Cartesian scale
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- This revision was by author genewardsmith and made on 2013-12-25 16:49:37 UTC.
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Original Wikitext content:
A //Cartesian sca;e// is a [[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 {ℇ^n g1^i1 ... gk^ik| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. Here the multiplicities are fixed positive integers, and n ranges over al integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is {nℇ^n + k1g1 + ... + ikgk| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}.
If the generators are odd primes and ℇ = 2, then the Cartesian scale is an [[Euler genera|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 [[http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic|fundamental theorem of arithmetic]], the odd prime generators define an [[http://en.wikipedia.org/wiki/Integer_lattice|integer lattice]], the points of which define unique representatives of the pitch classes of the scale. These ℇ-equivalence pitch classes form a [[http://en.wikipedia.org/wiki/Convex_lattice_polytope|Z-polytope]] which consists of the set of lattice points contained in an [[http://en.wikipedia.org/wiki/Hyperrectangle|orthotope]] aligned with the lattice. The same is true more generally for any [[http://planetmath.org/multiplicativelyindependent|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 notes on distunt lattice points with the same numerical value, we may call the scale "dependent".
[[math]]
{\Sigma^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]]Original HTML content:
<html><head><title>Cartesian scales</title></head><body>A <em>Cartesian sca;e</em> is a <a class="wiki_link" href="/periodic%20scale">periodic scale</a> 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 {ℇ^n g1^i1 ... gk^ik| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. Here the multiplicities are fixed positive integers, and n ranges over al integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is {nℇ^n + k1g1 + ... + ikgk| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. <br />
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If the generators are odd primes and ℇ = 2, then the Cartesian scale is an <a class="wiki_link" href="/Euler%20genera">Euler genus</a>; 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic" rel="nofollow">fundamental theorem of arithmetic</a>, the odd prime generators define an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integer_lattice" rel="nofollow">integer lattice</a>, the points of which define unique representatives of the pitch classes of the scale. These ℇ-equivalence pitch classes form a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_lattice_polytope" rel="nofollow">Z-polytope</a> which consists of the set of lattice points contained in an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hyperrectangle" rel="nofollow">orthotope</a> aligned with the lattice. The same is true more generally for any <a class="wiki_link_ext" href="http://planetmath.org/multiplicativelyindependent" rel="nofollow">multiplicatively independent</a> 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 notes on distunt lattice points with the same numerical value, we may call the scale "dependent".<br />
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