Just intonation classes

This module provides pure Python classes for experimenting with Just intonation (music made from frequency ratio relationships rather than equal divisions of the octave):

• Interval(numerator, denominator) - Represents musical interval ratios/rational intervals, a relative step between pitches
  • .numerator - Numerator after reducing to simplest form
  • .denominator - Denominator after reducing to simplest form
  • .complement - The inversion or octave complement of the interval (P4 → P5, etc.)
  • .odd_limit - The smallest odd limit of an interval
  • .prime_limit - Prime limit or harmonic limit of an interval
  • .kees_height - Kees semi-height
  • .benedetti_height - Benedetti height
  • .tenney_height - Tenney height
  • Mathematical operations like addition of intervals, multiplication by integers, etc.
• Convenience intervals
  • P1 - Unison
  • m2 - Minor second
  • M2 - Major second
  • m3 - Minor third
  • M3 - Major third
  • P4 - Perfect fourth
  • P5 - Perfect fifth
  • etc.
• Chord(4, 5, 6), Chord(Interval('M3'), Interval('P5')), etc. - Class that represents chords/triads, a combination of multiple intervals
  • .terms - List of terms in the frequency ratio that makes up the chord
  • .intervals - List of musical intervals that make up the chord, relative to the root
  • .steps - List of musical intervals which, stacked together, produce the chord
  • .all_steps - Set of all music intervals that can be made by any tone in the chord with any other tone
  • .odd_limit - The intervallic odd-limit
  • .prime_limit - The highest prime limit of any interval found in the Chord
  • .inversion(n) - The nth inversion of a chord
• Pitch(frequency) - Class that represents absolute frequencies
  • .frequency - Value of the frequency in hertz

Probably all of this is redundant with Scala, but

1. I don't know how to use it.
2. I wanted to learn by doing:
   • Just Intonation
   • Object-oriented Python
   • Unit testing

Usage

>>> Chord(4, 5, 6).intervals
(Interval(5, 4), Interval(3, 2))

>>> M3 + m3
Interval(3, 2)

>>> Interval('P5').complement == P4
True

>>> Pitch(440) + P5
Pitch(660)

Installation

One possibility is to install with pip from GitHub:

pip install git+https://github.com/endolith/just_intonation.git

Examples

• Everything is a power chord in just intonation - Dyads made from the harmonic series, first in piano, then undistorted guitar, then distorted guitar.
• Enumerating the rationals - A small portion of the Calkin-Wilf sequence, played on a fractal piano with an infinite number of keys.