Cartesian scales

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A Cartesian scale is a monotone periodic scale with an interval of equivalence ℇ (normally 2 or 1200.0 cents or an approximation to the just octave) and k generators G = [g1, g2 ... gk] with k multiplicities M = [m1,m2 ... mk], leading to a scale Descartes(ℇ, G, M) which if ℇ and g are given multplicatively is

[math]\{{ℇ^n {g_1}^{i_1} {g_2}^{i_2} \ldots {g_k}^{i_k}| 0 \leq i_1 \leq m_1, 0 \leq i_2 \leq m_2, \ldots, 0 \leq i_k \leq m_k}\}.[/math]

Here the multiplicities are fixed positive integers, and n ranges over all integers, with the scale sorted by ascending size and with all duplicates removed. If intervals are written additively as cents, then Descartes(ℇ, g, m) is

[math]\{{nℇ + i_1g_1 + \ldots + i_kg_k| 0 \leq i_1 \leq m_1 \ldots 0 \leq i_k \leq m_k}\}.[/math]

If the generators are odd primes and ℇ = 2, then the Cartesian scale is an Euler genus; if G = [p1, p2 ... pk] are the generators and M = [m1, m2 .. mk] the multiplicities, then Genus(p1^m1 p2^m2 ... pk^mk) = Descartes(2, G, M). By the fundamental theorem of arithmetic, the odd prime generators define an integer lattice, the points of which define unique representatives of the pitch classes of the scale. These ℇ-equivalence pitch classes form a Z-polytope which consists of the set of lattice points contained in an orthotope aligned with the lattice. The same is true more generally for any multiplicatively independent set {ℇ}∪G of generators; a Cartesian scale defined in terms of these may be called "independent". On the other hand if we expand the scale by increasing each of the multiplicities by one, and if in this expanded scale there are two distinct products of generators with the same ℇ-reduced numerical value, we may call the scale "redundant". An example of a redundant Cartesian scale is the octatonic scale, Descartes(1200, [300, 100], [3, 1]). Expanding that to Descartes(1200, [300, 100], [4,2]) gives a scale which rather than having (4+1)*(2+1) = 15 notes to the octave, has just 12, the 12 notes of 12edo.

Margo Schulter suggested the name "Cartesian" in a 2002 article on the Yahoo tuning list. Also, under the name "Euler-Fokker genus", Manuel Op de Coul gave Scala the capacity to construct Cartesian scales. Perhaps the first person to consider a Cartesian scale was Nicola Vicentino; his original conception for his second tuning of 1555 was for two 19 note 1/4 comma meantone scales (Meantone[19] in 1/4 comma tuning), separated by an interval of 1/4 of a syntonic comma, ie, (81/80)^(1/4); he only changed this to a 19+17 version because of physical limitations.