  # unit of interval measurement

[Joe Monzo]

In tuning theory, it is useful to have a small pitch interval which can be used as a logarithmic unit of measurement, to compare the sizes of different microtonal intervals. The unit most often used today is the cent, which is logarithmically 1/100 of a 12-edo semitone and 1/1200 of an octave of 2:1 ratio. Several other proposals have been advocated throughout history and in various different musical cultures, including the following (ordered by size from largest to smallest):

• (12-edo) semitone = 1/12 octave = exactly 100 cents
• sruti = approximately 1/22 octave (sizes vary widely, generally not considered equal); ~90, 70, and 22 cents
• quarter-tone = 1/24 octave = exactly 50 cents
• 31-edo diesis = 1/31 octave = exactly 3822/31 (38.709677419354838...) cents
• méride = 1/43 octave = exactly 2739/43 (27.906976744186046511627...) cents
• cleonides-morion = 1/30 perfect-4th (4:3 ratio) = ~16.6 cents
• 72-edo morion = 1/72 octave = exactly 162/3 (=16.6...) cents
• farab = 1/144 octave = exactly 81/3 (= 8.3...) cents
• mem = 1/205 octave = exactly 535/41 (= 5.85365...) cents
• savart = 1/300 octave = exactly 4 cents
• heptaméride = 1/301 octave = exactly 3297/301 (= 3.986710963455149501661129568106312292358803...) cents
• tredek = 1/270 octave = exactly 4 + 4/9 (= ~4.4...) cents
• gene = 1/311 octave = exactly 3267/311 (= ~3.858520900321543408360128617363344051446945337620 [the denominator is a prime so the decimal part is huge and continues from here] = ~3 6/7 = ~3.86) cents
• centitone / iring = 1/600 octave = exactly 2.0 cents
• sk (612-edo schisma) = 1/612 octave = exactly 149/51 (= 1.9607843137254901...) cents
• grad = 1/12 pythagorean-comma = ~1.955 cents
• woolhouse-unit = 1/730 octave = exactly 147/73 (= 1.64383561...) cents
• millioctave = 1/1000 octave = exactly 11/5 (= 1.2) cents
• yamaha-unit = 1/1024 octave = exactly 111/64 (= 1.171875 = ~11/6) cents
• cent = 1/1200 octave = exactly 1.0 cent
• mina = 1/2460 octave = exactly 20/41 (= 0.48780... = ~1/2) cent
• tina = 1/8539 octave = exactly 1200/8539 (= ~0.14053167818245696217355662255533434828434242885 [the denominator is a prime so the decimal part is huge and continues from here] = ~1/7) cent
• türk-sent = 1/10600 octave = exactly 6/53 (= 0.1132075471698... = ~1/9) cent
• jot = 1/30103 octave = exactly 1200/30103 (= ~0.0398631365644620137527821147393947447098295850911869248912068564594890874663654785237351758960900906 [the denominator is a prime so the decimal part is huge and continues from here] = ~1/25 = ~0.04) cent
• tuning-unit = 1/720 pythagorean-comma = ~0.0326 (~1/31) cent
• flu = 1/46032 octave = 25/959 (= 0.026068821689259645464025... = ~1/38) cent

In addition, there is the whole series of MIDI-units (mu).

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