Triadic diamonds

Given a triad 1/1, M, D, a triadic diamond is the seven notes

1/1, M, D, 2/M, 2/D, D/M, M/D

octave reduced and sorted. For example, for M = 5/4, D = 3/2, we get

1/1, 5/4, 3/2, 8/5, 4/3, 6/5, 5/3

giving the sorted scale

1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3

On a lattice where a factor of M moves one step up, a factor of D one step to the right, the triadic diamond appears as

M/D       M

2/D       1/1       D

          2/M       D/M

A triadic diamond is the diamond construction applied to the triad 1, M, D.

Triadic reversed diamonds

A triadic reversed diamond is the seven notes

1/1, M, D, 2/M, 2/D, D*M, 2/(D*M)

octave reduced and sorted. On the lattice these notes are

          M         D*M

2/D       1/1       D

2/(D*M)   2/M

For M = 5/4, D = 3/2 this gives the notes

1/1, 5/4, 3/2, 8/5, 4/3, 15/8, 16/15

and sorting gives the scale

1/1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8

Further reading

• John H. Chalmers, Jr., The Triadic Diamond, the Triadic Reversed Diamond, and their Constituent Tetrachords when D=3/2, Xenharmonikon 15 (1993), p.64

from fractions import Fraction
from math import floor, log2


def reduce(x):
    return x * Fraction(2) ** (-floor(log2(x)))


def triadic_diamond(M, D):
    """
    >>> triadic_diamond(Fraction(5, 4), Fraction(3, 2))
    [Fraction(1, 1), Fraction(6, 5), Fraction(5, 4), Fraction(4, 3), Fraction(3, 2), Fraction(8, 5), Fraction(5, 3)]
    """
    notes = [1, M, D, 2 / M, 2 / D, D / M, M / D]
    return sorted(reduce(x) for x in notes)


def triadic_reversed_diamond(M, D):
    """
    >>> triadic_reversed_diamond(Fraction(5, 4), Fraction(3, 2))
    [Fraction(1, 1), Fraction(16, 15), Fraction(5, 4), Fraction(4, 3), Fraction(3, 2), Fraction(8, 5), Fraction(15, 8)]
    """
    notes = [1, M, D, 2 / M, 2 / D, D * M, 2 / (D * M)]
    return sorted(reduce(x) for x in notes)