[Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 74-75]

An arbitrary arrangement of the Monophonic [musical frequency] ratios designed to constitute *prima facie* proof of the at least dual identity of each ratio, and consequently of the capacity of a Monophonic system of Just Intonation for providing tones that may be taken in more than one sense each.

Partch's Incipient Tonality Diamond (5-Limit):

(for a different view of this diamond, see the Monzo 5-Limit Lattice)

. . . . . . . . .

[John Chalmers, Divisions of the Tetrachord]

A chordal complex consisting of interlocking harmonic and subharmonic chords sharing a single tonic. It may be constructed by building replicas of a harmonic chord on roots that are the components of the octave inversion of the original. The highest prime (3, 5, 7, 11, etc.) number appearing in the chords determines the "limit".

The Diamond may be understood as the harmonic expansion of a single complex sound.

See Partch, Harry (1949, 1974, 1977).

. . . . . . . . .

[Paul Erlich, private communication with Joe Monzo]

When interpreted as a set of pitches instead of a set of intervals, the `n`-odd-limit is known as the "`n`-limit tonality diamond".

. . . . . . . . .

[Joe Monzo]

Illustrated below is an idea Joe Monzo had for a keyboard instrument, back around 1993. The buttons of the keyboard are arranged in a diamond, and the 50-cent boundaries between the 12-EDO scale are used to paint a pattern on the background similar to the usual Halberstadt pattern of black and white piano keys.

19-limit Tonality Diamond Keyboard idea, ©1993 by Joe Monzo

The pitches of a 19-odd-limit tonality diamond are laid out in Partch's diamond format, but arranged so that the otonal and utonal decads form ascending and descending scales, respectively. The red lines are otonal and the blue lines are utonal. The proportions of the scales are thus 16:17:18:19:(20=10):11:12:13:14:15.

The horizontal axis represents pitch-height exactly, so that a vertical column of buttons, viewed against the background of either a white or black "key", gives a precise view of all the different versions of any given 12-EDO pitch-class.

For example, looking down the row of buttons over the black "C#/Db key" just to the right of "middle-C" (a "white key" which runs down the exact center of the picture), from top to bottom, one can see:

15/14 = the 15-odentity of 8/7-otonality and the 7-udentity of 15/8-utonality, 14/13 = the 7-odentity of 16/13-otonality and the 13-udentity of 7/4-utonality, 13/12 = the 13-odentity of 4/3-otonality and the 3-udentity of 13/8-utonality, 12/11 = the 3-odentity of 16/11-otonality and the 11-udentity of 3/2-utonality, 20/19 = the 5-odentity of 32/19-otonality and the 19-udentity of 5/4-utonality, 19/18 = the 19-odentity of 16/9-otonality and the 9-udentity of 19/16-utonality, 18/17 = the 9-odentity of 32/17-otonality and the 17-udentity of 9/8-utonality, and 17/16 = the 17-odentity of 1/1-otonality and the 1-identity of 17/16-utonality,and finally, if we look over at the lone C#/Db in the lower 2:1 ("octave") of the keyboard (on the left side), we see:

16/15 = the 1-odentity of 16/15-otonality and the 15-udentity of 1/1-utonality.

This gives a nice graphical presentation of all the different pitches falling between 0.50 and 1.50 Semitones (or 50 and 150 cents), which would have the following values in my 72-EDO-based HEWM notation, C = n^0 ,

Semitones ratio HEWM prime-factor ------- monzo ------- 3 5 7 11 13 17 19 0.89 20/19 Db- 5where the accidentals have the following meaning:^{1}19^{-1}| 0 1 0 0 0 0 -1| 0.94 19/18 C# 3^{-2}19^{1}|-2 0 0 0 0 0 1| 0.99 18/17 C# 3^{2}17^{-1}| 2 0 0 0 0 -1 0| 1.05 17/16 Db 17^{-1}| 0 0 0 0 0 -1 0| 1.12 16/15 Db+ 3^{-1}5^{-1}|-1 -1 0 0 0 0 0| 1.19 15/14 C#+ 3^{1}5^{1}7^{-1}| 1 1 -1 0 0 0 0| 1.28 14/13 Db> 7^{1}13^{-1}| 0 0 1 0 -1 0 0| 1.39 13/12 Db> 3^{-1}13^{1}|-1 0 0 0 1 0 0| 1.51 12/11 Dv 3^{1}11^{-1}| 1 0 0 -1 0 0 0|

^ +1/4-tone > + <6th-t>1/6-tone + + <12th-t>1/12-tone 0 - - 1/12-tone < - 1/6-tone v - 1/4-tone

And so on for all the other "black and white keys".
The only "key" or pitch-class having only one pitch
is C = `n`^{0}.

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

tredek (unit of interval measurement)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)