Tonality diamond: Difference between revisions

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~~The~~ ''q''-[[odd-limit]] ~~'''tonality diamond''' is~~ the [[~~diamond function~~]] ~~applied to the odd numbers from 1 to ''q''~~: ~~diamond ({1, 3, 5, … , ''q''})~~. ~~Another way of defining it is~~ in ~~terms~~ of the [[~~Weil height~~]]~~: <math>H\left(\frac{n}{d}\right) = max~~(|~~n|, |d|)</math> - as all rational numbers which are the quotient~~ of ~~two positive odd integers ''n''/''d'' with ''H''(''n''/''d'') ≤ ''q'',~~ [[~~octave-reduced~~]].

A '''tonality diamond''' is a symmetric organization of [[Otonality and utonality|otonal and utonal]] chords based around a central note and bounded by an [[Odd limit|odd-limit]]. First formalized in the [[7-odd-limit]] by [[wikipedia:Max_Friedrich_Meyer|Max F. Meyer]] in 1929, the idea became central to the music and theories of [[Harry Partch]], who built his tonal system around the [[11-odd-limit]] tonality diamond. Tonality diamonds have been used both conceptually (such as for [[Target tuning|targets]] of [[temperaments]]) and practically (such as for instrument layouts) in xenharmonics ever since.

== Construction ==

~~A generalized~~ tonality diamond ~~can be constructed given an equave '''E~~''' and ''n'~~' harmonics '''P1, P~~2~~, .~~.~~. Pn~~'''~~, sorted in increasing size ''after being equave-reduced'' so~~ as ~~to lie between 1~~ and ~~'''E'''. (In~~ the ~~''q''-odd-limit construction~~, the ~~harmonics are simply~~ the ~~octave-reduced odd harmonics up to~~ ''q'~~'.) The~~ tonality diamond ~~then consists of the harmonics '''P1, P2,~~ .~~.. Pn~~''', ~~their octave complements '''E/P~~1~~, E/P~~2~~~~, .~~.. E/Pn~~''' ~~alongside fractions of the harmonics amongst each other~~: '''~~Pi/Pj''' for every ''i'' > ''j''~~, ~~and~~ '''~~EPi~~</~~sub~~>~~/Pj''' for every ''i'' < ''j'' (in addition to~~ the ~~[[unison]]). If the harmonics are all linearly independent (as in~~ the 5-odd ~~or 7~~-~~odd limits), there~~ are ~~''n''(''n''+1) distinct consonances; however, if some fraction~~ of ~~two harmonics reduces to a different harmonic [e.g. (3/2)/(9/8) = 4/3] or is equivalent to another fraction [e.g. (15/8)/(9/8) = 5/3 = 2*(5/4)/(3/2)], this number reduces.~~

File:How to tonality diamond 1.png|'''Step 1: Take the numbers of an odd-limit and arrange them along two axes.'''

File:How to tonality diamond 2.png|'''Step 2: Using one axis as the numerator and the other as the denominator, fill in the cells with the ratios they form.'''

File:How to tonality diamond 3.png|'''Step 3: Octave-reduce the ratios (ie, make sure the decimal form of each ratio is between 1 and 2; if it is not, double one of the numbers until it is).'''

File:How to tonality diamond 4.png|'''Optional step: to make the rows play rooted chords, one half of the diamond (not including the middle unison row) must be lowered by an octave (represented by grey cells in image).'''

</gallery>Note: the numbers of the odd-limit are generally arranged in one of three ways:

=== ~~Relationship~~ to ~~subgroups ===~~

* numerically (ie, 1 3 5 7 9 11) as in Meyer's 7-limit diamond

~~While~~, ~~given any subgroup~~ of [[~~just intonation~~]], a tonality diamond ~~can be constructed from the equave~~ and the ~~higher primes in the subgroup~~, the ~~correspondence~~ is ~~not one~~-to-~~one~~: ~~an infinite number of possible tonality diamonds are constructible from a subgroup; for instance~~, the 2.3.~~7 subgroup would possess distinct diamonds for harmonics 3 and 7 to equave 2~~, and for ~~3 and 21 to 2~~, ~~or even for 3, 7~~, and ~~9 to 2 (to say nothing of 2 and 7~~/~~4 to 3)~~. ~~However, any tonality diamond~~ with ~~rational consonances to a rational equave defines a subgroup~~.

