## Definitions of tuning terms © 1998 by Joseph L. Monzo All definitions by Joe Monzo unless otherwise cited

schisma

• 1.
• The historically prior usage of this term is by Philolaus (fl. c. 400 BC), as quoted by Boethius, to refer to an interval that is considerably larger than the schisma we normally mean today.

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 * 312)(1/2)
= 2-(19/2) * 36
= ~11.73 cents.

```

• 2.
• In its usual modern sense, a term coined by Alexander Ellis in his translation of Helmholtz's On the Sensations of Tone, and originally spelled skhisma. It designates an extremely small interval, just barely discernible to human pitch-detection.

Unqualified, it is the difference between the 51 just "major 3rd" and the 3-8 Pythagorean "diminished 4th", and has an interval size of approximately 1/50 Semitone [= ~ 2 cents]:

```
prime-factor 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:

```
prime-factor 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] prime-factorization 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

```

• 3.
• As no combination of different prime numbers will ever produce ratios which have exactly the same interval size, if cycles of a particular ratio are calculated far enough, very small intervals like this eventually appear between ratios having different sets of prime factors. When these are under consideration, the term schisma is qualified with a latin word designating the higher prime, with the assumption that the other prime being compared is a more familiar one, almost always 3.

Thus, we get the septimal schisma, which is the difference between the 3-14 Pythagorean "doubly diminished 8ve" and the 71 harmonic "minor 7th", and has an interval size of approximately 1/26 Semitone:

```
prime-factor 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 191 harmonic "augmented 2nd" and the 'standard' 3-3 Pythagorean "minor 3rd", and which has an interval size of approximately 1/30 Semitone:

```
prime-factor 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.

anomaly
schismic
skhismic major 3rd
comma
kleisma
5-limit intervals, 100 cents and under

[from Joe Monzo, JustMusic: A New Harmony]