Tonality diamond: Difference between revisions

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no number theory in the intro please! (page needs a lot more work)
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merg in stuff from User:Nick_Vuci/TonalityDiamond
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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]] 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 ==
== 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 ==
== Examples of scales ==
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* [[diamond13]]
* [[diamond13]]
* [[diamond15]]
* [[diamond15]]
* [[diamond9plus-marvel]]


== Music ==
== Music ==
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== 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]]
== See also ==
* [[Cross-set scale]]
* [[Diamond function|Diamond Function]]
* [[Lattice]]
== References ==
<references/>


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

Revision as of 13:44, 11 May 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, 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.

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 (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).
    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).
    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

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."

Play with Partch's Diamond Marimba here.

Examples of scales

Music

External links

See also

References

  1. ↑ 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.
  2. ↑ Forster, Cris (2015). The Partch Hoax Doctrines. Self-published.
  3. ↑ Wilson, Erv. Letters on Diamond Lattices, 1965–1970 (PDF). Self-published.
  4. ↑ Wilson, Erv. The Partch Papers (collection of documents on Harry Partch’s 11-limit diamond and its extensions), 1964-2002 (PDF). Self-published.
  5. ↑ Secor, George (1975). “A New Look at the Partch Monophonic Fabric.” Xenharmonicon. Vol. 3
  6. ↑ Secor, George. "The Miracle Temperament and Decimal Keyboard". Xenharmonikon. Vol. 18. 2006. pp. 5–15. © 2003.
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