* tonally (ie, 1 9 5 11 3 7) as in Partch's 11-limit diamond

* chordally (ie, 1 5 3 7 9 11) as in the layout for the Diamond Marimba

== History ==

The tonality diamond was first formally explained by Max F. Meyer in his 1929 publication ''The Musician's Arithmetic'' using the 7-odd-limit.<ref>[https://archive.org/details/max-f-meyer-the-musicians-arithmetic/page/22/mode/2up Meyer, Max F. "The Musician’s Arithmetic: Drill Problems for an Introduction to the Scientific Study of Musical Composition". ''The University of Missouri Studies''. Vol. 4, no. 1. University of Missouri. January 1, 1929. p. 22.]</ref>

Harry Partch is the person most associated with the tonality diamond, and claimed to have invented it. However, it is likely that he plagarized the idea from Meyer.<ref>[https://www.chrysalis-foundation.org/wp-content/uploads/ThePartchHoaxDoctrines.pdf Forster, Cris (2015). ''The Partch Hoax Doctrines''. Self-published.]</ref> Regardless, his extending of the concept to the 11-odd-limit (as well as his other extensions and uses of it) was an extremely important and foundational moment in the history of xenharmonic music.

[[Erv Wilson]] in particular was inspired by Partch's use of the tonality diamond and it's extended form. He developed a number of "diamonds" himself,<ref>[https://anaphoria.com/diamond.pdf Wilson, Erv. ''Letters on Diamond Lattices, 1965–1970'' (PDF). Self-published.]</ref> as well as other concepts based on Partch's extended tonality diamond such as "[[constant structure]]."<ref>[https://www.anaphoria.com/Partchpapers.pdf Wilson, Erv. ''The Partch Papers (collection of documents on Harry Partch’s 11-limit diamond and its extensions), 1964-2002'' (PDF). Self-published.] </ref> A related idea of Wilson's is the "[[Cross-set scale|cross-set]]," of which the tonality diamond is a special case.

The first novel xenharmonic temperament — [[George Secor|George Secor's]] later-named "[[Miracle]]" temperament — was made to approximate Partch's 11-limit diamond.<ref>[https://www.anaphoria.com/SecorMiracle.pdf Secor, George (1975). “A New Look at the Partch Monophonic Fabric.” ''Xenharmonicon''. Vol. 3]</ref><ref>[https://www.anaphoria.com/SecorMiracle.pdf Secor, George. "The Miracle Temperament and Decimal Keyboard". ''Xenharmonikon''. Vol. 18. 2006. pp. 5–15. © 2003.]</ref>

== Uses ==

=== Instrument layout ===

The most famous example of the tonality diamond as a practical layout for an instrument is Harry Partch's "Diamond Marimba," which uses the 11-odd-limit tonality diamond exactly. This idea was explored further with Partch's "Quadrangularis Reversum," and by Cris Forster with his [[13-odd-limit]] "Diamond Marimba."

[https://sintel.website/posts/diamond_marimba.html Play with Partch's Diamond Marimba here.]

