August, 1990

--------Ivor Darreg---------------- Composer and instrument designer

DIAGRAM OF THE 24 JUST PITCHES IN THE FIRST JUST

INTONATION TABLE. ------- (7 LIMIT)

------The ratios of the frequencies of the various tones symbolized here involve one, two, or three of the prime factors 3, 5, and 7; hence, after Harry Partch, such a tonal array is known as a 7-limit scheme. Names of the notes are after Alexander J. Ellis in the English edition of Helmholtz's Sensations of Tone. Sub- and superscript numerals denote commas 81:80 below or above the chain-of-fifths meaning of the note-names, respectively. Other authors have used other means of denoting commas 81:80; no prejudice is intended at all.

Ellis used a superscript prefixed 7 to denote septimal intervals; we use a printer's obelisk or dagger here. The diagram above can be considered a small selection from an infinite 3-dimensional lattice. This or certain other possible arrays of 24 just pitches, are about the smallest number of tones per octave affording the possibility of playing existing music with some nearby modulations.

Even this array may not be ample enough for newly-composed music; but with 24 tones per octave the possibility exists of retuning certain electronic organs with two sets of 12 tone-generators per octave each.

In our opinion more harm than good will be done the cause of composing and performing in just intonation if the attempt is made to get along with fewer notes per octave than this. The awkward sudden jumps to avoid modulat ions beyond the array and even more awkward shifts of a comma up or down will soon annoy and prejudice listeners. So will the attempt to tune a piano to 12 notes of just intonation, which cannot be the best selection of 12 notes without breaking strings. In this Electronic Age there is no excuse for such unquiet desperation.