib/tonal/approximations.rb create mode 100644 lib/tonal/attributions.rb rename spec/tonal_tools/{approximations_spec.rb => approximation_spec.rb} (59%) diff --git a/docs/Array.html b/docs/Array.html index 141e625..3a2fc92 100644 --- a/docs/Array.html +++ b/docs/Array.html @@ -143,7 +143,7 @@

  • - #lcm ⇒ Integer + #denominize ⇒ Array @@ -158,7 +158,7 @@

    -

    Least common multiple of integer elements of self.

    +

    An array of ratios with equalized denominators.

    • @@ -167,7 +167,7 @@

  • - #mean ⇒ Float + #lcm ⇒ Integer @@ -182,7 +182,7 @@

    -

    The mean of the elements of self.

    +

    Least common multiple of integer elements of self.

    • @@ -191,7 +191,7 @@

  • - #modulo_translate(lower = 0, upper) ⇒ Array + #mean ⇒ Float @@ -206,7 +206,7 @@

    -

    Translated modularly.

    +

    The mean of the elements of self.

    • @@ -215,7 +215,7 @@

  • - #normalize ⇒ Array + #modulo_translate(lower = 0, upper) ⇒ Array @@ -230,7 +230,7 @@

    -

    An array of normalized ratios.

    +

    Translated modularly.

    • @@ -265,7 +265,7 @@

  • - #ratio_from_prime_divisions ⇒ Tonal::ReducedRatio + #ratio_from_prime_divisions(reduced: false) ⇒ Tonal::ReducedRatio @@ -508,6 +508,75 @@

    + + +
    +

    + + #denominize ⇒ Array + + + + + +

    +
    + +

    Returns an array of ratios with equalized denominators.

    + + +
    +
    +
    + +
    +

    Examples:

    + + + 
    [4/3r, 3/2r].denominize => [(8/6), (9/6)]
    + +
    + +

    Returns:

    +
      + +
    • + + + (Array) + + + + — +
      +

      an array of ratios with equalized denominators

      +
      + +
      • + +
    + +
    + + + + +
    + 
    +
+
+331
+332
+333
+334
    +     		+ 
    # File 'lib/tonal/extensions.rb', line 331
+
+def denominize
+  l = denominators.lcm
+  map{|r| Tonal::Ratio.new(l / r.denominator * r.numerator, l)}
+end
    +
    @@ -727,15 +796,15 @@ 

     
 
-386
 387
 388
 389
 390
-391
    +391 +392 - 
    # File 'lib/tonal/extensions.rb', line 386
+      
    # File 'lib/tonal/extensions.rb', line 387
 
 def modulo_translate(lower=0, upper)
   range = (upper - lower) == 0 ? 1 : upper - lower
@@ -746,75 +815,6 @@ 
    
     
   
 
-

    - -
    -

    - - #normalize ⇒ Array - - - - - -

    -
    - -

    Returns an array of normalized ratios.

    - - -
    -
    -
    - -
    -

    Examples:

    - - - 
    [4/3r, 3/2r].normalize => [(8/6), (9/6)]
    - -
    - -

    Returns:

    -
      - -
    • - - - (Array) - - - - — -
      -

      an array of normalized ratios

      -
      - -
      • - -
    - -
    - - - - -
    - 
    -
-
-331
-332
-333
-334
    -     		- 
    # File 'lib/tonal/extensions.rb', line 331
-
-def normalize
-  l = denominators.lcm
-  map{|r| Tonal::Ratio.new(l / r.denominator * r.numerator, l)}
-end
    -
    @@ -887,7 +887,7 @@

    - #ratio_from_prime_divisions ⇒ Tonal::ReducedRatio + #ratio_from_prime_divisions(reduced: false) ⇒ Tonal::ReducedRatio @@ -936,12 +936,12 @@ 

     
 
-353
    +354 - 
    # File 'lib/tonal/extensions.rb', line 353
+      
    # File 'lib/tonal/extensions.rb', line 354
 
-def ratio_from_prime_divisions = Tonal::Ratio.new(Prime.int_from_prime_division(self.first), Prime.int_from_prime_division(self.last))
    
+def ratio_from_prime_divisions(reduced: false) = reduced ? Tonal::ReducedRatio.new(Prime.int_from_prime_division(self.first), Prime.int_from_prime_division(self.last)) : Tonal::Ratio.new(Prime.int_from_prime_division(self.first), Prime.int_from_prime_division(self.last))
    @@ -1028,17 +1028,17 @@ 

     
 
-370
 371
 372
 373
 374
 375
 376
-377
    +377 +378 - 
    # File 'lib/tonal/extensions.rb', line 370
+      
    # File 'lib/tonal/extensions.rb', line 371
 
 def rescale(new_min=0, new_max)
   old_min = min
@@ -1447,10 +1447,10 @@ 
    
       
     
 
-360
    
+361
    
     
     
-      
    # File 'lib/tonal/extensions.rb', line 360
+      
    # File 'lib/tonal/extensions.rb', line 361
 
 def translate(value) = self.map{|e| e + value}
    
     
@@ -1463,7 +1463,7 @@

    diff --git a/docs/Integer.html b/docs/Integer.html index 53047af..d738776 100644 --- a/docs/Integer.html +++ b/docs/Integer.html @@ -808,7 +808,7 @@

    diff --git a/docs/Math.html b/docs/Math.html index a714c27..3177834 100644 --- a/docs/Math.html +++ b/docs/Math.html @@ -214,10 +214,10 @@ 

     
 
-411
    +412 - 
    # File 'lib/tonal/extensions.rb', line 411
+      
    # File 'lib/tonal/extensions.rb', line 412
 
 def self.factorial(limit) = (2..limit).reduce(1, :*)
    
     
@@ -230,7 +230,7 @@ 
    
 
 
       
diff --git a/docs/Numeric.html b/docs/Numeric.html
index 6d2ab4f..6550fd7 100644
--- a/docs/Numeric.html
+++ b/docs/Numeric.html
@@ -2468,7 +2468,7 @@ 
    
 
 
       
diff --git a/docs/Prime.html b/docs/Prime.html
index fe2e8d5..e23ccf8 100644
--- a/docs/Prime.html
+++ b/docs/Prime.html
@@ -253,7 +253,7 @@ 
    
 
 
       
diff --git a/docs/Rational.html b/docs/Rational.html
index 0bd9860..8680f51 100644
--- a/docs/Rational.html
+++ b/docs/Rational.html
@@ -482,7 +482,7 @@ 
    
 
 
       
diff --git a/docs/Tonal.html b/docs/Tonal.html
index d88e9c8..1cf38c6 100644
--- a/docs/Tonal.html
+++ b/docs/Tonal.html
@@ -78,7 +78,9 @@
   
   
    
     
    Defined in:
    
-    
    lib/tonal/tools.rb
    
+    
    lib/tonal/tools.rb,
    
+  lib/tonal/attributions.rb    
+
    
   
    
   
 
@@ -94,6 +96,27 @@ 
    Defined Under Namespace
    
   
 
    
 
+  
+    
    
+      Constant Summary
+      collapse
+    
    
+
+    
    
+      
+        
    TOOLS_PRODUCER =
+          
+        
    
+        
    "mTonal"
    
+      
+        
    TOOLS_VERSION =
+          
+        
    
+        
    "2.0.1"
    
+      
+    
    
+  
+
 
 
 
@@ -105,7 +128,7 @@ 
    Defined Under Namespace
    
 
 
       
diff --git a/docs/Tonal/Cents.html b/docs/Tonal/Cents.html
index 9256abd..034d8cc 100644
--- a/docs/Tonal/Cents.html
+++ b/docs/Tonal/Cents.html
@@ -143,11 +143,6 @@ 
    
         
         
    2
    
       
-        
    DEFAULT_ROUNDING_PRECISION =
-          
-        
    
-        
    2
    
-      
     
   
 
@@ -546,6 +541,7 @@ 
    
       
     
 
+23
 24
 25
 26
@@ -563,11 +559,10 @@ 
    
 38
 39
 40
-41
-42
    
+41
    - 
    # File 'lib/tonal/cents.rb', line 24
+      
    # File 'lib/tonal/cents.rb', line 23
 
 def initialize(cents: nil, log: nil, ratio: nil, precision: PRECISION)
   raise ArgumentError, "One of cents:, log: or ratio: must be provided" unless [cents, log, ratio].compact.count == 1
@@ -627,6 +622,7 @@ 
    
       
     
 
+129
 130
 131
 132
@@ -638,11 +634,10 @@ 
    
 138
 139
 140
-141
-142
    
+141
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 130
+      
    # File 'lib/tonal/cents.rb', line 129
 
 def method_missing(op, *args, &blk)
   rhs = args.collect do |arg|
@@ -695,12 +690,12 @@ 
    
       
     
 
+13
 14
-15
-16
    
+15
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 14
+      
    # File 'lib/tonal/cents.rb', line 13
 
 def log
   @log
@@ -738,12 +733,12 @@ 
    
       
     
 
+13
 14
-15
-16
    
+15
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 14
+      
    # File 'lib/tonal/cents.rb', line 13
 
 def ratio
   @ratio
@@ -812,12 +807,12 @@ 
    
       
     
 
+47
 48
-49
-50
    
+49
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 48
+      
    # File 'lib/tonal/cents.rb', line 47
 
 def self.default_tolerance
   self.new(cents: TOLERANCE)
@@ -859,15 +854,15 @@ 
    
       
     
 
+101
 102
-103
-104
    
+103
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 102
+      
    # File 'lib/tonal/cents.rb', line 101
 
 def <=>(rhs)
-  rhs.kind_of?(self.class) ? value.round(2) <=> rhs.value.round(2) : value.round(2) <=> rhs.round(2)
+  rhs.kind_of?(self.class) ? value.round(PRECISION) <=> rhs.value.round(PRECISION) : value.round(PRECISION) <=> rhs.round(PRECISION)
 end
    
     
   
@@ -928,12 +923,12 @@ 
    
       
     
 
+92
 93
-94
-95
    
+94
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 93
+      
    # File 'lib/tonal/cents.rb', line 92
 
 def inspect
   "#{value.round(@precision)}"
@@ -997,12 +992,12 @@ 
    
       
     
 
+65
 66
-67
-68
    
+67
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 66
+      
    # File 'lib/tonal/cents.rb', line 65
 
 def nearest_hundredth
   self.class.new(cents: value.round(Tonal::Cents::HUNDREDTHS_ROUNDOFF).to_f)
@@ -1066,12 +1061,12 @@ 
    
       
     
 
+74
 75
-76
-77
    
+76
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 75
+      
    # File 'lib/tonal/cents.rb', line 74
 
 def nearest_hundredth_difference
   self.class.new(cents: (value - nearest_hundredth))
@@ -1134,12 +1129,12 @@ 
    
       
     
 
+83
 84
-85
-86
    
+85
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 84
+      
    # File 'lib/tonal/cents.rb', line 83
 
 def plus_minus(limit = 5)
   [self - limit, self + limit]
@@ -1205,12 +1200,12 @@ 
    
       
     
 
+55
 56
-57
-58
    
+57
    
     
     
-      
    # File 'lib/tonal/cents.rb', line 56
+      
    # File 'lib/tonal/cents.rb', line 55
 
 def value(precision: @precision)
   @value.round(precision)
@@ -1225,7 +1220,7 @@ 
    
 
 
       
diff --git a/docs/Tonal/Comma.html b/docs/Tonal/Comma.html
index 3e64694..059d54a 100644
--- a/docs/Tonal/Comma.html
+++ b/docs/Tonal/Comma.html
@@ -504,7 +504,7 @@ 
    
 
 
       
diff --git a/docs/Tonal/Hertz.html b/docs/Tonal/Hertz.html
index fc0856a..b99da78 100644
--- a/docs/Tonal/Hertz.html
+++ b/docs/Tonal/Hertz.html
@@ -822,7 +822,7 @@ 
    
 
 
       
diff --git a/docs/Tonal/Interval.html b/docs/Tonal/Interval.html
index 039e896..fd1f17f 100644
--- a/docs/Tonal/Interval.html
+++ b/docs/Tonal/Interval.html
@@ -139,6 +139,8 @@ 
    Instance Attribute Summary collaps
     
 
     
+      (also: #ratio)
+    
   
   
   
@@ -261,15 +263,13 @@ 
    
         
  • 
   
     
-      #initialize(upper_ratio, lower_ratio)  ⇒ Interval 
+      #denominize  ⇒ Object 
     
 
     
   
   
   
-    constructor
-  
   
   
   
@@ -277,9 +277,7 @@ 
    
   
 
   
-    
    
-
    A new instance of Interval.
    
-
    
+    
    
   
 
    • 
 
@@ -287,13 +285,15 @@ 
    
         
  • 
   
     
-      #inspect  ⇒ Object 
+      #initialize(upper_ratio, lower_ratio)  ⇒ Interval 
     
 
     
   
   
   
+    constructor
+  
   
   
   
@@ -301,7 +301,9 @@ 
    
   
 
   
-    
    
+    
    
+
    A new instance of Interval.
    
+
    
   
 
    • 
 
@@ -309,7 +311,7 @@ 
    
         
  • 
   
     
-      #normalize  ⇒ Object 
+      #inspect  ⇒ Object 
     
 
     
@@ -417,6 +419,10 @@ 
    
   
 
   
+    Also known as:
+    ratio
+    
+  
 
   
 
    
@@ -567,12 +573,12 @@ 
    
       
     
 
-35
 36
-37
    
+37
+38
    • 
     
     
-      
    # File 'lib/tonal/interval.rb', line 35
+      
    # File 'lib/tonal/interval.rb', line 36
 
 def <=>(rhs)
   interval.to_r <=> rhs.interval.to_r
@@ -583,9 +589,9 @@ 
    
 
     
       
    
-  
    
+  
    
   
-    #inspect  ⇒ Object 
+    #denominize  ⇒ Object 
   
 
   
@@ -597,15 +603,19 @@ 
    
       
     
 
-31
-32
-33
    
+26
+27
+28
+29
+30
    
     
     
-      
    # File 'lib/tonal/interval.rb', line 31
+      
    # File 'lib/tonal/interval.rb', line 26
 
-def inspect
-  "#{self.to_r} (#{upper.to_r} / #{lower.to_r})"
+def denominize
+  ratios = to_a
+  lcm = ratios.denominators.lcm
+  ratios.map{|r| Tonal::Ratio.new(lcm / r.denominator * r.numerator, lcm)}
 end
    
     
   
@@ -613,9 +623,9 @@ 
    
 
     
       
    
-  
    
+  
    
   
-    #normalize  ⇒ Object 
+    #inspect  ⇒ Object 
   
 
   
@@ -627,19 +637,15 @@ 
    
       
     
 
-25
-26
-27
-28
-29
    
+32
+33
+34
    
     
     
-      
    # File 'lib/tonal/interval.rb', line 25
+      
    # File 'lib/tonal/interval.rb', line 32
 
-def normalize
-  ratios = to_a
-  lcm = ratios.denominators.lcm
-  ratios.map{|r| Tonal::Ratio.new(lcm / r.denominator * r.numerator, lcm)}
+def inspect
+  "#{self.to_r} (#{upper.to_r} / #{lower.to_r})"
 end
    
     
   
@@ -661,12 +667,12 @@ 
    
       
     
 
-21
 22
-23
    
+23
+24
    
     
     
-      
    # File 'lib/tonal/interval.rb', line 21
+      
    # File 'lib/tonal/interval.rb', line 22
 
 def to_a
   [lower_ratio, upper_ratio]
@@ -681,7 +687,7 @@ 
    
 
 
       
diff --git a/docs/Tonal/Log.html b/docs/Tonal/Log.html
index d874366..adced86 100644
--- a/docs/Tonal/Log.html
+++ b/docs/Tonal/Log.html
@@ -1016,7 +1016,7 @@ 
    
 
 
       
diff --git a/docs/Tonal/Log2.html b/docs/Tonal/Log2.html
index b581141..b8ff3b6 100644
--- a/docs/Tonal/Log2.html
+++ b/docs/Tonal/Log2.html
@@ -216,7 +216,7 @@ 
    
 
 
       
diff --git a/docs/Tonal/Ratio.html b/docs/Tonal/Ratio.html
index 5df8d70..c4410c2 100644
--- a/docs/Tonal/Ratio.html
+++ b/docs/Tonal/Ratio.html
@@ -104,7 +104,7 @@
   
   
    
     
    Defined in:
    
-    
    lib/tonal/approximations.rb,
    
+    
    lib/tonal/approximation.rb,
    
   lib/tonal/ratio.rb    
 
    
   
    
@@ -121,7 +121,7 @@ 
    Defined Under Namespace
    
     
   
     
-      Classes: Approximations
+      Classes: Approximation
     
   
 
    
@@ -321,7 +321,7 @@ 
    
         
  • 
   
     
-      .random_ratio(number_of_factors = 2, within: 100)  ⇒ Tonal::Ratio 
+      .random_ratio(number_of_factors = 2, within: 100, reduced: false)  ⇒ Tonal::Ratio 
     
 
     
@@ -345,7 +345,7 @@ 
    
         
  • 
   
     
-      .superparticular(n)  ⇒ Tonal::Ratio 
+      .superparticular(n, factor: 1, superpart: :upper)  ⇒ Tonal::Ratio 
     
 
     
@@ -360,7 +360,7 @@ 
    
 
   
     
    
-
    Ratio who’s numerator and denominator are 1 apart.
    
+
    Ratio who’s numerator and denominator are seperated by a difference of 1.
    
 
    
   
 
    • 
@@ -369,7 +369,7 @@ 
    
         
  • 
   
     
-      .superpartient(n, part)  ⇒ Tonal::Ratio 
+      .superpartient(n, summand:, factor: 1, superpart: :upper)  ⇒ Tonal::Ratio 
     
 
     
@@ -384,7 +384,7 @@ 
    
 
   
     
    
-
    Ratio who’s numerator and denominator are a partient apart.
    
+
    Ratio who’s numerator and denominator are separated by a summand difference.
    
 
    
   
 
    • 
@@ -558,7 +558,7 @@ 
    
         
  • 
   
     
-      #approx  ⇒ Tonal::Ratio::Approximations 
+      #approximate  ⇒ Tonal::Ratio::Approximation 
     
 
     
@@ -1072,28 +1072,6 @@ 
    
 
    The mirror of self along the axis (default 1/1).
    
