By Gavin Putland

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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Gavin Putland



Published: 20 Mar, 2015

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