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The idea

Let's represent tempo as a list of functions. These functions return tempo as BPM.

With this representation, you can get tempo at any time in a song.
You can also get the passed beats at any real time in the song, or real time at any beat in the song, via analytical integration of each function in sequence.

  • This representation makes it possible to transform between two tempo representations.

    This allows one to, for example, take a song with variable tempo and transform it to constant tempo (and vice-versa).

  • Another application allows rhythm game chart makers to map songs with variable tempo easily.
    Should be easy enough to just transform discrete note values of their map format between tempo representations.

  • A graphical editor for "tempo graphs" is considered. Could make this very accessible.

  • A long-term goal would be to have a database of "tempo graphs" to allow anyone to contribute or download them, like MusicBrainz for music information.

Types used for simplicity

OffsetUnit

a number with at most 6 decimals(?) that can represent either seconds or beats. This is denoted using a suffix.

Example:
0.166666 means 1/60th of a second.
0.1b means 1/10 of a beat.
(b suffix cannot be used for the first tempo change in a graph, as no prior tempo exists to integrate.)

Tempo functions

CONSTANT (C)

Describes a constant tempo bpm until the next tempo change.

  • beatsperminute (bpm): float
  • start_offset (sofs): OffsetUnit

LINE (L)

Describes a tempo that changes from start_bpm to end_bpm during length with a non-linearity of power. If length is left undefined, it will last until the next tempo change. If this is the last tempo change, length field must be set.

  • start_bpm (sbpm): float
  • end_bpm (ebpm): float
  • length (l): float
  • power (p): float
  • start_offset (sofs): OffsetUnit

POLYNOMIAL (P)

Describes a tempo that changes from start_bpm to end_bpm during length with respect to the [0,1] XY-axis range of the xnd-degree polynomial function defined in p.
If length is left undefined, it will last until the next tempo change.
If this is the last tempo change, length field must be set.

tempograph shall remap the X-axis from [0,1] to [0,length] (scaled) and the Y-axis from [0,1] to [s,e] automatically.

Polynomials are curves that can go up and down over time. They should be pretty performant, as they're easy to evaluate and calculate integrals for.

This is intended for tempo changes where you think other functions might still be too restrictive.

Users are not expected to manually set the terms; they're expected to map the beats by-hand and perform polynomial regression on them. The graphical editor should have this functionality.

  • start_bpm (s): float
  • end_bpm (e): float
  • length(l): float
  • terms: list of float
  • start_offset (sofs): OffsetUnit

Grammar

  • program → funcList
  • funcList → funcList func | func
  • func → constant | line | polynomial
  • constant → C bpm offsetUnit
  • line → L bpm bpm length power offsetUnit
  • polynomial → P bpm bpm length terms offsetUnit
  • terms → terms term | term
  • term → <float>
  • length → <float>
  • power → <float>
  • bpm → <float>
  • length → <float>
  • *power → <float>
  • offsetUnit → <float> | <float>b