 
   
-
    • 
-
-      
-        
  • 
-  
-    
-      #mirror2(ratio)  ⇒ Tonal::Ratio 
-    
-
-    
-  
-  
-  
-  
-  
-  
-  
-  
-
-  
-    
    
-  
 
    • 
 
       
@@ -1826,7 +1804,7 @@ 
    
   @equave = equave
   @reduced_antecedent, @reduced_consequent = _equave_reduce(equave)
   @label = label
-  @approximations = Approximations.new(ratio: self)
+  @approximation = Approximation.new(ratio: self)
 end
    • 
     
   
@@ -2160,12 +2138,12 @@ 
    
       
     
 
-80
-81
-82
    
+89
+90
+91
    
     
     
-      
    # File 'lib/tonal/ratio.rb', line 80
+      
    # File 'lib/tonal/ratio.rb', line 89
 
 def self.ed(modulo, step, equave: 2/1r)
   self.new(2**(step.to_f/modulo), equave: equave)
@@ -2178,7 +2156,7 @@ 
    
       
    
   
    
   
-    .random_ratio(number_of_factors = 2, within: 100)  ⇒ Tonal::Ratio 
+    .random_ratio(number_of_factors = 2, within: 100, reduced: false)  ⇒ Tonal::Ratio 
   
 
   
@@ -2257,21 +2235,21 @@ 
    
       
     
 
-61
-62
-63
-64
-65
-66
-67
-68
-69
-70
    
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
    
     
     
-      
    # File 'lib/tonal/ratio.rb', line 61
+      
    # File 'lib/tonal/ratio.rb', line 70
 
-def self.random_ratio(number_of_factors = 2, within: 100)
+def self.random_ratio(number_of_factors = 2, within: 100, reduced: false)
   primes = Prime.each(within).to_a
   nums = []
   dens = []
@@ -2279,7 +2257,7 @@ 
    
     nums << [primes[rand(10)], rand(3)]
     dens << [primes[rand(10)], rand(3)]
   end
-  [nums, dens].ratio_from_prime_divisions
+  [nums, dens].ratio_from_prime_divisions(reduced:)
 end
    
     
   
@@ -2289,7 +2267,7 @@ 
    
       
    
   
    
   
-    .superparticular(n)  ⇒ Tonal::Ratio 
+    .superparticular(n, factor: 1, superpart: :upper)  ⇒ Tonal::Ratio 
   
 
   
@@ -2298,7 +2276,7 @@ 
    
 
    
   
    
     
-
    Returns ratio who’s numerator and denominator are 1 apart.
    
+
    Returns ratio who’s numerator and denominator are seperated by a difference of 1.
    
 
 
   
    
@@ -2309,7 +2287,7 @@ 
    
     
    Examples:
    
     
       
-      
    Tonal::Ratio.superparticular(100) = (100/99)
    
+      
    Tonal::Ratio.superparticular(100) = (101/100)
    
     
   
    
 
    Parameters:
    
@@ -2320,13 +2298,49 @@ 
    
         n
       
       
-        
+        (Integer)
+      
+      
+      
+        —
+        
    
+
    number from which the superior part is calculated
    
+
    
+      
+

    • + +
  • + + factor + (Rational) + + + (defaults to: 1) + —
    -

    numerator of ratio

    +

    multiplied into the resulting ratio, default 1/1

    +
    + +
    • + +
  • + + superpart + + + (Symbol) + + + (defaults to: :upper) + + + — +
    +

    assigning the superior part to the antecedent or consequent

    • @@ -2345,7 +2359,7 @@

    —
    -

    ratio who’s numerator and denominator are 1 apart

    +

    ratio who’s numerator and denominator are seperated by a difference of 1

    @@ -2358,15 +2372,15 @@ 

     
 
-41
-42
-43
    +43 +44 +45 - 
    # File 'lib/tonal/ratio.rb', line 41
+      
    # File 'lib/tonal/ratio.rb', line 43
 
-def self.superparticular(n)
-  superpartient(n, 1)
+def self.superparticular(n, factor: 1/1r, superpart: :upper)
+  superpartient(n, summand: 1, factor:, superpart:)
 end
    
     
   
@@ -2376,7 +2390,7 @@ 
    
       
    
   
    
   
-    .superpartient(n, part)  ⇒ Tonal::Ratio 
+    .superpartient(n, summand:, factor: 1, superpart: :upper)  ⇒ Tonal::Ratio 
   
 
   
@@ -2385,7 +2399,7 @@ 
    
 
    
   
    
     
-
    Returns ratio who’s numerator and denominator are a partient apart.
    
+
    Returns ratio who’s numerator and denominator are separated by a summand difference.
    
 
 
   
    
@@ -2396,7 +2410,7 @@ 
    
     
    Examples:
    
     
       
-      
    Tonal::Ratio.superpartient(100, 5) => (100/95)
    
+      
    Tonal::Ratio.superpartient(23, summand: 3) => (26/23)
    
     
   
    
 
    Parameters:
    
@@ -2407,29 +2421,65 @@ 
    
         n
       
       
-        
+        (Integer)
       
       
       
         —
         
    
-
    numerator of ratio
    
+
    number from which the superior part is calculated
    
 
    
       
     
   
     
  • 
       
-        part
+        summand
       
       
-        
+        (Integer)
       
       
       
         —
         
    
-
    partient separating the numerator and denominator
    
+
    term added to the superior part
    
+
    
+      
+    
    • 
+  
+    
  • 
+      
+        factor
+      
+      
+        (Rational)
+      
+      
+        (defaults to: 1)
+      
+      
+        —
+        
    
+
    multiplied into the resulting ratio, default 1/1
    
+
    
+      
+    
    • 
+  
+    
  • 
+      
+        superpart
+      
+      
+        (Symbol)
+      
+      
+        (defaults to: :upper)
+      
+      
+        —
+        
    
+
    assigning the superior part to the antecedent or consequent
    
 
    
       
     
    • 
@@ -2448,7 +2498,7 @@ 
    
       
         —
         
    
-
    ratio who’s numerator and denominator are a partient apart
    
+
    ratio who’s numerator and denominator are separated by a summand difference
    
 
    
       
     
@@ -2461,15 +2511,25 @@ 
    
       
     
 
-51
-52
-53
    
+55
+56
+57
+58
+59
+60
+61
+62
    - 
    # File 'lib/tonal/ratio.rb', line 51
+      
    # File 'lib/tonal/ratio.rb', line 55
 
-def self.superpartient(n, part)
-  self.new(n, n-part)
+def self.superpartient(n, summand:, factor: 1/1r, superpart: :upper)
+  case superpart.to_sym.downcase
+  when :lower, :consequent, :denominator
+    self.new(n*factor.numerator, (n+summand)*factor.denominator)
+  else
+    self.new((n+summand)*factor.numerator, n*factor.denominator)
+  end
 end
    
     
   
@@ -2565,12 +2625,12 @@ 
    
       
     
 
-91
-92
-93
    
+100
+101
+102
    - 
    # File 'lib/tonal/ratio.rb', line 91
+      
    # File 'lib/tonal/ratio.rb', line 100
 
 def self.within_cents?(cents1, cents2, within)
   (cents1 - cents2).abs <= within
@@ -2771,9 +2831,9 @@ 
    
 
     
       
    
-  
    
+  
    
   
-    #approx  ⇒ Tonal::Ratio::Approximations 
+    #approximate  ⇒ Tonal::Ratio::Approximation 
   
 
   
@@ -2795,7 +2855,7 @@ 
    
     
  • 
       
       
-        (Tonal::Ratio::Approximations)
+        (Tonal::Ratio::Approximation)
       
       
       
@@ -2814,15 +2874,15 @@ 
    
       
     
 
-103
-104
-105
    
+112
+113
+114
    • 
     
     
-      
    # File 'lib/tonal/ratio.rb', line 103
+      
    # File 'lib/tonal/ratio.rb', line 112
 
-def approx
-  @approximations
+def approximate
+  @approximation
 end
    
     
   
@@ -3288,7 +3348,7 @@ 
    
       
    # File 'lib/tonal/ratio.rb', line 436
 
 def efficiency(modulo)
-  (Tonal::Cents::CENT_SCALE * step(modulo).step / modulo.to_f) - to_cents
+  ((Tonal::Cents::CENT_SCALE * step(modulo).step / modulo.to_f) - to_cents).round(Tonal::Cents::PRECISION)
 end
    
     
   
@@ -3373,12 +3433,12 @@ 
    
       
     
 
-212
-213
-214
    
+221
+222
+223
    
     
     
-      
    # File 'lib/tonal/ratio.rb', line 212
+      
    # File 'lib/tonal/ratio.rb', line 221
 
 def equave_reduce(equave=2/1r)
   self.class.new(*_equave_reduce(equave))
@@ -3466,13 +3526,13 @@ 
    
       
     
 
-223
-224
-225
-226
    
+232
+233
+234
+235
    
     
     
-      
    # File 'lib/tonal/ratio.rb', line 223
+      
    # File 'lib/tonal/ratio.rb', line 232
 
 def equave_reduce!(equave=2/1r)
   @antecedent, @consequent = _equave_reduce(equave)
@@ -3542,12 +3602,12 @@ 
    
       
     
 
-203
-204
-205
    
+212
+213
+214
    
     
     
-      
    # File 'lib/tonal/ratio.rb', line 203
+      
    # File 'lib/tonal/ratio.rb', line 212
 
 def fraction_reduce
   self.class.new(to_r)
@@ -3684,12 +3744,12 @@ 
    
       
     
 
-243
-244
-245
    
+252
+253
+254
    
     
     
-      
    # File 'lib/tonal/ratio.rb', line 243
+      
    # File 'lib/tonal/ratio.rb', line 252
 
 def invert
   self.class.new(consequent, antecedent)
@@ -3751,14 +3811,14 @@ 
    
       
     
 
-252
-253
-254
-255
-256
    
+261
+262
+263
+264
+265
    
     
     
-      
    # File 'lib/tonal/ratio.rb', line 252
+      
    # File 'lib/tonal/ratio.rb', line 261
 
 def invert!
   tmp = antecedent
@@ -3828,7 +3888,7 @@ 
    
   # Return label, if defined; or,
   # Return the "antecedent/consequent", if antecedent is less than 7 digits long; or
   # Return the floating point representation rounded to 2 digits of precision
-  (@label || ((Math.log10(antecedent).to_i + 1) <= 6 ? "#{antecedent}/#{consequent}" : to_f.round(2))).to_s
+  (@label || ((Math.log10(antecedent).to_i + 1) <= 6 ? "#{antecedent}/#{consequent}" : to_f.round(Tonal::Cents::PRECISION))).to_s
 end
    
     
   
@@ -4278,81 +4338,6 @@ 
    
       
     
 
-263
-264
-265
    
-    
-    
-      
    # File 'lib/tonal/ratio.rb', line 263
-
-def mirror(axis=1/1r)
-  (self.class.new(axis) ** 2) / self
-end
    
-    
-  
-
-
-    
-      
    
-  
    
-  
-    #mirror2(ratio)  ⇒ Tonal::Ratio 
-  
-
-  
-
-  
-
    
-  
    
-    
-
-  
    
-
    
-
    
-  
-  
    
-    
    Examples:
    
-    
-      
-      
    Tonal::Ratio.new(4/3r).mirror2(4/2r) => (3/8)
    
-    
-  
    
-
    Parameters:
    
-
      
-  
-    
    • 
-      
-        ratio
-      
-      
-        
-      
-      
-      
-    
      • 
-  
-
    
-
-
    Returns:
    
-
-
-
    
-  
-    
@@ -4504,12 +4489,12 @@ 
    
       
     
 
-185
-186
-187
    
+194
+195
+196
     
     
    
@@ -242,7 +242,7 @@ 
    
 
 
       
diff --git a/docs/_config.yml b/docs/_config.yml
deleted file mode 100644
index a25eb13..0000000
--- a/docs/_config.yml
+++ /dev/null
@@ -1 +0,0 @@
-theme: minima
diff --git a/docs/_index.html b/docs/_index.html
index f211bfe..f34ddfd 100644
--- a/docs/_index.html
+++ b/docs/_index.html
@@ -81,7 +81,7 @@ 
    Namespace Listing A-Z
      
             
               
    • 
-                Approximations
+                Approximation
                 
                   (Tonal::Ratio)
                 
@@ -248,6 +248,13 @@ 
      Namespace Listing A-Z
      
           
    • S
      • 
           
        
             
+              
      • 
+                Set
+                
+                  (Tonal::Ratio::Approximation)
+                
+              
        • 
+            
               
      • 
                 Step
                 
@@ -293,7 +300,7 @@ 
        Namespace Listing A-Z
        
 
 
       
diff --git a/docs/class_list.html b/docs/class_list.html
index d514395..30ff551 100644
--- a/docs/class_list.html
+++ b/docs/class_list.html
@@ -43,7 +43,7 @@ 
        Class List
        
 
       
     
diff --git a/docs/file.README.html b/docs/file.README.html
index e1ce386..8de7c05 100644
--- a/docs/file.README.html
+++ b/docs/file.README.html
@@ -79,7 +79,7 @@ 
        License
        
 
 
       
diff --git a/docs/index.html b/docs/index.html
index 18e24d0..521b5a1 100644
--- a/docs/index.html
+++ b/docs/index.html
@@ -79,7 +79,7 @@ 
        License
        
 
 
       
diff --git a/docs/method_list.html b/docs/method_list.html
index 1bf318f..a8ecf37 100644
--- a/docs/method_list.html
+++ b/docs/method_list.html
@@ -94,8 +94,8 @@ 
        Method List
        
 
   
      • 
     
        
-      #<=>
-      Tonal::Hertz
+      #<=>
+      Tonal::Cents
     
        
   
        • 
   
@@ -118,24 +118,24 @@ 
        Method List
        
 
   
      • 
     
        
-      #<=>
-      Tonal::Log
+      #<=>
+      Tonal::Step
     
        
   
        • 
   
 
   
      • 
     
        
-      #<=>
-      Tonal::Cents
+      #<=>
+      Tonal::Log
     
        
   
        • 
   
 
   
      • 
     
        
-      #<=>
-      Tonal::Step
+      #<=>
+      Tonal::Hertz
     
        
   
        • 
   
@@ -158,7 +158,7 @@ 
        Method List
        
 
   
      • 
     
        
-      #approx
+      #approximate
       Tonal::Ratio
     
        
   
        • 
@@ -166,23 +166,23 @@ 
        Method List
        
 
   
      • 
     
        
-      base
-      Tonal::Log
+      base
+      Tonal::Log2
     
        
   
        • 
   
 
   
      • 
     
        
-      base
-      Tonal::Log2
+      #base
+      Tonal::Log
     
        
   
        • 
   
 
   
      • 
     
        
-      #base
+      base
       Tonal::Log
     
        
   
        • 
@@ -190,53 +190,61 @@ 
        Method List
        
 
   
      • 
     
        
-      #benedetti_height
-      Tonal::Ratio
+      #benedetti_height
+      Numeric
     
        
   
        • 
   
 
   
      • 
     
        
-      #benedetti_height
-      Numeric
+      #benedetti_height
+      Tonal::Ratio
     
        
   
        • 
   
 
   
      • 
     
        
-      #by_continued_fraction
-      Tonal::Ratio::Approximations
+      #by_continued_fraction
+      Tonal::Ratio::Approximation
     
        
   
        • 
   
 
   
      • 
     
        
-      #by_neighborhood
-      Tonal::Ratio::Approximations
+      #by_neighborhood
+      Tonal::Ratio::Approximation
     
        
   
        • 
   
 
   
      • 
     
        
-      #by_quotient_walk
-      Tonal::Ratio::Approximations
+      #by_quotient_walk
+      Tonal::Ratio::Approximation
     
        
   
        • 
   
 
   
      • 
     
        
-      #by_tree_path
-      Tonal::Ratio::Approximations
+      #by_superparticular
+      Tonal::Ratio::Approximation
     
        
   
        • 
   
 
   
      • 
+    
        
+      #by_tree_path
+      Tonal::Ratio::Approximation
+    
        
+  
        • 
+  
+
+  
      • 
     
        
       #cent_diff
       Tonal::Ratio
@@ -244,7 +252,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #cents
       Numeric
@@ -252,7 +260,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #combination
       Tonal::Ratio
@@ -260,7 +268,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       commas
       Tonal::Comma
@@ -268,7 +276,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #consequent
       Tonal::Ratio
@@ -276,7 +284,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #convert
       Tonal::Step
@@ -284,7 +292,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #coprime?
       Integer
@@ -292,7 +300,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #coprimes
       Integer
@@ -300,7 +308,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       default_tolerance
       Tonal::Cents
@@ -308,7 +316,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #denominators
       Array
@@ -316,17 +324,25 @@ 
        Method List
        
   
        • 
   
 
+  
      • 
+    
        
+      #denominize
+      Array
+    
        
+  
        • 
+  
+
   
      • 
     
        
-      #difference
-      Tonal::Ratio
+      #denominize
+      Tonal::Interval
     
        
   
        • 
   
 
   
      • 
     
        
-      #div_times
+      #difference
       Tonal::Ratio
     
        
   
        • 
@@ -334,23 +350,23 @@ 
        Method List
        
 
   
      • 
     
        
-      #div_times
-      Numeric
+      #div_times
+      Tonal::Ratio
     
        
   
        • 
   
 
   
      • 
     
        
-      ed
-      Tonal::Ratio
+      #div_times
+      Numeric
     
        
   
        • 
   
 
   
      • 
     
        
-      #efficiency
+      ed
       Tonal::Ratio
     
        
   
        • 
@@ -366,7 +382,7 @@ 
        Method List
        
 
   
      • 
     
        
-      #equave
+      #efficiency
       Tonal::Ratio
     
        
   
        • 
@@ -374,7 +390,7 @@ 
        Method List
        
 
   
      • 
     
        
-      #equave_reduce
+      #equave
       Tonal::Ratio
     
        
   
        • 
@@ -382,7 +398,7 @@ 
        Method List
        
 
   
      • 
     
        
-      #equave_reduce!
+      #equave_reduce
       Tonal::Ratio
     
        
   
        • 
@@ -390,8 +406,8 @@ 
        Method List
        
 
   
      • 
     
        
-      #factorial
-      Integer
+      #equave_reduce!
+      Tonal::Ratio
     
        
   
        • 
   
@@ -405,6 +421,14 @@ 
        Method List
        
   
 
   
      • 
+    
        
+      #factorial
+      Integer
+    
        
+  
        • 
+  
+
+  
      • 
     
        
       #fraction_reduce
       Tonal::Ratio
@@ -412,7 +436,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #hz
       Numeric
@@ -420,7 +444,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       identity
       Tonal::ReducedRatio
@@ -428,6 +452,14 @@ 
        Method List
        
   
        • 
   
 
+  
      • 
+    
        
+      #initialize
+      Tonal::Ratio::Approximation::Set
+    
        
+  
        • 
+  
+
   
      • 
     
        
       #initialize
@@ -438,8 +470,8 @@ 
        Method List
        
 
   
      • 
     
        
-      #initialize
-      Tonal::Cents
+      #initialize
+      Tonal::Log
     
        
   
        • 
   
@@ -454,32 +486,32 @@ 
        Method List
        
 
   
      • 
     
        
-      #initialize
-      Tonal::ReducedRatio
+      #initialize
+      Tonal::Cents
     
