User:Nick Vuci/TonalityDiamond
WORK-IN-PROGRESS AS OF 07MAY2025
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.
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: 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. -
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: a) ascending numerically (ie, 2 3 5 7 9 11 etc) or, b) ascending by tonal order (ie 2 9 5 11 3 7), or c) ascending chordally (ie, 2 5 3 7 9 11).
History
The tonality diamond was first formally explained by Max F. Meyer in his 1929 publication The Musician's Arithmetic[1] 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.
Erv Wilson in particular was inspired by the tonality diamond and developed a number of "diamonds" himself.[2]
The first novel xenharmonic temperament — George Secor's later-named "Miracle" temperament — was made to approximate Partch's 11-limit diamond.
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."