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{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.
Schismic temperaments derivable from its S-expressions
Nestoria
As the schisma is expressible as S19/(S16/S18)^{2} and (S15/S20)/(S16/S18), we can derive the 12&53 temperament:
Subgroup: 2.3.5.19
Patent EDO tunings: 12, 17, 24, 29, 36, 41, 53, 65, 77, 82, 89, 94, 101, 106, 118, 130, 135, 142, 147, 154, 159, 171, 183, 195, 207, 219, 248, 260, 272
CTE generator: 701.684 ¢
Garibaldi
As the schisma is also equal to S15/(S8/S9), we can derive the 41&53 temperament:
Subgroup: 2.3.5.7
Patent EDO tunings: 12, 29, 41, 53, 82, 94, 106, 135, 147
CTE generator: 702.059 ¢
2.3.5.7.19[53&147] (garibaldi nestoria)
Adding Nestoria to Garibaldi (tempering 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.
Subgroup: 2.3.5.7.19
Patent EDO tunings: 12, 29, 41, 53, 82, 94, 106, 135, 147
CTE generator: 702.043 ¢
2.3.5.7.17[12&130&171] (unnamed)
As the schisma also equals S15/S16 * S18/S20, we can derive the extremely accurate 12&41 temperament:
Subgroup: 2.3.5.7.17
Patent EDO tunings < 300 (largest is 2548): 12, 29, 41, 53, 118, 130, 142, 159, 171, 183, 212, 224, 236, 289
CTE generators: (2/1,) 3/2 = 701.72 ¢, 7/4 = 968.831 ¢
2.3.5.7.17.19[12&130&171] (unnamed Nestoria)
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
Patent EDO tunings: 12, 29, 41, 53, 118, 130, 142, 159, 171, 183
CTE generators: (2/1,) 3/2 = 701.705 ¢, 7/4 = 968.928 ¢
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.