        
   
        • 
   
 
   
      • 
     
        
-      #initialize
-      Tonal::Ratio::Approximations
+      #initialize
+      Tonal::Ratio::Approximation
     
        
   
        • 
   
 
   
      • 
     
        
-      #initialize
-      Tonal::Log
+      #initialize
+      Tonal::Ratio
     
        
   
        • 
   
 
   
      • 
     
        
-      #initialize
-      Tonal::Ratio
+      #initialize
+      Tonal::ReducedRatio
     
        
   
        • 
   
@@ -494,16 +526,16 @@ 
        Method List
        
 
   
      • 
     
        
-      #inspect
-      Tonal::Log
+      #inspect
+      Tonal::Cents
     
        
   
        • 
   
 
   
      • 
     
        
-      #inspect
-      Tonal::Cents
+      #inspect
+      Tonal::Step
     
        
   
        • 
   
@@ -518,29 +550,37 @@ 
        Method List
        
 
   
      • 
     
        
-      #inspect
-      Tonal::Ratio
+      #inspect
+      Tonal::Interval
     
        
   
        • 
   
 
   
      • 
     
        
-      #inspect
-      Tonal::Interval
+      #inspect
+      Tonal::Ratio::Approximation::Set
     
        
   
        • 
   
 
   
      • 
     
        
-      #inspect
-      Tonal::Step
+      #inspect
+      Tonal::Log
     
        
   
        • 
   
 
   
      • 
+    
        
+      #inspect
+      Tonal::Ratio
+    
        
+  
        • 
+  
+
+  
      • 
     
        
       #interval
       Tonal::Interval
@@ -548,7 +588,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #interval_with
       Numeric
@@ -556,7 +596,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #interval_with
       Tonal::ReducedRatio
@@ -564,7 +604,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #invert
       Tonal::Ratio
@@ -572,7 +612,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #invert!
       Tonal::Ratio
@@ -580,7 +620,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       keys
       Tonal::Comma
@@ -588,7 +628,7 @@ 
        Method List
        
   
        • 
   
 
-  
      • 
+  
      • 
     
        
       #label
       Tonal::Ratio
@@ -596,6 +636,14 @@ 
        Method List
        
   
        • 
   
 
+  
      • 
+    
        
+      #lcm
+      Tonal::Ratio
+    
        
+  
        • 
+  
+
   
      • 
     
        
       #lcm
@@ -606,8 +654,8 @@ 
        Method List
        
 
   
      • 
     
        
-      #lcm
-      Tonal::Ratio
+      #log
+      Numeric
     
        
   
        • 
   
@@ -630,7 +678,7 @@ 
        Method List
        
 
   
      • 
     
        
-      #log
+      #log2
       Numeric
     
        
   
        • 
@@ -638,8 +686,8 @@ 
        Method List
        
 
   
      • 
     
        
-      #log2
-      Numeric
+      #log_weil_height
+      Tonal::Ratio
     
        
   
        • 
   
@@ -654,15 +702,15 @@ 
        Method List
        
 
   
      • 
     
        
-      #log_weil_height
-      Tonal::Ratio
+      #logarithm
+      Tonal::Log
     
        
   
        • 
   
 
   
      • 
     
        
-      #logarithm
+      #logarithmand
       Tonal::Log
     
        
   
        • 
@@ -670,24 +718,24 @@ 
        Method List
        
 
   
      • 
     
        
-      #logarithmand
-      Tonal::Log
+      #lower_ratio
+      Tonal::Interval
     
        
   
        • 
   
 
   
      • 
     
        
-      #lower_ratio
-      Tonal::Interval
+      #max_prime
+      Tonal::Ratio
     
        
   
        • 
   
 
   
      • 
     
        
-      #max_prime
-      Tonal::Ratio
+      #max_prime
+      Integer
     
        
   
        • 
   
@@ -710,24 +758,24 @@ 
        Method List
        
 
   
      • 
     
        
-      #max_prime
-      Integer
+      #mean
+      Array
     
        
   
        • 
   
 
   
      • 
     
        
-      #mean
-      Array
+      #mediant_sum
+      Tonal::Ratio
     
        
   
        • 
   
 
   
      • 
     
        
-      #mediant_sum
-      Tonal::Ratio
+      #method_missing
+      Tonal::Hertz
     
        
   
        • 
   
@@ -742,16 +790,16 @@ 
        Method List
        
 
   
      • 
     
        
-      #method_missing
-      Tonal::Hertz
+      #min_prime
+      Tonal::Ratio
     
        
   
        • 
   
 
   
      • 
     
        
-      #min_prime
-      Tonal::Ratio
+      #min_prime
+      Integer
     
        
   
        • 
   
@@ -772,22 +820,6 @@ 
        Method List
        
   
   
 
-  
      • 
-    
        
-      #min_prime
-      Integer
-    
        
-  
        • 
-  
-
-  
      • 
-    
        
-      #mirror
-      Tonal::Ratio
-    
        
-  
        • 
-  
-
   
      • 
     
        
       #mirror
@@ -798,7 +830,7 @@ 
        Method List
        
 
   
      • 
     
        
-      #mirror2
+      #mirror
       Tonal::Ratio
     
        
   
        • 
@@ -814,16 +846,16 @@ 
        Method List
        
 
   
      • 
     
        
-      #modulo_translate
-      Array
+      #modulo_translate
+      Numeric
     
        
   
        • 
   
 
   
      • 
     
        
-      #modulo_translate
-      Numeric
+      #modulo_translate
+      Array
     
        
   
        • 
   
@@ -862,32 +894,16 @@ 
        Method List
        
 
   
      • 
     
        
-      #neighborhood
-      Tonal::Ratio::Approximations
+      #neighborhood
+      Tonal::Ratio::Approximation
     
        
   
        • 
   
 
   
      • 
     
        
-      neighbors
-      Tonal::Ratio::Approximations
-    
        
-  
        • 
-  
-
-  
      • 
-    
        
-      #normalize
-      Array
-    
        
-  
        • 
-  
-
-  
      • 
-    
        
-      #normalize
-      Tonal::Interval
+      neighbors
+      Tonal::Ratio::Approximation
     
        
   
        • 
   
@@ -958,40 +974,40 @@ 
        Method List
        
 
   
      • 
     
        
-      #plus_minus
-      Tonal::Ratio
+      #plus_minus
+      Numeric
     
        
   
        • 
   
 
   
      • 
     
        
-      #plus_minus
-      Numeric
+      #plus_minus
+      Tonal::Cents
     
        
   
        • 
   
 
   
      • 
     
        
-      #plus_minus
-      Tonal::Cents
+      #plus_minus
+      Tonal::Ratio
     
        
   
        • 
   
 
   
      • 
     
        
-      #prime_divisions
-      Tonal::Ratio
+      #prime_divisions
+      Rational
     
        
   
        • 
   
 
   
      • 
     
        
-      #prime_divisions
-      Rational
+      #prime_divisions
+      Tonal::Ratio
     
        
   
        • 
   
@@ -1006,16 +1022,16 @@ 
        Method List
        
 
   
      • 
     
        
-      #prime_vector
-      Numeric
+      #prime_vector
+      Tonal::Ratio
     
        
   
        • 
   
 
   
      • 
     
        
-      #prime_vector
-      Tonal::Ratio
+      #prime_vector
+      Numeric
     
        
   
        • 
   
@@ -1030,16 +1046,16 @@ 
        Method List
        
 
   
      • 
     
        
-      #ratio
-      Tonal::Cents
+      #ratio
+      Tonal::Step
     
        
   
        • 
   
 
   
      • 
     
        
-      #ratio
-      Tonal::Ratio::Approximations
+      #ratio
+      Tonal::Cents
     
        
   
        • 
   
@@ -1054,13 +1070,21 @@ 
        Method List
        
 
   
      • 
     
        
-      #ratio
-      Tonal::Step
+      #ratio
+      Tonal::Ratio::Approximation::Set
     
        
   
        • 
   
 
   
      • 
+    
        
+      #ratio
+      Tonal::Ratio::Approximation
+    
        
+  
        • 
+  
+
+  
      • 
     
        
       #ratio_from_prime_divisions
       Array
@@ -1068,6 +1092,14 @@ 
        Method List
        
   
        • 
   
 
+  
      • 
+    
        
+      #ratios
+      Tonal::Ratio::Approximation::Set
+    
        
+  
        • 
+  
+
   
      • 
     
        
       #reduced_antecedent
@@ -1126,8 +1158,8 @@ 
        Method List
        
 
   
      • 
     
        
-      #step
-      Tonal::Log
+      #step
+      Tonal::Ratio
     
        
   
        • 
   
@@ -1150,8 +1182,8 @@ 
        Method List
        
 
   
      • 
     
        
-      #step
-      Tonal::Ratio
+      #step
+      Tonal::Log
     
        
   
        • 
   
@@ -1222,40 +1254,40 @@ 
        Method List
        
 
   
      • 
     
        
-      #to_cents
-      Tonal::Step
+      #to_cents
+      Numeric
     
        
   
        • 
   
 
   
      • 
     
        
-      #to_cents
-      Tonal::Log
+      #to_cents
+      Tonal::Ratio
     
        
   
        • 
   
 
   
      • 
     
        
-      #to_cents
-      Tonal::Ratio
+      #to_cents
+      Array
     
        
   
        • 
   
 
   
      • 
     
        
-      #to_cents
-      Array
+      #to_cents
+      Tonal::Step
     
        
   
        • 
   
 
   
      • 
     
        
-      #to_cents
-      Numeric
+      #to_cents
+      Tonal::Log
     
        
   
        • 
   
@@ -1294,8 +1326,8 @@ 
        Method List
        
 
   
      • 
     
        
-      #to_r
-      Tonal::Ratio
+      #to_r
+      Tonal::Step
     
        
   
        • 
   
@@ -1310,8 +1342,8 @@ 
        Method List
        
 
   
      • 
     
        
-      #to_r
-      Tonal::Step
+      #to_r
+      Tonal::Ratio
     
        
   
        • 
   
@@ -1350,16 +1382,16 @@ 
        Method List
        
 
   
      • 
     
        
-      #to_vector
-      Rational
+      #to_vector
+      Array
     
        
   
        • 
   
 
   
      • 
     
        
-      #to_vector
-      Array
+      #to_vector
+      Rational
     
        
   
        • 
   
diff --git a/docs/top-level-namespace.html b/docs/top-level-namespace.html
index 5ae45a1..22c14fb 100644
--- a/docs/top-level-namespace.html
+++ b/docs/top-level-namespace.html
@@ -102,7 +102,7 @@ 
        Defined Under Namespace
        
 
 