== Examples of scales ==

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* [[diamond13]]

* [[diamond15]]

* [[diamond9plus-marvel]]

== Music ==

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== External links ==

* [http://www.tonalsoft.com/enc/t/tonality-diamond.aspx Tonality diamond – arrangement of musical frequency ratios showing the dual identity of each ratio] on [[Tonalsoft Encyclopedia]]

== See also ==

* [[Cross-set scale]]

* [[Diamond function|Diamond Function]]

* [[Lattice]]

== References ==

[[Category:Diamond]]

[[Category:Pitch space]]

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 }}
 {{Wikipedia|Tonality diamond}}
-The ''q''-[[odd-limit]] '''tonality diamond''' is the [[diamond function]] applied to the odd numbers from 1 to ''q'': diamond ({1, 3, 5, … , ''q''}). Another way of defining it is in terms of the [[Weil height]]: <math>H\left(\frac{n}{d}\right) = max(|n|, |d|)</math> - as all rational numbers which are the quotient of two positive odd integers ''n''/''d'' with ''H''(''n''/''d'') ≤ ''q'', [[octave-reduced]].
+A '''tonality diamond''' is a symmetric organization of [[Otonality and utonality|otonal and utonal]] chords based around a central note and bounded by an [[Odd limit|odd-limit]]. First formalized in the [[7-odd-limit]] by [[wikipedia:Max_Friedrich_Meyer|Max F. Meyer]] in 1929, the idea became central to the music and theories of [[Harry Partch]], who built his tonal system around the [[11-odd-limit]] tonality diamond. Tonality diamonds have been used both conceptually (such as for [[Target tuning|targets]] of [[temperaments]]) and practically (such as for instrument layouts) in xenharmonics ever since.
 == Construction ==
-A generalized tonality diamond can be constructed given an equave '''E''' and ''n'' harmonics '''P<sub>1</sub>, P<sub>2</sub>, ... P<sub>n</sub>''', sorted in increasing size ''after being equave-reduced'' so as to lie between 1 and '''E'''. (In the ''q''-odd-limit construction, the harmonics are simply the octave-reduced odd harmonics up to ''q''.) The tonality diamond then consists of the harmonics '''P<sub>1</sub>, P<sub>2</sub>, ... P<sub>n</sub>''', their octave complements '''E/P<sub>1</sub>, E/P<sub>2</sub>, ... E/P<sub>n</sub>''' alongside fractions of the harmonics amongst each other: '''P<sub>i</sub>/P<sub>j</sub>''' for every ''i'' > ''j'', and '''EP<sub>i</sub>/P<sub>j</sub>''' for every ''i'' < ''j'' (in addition to the [[unison]]). If the harmonics are all linearly independent (as in the 5-odd or 7-odd limits), there are ''n''(''n''+1) distinct consonances; however, if some fraction of two harmonics reduces to a different harmonic [e.g. (3/2)/(9/8) = 4/3] or is equivalent to another fraction [e.g. (15/8)/(9/8) = 5/3 = 2*(5/4)/(3/2)], this number reduces.
+<gallery mode="nolines" widths="200" heights="200">
+File:How to tonality diamond 1.png|'''Step 1: Take the numbers of an odd-limit and arrange them along two axes.'''
+File:How to tonality diamond 2.png|'''Step 2: Using one axis as the numerator and the other as the denominator, fill in the cells with the ratios they form.'''
+File:How to tonality diamond 3.png|'''Step 3: Octave-reduce the ratios (ie, make sure the decimal form of each ratio is between 1 and 2; if it is not, double one of the numbers until it is).'''
+File:How to tonality diamond 4.png|'''Optional step: to make the rows play rooted chords, one half of the diamond (not including the middle unison row) must be lowered by an octave (represented by grey cells in image).'''
+</gallery>Note: the numbers of the odd-limit are generally arranged in one of three ways:
-=== Relationship to subgroups ===
+* numerically (ie, 1 3 5 7 9 11) as in Meyer's 7-limit diamond
