Schisma

Interval information

Ratio	32805/32768
Factorization	2-15 × 38 × 5
Monzo	[-15 8 1⟩
Size in cents	1.953721¢
Name	schisma
Color name	Ly-2, Layo comma
FJS name	[math]\displaystyle{ \text{d}{-2}^{5} }[/math]
Special properties	reduced, reduced harmonic
Tenney height (log2 nd)	30.0016
Weil height (log2 max(n, d))	30.0033
Wilson height (sopfr(nd))	59
Comma size	unnoticeable
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Schisma

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 (2²⁰) and 104976 (18⁴) look alike in decimal. The largest common power of two between these numbers is 2⁵, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768.

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 ¢

See also

• Unnoticeable comma

div_iv( mul_iv(s(15),s(15)), (3125**1,3087**1) )