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{{Wikipedia|Tonality diamond}}
{{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]] [[chord]]s based around a central note and bounded by an [[odd limit]]. First formalized in the [[7-odd-limit]] by {{w|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://tonalitydiamondapplet.nickvuci.com/ Play some tonality diamonds on your browser here.]


== Construction ==
== 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 (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).'''
</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-odd-limit diamond.
* Tonally: (1, 9, 5, 11, 3, 7) as in Partch's 11-odd-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.
 
Here is 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:
 
[[File:5-Limit Tonality Diamond original format.mp4|1000x400px]]
 
== 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 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 plagiarized the idea from Meyer.<ref>Forster, Cris (2015). [https://web.archive.org/web/20221207160002/https://www.chrysalis-foundation.org/the-partch-hoax-doctrines/ ''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.  


=== Relationship to subgroups ===
[[Erv Wilson]] in particular was inspired by Partch's use of the tonality diamond and its 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 inspired by Partch's use of the 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-odd-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.  
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.


== Examples of scales ==
The first novel xenharmonic temperament—[[George Secor]]'s later-named [[miracle]] temperament—was made to approximate Partch's 11-odd-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>
* [[diamond5]]
 
* [[diamond7]]
=== Instrument layout ===
* [[diamond9]]
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.<ref>[https://web.archive.org/web/20220901100217/https://www.chrysalis-foundation.org/instruments-and-music/diamond-marimba-i/ Diamond Marimba I – The Chrysalis Foundation]</ref>[[File:Diamond_marimba_layout.png|thumb|Layout of the Diamond Marimba. Ratios are shown unreduced to highlight the structure. [https://sintel.website/posts/diamond_marimba.html Click here to play the Diamond Marimba on your browser.]|488x488px|none]]
* [[diamond11]]
* [[diamond13]]
* [[diamond15]]
* [[diamond9plus-marvel]]


== Music ==
== Music ==
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Oldani/GWS%20Scale%20Study-ModernJazzAtTheCrystalBall%20.mp3 Modern Jazz at the Crystal Ball] by Norbert Oldani in the [[7-limit diamond]].
; [[Banjo Boogie]]
* [https://www.youtube.com/watch?v=1K227vkfBdc Demonstration and improvisation on a banjo drumset tuned to the 7-limit tonality diamond]
 
; [[Cris Forster]]
* [https://www.youtube.com/watch?v=jDFRGgWQp4I ''Dream Time'']
* [https://www.youtube.com/watch?v=6zyCZklRrnI ''The Harbor'']
* [https://www.youtube.com/watch?v=foqWB37nW7w ''Wild Flower'']
 
; [[Harry Partch]]
* [https://www.youtube.com/watch?v=kCuYcS_Lcro ''Castor & Pollux'']
* [https://www.youtube.com/watch?v=J_trV1AWU0Y "Diamond Marimba"] from ''The World of Harry Partch''
* [https://www.youtube.com/watch?v=qZybJAEPu18 ''Sonata Dementia''] (1950)
* [https://www.youtube.com/watch?v=gZiTiveqbDw ''Three Dances''] (1952)
 
; [[David Paulick]]
* [https://www.youtube.com/watch?v=-c3hYWunKps ''Improvisation using a Web MIDI Tonality Diamond''] (2022)
 
; [[T.J Troy]]
* [https://www.youtube.com/watch?v=4Q-sq9UwSgY ''Five-Corner Square'']
 
== Related scales ==
* [[Diamond5|5-odd-limit diamond]]
* [[Diamond7|7-odd-limit diamond]]
* [[Diamond9|9-odd-limit diamond]]
* [[Diamond11|11-odd-limit diamond]]
* [[Diamond13|13-odd-limit diamond]]
* [[Diamond15|15-odd-limit diamond]]
 
== See also ==
* [[Odd limit]]
* [[Cross-set scale]]
* [[Diamond function]]
* [[Lattice]]


== External links ==
== 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]]
* [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]]
* [https://www.youtube.com/watch?v=jsBsnNGkdcc Harry Partch's Diamond Marimba, as demonstrated by John Schneider]
* [https://www.youtube.com/watch?v=N57Wt0mpSu4 <nowiki>"What is the Tonality Diamond? (Harry Partch's Theories, Explained) [Harry Partch, Pt. 2/2]"</nowiki>] on [https://www.youtube.com/@ClassicalNerd Classical Nerd YouTube Channel]
* [https://web.archive.org/web/20221226001701/https://tonalitydiamond.com/ David Paulick's webapp of various tonality diamond inspired layouts for the Novation Launchpad] (now only available on archive.org)
== References ==
<references/>


[[Category:Diamond]]
[[Category:Diamond]]
[[Category:Pitch space]]
[[Category:Pitch space]]

Latest revision as of 05:17, 17 August 2025

English Wikipedia has an article on:

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,[1] the idea became central to the music and theories of Harry Partch,[2] 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.

Play some tonality diamonds on your browser here.

Construction

  • Step 1: Take the numbers of an odd limit and arrange them along two axes.
    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 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 (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).
    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).
  • 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).
    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: (1, 3, 5, 7, 9, 11) as in Meyer's 7-odd-limit diamond.
  • Tonally: (1, 9, 5, 11, 3, 7) as in Partch's 11-odd-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.

Here is 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:

Load video
Local File

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 plagiarized the idea from Meyer.[3] 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 its extended form. He developed a number of "diamonds" himself,[4] as well as other concepts inspired by Partch's use of the extended tonality diamond such as constant structure.[5] 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-odd-limit diamond.[6][7]

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.[8]

Layout of the Diamond Marimba. Ratios are shown unreduced to highlight the structure. Click here to play the Diamond Marimba on your browser.

Music

Banjo Boogie
Cris Forster
Harry Partch
David Paulick
T.J Troy

Related scales

See also

External links

References

  1. ↑ 1.0 1.1 Meyer, Max F. (1929) 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.
  2. ↑ Harry Partch (1949), Genesis of a Music, University of Wisconsin Press
  3. ↑ Forster, Cris (2015). The Partch Hoax Doctrines. Self-published.
  4. ↑ Wilson, Erv. (1965–1970) Letters on Diamond Lattices (PDF) Self-published.
  5. ↑ Wilson, Erv. (1964-2002) The Partch Papers (collection of documents on Harry Partch's 11-odd-limit diamond and its extensions, PDF). Self-published.
  6. ↑ Secor, George (1975). A New Look at the Partch Monophonic Fabric. Xenharmonicon. Vol. 3
  7. ↑ Secor, George. (2006) The Miracle Temperament and Decimal Keyboard. Xenharmonikon. Vol. 18. 2006. pp. 5–15
  8. ↑ Diamond Marimba I – The Chrysalis Foundation
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