1 Ben Johnston, in notes to the recording of his String Quartet No. 4 on Gasparo GS-205.

2 The cent is a logarithmic unit of measure. The octave is divided into 1,200 cents; thus every 12-equal semitone is 100 cents. Differences in pitch of only a few cents are often easily audible, and of less than that audible in certain environments. Even inaudible amounts become audible when they are ‘added’, however. We may not hear the difference between a fifth of 700.00000 cents and one of 701.88679, but a chain of 12 of one fifth will land in a very different place than a chain of 12 of the other.

3 Other composers such as Carrillo, Hába, and Vyshnegradskiy have written extensively in 24 equal (i.e. quartertones in terms of 12 equal), although further equal subdivision of the 12-equal tone or semitone provides relatively few musical gains.

4 Going back at least as far as the time of Nicola Vicentino (1511–1572).

5 The trademarked and patented Scalatron, made by Motorola Inc. of Chicago under the direction of Richard Harasek, is no longer in production.

6 Theoretical design by Erv Wilson.

7 There is still a question of how chords and keys can best be connected in just tunings, a fascinating problem which cannot be discussed here.

8 Special thanks are due to Richard Harasek, George Secor, Ken Stansfield, and Erv Wilson for their assistance in developing plans and materials for the Scalatron mentioned in this article.

9 Partch, Harry: Genesis of a Music, 2nd edition (New York: Da Capo, 1974), p. 125 Google Scholar.