Graham complexity - Revision history

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	The '''Graham complexity''' measures the [[complexity]] of a set of intervals in a [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]]. For a rank-2 temperament with [[~~2/1\|~~octave]]s, the Graham complexity of a set of [[pitch class]]es equals the number of [[period]]s per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider the [[just major triad]] in [[diaschismic]] temperament, with mapping {{mapping\| 2 0 11 \| 0 1 -2 }} corresponding to a generator of a [[3/1]]. That makes a [[5/4]] represented by {{monzo\| 7 -2 }} and [[3/2]] by {{monzo\| -2 1 }}, and [[1/1\|1]] of course by {{monzo\| 0 0 }}; so that the difference between the maximum and minimum number of generators is {{nowrap\| 1 - (-2) {{=}} 3 }}, and so the Graham complexity of a major triad is {{nowrap\| 2 × 3 {{=}} 6 }}.		The '''Graham complexity''' measures the [[complexity]] of a set of intervals in a [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]]. For a rank-2 temperament with [[octave]]s, the Graham complexity of a set of [[pitch class]]es equals the number of [[period]]s per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider the [[just major triad]] in [[diaschismic]] temperament, with mapping {{mapping\| 2 0 11 \| 0 1 -2 }} corresponding to a generator of a [[3/1]]. That makes a [[5/4]] represented by {{monzo\| 7 -2 }} and [[3/2]] by {{monzo\| -2 1 }}, and [[1/1\|1]] of course by {{monzo\| 0 0 }}; so that the difference between the maximum and minimum number of generators is {{nowrap\| 1 - (-2) {{=}} 3 }}, and so the Graham complexity of a major triad is {{nowrap\| 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 {{nowrap\|(''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 {{nowrap\| 7 - 4 {{=}} 3 }} major triads in the [[diatonic scale]], which is Meantone[7], and 3 minor triads. When the chord is a [[dyad]], the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore {{nowrap\| ''N'' - ''m'' }}, where ''m'' is the number of specific interval dyads in its interval class. So long as the mos is [[constant structure]], which it will be except in special cases, this is therefore definable entirely in terms of the mos without reference to a rank-2 temperament, and such a definition can be extended to any chord in the mos as the maximum complexity of its dyads. Such complexity could be called mos Graham complexity as distinct from temperament Graham complexity.		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 {{nowrap\|(''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 {{nowrap\| 7 - 4 {{=}} 3 }} major triads in the [[diatonic scale]], which is Meantone[7], and 3 minor triads. When the chord is a [[dyad]], the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore {{nowrap\| ''N'' - ''m'' }}, where ''m'' is the number of specific interval dyads in its interval class. So long as the mos is [[constant structure]], which it will be except in special cases, this is therefore definable entirely in terms of the mos without reference to a rank-2 temperament, and such a definition can be extended to any chord in the mos as the maximum complexity of its dyads. Such complexity could be called mos Graham complexity as distinct from temperament Graham complexity.

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	The '''Graham complexity''' of a set of [[~~pitch class~~]]~~es in a~~ [[rank-2 temperament]] is the number of [[period]]s per [[octave]] times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider the [[just major triad]] in [[diaschismic]] temperament, with mapping {{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.		The '''Graham complexity''' measures the [[complexity]] of a set of intervals in a [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]]. For a rank-2 temperament with [[2/1\|octave]]s, the Graham complexity of a set of [[pitch class]]es equals the number of [[period]]s per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider the [[just major triad]] in [[diaschismic]] temperament, with mapping {{mapping\| 2 0 11 \| 0 1 -2 }} corresponding to a generator of a [[3/1]]. That makes a [[5/4]] represented by {{monzo\| 7 -2 }} and [[3/2]] by {{monzo\| -2 1 }}, and [[1/1\|1]] of course by {{monzo\| 0 0 }}; so that the difference between the maximum and minimum number of generators is {{nowrap\| 1 - (-2) {{=}} 3 }}, and so the Graham complexity of a major triad is {{nowrap\| 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. When the chord is a [[dyad]], the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore ''N'' - ''m'', where ''m'' is the number of specific interval dyads in its interval class. So long as the mos is [[constant structure]], which it will be except in special cases, this is therefore definable entirely in terms of the mos without reference to a rank-2 temperament, and such a definition can be extended to any chord in the mos as the maximum complexity of its dyads. Such complexity could be called mos Graham complexity as distinct from temperament Graham complexity.		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 {{nowrap\|(''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 {{nowrap\| 7 - 4 {{=}} 3 }} major triads in the [[diatonic scale]], which is Meantone[7], and 3 minor triads. When the chord is a [[dyad]], the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore {{nowrap\| ''N'' - ''m'' }}, where ''m'' is the number of specific interval dyads in its interval class. So long as the mos is [[constant structure]], which it will be except in special cases, this is therefore definable entirely in terms of the mos without reference to a rank-2 temperament, and such a definition can be extended to any chord in the mos as the maximum complexity of its dyads. Such complexity could be called mos Graham complexity as distinct from temperament Graham complexity.