       
diff --git a/lib/tonal/approximation.rb b/lib/tonal/approximation.rb
new file mode 100644
index 0000000..a8527b0
--- /dev/null
+++ b/lib/tonal/approximation.rb
@@ -0,0 +1,189 @@
+class Tonal::Ratio
+  class Approximation
+    DEFAULT_MAX_PRIME = Float::INFINITY
+    DEFAULT_MAX_GRID_SCALE = 100
+    DEFAULT_MAX_GRID_BOUNDARY = 5
+    DEFAULT_DEPTH = Float::INFINITY
+    DEFAULT_COMPLEXITY_AMOUNT = 50.0
+    CONVERGENT_LIMIT = 10
+
+    extend Forwardable
+    def_delegators :@ratio, :antecedent, :consequent, :to_cents, :to_basic_ratio, :to_f
+
+    attr_reader :ratio
+
+    def initialize(ratio:)
+      raise ArgumentError, "Tonal::Ratio required" unless ratio.kind_of?(Tonal::Ratio)
+      @ratio = ratio
+    end
+
+    # @return [Tonal::Ratio::Approximation::Set] of ratios within cent tolerance of self found using continued fraction approximation
+    # @example
+    #   Tonal::Ratio.ed(12,1).approximate.by_continued_fraction
+    #   => (4771397596969315/4503599627370496): [(18/17), (196/185), (1657/1564), (7893/7450), (18904/17843), (3118/2943), (1461/1379), (89/84), (17/16)]
+    # @param cents_tolerance the cents tolerance used to scope the collection
+    # @param depth the maximum number of ratios in the collection
+    # @param max_prime the maximum prime number to allow in the collection
+    # @param conv_limit the number of convergents to limit the ContinuedFraction method
+    #
+    def by_continued_fraction(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)
+      self_in_cents = to_cents
+      within = cents_tolerance.kind_of?(Tonal::Cents) ? cents_tolerance : Tonal::Cents.new(cents: cents_tolerance)
+      Set.new(ratio: ratio).tap do |set|
+        ContinuedFraction.new(antecedent.to_f/consequent, conv_limit).convergents.each do |convergent|
+          ratio2 = ratio.class.new(convergent.numerator,convergent.denominator)
+          set.ratios << ratio2 if ratio.class.within_cents?(self_in_cents, ratio2.to_cents, within) && ratio2.within_prime?(max_prime)
+          break if set.length >= depth
+        end
+      end
+    end
+
+    # @return [Tonal::Ratio::Approximation::Set] of ratios within cent tolerance of self found using a quotient walk on the fraction tree
+    # @example
+    #   Tonal::Ratio.ed(12,1).approximate.by_quotient_walk(max_prime: 89)
+    #   => (4771397596969315/4503599627370496): [(18/17), (196/185), (89/84), (71/67), (53/50), (35/33), (17/16)]
+    # @param cents_tolerance the cents tolerance used to scope the collection
+    # @param depth the maximum number of ratios in the collection
+    # @param max_prime the maximum prime number to allow in the collection
+    # @param conv_limit the number of convergents to limit the ContinuedFraction method
+    #
+    def by_quotient_walk(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)
+      self_in_cents = to_cents
+      within = cents_tolerance.kind_of?(Tonal::Cents) ? cents_tolerance : Tonal::Cents.new(cents: cents_tolerance)
+
+      Set.new(ratio: ratio).tap do |set|
+        FractionTree.quotient_walk(to_f, limit: conv_limit).each do |node|
+          ratio2 = ratio.class.new(node.weight)
+          set.ratios << ratio2 if ratio.class.within_cents?(self_in_cents, ratio2.to_cents, within) && ratio2.within_prime?(max_prime)
+          break if set.length >= depth
+        end
+      end
+    end
+
+    # @return [Tonal::Ratio::Approximation::Set] of fraction tree ratios within cent tolerance of self
+    # @example
+    #   Tonal::Ratio.ed(12,1).approximate.by_tree_path(max_prime: 17)
+    #   => (4771397596969315/4503599627370496): [(18/17), (35/33), (17/16)]
+    # @param cents_tolerance the cents tolerance used to scope the collection
+    # @param depth the maximum number of ratios in the collection
+    # @param max_prime the maximum prime number to allow in the collection
+    #
+    def by_tree_path(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME)
+      self_in_cents = to_cents
+      within = cents_tolerance.kind_of?(Tonal::Cents) ? cents_tolerance : Tonal::Cents.new(cents: cents_tolerance)
+      Set.new(ratio: ratio).tap do |set|
+        FractionTree.path_to(to_f).each do |node|
+          ratio2 = ratio.class.new(node.weight)
+          set.ratios << ratio2 if ratio.class.within_cents?(self_in_cents, ratio2.to_cents, within) && ratio2.within_prime?(max_prime)
+          break if set.length >= depth
+        end
+      end
+    end
+
+    # @return [Tonal::Ratio::Approximation::Set] of superparticular approximations within cent tolerance of self
+    # @example
+    #   Tonal::Ratio.new(3/2r).approximate.by_superparticular
+    #   => (3/2): [(1098/730), (1095/728), (1092/726), (1089/724), (1086/722), (1083/720), (1080/718), (1077/716), (1074/714), (1071/712), (1068/710), (1065/708), (1062/706), (1059/704), (1056/702), (1053/700), (1050/698), (1047/696), (1044/694), (1041/692)]
+    # @param cents_tolerance the cents tolerance used to scope the collection
+    # @param depth the maximum number of ratios in the collection
+    # @param max_prime the maximum prime number to allow in the collection
+    # @param superpart if the superior part is the numerator or denominator
+    #
+    def by_superparticular(cents_tolerance: Tonal::Cents::TOLERANCE, depth: 20, max_prime: DEFAULT_MAX_PRIME, superpart: :upper)
+      self_in_cents = to_cents
+      within = cents_tolerance.kind_of?(Tonal::Cents) ? cents_tolerance : Tonal::Cents.new(cents: cents_tolerance)
+      Set.new(ratio: ratio).tap do |set|
+        n = 1
+        while true do
+          ratio2 = ratio.class.superparticular(n, factor: ratio.to_r, superpart:)
+          set.ratios << ratio2 if ratio.class.within_cents?(self_in_cents, ratio2.to_cents, within) && ratio2.within_prime?(max_prime) && ratio2 != ratio
+          break if set.length >= depth
+          n += 1
+        end
+      end
+    end
+
+    # @return [Array] of ratios within cent tolerance of self found on the ratio grid
+    # @example
+    #   Tonal::Ratio.new(3,2).approximate.by_neighborhood(max_prime: 23, cents_tolerance: 5, max_boundary: 10, max_scale: 60)
+    #   => (3/2): [(175/117), (176/117)]
+    # @param cents_tolerance the cents tolerance used to scope the collection
+    # @param depth the maximum number of ratios in the collection
+    # @param max_prime the maximum prime number to allow in the collection
+    # @param max_boundary the maximum distance grid ratios will be from the scaled ratio
+    # @param max_scale the maximum self will be scaled
+    #
+    def by_neighborhood(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, max_boundary: DEFAULT_MAX_GRID_BOUNDARY, max_scale: DEFAULT_MAX_GRID_SCALE)
+      self_in_cents = to_cents
+      within = cents_tolerance.kind_of?(Tonal::Cents) ? cents_tolerance : Tonal::Cents.new(cents: cents_tolerance)
+      Set.new(ratio: ratio).tap do |set|
+        scale = 1
+        boundary = 1
+
+        while set.length <= depth && scale <= max_scale do
+          while boundary <= max_boundary
+            vacinity = ratio.respond_to?(:to_basic_ratio) ? to_basic_ratio.scale(scale) : ratio.scale(scale)
+            self.class.neighbors(away: boundary, vacinity: vacinity).each do |neighbor|
+              set.ratios << neighbor if ratio.class.within_cents?(self_in_cents, neighbor.to_cents, within) && neighbor.within_prime?(max_prime) && neighbor != ratio
+            end
+            boundary += 1
+          end
+          boundary = 1
+          scale += 1
+        end
+      end
+    end
+
+    # @return [Array] of bounding ratios in the ratio grid vacinity of antecedent/consequent scaled by scale
+    # @example
+    #   Tonal::ReducedRatio.new(3,2).neighborhood(scale: 256, boundary: 2)
+    #     => [(768/514), (766/512), (768/513), (767/512), (768/512), (769/512), (768/511), (770/512), (768/510)]
+    # @param scale [Integer] used to scale antecedent/consequent on coordinate system
+    # @param boundary [Integer] limit within which to calculate neighboring ratios
+    #
+    def neighborhood(scale: 2**0, boundary: 1)
+      scale = scale.round
+      vacinity = ratio.respond_to?(:to_basic_ratio) ? to_basic_ratio.scale(scale) : ratio.scale(scale)
+      SortedSet.new([].tap do |ratio_list|
+                      1.upto(boundary) do |away|
+                        ratio_list << self.class.neighbors(away: away, vacinity: vacinity)
+                      end
+                    end.flatten).to_a
+    end
+
+    # @return [Array] an array of Tonal::Ratio neighbors in the scaled ratio's grid neighborhood
+    # @example
+    #   Tonal::Ratio::Approximation.neighbors(vacinity: (3/2r).ratio(reduced:false).scale(256), away: 1)
+    #     => [(768/513), (767/512), (768/512), (769/512), (768/511)]
+    # @param away [Integer] the neighbors distance away from self's antecedent and consequent
+    #
+    def self.neighbors(vacinity:, away: 1)
+      [vacinity,
+       vacinity.class.new(vacinity.antecedent+away, vacinity.consequent),
+       vacinity.class.new(vacinity.antecedent-away, vacinity.consequent),
+       vacinity.class.new(vacinity.antecedent, vacinity.consequent+away),
+       vacinity.class.new(vacinity.antecedent, vacinity.consequent-away),
+       vacinity.class.new(vacinity.antecedent+away, vacinity.consequent+away),
+       vacinity.class.new(vacinity.antecedent+away, vacinity.consequent-away),
+       vacinity.class.new(vacinity.antecedent-away, vacinity.consequent+away),
+       vacinity.class.new(vacinity.antecedent-away, vacinity.consequent-away)]
+    end
+
+    class Set
+      extend Forwardable
+      def_delegators :@ratios, :count, :length, :min, :max, :entries, :all?, :any?, :reject, :map
+
+      attr_reader :ratios, :ratio
+
+      def initialize(ratio:)
+        @ratio = ratio
+        @ratios = SortedSet.new
+      end
+      alias :approximations :entries
+
+      def inspect
+        "#{ratio}: #{entries}"
+      end
+    end
+  end
+end
\ No newline at end of file
diff --git a/lib/tonal/approximations.rb b/lib/tonal/approximations.rb
deleted file mode 100644
index 0558942..0000000
--- a/lib/tonal/approximations.rb
+++ /dev/null
@@ -1,148 +0,0 @@
-class Tonal::Ratio
-  class Approximations
-    DEFAULT_MAX_PRIME = Float::INFINITY
-    DEFAULT_MAX_GRID_SCALE = 100
-    DEFAULT_MAX_GRID_BOUNDARY = 5
-    DEFAULT_DEPTH = Float::INFINITY
-    DEFAULT_COMPLEXITY_AMOUNT = 50.0
-    CONVERGENT_LIMIT = 10
-
-    extend Forwardable
-    def_delegators :@ratio, :antecedent, :consequent, :to_cents, :to_basic_ratio, :to_f
-
-    attr_reader :ratio
-
-    def initialize(ratio:)
-      raise ArgumentError, "Tonal::Ratio required" unless ratio.kind_of?(Tonal::Ratio)
-      @ratio = ratio
-    end
-
-    # @return [Array] of ratios within cent tolerance of self found using continued fraction approximation
-    # @example
-    #   Tonal::Ratio.ed(12,1).by_continued_fraction
-    #   => [(18/17), (196/185), (1657/1564), (7893/7450), (18904/17843), (3118/2943), (1461/1379), (89/84), (17/16)]
-    # @param cents_tolerance
-    # @param depth
-    # @param max_prime
-    # @param conv_limit
-    #
-    def by_continued_fraction(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)
-      self_in_cents = to_cents
-      within = cents_tolerance.kind_of?(Tonal::Cents) ? cents_tolerance : Tonal::Cents.new(cents: cents_tolerance)
-      [].tap do |results|
-        ContinuedFraction.new(antecedent.to_f/consequent, conv_limit).convergents.each do |convergent|
-          ratio2 = ratio.class.new(convergent.numerator,convergent.denominator)
-          results << ratio2 if ratio.class.within_cents?(self_in_cents, ratio2.to_cents, within) && ratio2.within_prime?(max_prime)
-          break if results.length >= depth
-        end
-      end.sort
-    end
-
-    # @return [Array] of ratios within cent tolerance of self found using a quotient walk on the fraction tree
-    # @example
-    #   Tonal::Ratio.ed(12,1).by_quotient_walk(max_prime: 89)
-    #   => [(18/17), (196/185), (89/84), (71/67), (53/50), (35/33), (17/16)]
-    # @param cents_tolerance
-    # @param depth
-    # @param max_prime
-    #
-    def by_quotient_walk(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)
-      self_in_cents = to_cents
-      within = cents_tolerance.kind_of?(Tonal::Cents) ? cents_tolerance : Tonal::Cents.new(cents: cents_tolerance)
-
-      [].tap do |results|
-        FractionTree.quotient_walk(to_f, limit: conv_limit).each do |node|
-          ratio2 = ratio.class.new(node.weight)
-          results << ratio2 if ratio.class.within_cents?(self_in_cents, ratio2.to_cents, within) && ratio2.within_prime?(max_prime)
-          break if results.length >= depth
-        end
-      end.sort
-    end
-
-    # @return [Array] of fraction tree ratios within cent tolerance of self
-    # @example
-    #   Tonal::Ratio.ed(12,1).by_tree_path(max_prime: 17)
-    #   => [(18/17), (35/33), (17/16)]
-    # @param cents_tolerance
-    # @param depth
-    # @param max_prime
-    #
-    def by_tree_path(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME)
-      self_in_cents = to_cents
-      within = cents_tolerance.kind_of?(Tonal::Cents) ? cents_tolerance : Tonal::Cents.new(cents: cents_tolerance)
-      [].tap do |results|
-        FractionTree.path_to(to_f).each do |node|
-          ratio2 = ratio.class.new(node.weight)
-          results << ratio2 if ratio.class.within_cents?(self_in_cents, ratio2.to_cents, within) && ratio2.within_prime?(max_prime)
-          break if results.length >= depth
-        end
-      end.sort
-    end
-
-    # @return [Array] of ratios within cent tolerance of self found on the ratio grid
-    # @example
-    #   Tonal::Ratio.new(3,2).by_neighborhood(max_prime: 23, cents_tolerance: 5, max_boundary: 10, max_scale: 60)
-    #   => [(175/117), (176/117)]
-    # @param cents_tolerance the maximum cents self is allowed from grid ratios
-    # @param depth the maximum depth the array will get
-    # @param max_prime the maximum prime the grid ratios will contain
-    # @param max_boundary the maximum distance grid ratios will be from the scaled ratio
-    # @param max_scale the maximum self will be scaled
-    #
-    def by_neighborhood(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, max_boundary: DEFAULT_MAX_GRID_BOUNDARY, max_scale: DEFAULT_MAX_GRID_SCALE)
-      self_in_cents = to_cents
-      within = cents_tolerance.kind_of?(Tonal::Cents) ? cents_tolerance : Tonal::Cents.new(cents: cents_tolerance)
-      [].tap do |results|
-        scale = 1
-        boundary = 1
-
-        while results.length <= depth && scale <= max_scale do
-          while boundary <= max_boundary
-            vacinity = ratio.respond_to?(:to_basic_ratio) ? to_basic_ratio.scale(scale) : ratio.scale(scale)
-            self.class.neighbors(away: boundary, vacinity: vacinity).each do |neighbor|
-              results << neighbor if ratio.class.within_cents?(self_in_cents, neighbor.to_cents, within) && neighbor.within_prime?(max_prime)
-            end
-            boundary += 1
-          end
-          boundary = 1
-          scale += 1
-        end
-      end.uniq(&:to_r).reject{|r| r == ratio}.sort
-    end
-
-    # @return [Array] of bounding ratios in the ratio grid vacinity of antecedent/consequent scaled by scale
-    # @example
-    #   Tonal::ReducedRatio.new(3,2).neighborhood(scale: 256, boundary: 2)
-    #     => [(768/514), (766/512), (768/513), (767/512), (768/512), (769/512), (768/511), (770/512), (768/510)]
-    # @param scale [Integer] used to scale antecedent/consequent on coordinate system
-    # @param boundary [Integer] limit within which to calculate neighboring ratios
-    #
-    def neighborhood(scale: 2**0, boundary: 1)
-      scale = scale.round
-      vacinity = ratio.respond_to?(:to_basic_ratio) ? to_basic_ratio.scale(scale) : ratio.scale(scale)
-      SortedSet.new([].tap do |ratio_list|
-                      1.upto(boundary) do |away|
-                        ratio_list << self.class.neighbors(away: away, vacinity: vacinity)
-                      end
-                    end.flatten).to_a
-    end
-
-    # @return [Array] an array of Tonal::Ratio neighbors in the scaled ratio's grid neighborhood
-    # @example
-    #   Tonal::Ratio::Approximations.neighbors(vacinity: (3/2r).ratio(reduced:false).scale(256), away: 1)
-    #     => [(768/513), (767/512), (768/512), (769/512), (768/511)]
-    # @param away [Integer] the neighbors distance away from self's antecedent and consequent
-    #
-    def self.neighbors(vacinity:, away: 1)
-      [vacinity,
-       vacinity.class.new(vacinity.antecedent+away, vacinity.consequent),
-       vacinity.class.new(vacinity.antecedent-away, vacinity.consequent),
-       vacinity.class.new(vacinity.antecedent, vacinity.consequent+away),
-       vacinity.class.new(vacinity.antecedent, vacinity.consequent-away),
-       vacinity.class.new(vacinity.antecedent+away, vacinity.consequent+away),
-       vacinity.class.new(vacinity.antecedent+away, vacinity.consequent-away),
-       vacinity.class.new(vacinity.antecedent-away, vacinity.consequent+away),
-       vacinity.class.new(vacinity.antecedent-away, vacinity.consequent-away)]
-    end
-  end
-end
\ No newline at end of file
diff --git a/lib/tonal/attributions.rb b/lib/tonal/attributions.rb
new file mode 100644
index 0000000..64c4c32
--- /dev/null
+++ b/lib/tonal/attributions.rb
@@ -0,0 +1,4 @@
+module Tonal
+  TOOLS_PRODUCER = "mTonal"
+  TOOLS_VERSION = "2.0.1"
+end
diff --git a/lib/tonal/cents.rb b/lib/tonal/cents.rb
index 2563674..d3a779c 100644
--- a/lib/tonal/cents.rb
+++ b/lib/tonal/cents.rb
@@ -9,7 +9,6 @@ class Tonal::Cents
   CENT_SCALE = 1200.0
   TOLERANCE = 5
   PRECISION = 2
-  DEFAULT_ROUNDING_PRECISION = 2
 
   attr_reader :log, :ratio
 
@@ -100,7 +99,7 @@ def inspect
   # https://embeddeduse.com/2019/08/26/qt-compare-two-floats/
   #
   def <=>(rhs)
-    rhs.kind_of?(self.class) ? value.round(2) <=> rhs.value.round(2) : value.round(2) <=> rhs.round(2)
+    rhs.kind_of?(self.class) ? value.round(PRECISION) <=> rhs.value.round(PRECISION) : value.round(PRECISION) <=> rhs.round(PRECISION)
   end
 
   private
diff --git a/lib/tonal/extensions.rb b/lib/tonal/extensions.rb
index a875e39..30279f0 100644
--- a/lib/tonal/extensions.rb
+++ b/lib/tonal/extensions.rb
@@ -324,11 +324,11 @@ def numerators = self.map(&:numerator)
   def denominators = self.map(&:denominator)
   alias :consequents :denominators
 
-  # @return [Array] an array of normalized ratios
+  # @return [Array] an array of ratios with equalized denominators
   # @example
-  #   [4/3r, 3/2r].normalize => [(8/6), (9/6)]
+  #   [4/3r, 3/2r].denominize => [(8/6), (9/6)]
   #
-  def normalize
+  def denominize
     l = denominators.lcm
     map{|r| Tonal::Ratio.new(l / r.denominator * r.numerator, l)}
   end
@@ -349,8 +349,9 @@ def mean = self.sum / self.count.to_f
   # @return [Tonal::ReducedRatio] ratio reconstructed from the result of a prime factor decomposition
   # @example
   #   [[[3, 1]], [[2, 1]]].ratio_from_prime_divisions => (3/2)
+  # @reduced [Boolean] if a reduced or unreduced ratio is returned
   #
-  def ratio_from_prime_divisions = Tonal::Ratio.new(Prime.int_from_prime_division(self.first), Prime.int_from_prime_division(self.last))
+  def ratio_from_prime_divisions(reduced: false) = reduced ? Tonal::ReducedRatio.new(Prime.int_from_prime_division(self.first), Prime.int_from_prime_division(self.last)) : Tonal::Ratio.new(Prime.int_from_prime_division(self.first), Prime.int_from_prime_division(self.last))
 
   # @return [Array] translated by value
   # @example
diff --git a/lib/tonal/interval.rb b/lib/tonal/interval.rb
index 30e5132..12b183d 100644
--- a/lib/tonal/interval.rb
+++ b/lib/tonal/interval.rb
@@ -2,7 +2,7 @@ class Tonal::Interval
   extend Forwardable
   include Comparable
 
-  def_delegators :@interval, :to_r, :antecedent, :consequent
+  def_delegators :@interval, :to_r, :antecedent, :consequent, :to_cents
 
   attr_reader :lower_ratio, :upper_ratio, :interval
 
@@ -13,6 +13,7 @@ def initialize(upper_ratio, lower_ratio)
     @upper_ratio = upper_ratio.ratio
     @interval = @upper_ratio / @lower_ratio
   end
+  alias :ratio :interval
   alias :lower :lower_ratio
   alias :upper :upper_ratio
   alias :numerator :antecedent
@@ -22,7 +23,7 @@ def to_a
     [lower_ratio, upper_ratio]
   end
 
-  def normalize
+  def denominize
     ratios = to_a
     lcm = ratios.denominators.lcm
     ratios.map{|r| Tonal::Ratio.new(lcm / r.denominator * r.numerator, lcm)}
diff --git a/lib/tonal/ratio.rb b/lib/tonal/ratio.rb
index 201409e..af99713 100644
--- a/lib/tonal/ratio.rb
+++ b/lib/tonal/ratio.rb
@@ -2,7 +2,7 @@ class Tonal::Ratio
   extend Forwardable
   include Comparable
 
-  def_delegators :@approximations, :neighborhood
+  def_delegators :@approximation, :neighborhood
 
   attr_reader :antecedent, :consequent, :equave, :reduced_antecedent, :reduced_consequent
 
@@ -27,29 +27,38 @@ def initialize(antecedent, consequent=nil, label: nil, equave: 2/1r)
     @equave = equave
     @reduced_antecedent, @reduced_consequent = _equave_reduce(equave)
     @label = label
-    @approximations = Approximations.new(ratio: self)
+    @approximation = Approximation.new(ratio: self)
   end
 
   alias :numerator :antecedent
   alias :denominator :consequent
 
-  # @return [Tonal::Ratio] ratio who's numerator and denominator are 1 apart
+  # @return [Tonal::Ratio] ratio who's numerator and denominator are seperated by a difference of 1
   # @example
-  #   Tonal::Ratio.superparticular(100) = (100/99)
-  # @param n numerator of ratio
+  #   Tonal::Ratio.superparticular(100) = (101/100)
+  # @param n [Integer] number from which the superior part is calculated
+  # @param factor [Rational] multiplied into the resulting ratio, default 1/1
+  # @param superpart [Symbol] assigning the superior part to the antecedent or consequent
   #
-  def self.superparticular(n)
-    superpartient(n, 1)
+  def self.superparticular(n, factor: 1/1r, superpart: :upper)
+    superpartient(n, summand: 1, factor:, superpart:)
   end
 
-  # @return [Tonal::Ratio] ratio who's numerator and denominator are a partient apart
+  # @return [Tonal::Ratio] ratio who's numerator and denominator are separated by a summand difference
   # @example
-  #   Tonal::Ratio.superpartient(100, 5) => (100/95)
-  # @param n numerator of ratio
-  # @param part partient separating the numerator and denominator
+  #   Tonal::Ratio.superpartient(23, summand: 3) => (26/23)
+  # @param n [Integer] number from which the superior part is calculated
+  # @param summand [Integer] term added to the superior part
+  # @param factor [Rational] multiplied into the resulting ratio, default 1/1
+  # @param superpart [Symbol] assigning the superior part to the antecedent or consequent
   #
-  def self.superpartient(n, part)
-    self.new(n, n-part)
+  def self.superpartient(n, summand:, factor: 1/1r, superpart: :upper)
+    case superpart.to_sym.downcase
+    when :lower, :consequent, :denominator
+      self.new(n*factor.numerator, (n+summand)*factor.denominator)
+    else
+      self.new((n+summand)*factor.numerator, n*factor.denominator)
+    end
   end
 
   # @return [Tonal::Ratio] a randomly generated ratio
@@ -58,7 +67,7 @@ def self.superpartient(n, part)
   # @param number_of_factors
   # @param within
   #
-  def self.random_ratio(number_of_factors = 2, within: 100)
+  def self.random_ratio(number_of_factors = 2, within: 100, reduced: false)
     primes = Prime.each(within).to_a
     nums = []
     dens = []
@@ -66,7 +75,7 @@ def self.random_ratio(number_of_factors = 2, within: 100)
       nums << [primes[rand(10)], rand(3)]
       dens << [primes[rand(10)], rand(3)]
     end
-    [nums, dens].ratio_from_prime_divisions
+    [nums, dens].ratio_from_prime_divisions(reduced:)
   end
 
   # @return [Tonal::Ratio] the ratio of step in the modulo
@@ -98,10 +107,10 @@ def ratio
     self
   end
 
-  # @return [Tonal::Ratio::Approximations] self's approximation instance
+  # @return [Tonal::Ratio::Approximation] self's approximation instance
   #
-  def approx
-    @approximations
+  def approximate
+    @approximation
   end
 
   # ==================================
@@ -264,15 +273,6 @@ def mirror(axis=1/1r)
     (self.class.new(axis) ** 2) / self
   end
 
-  # @return [Tonal::Ratio]
-  # @example
-  #   Tonal::Ratio.new(4/3r).mirror2(4/2r) => (3/8)
-  # @param ratio
-  #
-  def mirror2(ratio)
-    self.class.new(invert.to_r / ratio)
-  end
-
   # @return Tonal::ReducedRatio the Ernst Levy negative of self
   # @example
   #   Tonal::ReducedRatio.new(7/4r).negative => (12/7)
@@ -434,7 +434,7 @@ def wilson_height(prime_rejects: [2])
   # @param modulo
   #
   def efficiency(modulo)
-    (Tonal::Cents::CENT_SCALE * step(modulo).step / modulo.to_f) - to_cents
+    ((Tonal::Cents::CENT_SCALE * step(modulo).step / modulo.to_f) - to_cents).round(Tonal::Cents::PRECISION)
   end
 
