## Definitions of tuning terms © 1998 by Joseph L. Monzo All definitions by Joe Monzo unless otherwise cited

Semitone

• 1. With the first letter capitalized and always with two decimal places in the number, a term used by me to delineate 1200 logarithmic divisions of the "octave", thus exactly analogus to Ellis's measurement of cents.

I feel that since the prime-factor or ratio notations give precise measurements, and 1/1200th of an "octave" is approximately the limit of human pitch discrimination, more precision than this is not ordinarily needed, and I prefer to use the decimal point so that the interval may be related immediately to the familiar 12edo scale. I use cents on occasion, when I feel that more precision is valuable.

• 2. With all letters in lower-case and no decimal places in the number, the term simply refers to a logarithmic division of 1/12 of an "octave", or one degree or "half-step" in the familiar 12edo scale.

In this specific sense, the Semitone is calculated as the 12th root of 2, or 2(1/12), an irrational proportion with the approximate ratio of 1:1.059463094359.

Successively closer rational approximations to the semitone are:

```
ratio    prime-factorization    approx. cents error from 2(1/12)

18:17      21 32 17-1             - 1.0454 (~ - 1 1/22 )
89:84      2-2 3-1 7-1 891        + 1/10
196:185     22 5-1 72 37-1         - 1/170
1657:1564    2-2 17-1 23-1 16571    - 1/3,400
7893:7450    32 5-2 149-1 8771      - 1/86,000
```

(For some base-60 approximations, see my Simplified sexagesimal approximation to 12edo and Speculations On Sumerian Tuning.)

• 3. The term "semitone" is also used loosely in a general sense to indicate any interval of approximately 100 cents, including the limma, apotome, and many others.

[from Joe Monzo, JustMusic: A New Harmony]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A Half Tone, a musical interval ranging from about 25/24 (71 cents [¢]) to 27/25 (133¢). Unless qualified by context, a semitone equals 100¢.

Semitones measuring less than 100¢ are technically microtones.

[from John Chalmers, Divisions of the Tetrachord]

updated:

2003.07.05 -- added defintion #3, and rational approximations to definition #2

 For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here