	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~~.		The Graham complexity of the [[odd limit\|''q''-odd-limit]] [[chord of nature]] is a complexity measure of the temperament itself, which is useful for comparing different temperaments.

	[[Category:Interval complexity measures]]		[[Category:Interval complexity measures]]
			[[Category:Temperament complexity measures]]
	[[Category:Terms]]		[[Category:Terms]]

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	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.		The '''Graham complexity''' of a set of [[pitch class]]es in a [[rank-2 temperament]] is the number of [[period]]s per [[octave]] times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider the [[just major triad]] in [[diaschismic]] temperament, with mapping {{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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. So long as the ~~MOS~~ is [[~~constant_structure\|~~constant structure]], which it will be except in special cases, this is therefore definable entirely in terms of the ~~MOS~~ without reference to a rank ~~two~~ temperament, and such a definition can be extended to any chord in the ~~MOS~~ as the maximum complexity of its dyads. Such complexity could be called ~~MOS~~ Graham complexity as distinct from temperament Graham complexity.		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. When the chord is a [[dyad]], the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore ''N'' - ''m'', where ''m'' is the number of specific interval dyads in its interval class. So long as the mos is [[constant structure]], which it will be except in special cases, this is therefore definable entirely in terms of the mos without reference to a rank-2 temperament, and such a definition can be extended to any chord in the mos as the maximum complexity of its dyads. Such complexity could be called mos Graham complexity as distinct from temperament Graham complexity.

	The Graham complexity of the q-limit [[~~chord_of_nature\|~~chord of nature]] is a complexity measure of the temperament itself, which is also sometimes called the Graham complexity.		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.
	[[Category:complexity]]
			[[Category:Interval complexity measures]]
	[[Category:Terms]]		[[Category:Terms]]
	~~[[Category:todo:add_examples]]~~
	~~[[Category:todo:simplify]]~~

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	[[Category:complexity]]		[[Category:complexity]]
	[[Category:Terms]]		[[Category:Terms]]
	~~[[Category:theory]]~~
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Keenan Pepper category edits: Moving from Category:Term to Category:Terms using Cat-a-lot

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	The Graham complexity of the q-limit [[chord_of_nature\|chord of nature]] is a complexity measure of the temperament itself, which is also sometimes called the Graham complexity.		The Graham complexity of the q-limit [[chord_of_nature\|chord of nature]] is a complexity measure of the temperament itself, which is also sometimes called the Graham complexity.
	[[Category:complexity]]		[[Category:complexity]]
	[[Category:~~term~~]]		[[Category:Terms]]
	[[Category:theory]]		[[Category:theory]]
	[[Category:todo:add_examples]]		[[Category:todo:add_examples]]
	[[Category:todo:simplify]]		[[Category:todo:simplify]]

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	~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~		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.
	~~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>2015-10-06 23:24:54 UTC</tt>.<br>~~
	~~: The original revision id was <tt>561574015</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>~~
	~~<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">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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. So long as the MOS is [[constant structure]], which it will be except in special cases, this is therefore definable entirely in terms of the MOS without reference to a rank two temperament, and such a definition can be extended to any chord in the MOS as the maximum complexity of its dyads. Such complexity could be called MOS Graham complexity as distinct from temperament Graham complexity.		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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. So long as the MOS is [[constant_structure\|constant structure]], which it will be except in special cases, this is therefore definable entirely in terms of the MOS without reference to a rank two temperament, and such a definition can be extended to any chord in the MOS as the maximum complexity of its dyads. Such complexity could be called MOS Graham complexity as distinct from temperament Graham complexity.