   # @return [Array] the results of ratio dividing and multiplying self
@@ -473,7 +473,7 @@ def label
     # Return label, if defined; or,
     # Return the "antecedent/consequent", if antecedent is less than 7 digits long; or
     # Return the floating point representation rounded to 2 digits of precision
-    (@label || ((Math.log10(antecedent).to_i + 1) <= 6 ? "#{antecedent}/#{consequent}" : to_f.round(2))).to_s
+    (@label || ((Math.log10(antecedent).to_i + 1) <= 6 ? "#{antecedent}/#{consequent}" : to_f.round(Tonal::Cents::PRECISION))).to_s
   end
 
   # @return [String] the string representation of Tonal::Ratio
diff --git a/lib/tonal/tools.rb b/lib/tonal/tools.rb
index f799db4..d44e375 100644
--- a/lib/tonal/tools.rb
+++ b/lib/tonal/tools.rb
@@ -11,7 +11,7 @@ module Tonal
   require "tonal/hertz"
   require "tonal/log"
   require "tonal/log2"
-  require "tonal/approximations"
+  require "tonal/approximation"
   require "tonal/ratio"
   require "tonal/reduced_ratio"
   require "tonal/interval"
diff --git a/spec/tonal_tools/approximations_spec.rb b/spec/tonal_tools/approximation_spec.rb
similarity index 59%
rename from spec/tonal_tools/approximations_spec.rb
rename to spec/tonal_tools/approximation_spec.rb
index 6f0f584..0cc4bf5 100644
--- a/spec/tonal_tools/approximations_spec.rb
+++ b/spec/tonal_tools/approximation_spec.rb
@@ -1,8 +1,6 @@
 require "spec_helper"
 
-RSpec.describe Tonal::Ratio::Approximations do
-  let(:ratio) { Tonal::Ratio.new(3/2r) }
-
+RSpec.describe Tonal::Ratio::Approximation do
   describe "class methods" do
     describe ".new" do
       it "requires a ratio" do
@@ -15,13 +13,17 @@
     end
 
     describe ".neighbors" do
-      it "returns the neighbors distance away from self's antecedent and consequent" do
+      let(:ratio) { Tonal::Ratio.new(3/2r) }
+
+      it "returns the neighbors distance away from ratio's antecedent and consequent" do
         expect(described_class.neighbors(vacinity: ratio, away: 2)).to eq [Tonal::Ratio.new(3,2), Tonal::Ratio.new(5,2), Tonal::Ratio.new(1,2), Tonal::Ratio.new(3,4), Tonal::Ratio.new(3,0), Tonal::Ratio.new(5,4), Tonal::Ratio.new(5,0), Tonal::Ratio.new(1,4), Tonal::Ratio.new(1,0)]
       end
     end
   end
 
   describe "instance methods" do
+    let(:ratio) { Tonal::Ratio.new(3/2r) }
+
     subject { described_class.new(ratio: ratio) }
 
     describe "#neighborhood" do
@@ -35,15 +37,15 @@
 
       context "with defaults" do
         it "returns ratios up to the default depth" do
-          expect(ratio.approx.by_continued_fraction.length).to be <= Tonal::Ratio::Approximations::CONVERGENT_LIMIT
+          expect(ratio.approximate.by_continued_fraction.length).to be <= Tonal::Ratio::Approximation::CONVERGENT_LIMIT
         end
 
         it "returns ratios that are within 5 cents of self" do
-          expect(ratio.approx.by_continued_fraction.all?{|r| r.cent_diff(ratio) <= Tonal::Cents::TOLERANCE}).to be true
+          expect(ratio.approximate.by_continued_fraction.all?{|r| r.cent_diff(ratio) <= Tonal::Cents::TOLERANCE}).to be true
         end
 
         it "returns ratios with maximum primes limited only by the depth of the search" do
-          expect(ratio.approx.by_continued_fraction.all?{|r| r.max_prime <= 2549 }).to be true
+          expect(ratio.approximate.by_continued_fraction.all?{|r| r.max_prime <= 2549 }).to be true
         end
       end
     end
@@ -53,7 +55,7 @@
       let(:max_prime) { 89 }
 
       it "returns ratios with max prime" do
-        expect(ratio.approx.by_quotient_walk(max_prime: max_prime)).to eq [(18/17r), (196/185r), (89/84r), (71/67r), (53/50r), (35/33r), (17/16r)]
+        expect(ratio.approximate.by_quotient_walk(max_prime: max_prime).entries).to eq [(18/17r), (196/185r), (89/84r), (71/67r), (53/50r), (35/33r), (17/16r)]
       end
     end
 
@@ -62,7 +64,22 @@
       let(:depth) { 10 }
 
       it "returns 10 ratios" do
-        expect(ratio.approx.by_tree_path(depth: depth).count).to eq 10
+        expect(ratio.approximate.by_tree_path(depth: depth).count).to eq depth
+      end
+    end
+
+    describe "#by_superparticular" do
+      let(:ratio) { Tonal::Ratio.new(3/2r)}
+      let(:depth) { 5 }
+
+      it "returns approximations by descending superparticulars factored by ratio" do
+        expect(ratio.approximate.by_superparticular(depth: depth).entries).to eq [(351/350r), (350/349r), (349/348r), (348/347r), (347/346r)].map{|r| r * 3/2r}
+      end
+
+      context "when the superpart is the denomimator" do
+        it "returns approximations by ascending superparticulars factored by ratio" do
+          expect(ratio.approximate.by_superparticular(superpart: :lower, depth: depth).entries).to eq [(346/347r), (347/348r), (348/349r), (349/350r), (350/351r)].map{|r| r * 3/2r}
+        end
       end
     end
 
@@ -74,11 +91,11 @@
         let(:max_prime) { 23 }
         let(:max_boundary) { 10 }
         let(:max_scale) { 60 }
-        let(:ratios) { ratio.approx.by_neighborhood(max_prime: max_prime, max_boundary: max_boundary, max_scale: max_scale) }
+        let(:ratios) { ratio.approximate.by_neighborhood(max_prime: max_prime, max_boundary: max_boundary, max_scale: max_scale) }
         let(:ratio_in_cents) { ratio.to_cents }
 
         it "returns a set of ratios within 5¢ of ratio and with max prime of 23" do
-          expect(ratios).to eq [(175/117r), (176/117r)]
+          expect(ratios.entries).to eq [(175/117r), (176/117r)]
           expect(ratios.all?{|r| Tonal::Ratio.within_cents?(r.to_cents, ratio_in_cents, cents_tolerance)}).to be true
           expect(ratios.all?{|r| r.within_prime?(max_prime) }).to be true
         end
@@ -86,3 +103,7 @@
     end
   end
 end
+
+RSpec.describe Tonal::Ratio::Approximation::Set do
+  
+end
\ No newline at end of file
diff --git a/spec/tonal_tools/extensions_spec.rb b/spec/tonal_tools/extensions_spec.rb
index 7215743..c843e00 100644
--- a/spec/tonal_tools/extensions_spec.rb
+++ b/spec/tonal_tools/extensions_spec.rb
@@ -262,9 +262,9 @@
       end
     end
 
-    describe "#normalize" do
-      it "returns an array of normalized ratios" do
-        expect([4/3r, 3/2r].normalize).to eq [Tonal::Ratio.new(8,6), Tonal::Ratio.new(9,6)]
+    describe "#denominize" do
+      it "returns an array of ratios with equalized denominators" do
+        expect([4/3r, 3/2r].denominize).to eq [Tonal::Ratio.new(8,6), Tonal::Ratio.new(9,6)]
       end
     end
 
diff --git a/spec/tonal_tools/interval_spec.rb b/spec/tonal_tools/interval_spec.rb
index d6f4faa..adb35b1 100644
--- a/spec/tonal_tools/interval_spec.rb
+++ b/spec/tonal_tools/interval_spec.rb
@@ -55,12 +55,22 @@
     end
   end
 
-  describe "#normalize" do
+  describe "#denominize" do
     let(:lower_ratio) { 3/2r }
     let(:upper_ratio) { 7/4r }
 
-    it "returns the interval endpoints normalized to their lcm" do
-      expect(described_class.new(lower_ratio, upper_ratio).normalize).to eq [7/4r, 6/4r]
+    it "returns the interval endpoints with their denominators equalized" do
+      expect(described_class.new(lower_ratio, upper_ratio).denominize).to eq [7/4r, 6/4r]
+    end
+  end
+
+  describe "#ratio" do
+    let(:lower_ratio) { 3/2r }
+    let(:upper_ratio) { 7/4r }
+
+    it "returns the interval as a Tonal::ReducedRatio" do
+      expect(described_class.new(lower_ratio, upper_ratio).ratio).to be_a_kind_of(Tonal::ReducedRatio)
+      expect(described_class.new(lower_ratio, upper_ratio).ratio).to eq 12/7r
     end
   end
 
diff --git a/spec/tonal_tools/ratio_spec.rb b/spec/tonal_tools/ratio_spec.rb
index 5988677..1b73c8d 100644
--- a/spec/tonal_tools/ratio_spec.rb
+++ b/spec/tonal_tools/ratio_spec.rb
@@ -59,7 +59,7 @@
     let(:n) { rand(100)+1 }
 
     it "returns a ratio who's numerator and denominator are one apart" do
-      expect(described_class.superparticular(n)).to eq described_class.new(n, n-1)
+      expect(described_class.superparticular(n)).to eq described_class.new(n+1, n)
     end
   end
 
@@ -68,12 +68,16 @@
     let(:partient) { rand(10) }
 
     it "returns a ratio who's numerator and denominator are a partient apart" do
-      expect(described_class.superpartient(n, partient)).to eq described_class.new(n, n-partient)
+      expect(described_class.superpartient(n, summand: partient)).to eq described_class.new(n+partient, n)
     end
   end
 
   describe ".random_ratio" do
     it { expect(described_class.random_ratio).to be_a_kind_of(Tonal::Ratio) }
+
+    context "when reduced is true" do
+      it { expect(described_class.random_ratio(reduced: true)).to be_a_kind_of(Tonal::ReducedRatio) }
+    end
   end
 
   describe ".ed" do
@@ -354,7 +358,7 @@
 
     describe "#efficiency" do
       it "returns the difference between self and its step for a given modulo" do
-        expect(subject.efficiency(12)).to eq -1.9600000000000364
+        expect(subject.efficiency(12)).to eq -1.96
       end
     end
 
diff --git a/tonal-tools.gemspec b/tonal-tools.gemspec
index e703f3a..79a16be 100644
--- a/tonal-tools.gemspec
+++ b/tonal-tools.gemspec
@@ -1,8 +1,9 @@
 require "date"
+require_relative "lib/tonal/attributions"
 
 Gem::Specification.new do |spec|
   spec.name        = "tonal-tools"
-  spec.version     = "1.3.1"
+  spec.version     = Tonal::TOOLS_VERSION
   spec.summary     = "Tonal tools"
   spec.description = "Basic tools, utilities and conveniences for microtonal music making and analysis"
   spec.authors     = ["Jose Hales-Garcia"]

    
-      
    -
-
 272
 273
 274
    
@@ -4360,8 +4345,8 @@ 
    
     
    		
       
    # File 'lib/tonal/ratio.rb', line 272
 
-def mirror2(ratio)
-  self.class.new(invert.to_r / ratio)
+def mirror(axis=1/1r)
+  (self.class.new(axis) ** 2) / self
 end
    
         		
   
    
-      
    # File 'lib/tonal/ratio.rb', line 185
+      
    # File 'lib/tonal/ratio.rb', line 194
 
 def period_degrees(round: 2)
   (360.0 * Math.log(to_f, equave)).round(round)
@@ -4588,12 +4573,12 @@ 
    
       
     
 
-194
-195
-196
    
+203
+204
+205
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 194
+      
    # File 'lib/tonal/ratio.rb', line 203
 
 def period_radians(round: 2)
   (2 * Math::PI * Math.log(to_f, equave)).round(round)
@@ -5059,12 +5044,12 @@ 
    
       
     
 
-97
-98
-99
    
+106
+107
+108
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 97
+      
    # File 'lib/tonal/ratio.rb', line 106
 
 def ratio
   self
@@ -5320,12 +5305,12 @@ 
    
       
     
 
-176
-177
-178
    
+185
+186
+187
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 176
+      
    # File 'lib/tonal/ratio.rb', line 185
 
 def step(modulo=12)
   to_log2.step(modulo)
@@ -5458,12 +5443,12 @@ 
    
       
     
 
-115
-116
-117
    
+124
+125
+126
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 115
+      
    # File 'lib/tonal/ratio.rb', line 124
 
 def to_a
   [antecedent, consequent]
@@ -5529,12 +5514,12 @@ 
    
       
     
 
-167
-168
-169
    
+176
+177
+178
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 167
+      
    # File 'lib/tonal/ratio.rb', line 176
 
 def to_cents
   Tonal::Cents.new(ratio: to_r)
@@ -5596,12 +5581,12 @@ 
    
       
     
 
-140
-141
-142
    
+149
+150
+151
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 140
+      
    # File 'lib/tonal/ratio.rb', line 149
 
 def to_f
   antecedent.to_f / consequent.to_f
@@ -5684,12 +5669,12 @@ 
    
       
     
 
-149
-150
-151
    
+158
+159
+160
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 149
+      
    # File 'lib/tonal/ratio.rb', line 158
 
 def to_log(base=2)
   Tonal::Log.new(logarithmand: to_r, base: base)
@@ -5755,12 +5740,12 @@ 
    
       
     
 
-158
-159
-160
    
+167
+168
+169
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 158
+      
    # File 'lib/tonal/ratio.rb', line 167
 
 def to_log2
   Tonal::Log2.new(logarithmand: to_r)
@@ -5822,13 +5807,13 @@ 
    
       
     
 
-131
-132
-133
-134
    
+140
+141
+142
+143
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 131
+      
    # File 'lib/tonal/ratio.rb', line 140
 
 def to_r
   return Float::INFINITY if consequent.zero? || !antecedent.finite?
@@ -5895,12 +5880,12 @@ 
    
       
     
 
-234
-235
-236
    
+243
+244
+245
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 234
+      
    # File 'lib/tonal/ratio.rb', line 243
 
 def to_reduced_ratio
   Tonal::ReducedRatio.new(reduced_antecedent, reduced_consequent, equave: equave)
@@ -5962,12 +5947,12 @@ 
    
       
     
 
-123
-124
-125
    
+132
+133
+134
    
     
    		
     
-      
    # File 'lib/tonal/ratio.rb', line 123
+      
    # File 'lib/tonal/ratio.rb', line 132
 
 def to_v
   Vector[antecedent, consequent]
@@ -6271,7 +6256,7 @@ 
    
 
 
       
diff --git a/docs/Tonal/Ratio/Approximations.html b/docs/Tonal/Ratio/Approximation.html
similarity index 65%
rename from docs/Tonal/Ratio/Approximations.html
rename to docs/Tonal/Ratio/Approximation.html
index 2f1e62b..c82252f 100644
--- a/docs/Tonal/Ratio/Approximations.html
+++ b/docs/Tonal/Ratio/Approximation.html
@@ -4,7 +4,7 @@
     
 
 
-  Class: Tonal::Ratio::Approximations
+  Class: Tonal::Ratio::Approximation
   
     — Documentation by YARD 0.9.34
   
@@ -15,7 +15,7 @@
   <link rel="stylesheet" href="../../css/common.css" type="text/css" />
 
 <script type="text/javascript">
-  pathId = "Tonal::Ratio::Approximations";
+  pathId = "Tonal::Ratio::Approximation";
   relpath = '../../';
 </script>
 
@@ -39,7 +39,7 @@
     <a href="../../_index.html">Index (A)</a> »
     <span class='title'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span> » <span class='title'><span class='object_link'><a href="../Ratio.html" title="Tonal::Ratio (class)">Ratio</a></span></span>
      » 
-    <span class="title">Approximations</span>
+    <span class="title">Approximation</span>
   
 </div>
 
@@ -59,7 +59,7 @@
         <div class="clear"></div>
       </div>
 
-      <div id="content"><h1>Class: Tonal::Ratio::Approximations
+      <div id="content"><h1>Class: Tonal::Ratio::Approximation
   
   
   
@@ -74,7 +74,7 @@
         <ul class="fullTree">
           <li>Object</li>
           
-            <li class="next">Tonal::Ratio::Approximations</li>
+            <li class="next">Tonal::Ratio::Approximation</li>
           
         </ul>
         <a href="#" class="inheritanceTree">show all</a>
@@ -99,11 +99,21 @@
   
   <dl>
     <dt>Defined in:</dt>
-    <dd>lib/tonal/approximations.rb</dd>
+    <dd>lib/tonal/approximation.rb</dd>
   </dl>
   
 </div>
 
+<h2>Defined Under Namespace</h2>
+<p class="children">
+  
+    
+  
+    
+      <strong class="classes">Classes:</strong> <span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Set</a></span>
+    
+  
+</p>
 
   
     <h2>
@@ -230,7 +240,7 @@ <h2>
         <li class="public ">
   <span class="summary_signature">
     
-      <a href="#by_continued_fraction-instance_method" title="#by_continued_fraction (instance method)">#<strong>by_continued_fraction</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)  ⇒ Array </a>
+      <a href="#by_continued_fraction-instance_method" title="#by_continued_fraction (instance method)">#<strong>by_continued_fraction</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)  ⇒ Tonal::Ratio::Approximation::Set </a>
     
 
     
@@ -278,7 +288,7 @@ <h2>
         <li class="public ">
   <span class="summary_signature">
     
-      <a href="#by_quotient_walk-instance_method" title="#by_quotient_walk (instance method)">#<strong>by_quotient_walk</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)  ⇒ Array </a>
+      <a href="#by_quotient_walk-instance_method" title="#by_quotient_walk (instance method)">#<strong>by_quotient_walk</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)  ⇒ Tonal::Ratio::Approximation::Set </a>
     
 
     
@@ -302,7 +312,31 @@ <h2>
         <li class="public ">
   <span class="summary_signature">
     
