Graham complexity: Difference between revisions

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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''' 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|~~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 {{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|~~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:complexity]]

[[Category:Interval complexity measures]]

[[Category:Temperament complexity measures]]

[[Category:Terms]]

~~[[Category:theory]]~~

~~[[Category:todo:add_examples]]~~

~~[[Category:todo:simplify]]~~

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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 [&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.
+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|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 {{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|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:complexity]]
+[[Category:Interval complexity measures]]
+[[Category:Temperament complexity measures]]
 [[Category:Terms]]
-[[Category:theory]]
-[[Category:todo:add_examples]]
-[[Category:todo:simplify]]