	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>~~		The Graham complexity of the q-limit [[chord_of_nature\|chord of nature]] is a complexity measure of the temperament itself, which is also sometimes called the Graham complexity.
	~~<h4>Original HTML content~~:~~</h4>~~		[[Category:complexity]]
	~~<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 />~~		[[Category:term]]
	~~<br />~~		[[Category:theory]]
	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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. So long as the MOS is <a class="wiki_link" href="/constant%20structure">constant structure</a>, which it will be except in special cases, this is therefore definable entirely in terms of the MOS without reference to a rank two temperament, and such a definition can be extended to any chord in the MOS as the maximum complexity of its dyads. Such complexity could be called MOS Graham complexity as distinct from temperament Graham complexity.<br />		[[Category:todo:add_examples]]
	~~<br />~~		[[Category:todo:simplify]]
	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>

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	: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>~~2011~~-07-~~19 19~~:55:31 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2012-01-24 18:49:46 UTC</tt>.<br>
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	<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>

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	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>
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	<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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. ~~This~~ is therefore definable entirely in terms of the MOS without reference to a rank two temperament, and such a definition can be extended to any chord in the MOS as the maximum complexity of its dyads. Such complexity could be called MOS Graham complexity as distinct from temperament Graham complexity.		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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. So long as the MOS is [[constant structure]], which it will be except in special cases, this is therefore definable entirely in terms of the MOS without reference to a rank two temperament, and such a definition can be extended to any chord in the MOS as the maximum complexity of its dyads. Such complexity could be called MOS Graham complexity as distinct from temperament Graham complexity.

	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>		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>
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	<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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. ~~This~~ is therefore definable entirely in terms of the MOS without reference to a rank two temperament, and such a definition can be extended to any chord in the MOS as the maximum complexity of its dyads. Such complexity could be called MOS Graham complexity as distinct from temperament Graham complexity.<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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. So long as the MOS is <a class="wiki_link" href="/constant%20structure">constant structure</a>, which it will be except in special cases, this is therefore definable entirely in terms of the MOS without reference to a rank two temperament, and such a definition can be extended to any chord in the MOS as the maximum complexity of its dyads. Such complexity could be called MOS Graham complexity as distinct from temperament Graham complexity.<br />
	<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>		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>

← Older revision		Revision as of 01:37, 5 October 2015
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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>~~2012~~-01~~-24 18~~:49:46 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2015-10-05 01:37:29 UTC</tt>.<br>
	: The original revision id was <tt>~~295024044~~</tt>.<br>		: The original revision id was <tt>561301039</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>
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	<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.		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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. This is therefore definable entirely in terms of the MOS without reference to a rank two temperament, and such a definition can be extended to any chord in the MOS as the maximum complexity of its dyads. Such complexity could be called MOS Graham complexity as distinct from temperament Graham complexity.

	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>		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>
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	<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. <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. When the chord is a dyad, the Graham complexity of the dyad, which can be regarded as the Graham complexity of an interval, is therefore N-m, where m is the number of specific interval dyads in its interval class. This is therefore definable entirely in terms of the MOS without reference to a rank two temperament, and such a definition can be extended to any chord in the MOS as the maximum complexity of its dyads. Such complexity could be called MOS Graham complexity as distinct from temperament Graham complexity.<br />
	<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>		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>

← Older revision		Revision as of 19:55, 19 July 2011
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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>2011-07-19 18:34:40 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-07-19 19:55:31 UTC</tt>.<br>
	: The original revision id was <tt>~~242008691~~</tt>.<br>		: The original revision id was <tt>242020965</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">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.</pre></div>		<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>
	<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.</body></html></pre></div>		<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 />
			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>