User:Nick Vuci/TonalityDiamond: Difference between revisions

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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.

~~= Tonality Diamond =~~

The 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, 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 intervals 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 here ~~to see how they sound~~.] ~~===~~

'''[https://nickvuci.github.io/wiki-applets/TonalityDiamond/tonalityDiamond.html Play with tonality diamonds in your browser here.]'''

== How to ~~make a~~ tonality diamond ==

== Construction ==

[[File:~~NickVuci-~~How to ~~make a~~ tonality diamond.png|~~right~~|~~750x750px]]~~

~~Making a~~ tonality diamond ~~involves a few simple steps~~:

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:

~~# ''~~'~~Take the numbers of an odd~~-limit ~~and arrange them along two axes.'''~~

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

~~# '''Using one row as the numerator and the other~~ as ~~the denominator, fill~~ in ~~the cells with the ratios they form.~~'''

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

~~# '''Make sure the decimal form of the ratio is between 1 and 2. If it is not, double one of the numbers until it is.'''~~

~~Some finer points:~~

~~- the numbers of the odd-limit are generally arranged in one of two ways: a) ascending numerically~~ (ie, ~~2 3 5 7 etc) or, b) ascending by tonal order (ie 2~~ 5 3 7)

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

~~- in order to make~~ the ~~rows play rooted~~ chords~~, one half~~ of the diamond ~~(not including the middle unison row) must be lowered by an octave~~.

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:

[[File:5-Limit_Tonality_Diamond_original_format.mp4|1000x400px|center]]

== History ==

The tonality diamond was first formally explained by [https://archive.org/details/max-f-meyer-the-musicians-arithmetic Max F ~~Meyer in his 1929 publication~~ ''The ~~Musician~~'~~s Arithmetic~~'~~'] using the 7-odd-limit~~. ~~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~~.

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>

The first novel xenharmonic temperament — George Secor's later-named "Miracle" temperament — was made to approximate Partch's 11-limit diamond.

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."

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.''']

== See also ==

* [[Cross-set scale]]

* [[Diamond function|Diamond Function]]

* [[Lattice]]

== References ==

== External links ==

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

* [[wikipedia:Tonality_diamond|Wikipedia page on tonality diamonds]]

* [http://www.tonalsoft.com/enc/t/tonality-diamond.aspx Tonalsoft page on tonality diamonds]

* [https://www.chrysalis-foundation.org/musical-mathematics-pages/meyer-diamond/ Cris Forster's site on Meyer's tonality diamond]

@@ Line 1: / Line 1: @@
-  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.
-= Tonality Diamond =
-The 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, 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 intervals 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 here to see how they sound.] ===
+'''[https://nickvuci.github.io/wiki-applets/TonalityDiamond/tonalityDiamond.html Play with tonality diamonds in your browser here.]'''
-== How to make a tonality diamond ==
+== Construction ==
-[[File:NickVuci-How to make a tonality diamond.png|right|750x750px]]
+<gallery mode="nolines" widths="200" heights="200">
-Making a tonality diamond involves a few simple steps:
+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:
-# '''Take the numbers of an odd-limit and arrange them along two axes.'''
+* numerically (ie, 1 3 5 7 9 11) as in Meyer's 7-limit diamond
-# '''Using one row as the numerator and the other as the denominator, fill in the cells with the ratios they form.'''
+* tonally (ie, 1 9 5 11 3 7) as in Partch's 11-limit diamond
-# '''Make sure the decimal form of the ratio is between 1 and 2. If it is not, double one of the numbers until it is.'''
-Some finer points:
-- the numbers of the odd-limit are generally arranged in one of two ways: a) ascending numerically (ie, 2 3 5 7 etc) or, b) ascending by tonal order (ie 2 5 3 7)
+* chordally (ie, 1 5 3 7 9 11) as in the layout for the Diamond Marimba
-- in order to make the rows play rooted chords, one half of the diamond (not including the middle unison row) must be lowered by an octave.
+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:
+[[File:5-Limit_Tonality_Diamond_original_format.mp4|1000x400px|center]]
 == History ==
-The tonality diamond was first formally explained by [https://archive.org/details/max-f-meyer-the-musicians-arithmetic Max F Meyer in his 1929 publication ''The Musician's Arithmetic''] using the 7-odd-limit. 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.
+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>
-The first novel xenharmonic temperament — George Secor's later-named "Miracle" temperament — was made to approximate Partch's 11-limit diamond.
+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."
+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.''']
+== See also ==
+* [[Cross-set scale]]
+* [[Diamond function|Diamond Function]]
+* [[Lattice]]
+== References ==
+<references />
+== External links ==
-[https://sintel.website/posts/diamond_marimba.html Play with Partch’s Diamond Marimba here.]
+* [[wikipedia:Tonality_diamond|Wikipedia page on tonality diamonds]]
+* [http://www.tonalsoft.com/enc/t/tonality-diamond.aspx Tonalsoft page on tonality diamonds]
+* [https://www.chrysalis-foundation.org/musical-mathematics-pages/meyer-diamond/ Cris Forster's site on Meyer's tonality diamond]