Schisma
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} nd) | 30.0016 |
Weil height (log_{2} max(n, d)) | 30.0033 |
Wilson height (sopfr (nd)) | 59 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~1.23468 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).
Temperaments
Tempering out this comma gives a 5-limit microtemperament called schismatic, schismic or helmholtz, which if extended to larger subgroups leads to the schismatic family of temperaments.
Nestoria
Nestoria tempers out 361/360 (S19) and 513/512 (S15/S20), and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as S19/(S16/S18)^{2} and (S15/S20)/(S16/S18).
Garibaldi
Garibaldi tempers out 225/224 (S15) and 5120/5103 (S8/S9), and can be described as the 41 & 53 temperament in the 7-limit. This is derived since the schisma is also equal to S15/(S8/S9).
2.3.5.7.19 subgroup
Adding nestoria to garibaldi (tempering 400/399 (S20)) results in an extremely elegant temperament which has all of the same patent tunings that garibaldi has but which includes a mapping for 19 through nestoria.
2.3.5.7.17 12 & 118 & 171 (unnamed)
As the schisma also equals S15/S16 * S18/S20, we can derive the extremely accurate 12 & 118 & 171 temperament:
Subgroup: 2.3.5.7.17
Comma list: 1701/1700, 32805/32768
Mapping: [⟨1 0 15 0 -32], ⟨0 1 -8 0 21], ⟨0 0 0 1 1]]
- mapping generators: ~2, ~3, ~7
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7197, ~7/4 = 968.8307
Optimal ET sequence: 12, 29, 41, 53, 106d, 118, 171, 472, 525, 643, 814, 985, 1799, 2324, 2495, 3138b, 3309bd, 4294bdg
2.3.5.7.17.19 12 & 118 & 171 (unnamed)
By tempering S16/S18 we equate S15 with S20 (tempering the other comma of Nestoria) because of S15~S16~S18~S20, leading to:
Subgroup: 2.3.5.7.17.19
Comma list: 361/360, 513/512, 1701/1700
Mapping: [⟨1 0 15 0 -32 9], ⟨0 1 -8 0 21 -3], ⟨0 0 0 1 1 0]]
- mapping generators: ~2, ~3, ~7
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7053, ~7/4 = 968.9281
Optimal ET sequence: 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh
2.3.5.41 53 & 65 (unnamed)
The schisma can additionally split into two superparticular commas in the 41-limit: 32805/32768 = (1025/1024)*(6561/6560). Tempering both of these out provides a natural mapping for prime 41, if a little less practical than those for 19 or 7.
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.