Tonality diamond: Difference between revisions
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{{Wikipedia|Tonality diamond}} | {{Wikipedia|Tonality diamond}} | ||
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. | 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,<ref name="meyer1929">Meyer, Max F. (1929) [https://archive.org/details/max-f-meyer-the-musicians-arithmetic/page/22/mode/2up ''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> the idea became central to the music and theories of [[Harry Partch]],<ref>Harry Partch (1949), ''Genesis of a Music'', University of Wisconsin Press</ref> 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. | |||
[https://nickvuci.github.io/wiki-applets/tonalityDiamond.html Play some tonality diamonds to hear how they sound.] | |||
== Construction == | == Construction == | ||
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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 axis 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: Octave-reduce the ratios ( | File:How to tonality diamond 3.png|'''Step 3: Octave-reduce the ratios (i.e. 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: (1, 3, 5, 7, 9, 11) as in Meyer's 7-limit diamond. | ||
* | * Tonally: (1, 9, 5, 11, 3, 7) as in Partch's 11-limit diamond. | ||
* | * Chordally: (1, 5, 3, 7, 9, 11) as in the layout for the Diamond Marimba. This creates a 4:5:6:7:9:11 extended 11th chord on the diagonal, arranged in thirds. | ||
== History == | == 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 | 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 name="meyer1929"/> | ||
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 | 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>Forster, Cris (2015). [https://www.chrysalis-foundation.org/wp-content/uploads/ThePartchHoaxDoctrines.pdf ''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 | [[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>Wilson, Erv. (1965-1970) [https://anaphoria.com/diamond.pdf ''Letters on Diamond Lattices''] (PDF) Self-published.</ref> as well as other concepts based on Partch's extended tonality diamond such as [[constant structure]].<ref>Wilson, Erv. (1964-2002) [https://www.anaphoria.com/Partchpapers.pdf ''The Partch Papers''] (collection of documents on Harry Partch's 11-limit diamond and its extensions, 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 | The first novel xenharmonic temperament—[[George Secor|George Secor's]] later-named [[Miracle]] temperament—was made to approximate Partch's 11-limit diamond.<ref>Secor, George (1975). [https://www.anaphoria.com/SecorMiracle.pdf ''A New Look at the Partch Monophonic Fabric.''] Xenharmonicon. Vol. 3</ref><ref>Secor, George. (2006) [https://www.anaphoria.com/SecorMiracle.pdf ''The Miracle Temperament and Decimal Keyboard'']. Xenharmonikon. Vol. 18. 2006. pp. 5–15</ref> | ||
=== Instrument layout === | === 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]] | 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.] | [https://sintel.website/posts/diamond_marimba.html Play with Partch's Diamond Marimba here.] | ||
== Examples of scales == | == Examples of scales == | ||
* [[diamond5]] | * [[diamond5|5-limit diamond]] | ||
* [[diamond7]] | * [[diamond7|7-limit diamond]] | ||
* [[diamond9]] | * [[diamond9|9-limit diamond]] | ||
* [[diamond11]] | * [[diamond11|11-limit diamond]] | ||
* [[diamond13]] | * [[diamond13|13-limit diamond]] | ||
* [[diamond15 | * [[diamond15|15-limit diamond]] | ||
== External links == | == External links == | ||
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== See also == | == See also == | ||
* [[Odd limit]] | |||
* [[Cross-set scale]] | * [[Cross-set scale]] | ||
* [[Diamond function|Diamond Function]] | * [[Diamond function|Diamond Function]] | ||