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Encyclopedia of Microtonal Music Theory

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Semitone / semitone / 1/2-tone

[Joe Monzo]
1. Musical interval measurement unit

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

Monzo writes: "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 12-edo scale. I use cents on occasion, when I feel that more precision is valuable."

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Semitones calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = Semitones

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2. One degree of 12-edo

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 -- 12√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 Monzo, Simplified sexagesimal approximation to 12edo and Monzo, Speculations On Sumerian Tuning.)

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3. Approximate half-tone

The term "semitone" is also used loosely in a general sense to indicate any interval of approximately 100 cents -- or even more generally, approximately half of a whole-tone -- including the limma, apotome, and many others.

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[John Chalmers, Divisions of the Tetrachord]

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.

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