Cartesian scale
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2013-12-25 13:50:13 UTC.
- The original revision id was 479373586.
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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]].
[[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>.<br />
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{\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}<br/>[[math]]
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