Schisma
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Ratio | 32805/32768 |
Factorization | 2^{-15} × 3^{8} × 5 |
Monzo | [-15 8 1⟩ |
Size in cents | 1.9537208¢ |
Name | schisma |
Color name | Ly-2, Layo comma |
FJS name | [math]\text{d}{-2}^{5}[/math] |
Special properties | reduced, reduced harmonic |
Tenney height (log_{2} n⋅d) | 30.0016 |
Weil height (max(n, d)) | 32805 |
Benedetti height (n⋅d) | 1074954240 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.41502 bits |
Comma size | unnoticeable |
open this interval in xen-calc |
The schisma, 32805/32768, is the difference between the Pythagorean comma and the syntonic comma. It is equal to (9/8)^{4}/(8/5) and to (135/128)/(256/243) and also to (9/8)^{3}/(64/45). Tempering it out gives a 5-limit microtemperament called schismatic, schismic or Helmholtz, which if extended to larger subgroups leads to the schismatic family of temperaments.
Trivia
The schisma explains how the greatly composite numbers 1048576 (2^{20}) and 104976 (18^{4}) look alike in decimal. The largest common power of two between these numbers is 2^{5}, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768.