-While, given any subgroup of [[just intonation]], a tonality diamond can be constructed from the equave and the higher primes in the subgroup, the correspondence is not one-to-one: an infinite number of possible tonality diamonds are constructible from a subgroup; for instance, the 2.3.7 subgroup would possess distinct diamonds for harmonics 3 and 7 to equave 2, and for 3 and 21 to 2, or even for 3, 7, and 9 to 2 (to say nothing of 2 and 7/4 to 3). However, any tonality diamond with rational consonances to a rational equave defines a subgroup.
+* tonally (ie, 1 9 5 11 3 7) as in Partch's 11-limit diamond
+* chordally (ie, 1 5 3 7 9 11) as in the layout for the Diamond Marimba
+== History ==
+The tonality diamond was first formally explained by Max F. Meyer in his 1929 publication ''The Musician's Arithmetic'' using the 7-odd-limit.<ref>[https://archive.org/details/max-f-meyer-the-musicians-arithmetic/page/22/mode/2up Meyer, Max F. "The Musician’s Arithmetic: Drill Problems for an Introduction to the Scientific Study of Musical Composition". ''The University of Missouri Studies''. Vol. 4, no. 1. University of Missouri. January 1, 1929. p. 22.]</ref>
+Harry Partch is the person most associated with the tonality diamond, and claimed to have invented it. However, it is likely that he plagarized the idea from Meyer.<ref>[https://www.chrysalis-foundation.org/wp-content/uploads/ThePartchHoaxDoctrines.pdf Forster, Cris (2015). ''The Partch Hoax Doctrines''. Self-published.]</ref> Regardless, his extending of the concept to the 11-odd-limit (as well as his other extensions and uses of it) was an extremely important and foundational moment in the history of xenharmonic music.
+[[Erv Wilson]] in particular was inspired by Partch's use of the tonality diamond and it's extended form. He developed a number of "diamonds" himself,<ref>[https://anaphoria.com/diamond.pdf Wilson, Erv. ''Letters on Diamond Lattices, 1965–1970'' (PDF). Self-published.]</ref> as well as other concepts based on Partch's extended tonality diamond such as "[[constant structure]]."<ref>[https://www.anaphoria.com/Partchpapers.pdf Wilson, Erv. ''The Partch Papers (collection of documents on Harry Partch’s 11-limit diamond and its extensions), 1964-2002'' (PDF). Self-published.] </ref> A related idea of Wilson's is the "[[Cross-set scale|cross-set]]," of which the tonality diamond is a special case.
+The first novel xenharmonic temperament — [[George Secor|George Secor's]] later-named "[[Miracle]]" temperament — was made to approximate Partch's 11-limit diamond.<ref>[https://www.anaphoria.com/SecorMiracle.pdf Secor, George (1975). “A New Look at the Partch Monophonic Fabric.” ''Xenharmonicon''. Vol. 3]</ref><ref>[https://www.anaphoria.com/SecorMiracle.pdf Secor, George. "The Miracle Temperament and Decimal Keyboard". ''Xenharmonikon''. Vol. 18. 2006. pp. 5–15. © 2003.]</ref>
+== Uses ==
+=== Instrument layout ===
+The most famous example of the tonality diamond as a practical layout for an instrument is Harry Partch's "Diamond Marimba," which uses the 11-odd-limit tonality diamond exactly. This idea was explored further with Partch's "Quadrangularis Reversum," and by Cris Forster with his [[13-odd-limit]] "Diamond Marimba."
+[https://sintel.website/posts/diamond_marimba.html Play with Partch's Diamond Marimba here.]
 == Examples of scales ==
@@ Line 19: / Line 41: @@
 * [[diamond13]]
 * [[diamond15]]
-* [[diamond9plus-marvel]]
 == Music ==
@@ Line 26: / Line 47: @@
 == External links ==
 * [http://www.tonalsoft.com/enc/t/tonality-diamond.aspx Tonality diamond – arrangement of musical frequency ratios showing the dual identity of each ratio] on [[Tonalsoft Encyclopedia]]
+== See also ==
+* [[Cross-set scale]]
+* [[Diamond function|Diamond Function]]
+* [[Lattice]]
+== References ==
+<references/>
 [[Category:Diamond]]
 [[Category:Pitch space]]