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WORK-IN-PROGRESS AS OF 07MAY2025
Moved to [[Tonality diamond|main space]] 11 MAY 2025 by [[User:Sintel|Sintel]]
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.
The 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, they became central to the music and theories of [[Harry Partch]], who built his tonal system around the [[11-odd-limit]] tonality diamond. The principle has been used both conceptually (such as for [[Target tuning|targets]] of [[temperaments]]) and practically (such as for instrument layouts) in xenharmonics ever since.
[https://nickvuci.github.io/wiki-applets/tonalityDiamond.html '''Play some tonality diamonds to hear how they sound.''']
'''[https://nickvuci.github.io/wiki-applets/TonalityDiamond/tonalityDiamond.html Play with tonality diamonds in your browser here.]'''
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 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 row as the numerator and the other as the denominator, fill in the cells with the ratios they form.'''
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: Make sure the decimal form of the ratios is between 1 and 2. If it is not, double one of the numbers until it is.'''
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).'''
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:
</gallery>Note: the numbers of the odd-limit are generally arranged in one of three ways:
* numerically (ie, 2 3 5 7 9 11) as in Meyer's 7-limit diamond
* numerically (ie, 1 3 5 7 9 11) as in Meyer's 7-limit diamond
* by tonal order (ie, 2 9 5 11 3 7) as in Partch's 11-limit diamond
* tonally (ie, 1 9 5 11 3 7) as in Partch's 11-limit diamond
* chordally (ie, 2 5 3 7 9 11) as in the layout for the Diamond Marimba
* chordally (ie, 1 5 3 7 9 11) as in the layout for the Diamond Marimba
Here's a short video illustrating the interlocking nature of the otonal and utonal chords and constant presence of the 1/1 interval in the 5-limit tonality diamond:
The tonality diamond was first formally explained by Max F. Meyer in his 1929 publication ''The Musician's Arithmetic''<ref>https://archive.org/details/max-f-meyer-the-musicians-arithmetic/page/22/mode/2up</ref> using the 7-odd-limit.
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>
Even though Harry Partch gives a different story for how he discovered the concept, it is likely this source that gave him the idea, which he then extended to the 11-odd-limit and made the basis of his tonal system.
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 the tonality diamond and developed a number of "diamonds" himself.<ref>https://anaphoria.com/diamond.pdf</ref>
[[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.
A tonality diamond is a symmetric organization of otonal and utonal chords based around a central note and bounded by an odd-limit. First formalized in the 7-odd-limit by 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 targets of temperaments) and practically (such as for instrument layouts) in xenharmonics ever since.
Step 1: Take the numbers of an odd-limit and arrange them along two axes.
Step 2: Using one axis as the numerator and the other as the denominator, fill in the cells with the ratios they form.
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).
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).
Note: the numbers of the odd-limit are generally arranged in one of three ways:
numerically (ie, 1 3 5 7 9 11) as in Meyer's 7-limit diamond
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
Here's a short video illustrating the interlocking nature of the otonal and utonal chords and constant presence of the 1/1 interval in the 5-limit tonality diamond:
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.[1]
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.[2] 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,[3] as well as other concepts based on Partch's extended tonality diamond such as "constant structure."[4] A related idea of Wilson's is the "cross-set," of which the tonality diamond is a special case.
The first novel xenharmonic temperament — George Secor's later-named "Miracle" temperament — was made to approximate Partch's 11-limit diamond.[5][6]
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."
Step 1: Take the numbers of an odd-limit and arrange them along two axes.
Step 2: Using one axis as the numerator and the other as the denominator, fill in the cells with the ratios they form.
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).
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).
