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:
|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).