The mil: 1/1023 of an equal-tempered semitone
In musical temperament, intervals are usually stated in cents, i.e. hundredths of a 12-EDO semitone. A more convenient unit would be 1/12276 of an octave, i.e. 1/1023 of a 12-EDO semitone. I call this a mil because it is nearly 1/1000 of a semitone (cf. 1KB = 1024 bytes). Thus the frequency ratio f2/f1 is expressed in mils as 12276 log2(f2/f1). Advantages of the mil include: (i) its size is almost self-explanatory; (ii) rounding to the nearest mil is accurate enough for practical purposes; (iii) numerous important intervals are very close to whole numbers of mils:
| Interval | mils |
|---|---|
| Syntonic comma | 220.009 |
| Ditonic comma | 239.996 |
| Schisma | 19.987 |
| Lesser diesis | 420.03 |
| Greater diesis | 640.04 |
| Pythagorean limma | 923.002 |
| Diaschisma | 200.02 |
| Just fifth | 7180.9997 |
| Just major third | 3951.989 |
| Just minor third | 3229.01 |
| 31-EDO diatonic semitone (3/31 octave) | 1188* |
| 31-EDO chromatic semitone (2/31 octave) | 792* |
| * Exact because 1023 is divisible by 31. |
At Xenharmonic, the same unit has been named the prima but not (yet) further discussed. It was Brombaugh's temperament-unit (tu) that led me to the idea (1 mil ≈ 3.00005 tu).
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