User:Nick Vuci/TonalityDiamond
WORK-IN-PROGRESS AS OF 10MAY2025
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
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Step 1: Take the numbers of an odd-limit and arrange them along two axes. -
Step 2: Using one row 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
- by tonal order (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 explains that he 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.
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,[2] as well as other concepts based on Partch's extended tonality diamond such as "constant structure."[3]
The first novel xenharmonic temperament — George Secor's later-named "Miracle" temperament — was made to approximate Partch's 11-limit diamond.[4][5]
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."
See also
References
- ↑ 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.
- ↑ Wilson, Erv. Letters on Diamond Lattices, 1965–1970 (PDF). Self-published.
- ↑ Wilson, Erv. The Partch Papers (collection of documents on Harry Partch’s 11-limit diamond and its extensions), 1964-2002 (PDF). Self-published.
- ↑ Secor, George (1975). “A New Look at the Partch Monophonic Fabric.” Xenharmonicon. Vol. 3
- ↑ Secor, George. "The Miracle Temperament and Decimal Keyboard". Xenharmonikon. Vol. 18. 2006. pp. 5–15. © 2003.