-      <a href="#by_tree_path-instance_method" title="#by_tree_path (instance method)">#<strong>by_tree_path</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME)  ⇒ Array </a>
+      <a href="#by_superparticular-instance_method" title="#by_superparticular (instance method)">#<strong>by_superparticular</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: 20, max_prime: DEFAULT_MAX_PRIME, superpart: :upper)  ⇒ Tonal::Ratio::Approximation::Set </a>
+    
+
+    
+  </span>
+  
+  
+  
+  
+  
+  
+  
+
+  
+    <span class="summary_desc"><div class='inline'>
+<p>Of superparticular approximations within cent tolerance of self.</p>
+</div></span>
+  
+</li>
+
+      
+        <li class="public ">
+  <span class="summary_signature">
+    
+      <a href="#by_tree_path-instance_method" title="#by_tree_path (instance method)">#<strong>by_tree_path</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME)  ⇒ Tonal::Ratio::Approximation::Set </a>
     
 
     
@@ -326,7 +360,7 @@ <h2>
         <li class="public ">
   <span class="summary_signature">
     
-      <a href="#initialize-instance_method" title="#initialize (instance method)">#<strong>initialize</strong>(ratio:)  ⇒ Approximations </a>
+      <a href="#initialize-instance_method" title="#initialize (instance method)">#<strong>initialize</strong>(ratio:)  ⇒ Approximation </a>
     
 
     
@@ -343,7 +377,7 @@ <h2>
 
   
     <span class="summary_desc"><div class='inline'>
-<p>A new instance of Approximations.</p>
+<p>A new instance of Approximation.</p>
 </div></span>
   
 </li>
@@ -383,7 +417,7 @@ <h2>Constructor Details</h2>
     <div class="method_details first">
   <h3 class="signature first" id="initialize-instance_method">
   
-    #<strong>initialize</strong>(ratio:)  ⇒ <tt><span class='object_link'><a href="" title="Tonal::Ratio::Approximations (class)">Approximations</a></span></tt> 
+    #<strong>initialize</strong>(ratio:)  ⇒ <tt><span class='object_link'><a href="" title="Tonal::Ratio::Approximation (class)">Approximation</a></span></tt> 
   
 
   
@@ -392,7 +426,7 @@ <h3 class="signature first" id="initialize-instance_method">
 </h3><div class="docstring">
   <div class="discussion">
     
-<p>Returns a new instance of Approximations.</p>
+<p>Returns a new instance of Approximation.</p>
 
 
   </div>
@@ -425,7 +459,7 @@ <h3 class="signature first" id="initialize-instance_method">
 18</pre>
     </td>
     <td>
-      <pre class="code"><span class="info file"># File 'lib/tonal/approximations.rb', line 15</span>
+      <pre class="code"><span class="info file"># File 'lib/tonal/approximation.rb', line 15</span>
 
 <span class='kw'>def</span> <span class='id identifier rubyid_initialize'>initialize</span><span class='lparen'>(</span><span class='label'>ratio:</span><span class='rparen'>)</span>
   <span class='id identifier rubyid_raise'>raise</span> <span class='const'>ArgumentError</span><span class='comma'>,</span> <span class='tstring'><span class='tstring_beg'>"</span><span class='tstring_content'>Tonal::Ratio required</span><span class='tstring_end'>"</span></span> <span class='kw'>unless</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_kind_of?'>kind_of?</span><span class='lparen'>(</span><span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Ratio.html" title="Tonal::Ratio (class)">Ratio</a></span></span><span class='rparen'>)</span>
@@ -474,7 +508,7 @@ <h3 class="signature first" id="ratio-instance_method">
 15</pre>
     </td>
     <td>
-      <pre class="code"><span class="info file"># File 'lib/tonal/approximations.rb', line 13</span>
+      <pre class="code"><span class="info file"># File 'lib/tonal/approximation.rb', line 13</span>
 
 <span class='kw'>def</span> <span class='id identifier rubyid_ratio'>ratio</span>
   <span class='ivar'>@ratio</span>
@@ -514,7 +548,7 @@ <h3 class="signature first" id="neighbors-class_method">
     <p class="tag_title">Examples:</p>
     
       
-      <pre class="example code"><code>Tonal::Ratio::Approximations.neighbors(vacinity: (3/2r).ratio(reduced:false).scale(256), away: 1)
+      <pre class="example code"><code>Tonal::Ratio::Approximation.neighbors(vacinity: (3/2r).ratio(reduced:false).scale(256), away: 1)
   => [(768/513), (767/512), (768/512), (769/512), (768/511)]</code></pre>
     
   </div>
@@ -566,20 +600,20 @@ <h3 class="signature first" id="neighbors-class_method">
       <pre class="lines">
 
 
-136
-137
-138
-139
-140
-141
-142
-143
-144
-145
-146</pre>
+160
+161
+162
+163
+164
+165
+166
+167
+168
+169
+170</pre>
     </td>
     <td>
-      <pre class="code"><span class="info file"># File 'lib/tonal/approximations.rb', line 136</span>
+      <pre class="code"><span class="info file"># File 'lib/tonal/approximation.rb', line 160</span>
 
 <span class='kw'>def</span> <span class='kw'>self</span><span class='period'>.</span><span class='id identifier rubyid_neighbors'>neighbors</span><span class='lparen'>(</span><span class='label'>vacinity:</span><span class='comma'>,</span> <span class='label'>away:</span> <span class='int'>1</span><span class='rparen'>)</span>
   <span class='lbracket'>[</span><span class='id identifier rubyid_vacinity'>vacinity</span><span class='comma'>,</span>
@@ -606,7 +640,7 @@ <h2>Instance Method Details</h2>
       <div class="method_details first">
   <h3 class="signature first" id="by_continued_fraction-instance_method">
   
-    #<strong>by_continued_fraction</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)  ⇒ <tt><span class='object_link'><a href="../../Array.html" title="Array (class)">Array</a></span></tt> 
+    #<strong>by_continued_fraction</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)  ⇒ <tt><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Tonal::Ratio::Approximation::Set</a></span></tt> 
   
 
   
@@ -626,8 +660,8 @@ <h3 class="signature first" id="by_continued_fraction-instance_method">
     <p class="tag_title">Examples:</p>
     
       
-      <pre class="example code"><code>Tonal::Ratio.ed(12,1).by_continued_fraction
-=> [(18/17), (196/185), (1657/1564), (7893/7450), (18904/17843), (3118/2943), (1461/1379), (89/84), (17/16)]</code></pre>
+      <pre class="example code"><code>Tonal::Ratio.ed(12,1).approximate.by_continued_fraction
+=> (4771397596969315/4503599627370496): [(18/17), (196/185), (1657/1564), (7893/7450), (18904/17843), (3118/2943), (1461/1379), (89/84), (17/16)]</code></pre>
     
   </div>
 <p class="tag_title">Parameters:</p>
@@ -644,6 +678,11 @@ <h3 class="signature first" id="by_continued_fraction-instance_method">
         <em class="default">(defaults to: <tt>Tonal::Cents::TOLERANCE</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the cents tolerance used to scope the collection</p>
+</div>
+      
     </li>
   
     <li>
@@ -657,6 +696,11 @@ <h3 class="signature first" id="by_continued_fraction-instance_method">
         <em class="default">(defaults to: <tt>DEFAULT_DEPTH</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the maximum number of ratios in the collection</p>
+</div>
+      
     </li>
   
     <li>
@@ -670,6 +714,11 @@ <h3 class="signature first" id="by_continued_fraction-instance_method">
         <em class="default">(defaults to: <tt>DEFAULT_MAX_PRIME</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the maximum prime number to allow in the collection</p>
+</div>
+      
     </li>
   
     <li>
@@ -683,6 +732,11 @@ <h3 class="signature first" id="by_continued_fraction-instance_method">
         <em class="default">(defaults to: <tt>CONVERGENT_LIMIT</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the number of convergents to limit the ContinuedFraction method</p>
+</div>
+      
     </li>
   
 </ul>
@@ -693,7 +747,7 @@ <h3 class="signature first" id="by_continued_fraction-instance_method">
     <li>
       
       
-        <span class='type'>(<tt><span class='object_link'><a href="../../Array.html" title="Array (class)">Array</a></span></tt>)</span>
+        <span class='type'>(<tt><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Tonal::Ratio::Approximation::Set</a></span></tt>)</span>
       
       
       
@@ -725,18 +779,18 @@ <h3 class="signature first" id="by_continued_fraction-instance_method">
 39</pre>
     </td>
     <td>
-      <pre class="code"><span class="info file"># File 'lib/tonal/approximations.rb', line 29</span>
+      <pre class="code"><span class="info file"># File 'lib/tonal/approximation.rb', line 29</span>
 
-<span class='kw'>def</span> <span class='id identifier rubyid_by_continued_fraction'>by_continued_fraction</span><span class='lparen'>(</span><span class='label'>cents_tolerance:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html#TOLERANCE-constant" title="Tonal::Cents::TOLERANCE (constant)">TOLERANCE</a></span></span><span class='comma'>,</span> <span class='label'>depth:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_DEPTH-constant" title="Tonal::Ratio::Approximations::DEFAULT_DEPTH (constant)">DEFAULT_DEPTH</a></span></span><span class='comma'>,</span> <span class='label'>max_prime:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_PRIME-constant" title="Tonal::Ratio::Approximations::DEFAULT_MAX_PRIME (constant)">DEFAULT_MAX_PRIME</a></span></span><span class='comma'>,</span> <span class='label'>conv_limit:</span> <span class='const'><span class='object_link'><a href="#CONVERGENT_LIMIT-constant" title="Tonal::Ratio::Approximations::CONVERGENT_LIMIT (constant)">CONVERGENT_LIMIT</a></span></span><span class='rparen'>)</span>
+<span class='kw'>def</span> <span class='id identifier rubyid_by_continued_fraction'>by_continued_fraction</span><span class='lparen'>(</span><span class='label'>cents_tolerance:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html#TOLERANCE-constant" title="Tonal::Cents::TOLERANCE (constant)">TOLERANCE</a></span></span><span class='comma'>,</span> <span class='label'>depth:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_DEPTH-constant" title="Tonal::Ratio::Approximation::DEFAULT_DEPTH (constant)">DEFAULT_DEPTH</a></span></span><span class='comma'>,</span> <span class='label'>max_prime:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_PRIME-constant" title="Tonal::Ratio::Approximation::DEFAULT_MAX_PRIME (constant)">DEFAULT_MAX_PRIME</a></span></span><span class='comma'>,</span> <span class='label'>conv_limit:</span> <span class='const'><span class='object_link'><a href="#CONVERGENT_LIMIT-constant" title="Tonal::Ratio::Approximation::CONVERGENT_LIMIT (constant)">CONVERGENT_LIMIT</a></span></span><span class='rparen'>)</span>
   <span class='id identifier rubyid_self_in_cents'>self_in_cents</span> <span class='op'>=</span> <span class='id identifier rubyid_to_cents'>to_cents</span>
   <span class='id identifier rubyid_within'>within</span> <span class='op'>=</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='period'>.</span><span class='id identifier rubyid_kind_of?'>kind_of?</span><span class='lparen'>(</span><span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='rparen'>)</span> <span class='op'>?</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span> <span class='op'>:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="../Cents.html#initialize-instance_method" title="Tonal::Cents#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>cents:</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='rparen'>)</span>
-  <span class='lbracket'>[</span><span class='rbracket'>]</span><span class='period'>.</span><span class='id identifier rubyid_tap'>tap</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_results'>results</span><span class='op'>|</span>
+  <span class='const'><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Set</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="Approximation/Set.html#initialize-instance_method" title="Tonal::Ratio::Approximation::Set#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>ratio:</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_tap'>tap</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_set'>set</span><span class='op'>|</span>
     <span class='const'>ContinuedFraction</span><span class='period'>.</span><span class='id identifier rubyid_new'>new</span><span class='lparen'>(</span><span class='id identifier rubyid_antecedent'>antecedent</span><span class='period'>.</span><span class='id identifier rubyid_to_f'>to_f</span><span class='op'>/</span><span class='id identifier rubyid_consequent'>consequent</span><span class='comma'>,</span> <span class='id identifier rubyid_conv_limit'>conv_limit</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_convergents'>convergents</span><span class='period'>.</span><span class='id identifier rubyid_each'>each</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_convergent'>convergent</span><span class='op'>|</span>
       <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='op'>=</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_new'>new</span><span class='lparen'>(</span><span class='id identifier rubyid_convergent'>convergent</span><span class='period'>.</span><span class='id identifier rubyid_numerator'>numerator</span><span class='comma'>,</span><span class='id identifier rubyid_convergent'>convergent</span><span class='period'>.</span><span class='id identifier rubyid_denominator'>denominator</span><span class='rparen'>)</span>
-      <span class='id identifier rubyid_results'>results</span> <span class='op'><<</span> <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='kw'>if</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_within_cents?'>within_cents?</span><span class='lparen'>(</span><span class='id identifier rubyid_self_in_cents'>self_in_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_to_cents'>to_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_within'>within</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_within_prime?'>within_prime?</span><span class='lparen'>(</span><span class='id identifier rubyid_max_prime'>max_prime</span><span class='rparen'>)</span>
-      <span class='kw'>break</span> <span class='kw'>if</span> <span class='id identifier rubyid_results'>results</span><span class='period'>.</span><span class='id identifier rubyid_length'>length</span> <span class='op'>>=</span> <span class='id identifier rubyid_depth'>depth</span>
+      <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_ratios'>ratios</span> <span class='op'><<</span> <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='kw'>if</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_within_cents?'>within_cents?</span><span class='lparen'>(</span><span class='id identifier rubyid_self_in_cents'>self_in_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_to_cents'>to_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_within'>within</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_within_prime?'>within_prime?</span><span class='lparen'>(</span><span class='id identifier rubyid_max_prime'>max_prime</span><span class='rparen'>)</span>
+      <span class='kw'>break</span> <span class='kw'>if</span> <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_length'>length</span> <span class='op'>>=</span> <span class='id identifier rubyid_depth'>depth</span>
     <span class='kw'>end</span>
-  <span class='kw'>end</span><span class='period'>.</span><span class='id identifier rubyid_sort'>sort</span>
+  <span class='kw'>end</span>
 <span class='kw'>end</span></pre>
     </td>
   </tr>
@@ -766,8 +820,8 @@ <h3 class="signature " id="by_neighborhood-instance_method">
     <p class="tag_title">Examples:</p>
     
       
-      <pre class="example code"><code>Tonal::Ratio.new(3,2).by_neighborhood(max_prime: 23, cents_tolerance: 5, max_boundary: 10, max_scale: 60)
-=> [(175/117), (176/117)]</code></pre>
+      <pre class="example code"><code>Tonal::Ratio.new(3,2).approximate.by_neighborhood(max_prime: 23, cents_tolerance: 5, max_boundary: 10, max_scale: 60)
+=> (3/2): [(175/117), (176/117)]</code></pre>
     
   </div>
 <p class="tag_title">Parameters:</p>
@@ -786,7 +840,7 @@ <h3 class="signature " id="by_neighborhood-instance_method">
       
         —
         <div class='inline'>
-<p>the maximum cents self is allowed from grid ratios</p>
+<p>the cents tolerance used to scope the collection</p>
 </div>
       
     </li>
@@ -804,7 +858,7 @@ <h3 class="signature " id="by_neighborhood-instance_method">
       
         —
         <div class='inline'>
-<p>the maximum depth the array will get</p>
+<p>the maximum number of ratios in the collection</p>
 </div>
       
     </li>
@@ -822,7 +876,7 @@ <h3 class="signature " id="by_neighborhood-instance_method">
       
         —
         <div class='inline'>
-<p>the maximum prime the grid ratios will contain</p>
+<p>the maximum prime number to allow in the collection</p>
 </div>
       
     </li>
@@ -890,49 +944,49 @@ <h3 class="signature " id="by_neighborhood-instance_method">
       <pre class="lines">
 
 
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+116
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     </td>
     <td>
-      <pre class="code"><span class="info file"># File 'lib/tonal/approximations.rb', line 92</span>
+      <pre class="code"><span class="info file"># File 'lib/tonal/approximation.rb', line 116</span>
 
-<span class='kw'>def</span> <span class='id identifier rubyid_by_neighborhood'>by_neighborhood</span><span class='lparen'>(</span><span class='label'>cents_tolerance:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html#TOLERANCE-constant" title="Tonal::Cents::TOLERANCE (constant)">TOLERANCE</a></span></span><span class='comma'>,</span> <span class='label'>depth:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_DEPTH-constant" title="Tonal::Ratio::Approximations::DEFAULT_DEPTH (constant)">DEFAULT_DEPTH</a></span></span><span class='comma'>,</span> <span class='label'>max_prime:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_PRIME-constant" title="Tonal::Ratio::Approximations::DEFAULT_MAX_PRIME (constant)">DEFAULT_MAX_PRIME</a></span></span><span class='comma'>,</span> <span class='label'>max_boundary:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_GRID_BOUNDARY-constant" title="Tonal::Ratio::Approximations::DEFAULT_MAX_GRID_BOUNDARY (constant)">DEFAULT_MAX_GRID_BOUNDARY</a></span></span><span class='comma'>,</span> <span class='label'>max_scale:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_GRID_SCALE-constant" title="Tonal::Ratio::Approximations::DEFAULT_MAX_GRID_SCALE (constant)">DEFAULT_MAX_GRID_SCALE</a></span></span><span class='rparen'>)</span>
+<span class='kw'>def</span> <span class='id identifier rubyid_by_neighborhood'>by_neighborhood</span><span class='lparen'>(</span><span class='label'>cents_tolerance:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html#TOLERANCE-constant" title="Tonal::Cents::TOLERANCE (constant)">TOLERANCE</a></span></span><span class='comma'>,</span> <span class='label'>depth:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_DEPTH-constant" title="Tonal::Ratio::Approximation::DEFAULT_DEPTH (constant)">DEFAULT_DEPTH</a></span></span><span class='comma'>,</span> <span class='label'>max_prime:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_PRIME-constant" title="Tonal::Ratio::Approximation::DEFAULT_MAX_PRIME (constant)">DEFAULT_MAX_PRIME</a></span></span><span class='comma'>,</span> <span class='label'>max_boundary:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_GRID_BOUNDARY-constant" title="Tonal::Ratio::Approximation::DEFAULT_MAX_GRID_BOUNDARY (constant)">DEFAULT_MAX_GRID_BOUNDARY</a></span></span><span class='comma'>,</span> <span class='label'>max_scale:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_GRID_SCALE-constant" title="Tonal::Ratio::Approximation::DEFAULT_MAX_GRID_SCALE (constant)">DEFAULT_MAX_GRID_SCALE</a></span></span><span class='rparen'>)</span>
   <span class='id identifier rubyid_self_in_cents'>self_in_cents</span> <span class='op'>=</span> <span class='id identifier rubyid_to_cents'>to_cents</span>
   <span class='id identifier rubyid_within'>within</span> <span class='op'>=</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='period'>.</span><span class='id identifier rubyid_kind_of?'>kind_of?</span><span class='lparen'>(</span><span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='rparen'>)</span> <span class='op'>?</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span> <span class='op'>:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="../Cents.html#initialize-instance_method" title="Tonal::Cents#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>cents:</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='rparen'>)</span>
-  <span class='lbracket'>[</span><span class='rbracket'>]</span><span class='period'>.</span><span class='id identifier rubyid_tap'>tap</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_results'>results</span><span class='op'>|</span>
+  <span class='const'><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Set</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="Approximation/Set.html#initialize-instance_method" title="Tonal::Ratio::Approximation::Set#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>ratio:</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_tap'>tap</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_set'>set</span><span class='op'>|</span>
     <span class='id identifier rubyid_scale'>scale</span> <span class='op'>=</span> <span class='int'>1</span>
     <span class='id identifier rubyid_boundary'>boundary</span> <span class='op'>=</span> <span class='int'>1</span>
 
