Graham complexity: Difference between revisions
Jump to navigation
Jump to search
Wikispaces>genewardsmith **Imported revision 242020965 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 295024044 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-24 18:49:46 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>295024044</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> | ||
| Line 8: | Line 8: | ||
<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">The //Graham complexity// of a set of pitch classes in a rank two temperament is the number of periods per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider a major triad in diaschismic temperament, with mapping [<2 0 11|, <0 1 -2|] corresponding to a generator of a '3'. That makes a 5/4 represented by [7 -2] and 3/2 by [-2 1], and 1 of course by [0 0]; so that the difference between the maximum and minimum number of generators is 1 - (-2) = 3, and so the Graham complexity of a major triad is 2*3 = 6. | <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">The //Graham complexity// of a set of pitch classes in a rank two temperament is the number of periods per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider a major triad in diaschismic temperament, with mapping [<2 0 11|, <0 1 -2|] corresponding to a generator of a '3'. That makes a 5/4 represented by [7 -2] and 3/2 by [-2 1], and 1 of course by [0 0]; so that the difference between the maximum and minimum number of generators is 1 - (-2) = 3, and so the Graham complexity of a major triad is 2*3 = 6. | ||
Given a MOS (or any generated scale) with N notes in a temperament where a given chord has a Graham complexity of C results in N-C chords in the MOS. For instance, the Graham complexity for both a major or a minor triad in meantone is 4, and hence there are 7-4 = 3 major triads in the diatonic scale, which is Meantone[7], and 3 minor triads. </pre></div> | Given a MOS (or any generated scale) with N notes in a temperament where a given chord has a Graham complexity of C results in N-C chords in the MOS. For instance, the Graham complexity for both a major or a minor triad in meantone is 4, and hence there are 7-4 = 3 major triads in the diatonic scale, which is Meantone[7], and 3 minor triads. | ||
The Graham complexity of the q-limit [[chord of nature]] is a complexity measure of the temperament itself, which is also sometimes called the Graham complexity.</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>Graham complexity</title></head><body>The <em>Graham complexity</em> of a set of pitch classes in a rank two temperament is the number of periods per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider a major triad in diaschismic temperament, with mapping [&lt;2 0 11|, &lt;0 1 -2|] corresponding to a generator of a '3'. That makes a 5/4 represented by [7 -2] and 3/2 by [-2 1], and 1 of course by [0 0]; so that the difference between the maximum and minimum number of generators is 1 - (-2) = 3, and so the Graham complexity of a major triad is 2*3 = 6.<br /> | <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>Graham complexity</title></head><body>The <em>Graham complexity</em> of a set of pitch classes in a rank two temperament is the number of periods per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider a major triad in diaschismic temperament, with mapping [&lt;2 0 11|, &lt;0 1 -2|] corresponding to a generator of a '3'. That makes a 5/4 represented by [7 -2] and 3/2 by [-2 1], and 1 of course by [0 0]; so that the difference between the maximum and minimum number of generators is 1 - (-2) = 3, and so the Graham complexity of a major triad is 2*3 = 6.<br /> | ||
<br /> | <br /> | ||
Given a MOS (or any generated scale) with N notes in a temperament where a given chord has a Graham complexity of C results in N-C chords in the MOS. For instance, the Graham complexity for both a major or a minor triad in meantone is 4, and hence there are 7-4 = 3 major triads in the diatonic scale, which is Meantone[7], and 3 minor triads.</body></html></pre></div> | Given a MOS (or any generated scale) with N notes in a temperament where a given chord has a Graham complexity of C results in N-C chords in the MOS. For instance, the Graham complexity for both a major or a minor triad in meantone is 4, and hence there are 7-4 = 3 major triads in the diatonic scale, which is Meantone[7], and 3 minor triads. <br /> | ||
<br /> | |||
The Graham complexity of the q-limit <a class="wiki_link" href="/chord%20of%20nature">chord of nature</a> is a complexity measure of the temperament itself, which is also sometimes called the Graham complexity.</body></html></pre></div> | |||
Revision as of 18:49, 24 January 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-01-24 18:49:46 UTC.
- The original revision id was 295024044.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
The //Graham complexity// of a set of pitch classes in a rank two temperament is the number of periods per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider a major triad in diaschismic temperament, with mapping [<2 0 11|, <0 1 -2|] corresponding to a generator of a '3'. That makes a 5/4 represented by [7 -2] and 3/2 by [-2 1], and 1 of course by [0 0]; so that the difference between the maximum and minimum number of generators is 1 - (-2) = 3, and so the Graham complexity of a major triad is 2*3 = 6. Given a MOS (or any generated scale) with N notes in a temperament where a given chord has a Graham complexity of C results in N-C chords in the MOS. For instance, the Graham complexity for both a major or a minor triad in meantone is 4, and hence there are 7-4 = 3 major triads in the diatonic scale, which is Meantone[7], and 3 minor triads. The Graham complexity of the q-limit [[chord of nature]] is a complexity measure of the temperament itself, which is also sometimes called the Graham complexity.
Original HTML content:
<html><head><title>Graham complexity</title></head><body>The <em>Graham complexity</em> of a set of pitch classes in a rank two temperament is the number of periods per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider a major triad in diaschismic temperament, with mapping [<2 0 11|, <0 1 -2|] corresponding to a generator of a '3'. That makes a 5/4 represented by [7 -2] and 3/2 by [-2 1], and 1 of course by [0 0]; so that the difference between the maximum and minimum number of generators is 1 - (-2) = 3, and so the Graham complexity of a major triad is 2*3 = 6.<br /> <br /> Given a MOS (or any generated scale) with N notes in a temperament where a given chord has a Graham complexity of C results in N-C chords in the MOS. For instance, the Graham complexity for both a major or a minor triad in meantone is 4, and hence there are 7-4 = 3 major triads in the diatonic scale, which is Meantone[7], and 3 minor triads. <br /> <br /> The Graham complexity of the q-limit <a class="wiki_link" href="/chord%20of%20nature">chord of nature</a> is a complexity measure of the temperament itself, which is also sometimes called the Graham complexity.</body></html>