Cartesian scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 479457420 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 479457478 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-12-26 10: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-12-26 10:08:30 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>479457478</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 all 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 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. | 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 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. | ||
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[[math]]</pre></div> | [[math]]</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 all 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 /> | ||
<br /> | <br /> | ||
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 &quot;independent&quot;. 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 &quot;redundant&quot;. 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.<br /> | 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 &quot;independent&quot;. 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 &quot;redundant&quot;. 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.<br /> | ||