mouseover diamond to hear tones

This site is a work in progress. For the moment it works in Chrome and Edge, but may be unpredictable in other browsers. In the fall of 2022 will be adding other synth possibilities, a looper, and a better user interface. The information about MIDI mapping is at the bottom of this page.

The Tonality Diamond was used by the American composer Harry Partch as the foundation of his 11 limit Just Intonation scale, showing possibilities for ratio-based harmony. The diamond had previously been presented in the research of the Mexican composer and theorist Augusto Novaro and that of the American psychologist Max Meyer. In Just Intonation terms, a limit is the largest prime number used as a factor in a series of frequency ratios. Ratios in which the identity (odd numbers) are the numerator are called Otonalities by Partch, these are the basis of major scales, whereas ratios with the identity as the denominator are called Utonalities, the origin of minor. In the Tonality Diamonds presented here, Otonalities, intervals derived from the harmonic series, are displayed from left to right. Utonalities, from the subharmonic series, are displayed from right to left. We have kept Partch's convention of labelling the intervals with their octave reduced form, rather than the actual frequency.

Included on this page are playable versions of:

the five limit diamond: including just major thirds and sixths, and their inversions the just minor sixths and thirds. These intervals are the foundation of the post-renaissance western tonal system.

the seven limit diamond: which adds ratios related to the natural seventh harmonic.

the eleven limit diamond: adding ratios of nine and eleven. The basis of Partch's 43 tone scale.

the fifteen limit diamond diamond: Fifteen is not technically a limit as it is not a prime number, but rather a product of 3 and 5, yet it increases musical possibilities by bringing in ratios of both 13 and 15. The 13/8 is 40.53 cent sharper than an equal tempered minor sixth, and therefore offers a flavour not found in 12-TET. The ratios of 15 are more familiar, Just Intonation major sevenths as otonalities or Just Intonation minor seconds as utonalities.

A number of composers have experimented with more complex intervals and higher harmonics. La Monte Young in his sine-tone installations has gone as far as the 2304th harmonic and perhaps further. Ben Johnston's 7th string quartet uses more than 176 Just Intonation pitches. Johnny Reinhard, organiser of the American Festival of Microtonal Music and the online Microtonal University, has worked extensively with 128, a tuning based on the eighth octave of the harmonic series, avoiding the lower harmonics which strongly define the tonality. Zheanna Erose's theory of primodality, creating modes out of subsections of the harmonic series, is a notably accessible approach. For the purpose of experimenting with intervals derived from higher harmonics, the 8 by 8 playable MIDI diamond has been adapted to use prime numbers between 17 and 41. This is one of many selective higher harmonic approaches that could be considered.

Resources and essential reading

Some resources used in creating this site:

Genesis of a Music

Partch, Harry

Da Capo Press New York, 1949, 1974

The Arithmetic of Listening

Gann, Kyle

University of Illinois Press, 2019

Reinhard, Johnny.

American Festival of Microtonal Music

Microtonal University


Microtonality and the Tuning Systems of Erv Wilson

Narushima, Terumi

Routledge, London and New York, 2018

Erose Zheanna

Theory of Primodality

MIDI and technical information

The fifteen limit diamond is available here as mapped to the original Novation Launchpad Mini MIDI controller. Other diamonds are featured with more range to fit the 8 X 8 grid of the Launchpad. It is also possible to select the Legacy mapping for the Launchpad Pro with the Select Midi Device button. To activate Legacy Mode on the more recent Launchpad Pro Mk3 press the setup button then Mutation, if the lights go off Legacy Mode is activated - if the device is disconnected it will return to the original setting. The MIDI diamond can be played with other MIDI controllers configured to the following mapping of midi numbers to ratios, first the original Launchpad mini, followed by the Launpad Pro Legacy mapping:

We will be adding more synth controls soon.