Combination product sets

Given the list of numbers [1, 3, 5, 7], take all products of two numbers,

1*3, 1*5, 1*7, 3*5, 3*7, 5*7

Now pick one of these products as the root, say 3. Divide each number by the root

1, 5/3, 7/3, 5, 7, 35/3

and reduce within one octave

1, 5/3, 7/6, 5/4, 7/4, 35/24

These ratios 1, 7/6, 5/4, 35/24, 5/3, 7/4 form the 1-3-5-7 hexany.

Choosing different products as root gives different modes of the same scale.

In general, given a list of n numbers, taking products of k numbers gives a CPS written nCk. The scale gets a name based on its number of tones:

Name	Tones
monany	1
dyany	2
triany	3
tetrany	4
pentany	5
hexany	6
dekany	10
pentadekany	15
eikosany	20

Further reading

• The Hexany, the Eikosany, and the Other Combination Product Sets, The Wilson Archives

from itertools import combinations
from fractions import Fraction
from math import floor, log2, prod


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


def cps(A, k, *, root=None):
    """
    >>> cps([1, 3, 5, 7], 2)
    [Fraction(1, 1), Fraction(7, 6), Fraction(5, 4), Fraction(35, 24), Fraction(5, 3), Fraction(7, 4)]
    """
    products = [Fraction(prod(a)) for a in combinations(A, k)]
    if root is None:
        root = products[0]
    return sorted(reduce(x / root) for x in products)