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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English Wikipedia has an article on:

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).

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

See also: No-sevens subgroup 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

Main article: 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

Main article: Garibaldi

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

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.

See also

• Unnoticeable comma