-    <span class='kw'>while</span> <span class='id identifier rubyid_results'>results</span><span class='period'>.</span><span class='id identifier rubyid_length'>length</span> <span class='op'><=</span> <span class='id identifier rubyid_depth'>depth</span> <span class='op'>&&</span> <span class='id identifier rubyid_scale'>scale</span> <span class='op'><=</span> <span class='id identifier rubyid_max_scale'>max_scale</span> <span class='kw'>do</span>
+    <span class='kw'>while</span> <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_length'>length</span> <span class='op'><=</span> <span class='id identifier rubyid_depth'>depth</span> <span class='op'>&&</span> <span class='id identifier rubyid_scale'>scale</span> <span class='op'><=</span> <span class='id identifier rubyid_max_scale'>max_scale</span> <span class='kw'>do</span>
       <span class='kw'>while</span> <span class='id identifier rubyid_boundary'>boundary</span> <span class='op'><=</span> <span class='id identifier rubyid_max_boundary'>max_boundary</span>
         <span class='id identifier rubyid_vacinity'>vacinity</span> <span class='op'>=</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_respond_to?'>respond_to?</span><span class='lparen'>(</span><span class='symbol'>:to_basic_ratio</span><span class='rparen'>)</span> <span class='op'>?</span> <span class='id identifier rubyid_to_basic_ratio'>to_basic_ratio</span><span class='period'>.</span><span class='id identifier rubyid_scale'>scale</span><span class='lparen'>(</span><span class='id identifier rubyid_scale'>scale</span><span class='rparen'>)</span> <span class='op'>:</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_scale'>scale</span><span class='lparen'>(</span><span class='id identifier rubyid_scale'>scale</span><span class='rparen'>)</span>
         <span class='kw'>self</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_neighbors'>neighbors</span><span class='lparen'>(</span><span class='label'>away:</span> <span class='id identifier rubyid_boundary'>boundary</span><span class='comma'>,</span> <span class='label'>vacinity:</span> <span class='id identifier rubyid_vacinity'>vacinity</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_each'>each</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_neighbor'>neighbor</span><span class='op'>|</span>
-          <span class='id identifier rubyid_results'>results</span> <span class='op'><<</span> <span class='id identifier rubyid_neighbor'>neighbor</span> <span class='kw'>if</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_within_cents?'>within_cents?</span><span class='lparen'>(</span><span class='id identifier rubyid_self_in_cents'>self_in_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_neighbor'>neighbor</span><span class='period'>.</span><span class='id identifier rubyid_to_cents'>to_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_within'>within</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_neighbor'>neighbor</span><span class='period'>.</span><span class='id identifier rubyid_within_prime?'>within_prime?</span><span class='lparen'>(</span><span class='id identifier rubyid_max_prime'>max_prime</span><span class='rparen'>)</span>
+          <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_ratios'>ratios</span> <span class='op'><<</span> <span class='id identifier rubyid_neighbor'>neighbor</span> <span class='kw'>if</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_within_cents?'>within_cents?</span><span class='lparen'>(</span><span class='id identifier rubyid_self_in_cents'>self_in_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_neighbor'>neighbor</span><span class='period'>.</span><span class='id identifier rubyid_to_cents'>to_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_within'>within</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_neighbor'>neighbor</span><span class='period'>.</span><span class='id identifier rubyid_within_prime?'>within_prime?</span><span class='lparen'>(</span><span class='id identifier rubyid_max_prime'>max_prime</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_neighbor'>neighbor</span> <span class='op'>!=</span> <span class='id identifier rubyid_ratio'>ratio</span>
         <span class='kw'>end</span>
         <span class='id identifier rubyid_boundary'>boundary</span> <span class='op'>+=</span> <span class='int'>1</span>
       <span class='kw'>end</span>
       <span class='id identifier rubyid_boundary'>boundary</span> <span class='op'>=</span> <span class='int'>1</span>
       <span class='id identifier rubyid_scale'>scale</span> <span class='op'>+=</span> <span class='int'>1</span>
     <span class='kw'>end</span>
-  <span class='kw'>end</span><span class='period'>.</span><span class='id identifier rubyid_uniq'>uniq</span><span class='lparen'>(</span><span class='op'>&</span><span class='symbol'>:to_r</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_reject'>reject</span><span class='lbrace'>{</span><span class='op'>|</span><span class='id identifier rubyid_r'>r</span><span class='op'>|</span> <span class='id identifier rubyid_r'>r</span> <span class='op'>==</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='rbrace'>}</span><span class='period'>.</span><span class='id identifier rubyid_sort'>sort</span>
+  <span class='kw'>end</span>
 <span class='kw'>end</span></pre>
     </td>
   </tr>
@@ -942,7 +996,7 @@ <h3 class="signature " id="by_neighborhood-instance_method">
       <div class="method_details ">
   <h3 class="signature " id="by_quotient_walk-instance_method">
   
-    #<strong>by_quotient_walk</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)  ⇒ <tt><span class='object_link'><a href="../../Array.html" title="Array (class)">Array</a></span></tt> 
+    #<strong>by_quotient_walk</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME, conv_limit: CONVERGENT_LIMIT)  ⇒ <tt><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Tonal::Ratio::Approximation::Set</a></span></tt> 
   
 
   
@@ -962,8 +1016,8 @@ <h3 class="signature " id="by_quotient_walk-instance_method">
     <p class="tag_title">Examples:</p>
     
       
-      <pre class="example code"><code>Tonal::Ratio.ed(12,1).by_quotient_walk(max_prime: 89)
-=> [(18/17), (196/185), (89/84), (71/67), (53/50), (35/33), (17/16)]</code></pre>
+      <pre class="example code"><code>Tonal::Ratio.ed(12,1).approximate.by_quotient_walk(max_prime: 89)
+=> (4771397596969315/4503599627370496): [(18/17), (196/185), (89/84), (71/67), (53/50), (35/33), (17/16)]</code></pre>
     
   </div>
 <p class="tag_title">Parameters:</p>
@@ -980,6 +1034,11 @@ <h3 class="signature " id="by_quotient_walk-instance_method">
         <em class="default">(defaults to: <tt>Tonal::Cents::TOLERANCE</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the cents tolerance used to scope the collection</p>
+</div>
+      
     </li>
   
     <li>
@@ -993,6 +1052,11 @@ <h3 class="signature " id="by_quotient_walk-instance_method">
         <em class="default">(defaults to: <tt>DEFAULT_DEPTH</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the maximum number of ratios in the collection</p>
+</div>
+      
     </li>
   
     <li>
@@ -1006,6 +1070,29 @@ <h3 class="signature " id="by_quotient_walk-instance_method">
         <em class="default">(defaults to: <tt>DEFAULT_MAX_PRIME</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the maximum prime number to allow in the collection</p>
+</div>
+      
+    </li>
+  
+    <li>
+      
+        <span class='name'>conv_limit</span>
+      
+      
+        <span class='type'></span>
+      
+      
+        <em class="default">(defaults to: <tt>CONVERGENT_LIMIT</tt>)</em>
+      
+      
+        —
+        <div class='inline'>
+<p>the number of convergents to limit the ContinuedFraction method</p>
+</div>
+      
     </li>
   
 </ul>
@@ -1016,7 +1103,7 @@ <h3 class="signature " id="by_quotient_walk-instance_method">
     <li>
       
       
-        <span class='type'>(<tt><span class='object_link'><a href="../../Array.html" title="Array (class)">Array</a></span></tt>)</span>
+        <span class='type'>(<tt><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Tonal::Ratio::Approximation::Set</a></span></tt>)</span>
       
       
       
@@ -1035,7 +1122,6 @@ <h3 class="signature " id="by_quotient_walk-instance_method">
       <pre class="lines">
 
 
-49
 50
 51
 52
@@ -1046,22 +1132,187 @@ <h3 class="signature " id="by_quotient_walk-instance_method">
 57
 58
 59
-60</pre>
+60
+61</pre>
     </td>
     <td>
-      <pre class="code"><span class="info file"># File 'lib/tonal/approximations.rb', line 49</span>
+      <pre class="code"><span class="info file"># File 'lib/tonal/approximation.rb', line 50</span>
 
-<span class='kw'>def</span> <span class='id identifier rubyid_by_quotient_walk'>by_quotient_walk</span><span class='lparen'>(</span><span class='label'>cents_tolerance:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html#TOLERANCE-constant" title="Tonal::Cents::TOLERANCE (constant)">TOLERANCE</a></span></span><span class='comma'>,</span> <span class='label'>depth:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_DEPTH-constant" title="Tonal::Ratio::Approximations::DEFAULT_DEPTH (constant)">DEFAULT_DEPTH</a></span></span><span class='comma'>,</span> <span class='label'>max_prime:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_PRIME-constant" title="Tonal::Ratio::Approximations::DEFAULT_MAX_PRIME (constant)">DEFAULT_MAX_PRIME</a></span></span><span class='comma'>,</span> <span class='label'>conv_limit:</span> <span class='const'><span class='object_link'><a href="#CONVERGENT_LIMIT-constant" title="Tonal::Ratio::Approximations::CONVERGENT_LIMIT (constant)">CONVERGENT_LIMIT</a></span></span><span class='rparen'>)</span>
+<span class='kw'>def</span> <span class='id identifier rubyid_by_quotient_walk'>by_quotient_walk</span><span class='lparen'>(</span><span class='label'>cents_tolerance:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html#TOLERANCE-constant" title="Tonal::Cents::TOLERANCE (constant)">TOLERANCE</a></span></span><span class='comma'>,</span> <span class='label'>depth:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_DEPTH-constant" title="Tonal::Ratio::Approximation::DEFAULT_DEPTH (constant)">DEFAULT_DEPTH</a></span></span><span class='comma'>,</span> <span class='label'>max_prime:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_PRIME-constant" title="Tonal::Ratio::Approximation::DEFAULT_MAX_PRIME (constant)">DEFAULT_MAX_PRIME</a></span></span><span class='comma'>,</span> <span class='label'>conv_limit:</span> <span class='const'><span class='object_link'><a href="#CONVERGENT_LIMIT-constant" title="Tonal::Ratio::Approximation::CONVERGENT_LIMIT (constant)">CONVERGENT_LIMIT</a></span></span><span class='rparen'>)</span>
   <span class='id identifier rubyid_self_in_cents'>self_in_cents</span> <span class='op'>=</span> <span class='id identifier rubyid_to_cents'>to_cents</span>
   <span class='id identifier rubyid_within'>within</span> <span class='op'>=</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='period'>.</span><span class='id identifier rubyid_kind_of?'>kind_of?</span><span class='lparen'>(</span><span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='rparen'>)</span> <span class='op'>?</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span> <span class='op'>:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="../Cents.html#initialize-instance_method" title="Tonal::Cents#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>cents:</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='rparen'>)</span>
 
-  <span class='lbracket'>[</span><span class='rbracket'>]</span><span class='period'>.</span><span class='id identifier rubyid_tap'>tap</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_results'>results</span><span class='op'>|</span>
+  <span class='const'><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Set</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="Approximation/Set.html#initialize-instance_method" title="Tonal::Ratio::Approximation::Set#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>ratio:</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_tap'>tap</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_set'>set</span><span class='op'>|</span>
     <span class='const'>FractionTree</span><span class='period'>.</span><span class='id identifier rubyid_quotient_walk'>quotient_walk</span><span class='lparen'>(</span><span class='id identifier rubyid_to_f'>to_f</span><span class='comma'>,</span> <span class='label'>limit:</span> <span class='id identifier rubyid_conv_limit'>conv_limit</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_each'>each</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_node'>node</span><span class='op'>|</span>
       <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='op'>=</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_new'>new</span><span class='lparen'>(</span><span class='id identifier rubyid_node'>node</span><span class='period'>.</span><span class='id identifier rubyid_weight'>weight</span><span class='rparen'>)</span>
-      <span class='id identifier rubyid_results'>results</span> <span class='op'><<</span> <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='kw'>if</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_within_cents?'>within_cents?</span><span class='lparen'>(</span><span class='id identifier rubyid_self_in_cents'>self_in_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_to_cents'>to_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_within'>within</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_within_prime?'>within_prime?</span><span class='lparen'>(</span><span class='id identifier rubyid_max_prime'>max_prime</span><span class='rparen'>)</span>
-      <span class='kw'>break</span> <span class='kw'>if</span> <span class='id identifier rubyid_results'>results</span><span class='period'>.</span><span class='id identifier rubyid_length'>length</span> <span class='op'>>=</span> <span class='id identifier rubyid_depth'>depth</span>
+      <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_ratios'>ratios</span> <span class='op'><<</span> <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='kw'>if</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_within_cents?'>within_cents?</span><span class='lparen'>(</span><span class='id identifier rubyid_self_in_cents'>self_in_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_to_cents'>to_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_within'>within</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_within_prime?'>within_prime?</span><span class='lparen'>(</span><span class='id identifier rubyid_max_prime'>max_prime</span><span class='rparen'>)</span>
+      <span class='kw'>break</span> <span class='kw'>if</span> <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_length'>length</span> <span class='op'>>=</span> <span class='id identifier rubyid_depth'>depth</span>
     <span class='kw'>end</span>
-  <span class='kw'>end</span><span class='period'>.</span><span class='id identifier rubyid_sort'>sort</span>
+  <span class='kw'>end</span>
+<span class='kw'>end</span></pre>
+    </td>
+  </tr>
+</table>
+</div>
+    
+      <div class="method_details ">
+  <h3 class="signature " id="by_superparticular-instance_method">
+  
+    #<strong>by_superparticular</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: 20, max_prime: DEFAULT_MAX_PRIME, superpart: :upper)  ⇒ <tt><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Tonal::Ratio::Approximation::Set</a></span></tt> 
+  
+
+  
+
+  
+</h3><div class="docstring">
+  <div class="discussion">
+    
+<p>Returns of superparticular approximations within cent tolerance of self.</p>
+
+
+  </div>
+</div>
+<div class="tags">
+  
+  <div class="examples">
+    <p class="tag_title">Examples:</p>
+    
+      
+      <pre class="example code"><code>Tonal::Ratio.new(3/2r).approximate.by_superparticular
+=> (3/2): [(1098/730), (1095/728), (1092/726), (1089/724), (1086/722), (1083/720), (1080/718), (1077/716), (1074/714), (1071/712), (1068/710), (1065/708), (1062/706), (1059/704), (1056/702), (1053/700), (1050/698), (1047/696), (1044/694), (1041/692)]</code></pre>
+    
+  </div>
+<p class="tag_title">Parameters:</p>
+<ul class="param">
+  
+    <li>
+      
+        <span class='name'>cents_tolerance</span>
+      
+      
+        <span class='type'></span>
+      
+      
+        <em class="default">(defaults to: <tt>Tonal::Cents::TOLERANCE</tt>)</em>
+      
+      
+        —
+        <div class='inline'>
+<p>the cents tolerance used to scope the collection</p>
+</div>
+      
+    </li>
+  
+    <li>
+      
+        <span class='name'>depth</span>
+      
+      
+        <span class='type'></span>
+      
+      
+        <em class="default">(defaults to: <tt>20</tt>)</em>
+      
+      
+        —
+        <div class='inline'>
+<p>the maximum number of ratios in the collection</p>
+</div>
+      
+    </li>
+  
+    <li>
+      
+        <span class='name'>max_prime</span>
+      
+      
+        <span class='type'></span>
+      
+      
+        <em class="default">(defaults to: <tt>DEFAULT_MAX_PRIME</tt>)</em>
+      
+      
+        —
+        <div class='inline'>
+<p>the maximum prime number to allow in the collection</p>
+</div>
+      
+    </li>
+  
+    <li>
+      
+        <span class='name'>superpart</span>
+      
+      
+        <span class='type'></span>
+      
+      
+        <em class="default">(defaults to: <tt>:upper</tt>)</em>
+      
+      
+        —
+        <div class='inline'>
+<p>if the superior part is the numerator or denominator</p>
+</div>
+      
+    </li>
+  
+</ul>
+
+<p class="tag_title">Returns:</p>
+<ul class="return">
+  
+    <li>
+      
+      
+        <span class='type'>(<tt><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Tonal::Ratio::Approximation::Set</a></span></tt>)</span>
+      
+      
+      
+        —
+        <div class='inline'>
+<p>of superparticular approximations within cent tolerance of self</p>
+</div>
+      
+    </li>
+  
+</ul>
+
+</div><table class="source_code">
+  <tr>
+    <td>
+      <pre class="lines">
+
+
+92
+93
+94
+95
+96
+97
+98
+99
+100
+101
+102
+103
+104</pre>
+    </td>
+    <td>
+      <pre class="code"><span class="info file"># File 'lib/tonal/approximation.rb', line 92</span>
+
+<span class='kw'>def</span> <span class='id identifier rubyid_by_superparticular'>by_superparticular</span><span class='lparen'>(</span><span class='label'>cents_tolerance:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html#TOLERANCE-constant" title="Tonal::Cents::TOLERANCE (constant)">TOLERANCE</a></span></span><span class='comma'>,</span> <span class='label'>depth:</span> <span class='int'>20</span><span class='comma'>,</span> <span class='label'>max_prime:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_PRIME-constant" title="Tonal::Ratio::Approximation::DEFAULT_MAX_PRIME (constant)">DEFAULT_MAX_PRIME</a></span></span><span class='comma'>,</span> <span class='label'>superpart:</span> <span class='symbol'>:upper</span><span class='rparen'>)</span>
+  <span class='id identifier rubyid_self_in_cents'>self_in_cents</span> <span class='op'>=</span> <span class='id identifier rubyid_to_cents'>to_cents</span>
+  <span class='id identifier rubyid_within'>within</span> <span class='op'>=</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='period'>.</span><span class='id identifier rubyid_kind_of?'>kind_of?</span><span class='lparen'>(</span><span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='rparen'>)</span> <span class='op'>?</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span> <span class='op'>:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="../Cents.html#initialize-instance_method" title="Tonal::Cents#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>cents:</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='rparen'>)</span>
+  <span class='const'><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Set</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="Approximation/Set.html#initialize-instance_method" title="Tonal::Ratio::Approximation::Set#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>ratio:</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_tap'>tap</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_set'>set</span><span class='op'>|</span>
+    <span class='id identifier rubyid_n'>n</span> <span class='op'>=</span> <span class='int'>1</span>
+    <span class='kw'>while</span> <span class='kw'>true</span> <span class='kw'>do</span>
+      <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='op'>=</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_superparticular'>superparticular</span><span class='lparen'>(</span><span class='id identifier rubyid_n'>n</span><span class='comma'>,</span> <span class='label'>factor:</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_to_r'>to_r</span><span class='comma'>,</span> <span class='label'>superpart:</span><span class='rparen'>)</span>
+      <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_ratios'>ratios</span> <span class='op'><<</span> <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='kw'>if</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_within_cents?'>within_cents?</span><span class='lparen'>(</span><span class='id identifier rubyid_self_in_cents'>self_in_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_to_cents'>to_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_within'>within</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_within_prime?'>within_prime?</span><span class='lparen'>(</span><span class='id identifier rubyid_max_prime'>max_prime</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='op'>!=</span> <span class='id identifier rubyid_ratio'>ratio</span>
+      <span class='kw'>break</span> <span class='kw'>if</span> <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_length'>length</span> <span class='op'>>=</span> <span class='id identifier rubyid_depth'>depth</span>
+      <span class='id identifier rubyid_n'>n</span> <span class='op'>+=</span> <span class='int'>1</span>
+    <span class='kw'>end</span>
+  <span class='kw'>end</span>
 <span class='kw'>end</span></pre>
     </td>
   </tr>
@@ -1071,7 +1322,7 @@ <h3 class="signature " id="by_quotient_walk-instance_method">
       <div class="method_details ">
   <h3 class="signature " id="by_tree_path-instance_method">
   
