Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
Philolaus described the schisma as an integral 1/2 of the comma (which today we call the Pythagorean comma):
Philolaus's schisma = comma^{(1/2)} = (2^{19} * 3^{12})^{(1/2)} = 2^{(19/2)} * 3^{6} = ~11.73 cents.
Unqualified, it is the difference between the 5^{1} just "major 3rd" and the 3^{8} Pythagorean "diminished 4th", and has an interval size of approximately ^{1}/_{50} Semitone [= ~ 2 cents]:
primefactor vector ratio Semitones ~cents 2 3 5 [ 2 0 1] 5/4 3.86 386.3137139 just "major 3rd"  [ 13 8 0] 8192/6561 3.84 384.3599931 Pythagorean "diminished 4th"  [15 8 1] 32805/32768 0.02 1.953720788 skhisma
The skhisma may also be found as the difference between the Pythagorean comma and the syntonic comma:
primefactor vector ratio Semitones ~cents 2 3 5 [19 12 0] 531441/524288 0.235 23.46001038 Pythagorean comma  [ 4 4 1] 81/80 0.215 21.5062896 syntonic comma  [15 8 1] 32805/32768 0.02 1.953720788 skhisma
Below is a Monzo lattice illustrating the [3 5] primefactorization of the skhisma:
Note that the skhisma is nearly the same size as the
grad,
the difference between them being only ~0.001280077
(= ~1/781) cent:
2^x 3^y 5^z [ 19/12 1 0 ] grad  [ 15 8 1 ] skhisma  [ 161/12 7 1 ] difference between grad and skhisma
Thus, we get the septimal schisma, which is the difference between the 3^{14} Pythagorean "doubly diminished 8ve" and the 7^{1} harmonic "minor 7th", and has an interval size of approximately ^{1}/_{26} Semitone:
primefactor vector ratio Semitones ~cents 2 3 7 [23 14 0] 8388608/4782969 9.73 972.6299879 Pythagorean "doubly diminished 8ve"  [2 0 1] 7/4 9.69 968.8259065 harmonic "minor 7th"  [25 14 1] 33554432/33480783 0.04 3.804081415 septimal skhisma
Likewise, there is the nondecimal schisma, which is the difference between the 19^{1} harmonic "augmented 2nd" and the 'standard' 3^{3} Pythagorean "minor 3rd", and which has an interval size of approximately ^{1}/_{30} Semitone:
primefactor vector ratio Semitones ~cents 2 3 19 [4 0 1] 19/16 2.98 297.5130161 harmonic "augmented 2nd"  [ 5 3 0] 32/27 2.94 294.1349974 Pythagorean "minor 3rd"  [9 3 1] 513/512 0.04 3.378018728 nondecimal skhisma
Schismas are a key element in my concept of bridging.
I refer to the three schismas explained here as the 3==5, 3==7, and 3==19 bridges,
respectively.
see also:
anomaly
schismic
skhismic major 3rd
comma
kleisma
5limit intervals, 100 cents and under
[from Joe Monzo, JustMusic: A New Harmony]
Updates:
2002.09.28  added vector notation describing commatic derivation, in #2
2002.09.23  added descriptive interval names and vector addition, numbered #3 as such
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