Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
Commas are small musical intervals between a note formed by one cycle of just intervals and the same note formed by another cycle of different just intervals. These commas prove important in constructing musical scales, and in notating musical chords and musical pitches.
The Pythagorean comma is the difference between 12 just perfect fifths up and 7 octaves up:
[from Brian McLaren]
McLaren's description can be shown in both vector notation and in regular fractional math thus:
2 3 [12 12] (3/2)^{12} = 3^{12}/2^{12} = 531441/4096 531441/4096 12 "5ths"  [ 7 0] ÷ (2/1)^{7} = 128 * 1/128 7 "8ves"  =  =  [19 12] 531441/524288 Pythagorean comma
The ratio 531441/ 524288, in JustMusic primefactor notation designated as 3^{12}, with an interval size of approximately 0.23 Semitones [= 23.460+ cents].
It is the difference between the Pythagorean or 3Limit "tritone" or "augmented 4th" of approximately 6.12 Semitones [= 729/512 = 3^{6} = 611.730+ cents] and the Pythagorean or 3Limit "diminished 5th" of approximately 5.88 Semitones [= 1024/729 = 3^{6} = ~588.2699948 cents], shown below in both vector notation and in regular fractional math:
for C = n^{0} = 1/1 : 2 3 [ 9 6] (9/8)^{3} = 9^{3}/8^{3} = 729/512 729/512 tritone: F:B  [ 10 6] ÷ (2^{2}3^{1})/(2^{8}3^{5}) = (4/3)/(243/256) = 1024/729 * 729/1024 "diminished 5th": B:F  =  =  [19 12] 531441/524288 Pythagorean comma
It was first described c. 300 BC by pseudoEuclid in Divisions of the Canon.
Below are some other interval measurements for the Pythagorean comma:
~0.234600104 (~^{1}/_{4}) Semitone ~1.407600623 (~1 ^{2}/_{5}) 72edomoria ~5.865002596 (~5 ^{6}/_{7}) savarts 12 grads ~14.27150632 (~14 ^{1}/_{4}) 730edo Woolhouseunits ~19.55000865 (~19 ^{5}/_{9}) milli8ves ~20.72300917 (~20 ^{5}/_{7}) Türk cents ~23.46001038 (~23 ^{1}/_{2}) cents ~588.5139105 (~588 ^{1}/_{2}) jots 720 tuning units ~960.9220254 (~961 ^{71}/_{77}) cawapus ~3843.688101 (~3843 ^{2}/_{3}) midipus
[from Joe Monzo, JustMusic: A New Harmony]
updated:
2002.09.28  added table of interval measurements
2002.09.22  added vector notation for tritone explanation
2002.09.07  added vector notation for McLaren's explanation
1999.12.16
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