-    #<strong>by_tree_path</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME)  ⇒ <tt><span class='object_link'><a href="../../Array.html" title="Array (class)">Array</a></span></tt> 
+    #<strong>by_tree_path</strong>(cents_tolerance: Tonal::Cents::TOLERANCE, depth: DEFAULT_DEPTH, max_prime: DEFAULT_MAX_PRIME)  ⇒ <tt><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Tonal::Ratio::Approximation::Set</a></span></tt> 
   
 
   
@@ -1091,8 +1342,8 @@ <h3 class="signature " id="by_tree_path-instance_method">
     <p class="tag_title">Examples:</p>
     
       
-      <pre class="example code"><code>Tonal::Ratio.ed(12,1).by_tree_path(max_prime: 17)
-=> [(18/17), (35/33), (17/16)]</code></pre>
+      <pre class="example code"><code>Tonal::Ratio.ed(12,1).approximate.by_tree_path(max_prime: 17)
+=> (4771397596969315/4503599627370496): [(18/17), (35/33), (17/16)]</code></pre>
     
   </div>
 <p class="tag_title">Parameters:</p>
@@ -1109,6 +1360,11 @@ <h3 class="signature " id="by_tree_path-instance_method">
         <em class="default">(defaults to: <tt>Tonal::Cents::TOLERANCE</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the cents tolerance used to scope the collection</p>
+</div>
+      
     </li>
   
     <li>
@@ -1122,6 +1378,11 @@ <h3 class="signature " id="by_tree_path-instance_method">
         <em class="default">(defaults to: <tt>DEFAULT_DEPTH</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the maximum number of ratios in the collection</p>
+</div>
+      
     </li>
   
     <li>
@@ -1135,6 +1396,11 @@ <h3 class="signature " id="by_tree_path-instance_method">
         <em class="default">(defaults to: <tt>DEFAULT_MAX_PRIME</tt>)</em>
       
       
+        —
+        <div class='inline'>
+<p>the maximum prime number to allow in the collection</p>
+</div>
+      
     </li>
   
 </ul>
@@ -1145,7 +1411,7 @@ <h3 class="signature " id="by_tree_path-instance_method">
     <li>
       
       
-        <span class='type'>(<tt><span class='object_link'><a href="../../Array.html" title="Array (class)">Array</a></span></tt>)</span>
+        <span class='type'>(<tt><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Tonal::Ratio::Approximation::Set</a></span></tt>)</span>
       
       
       
@@ -1164,7 +1430,6 @@ <h3 class="signature " id="by_tree_path-instance_method">
       <pre class="lines">
 
 
-70
 71
 72
 73
@@ -1174,21 +1439,22 @@ <h3 class="signature " id="by_tree_path-instance_method">
 77
 78
 79
-80</pre>
+80
+81</pre>
     </td>
     <td>
-      <pre class="code"><span class="info file"># File 'lib/tonal/approximations.rb', line 70</span>
+      <pre class="code"><span class="info file"># File 'lib/tonal/approximation.rb', line 71</span>
 
-<span class='kw'>def</span> <span class='id identifier rubyid_by_tree_path'>by_tree_path</span><span class='lparen'>(</span><span class='label'>cents_tolerance:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html#TOLERANCE-constant" title="Tonal::Cents::TOLERANCE (constant)">TOLERANCE</a></span></span><span class='comma'>,</span> <span class='label'>depth:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_DEPTH-constant" title="Tonal::Ratio::Approximations::DEFAULT_DEPTH (constant)">DEFAULT_DEPTH</a></span></span><span class='comma'>,</span> <span class='label'>max_prime:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_PRIME-constant" title="Tonal::Ratio::Approximations::DEFAULT_MAX_PRIME (constant)">DEFAULT_MAX_PRIME</a></span></span><span class='rparen'>)</span>
+<span class='kw'>def</span> <span class='id identifier rubyid_by_tree_path'>by_tree_path</span><span class='lparen'>(</span><span class='label'>cents_tolerance:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html#TOLERANCE-constant" title="Tonal::Cents::TOLERANCE (constant)">TOLERANCE</a></span></span><span class='comma'>,</span> <span class='label'>depth:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_DEPTH-constant" title="Tonal::Ratio::Approximation::DEFAULT_DEPTH (constant)">DEFAULT_DEPTH</a></span></span><span class='comma'>,</span> <span class='label'>max_prime:</span> <span class='const'><span class='object_link'><a href="#DEFAULT_MAX_PRIME-constant" title="Tonal::Ratio::Approximation::DEFAULT_MAX_PRIME (constant)">DEFAULT_MAX_PRIME</a></span></span><span class='rparen'>)</span>
   <span class='id identifier rubyid_self_in_cents'>self_in_cents</span> <span class='op'>=</span> <span class='id identifier rubyid_to_cents'>to_cents</span>
   <span class='id identifier rubyid_within'>within</span> <span class='op'>=</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='period'>.</span><span class='id identifier rubyid_kind_of?'>kind_of?</span><span class='lparen'>(</span><span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='rparen'>)</span> <span class='op'>?</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span> <span class='op'>:</span> <span class='const'><span class='object_link'><a href="../../Tonal.html" title="Tonal (module)">Tonal</a></span></span><span class='op'>::</span><span class='const'><span class='object_link'><a href="../Cents.html" title="Tonal::Cents (class)">Cents</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="../Cents.html#initialize-instance_method" title="Tonal::Cents#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>cents:</span> <span class='id identifier rubyid_cents_tolerance'>cents_tolerance</span><span class='rparen'>)</span>
-  <span class='lbracket'>[</span><span class='rbracket'>]</span><span class='period'>.</span><span class='id identifier rubyid_tap'>tap</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_results'>results</span><span class='op'>|</span>
+  <span class='const'><span class='object_link'><a href="Approximation/Set.html" title="Tonal::Ratio::Approximation::Set (class)">Set</a></span></span><span class='period'>.</span><span class='id identifier rubyid_new'><span class='object_link'><a href="Approximation/Set.html#initialize-instance_method" title="Tonal::Ratio::Approximation::Set#initialize (method)">new</a></span></span><span class='lparen'>(</span><span class='label'>ratio:</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_tap'>tap</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_set'>set</span><span class='op'>|</span>
     <span class='const'>FractionTree</span><span class='period'>.</span><span class='id identifier rubyid_path_to'>path_to</span><span class='lparen'>(</span><span class='id identifier rubyid_to_f'>to_f</span><span class='rparen'>)</span><span class='period'>.</span><span class='id identifier rubyid_each'>each</span> <span class='kw'>do</span> <span class='op'>|</span><span class='id identifier rubyid_node'>node</span><span class='op'>|</span>
       <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='op'>=</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_new'>new</span><span class='lparen'>(</span><span class='id identifier rubyid_node'>node</span><span class='period'>.</span><span class='id identifier rubyid_weight'>weight</span><span class='rparen'>)</span>
-      <span class='id identifier rubyid_results'>results</span> <span class='op'><<</span> <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='kw'>if</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_within_cents?'>within_cents?</span><span class='lparen'>(</span><span class='id identifier rubyid_self_in_cents'>self_in_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_to_cents'>to_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_within'>within</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_within_prime?'>within_prime?</span><span class='lparen'>(</span><span class='id identifier rubyid_max_prime'>max_prime</span><span class='rparen'>)</span>
-      <span class='kw'>break</span> <span class='kw'>if</span> <span class='id identifier rubyid_results'>results</span><span class='period'>.</span><span class='id identifier rubyid_length'>length</span> <span class='op'>>=</span> <span class='id identifier rubyid_depth'>depth</span>
+      <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_ratios'>ratios</span> <span class='op'><<</span> <span class='id identifier rubyid_ratio2'>ratio2</span> <span class='kw'>if</span> <span class='id identifier rubyid_ratio'>ratio</span><span class='period'>.</span><span class='id identifier rubyid_class'>class</span><span class='period'>.</span><span class='id identifier rubyid_within_cents?'>within_cents?</span><span class='lparen'>(</span><span class='id identifier rubyid_self_in_cents'>self_in_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_to_cents'>to_cents</span><span class='comma'>,</span> <span class='id identifier rubyid_within'>within</span><span class='rparen'>)</span> <span class='op'>&&</span> <span class='id identifier rubyid_ratio2'>ratio2</span><span class='period'>.</span><span class='id identifier rubyid_within_prime?'>within_prime?</span><span class='lparen'>(</span><span class='id identifier rubyid_max_prime'>max_prime</span><span class='rparen'>)</span>
+      <span class='kw'>break</span> <span class='kw'>if</span> <span class='id identifier rubyid_set'>set</span><span class='period'>.</span><span class='id identifier rubyid_length'>length</span> <span class='op'>>=</span> <span class='id identifier rubyid_depth'>depth</span>
     <span class='kw'>end</span>
-  <span class='kw'>end</span><span class='period'>.</span><span class='id identifier rubyid_sort'>sort</span>
+  <span class='kw'>end</span>
 <span class='kw'>end</span></pre>
     </td>
   </tr>
@@ -1288,18 +1554,18 @@ <h3 class="signature " id="neighborhood-instance_method">
       <pre class="lines">
 
 
-120
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+144
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     </td>
     <td>
-      <pre class="code"><span class="info file"># File 'lib/tonal/approximations.rb', line 120</span>
+      <pre class="code"><span class="info file"># File 'lib/tonal/approximation.rb', line 144</span>
 
 <span class='kw'>def</span> <span class='id identifier rubyid_neighborhood'>neighborhood</span><span class='lparen'>(</span><span class='label'>scale:</span> <span class='int'>2</span><span class='op'>**</span><span class='int'>0</span><span class='comma'>,</span> <span class='label'>boundary:</span> <span class='int'>1</span><span class='rparen'>)</span>
   <span class='id identifier rubyid_scale'>scale</span> <span class='op'>=</span> <span class='id identifier rubyid_scale'>scale</span><span class='period'>.</span><span class='id identifier rubyid_round'>round</span>
@@ -1320,7 +1586,7 @@ <h3 class="signature " id="neighborhood-instance_method">
 </div>
 
       <div id="footer">
-  Generated on Tue Jan 23 15:49:04 2024 by
+  Generated on Tue Feb 13 15:15:39 2024 by
   <a href="https://yardoc.org" title="Yay! A Ruby Documentation Tool" target="_parent">yard</a>
   0.9.34 (ruby-3.2.2).
 </div>
diff --git a/docs/Tonal/Ratio/Approximation/Set.html b/docs/Tonal/Ratio/Approximation/Set.html
new file mode 100644
index 0000000..458d101
--- /dev/null
+++ b/docs/Tonal/Ratio/Approximation/Set.html
@@ -0,0 +1,423 @@
+<!DOCTYPE html>
+<html>
+  <head>
+    <meta charset="utf-8">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<title>
+  Class: Tonal::Ratio::Approximation::Set
+  
+    — Documentation by YARD 0.9.34
+  
+
+
+  
+
+  
+
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+
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+  
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+  
+
+
+  
+  
+    
    
+      
+      
    
+    
    
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+    
    
+      
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+      
    Class: Tonal::Ratio::Approximation::Set
+  
+  
+  
+
    
+
    
+  
+  
    
+    
    Inherits:
    
+    
    
+      Object
+      
+        
      
+          
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+            
+          
+        
    
+        show all
+      
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+  
    
+      
    Extended by:
    
+      
    Forwardable
    
+  
    
+  
+  
+  
+  
+
+  
+
+  
+  
    
+    
    Defined in:
    
+    
    lib/tonal/approximation.rb
    
+  
    
+  
+
    
+
+
+
+
+
+  
    Instance Attribute Summary collapse
    
+  
      
+    
+      
    • 
+  
+    
+      #ratio  ⇒ Object 
+    
+
+    
+  
+  
+  
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+    
+      readonly
+    
+    
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+    
      
+
      Returns the value of attribute ratio.
      
+
      
+  
+
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+
+    
+      
    • 
+  
+    
+      #ratios  ⇒ Object 
+    
+
+    
+  
+  
+  
+  
+    
+      readonly
+    
+    
+  
+  
+  
+  
+  
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+  
+    
      
+
      Returns the value of attribute ratios.
      
+
      
+  
+
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+
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+  
    
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+
+  
+    
    
+      Instance Method Summary
+      collapse
+    
    
+
+    
+  
+
+
+  
    
+  
    Constructor Details
    
+  
+    
    
+  
    
+  
+    #initialize(ratio:)  ⇒ Set 
+  
+
+  
+
+  
+
    
+  
    
+    
+
    Returns a new instance of Set.
    
+
+
+  
    
+
    
+
    
+  
+
+
    
+  
+    
+    
+  
+
    
+      
    +
+
+178
+179
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+181
    
+        		
+      
    # File 'lib/tonal/approximation.rb', line 178
+
+def initialize(ratio:)
+  @ratio = ratio
+  @ratios = SortedSet.new
+end
    
+    
    
+
    
+  
+
    
+
+  
    
+    
    Instance Attribute Details
    
+    
+      
+      
+      
    
+  
    
+  
+    #ratio  ⇒ Object  (readonly)
+  
+
+  
+
+  
+
    
+  
    
+    
+
    Returns the value of attribute ratio.
    
+
+
+  
    
+
    
+
    
+  
+
+
    
+  
+    
+    
+  
+
    
+      
    +
+
+176
+177
+178
    
+        		
+      
    # File 'lib/tonal/approximation.rb', line 176
+
+def ratio
+  @ratio
+end
    
+    
    
+
    
+    
+      
+      
+      
    
+  
    
+  
+    #ratios  ⇒ Object  (readonly)
+  
+
+  
+
+  
+
    
+  
    
+    
+
    Returns the value of attribute ratios.
    
+
+
+  
    
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+  
+    
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+      
    +
+
+176
+177
+178
    
+        		
+      
    # File 'lib/tonal/approximation.rb', line 176
+
+def ratios
+  @ratios
+end
    
+    
    
+
    
+    
+  
    
+
+
+  
    
+    
    Instance Method Details
    
+
+    
+      
    
+  
    
+  
+    #inspect  ⇒ Object 
+  
+
+  
+
+  
+
    
+  
+    
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+
    
+      
    +
+
+184
+185
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+        		
+      
    # File 'lib/tonal/approximation.rb', line 184
+
+def inspect
+  "#{ratio}: #{entries}"
+end
    
+    
    
+
    
+    
+  
    
+
+
    
+
+      
+
+    
    
+  
+
\ No newline at end of file
diff --git a/docs/Tonal/ReducedRatio.html b/docs/Tonal/ReducedRatio.html
index 5719a57..c784864 100644
--- a/docs/Tonal/ReducedRatio.html
+++ b/docs/Tonal/ReducedRatio.html
@@ -253,7 +253,7 @@ 
    
   
   
   
    Methods inherited from Ratio
    
-  
    #*, #**, #+, #-, #/, #<=>, #approx, #benedetti_height, #cent_diff, #combination, #difference, #div_times, ed, #efficiency, #equave_reduce, #equave_reduce!, #fraction_reduce, #inspect, #invert, #invert!, #label, #lcm, #log_weil_height, #max_prime, #mediant_sum, #min_prime, #mirror, #mirror2, #negative, #period_degrees, #period_radians, #planar_degrees, #planar_radians, #plus_minus, #prime_divisions, #prime_vector, random_ratio, #ratio, #scale, #shear, #step, superparticular, superpartient, #tenney_height, #to_a, #to_cents, #to_f, #to_log, #to_log2, #to_r, #to_reduced_ratio, #to_v, #translate, #weil_height, #wilson_height, within_cents?, #within_prime?
    
+  
    #*, #**, #+, #-, #/, #<=>, #approximate, #benedetti_height, #cent_diff, #combination, #difference, #div_times, ed, #efficiency, #equave_reduce, #equave_reduce!, #fraction_reduce, #inspect, #invert, #invert!, #label, #lcm, #log_weil_height, #max_prime, #mediant_sum, #min_prime, #mirror, #negative, #period_degrees, #period_radians, #planar_degrees, #planar_radians, #plus_minus, #prime_divisions, #prime_vector, random_ratio, #ratio, #scale, #shear, #step, superparticular, superpartient, #tenney_height, #to_a, #to_cents, #to_f, #to_log, #to_log2, #to_r, #to_reduced_ratio, #to_v, #translate, #weil_height, #wilson_height, within_cents?, #within_prime?
    
 
   
   
    
@@ -553,7 +553,7 @@ 
    
 
    
 
       
diff --git a/docs/Tonal/Step.html b/docs/Tonal/Step.html
index 8ec87c7..882bf4f 100644
--- a/docs/Tonal/Step.html
+++ b/docs/Tonal/Step.html
@@ -944,7 +944,7 @@ 
    
 
 
       
diff --git a/docs/Vector.html b/docs/Vector.html
index fb61512..2146f96 100644
--- a/docs/Vector.html
+++ b/docs/Vector.html
@@ -226,10 +226,10 @@ 
    
       
     
 
-401
    
+402
    
     
    		
     
-      
    # File 'lib/tonal/extensions.rb', line 401
+      
    # File 'lib/tonal/extensions.rb', line 402
 
 def to_ratio(reduced: true, equave: 2/1r) = reduced ? Tonal::ReducedRatio.new(*self, equave: equave) : Tonal::Ratio.new(*self, equave: equave)