Schisma: Difference between revisions
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{{Wikipedia| Schisma }} | {{Wikipedia| Schisma }} | ||
The '''schisma''', '''32805/32768''', is | The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]). | ||
== Other intervals == | |||
Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]], [[Alpharabian schisma]] and [[tridecaschisma]]. | |||
== Temperaments == | == Temperaments == | ||
Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments. | Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s 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 [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]). This corresponds to making 19/16 the minor third and 24/19~19/15 the major third; a good tuning for this is [[65edo]], or if you prefer a more accurate [[19/16]], [[77edo]]. | |||
=== Garibaldi === | |||
{{Main| Garibaldi }} | |||
Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] ([[64/63|S8]]/[[81/80|S9]]), and can be described as the 41 & 53 temperament in the 7-limit that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)<sup>6</sup> / (2/1). This is derived as the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]). | |||
==== 2.3.5.7.19 subgroup ==== | |||
{{Main| 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 [[57375/57344|S15/S16]] * [[1701/1700|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|legend=1| 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|legend=1| 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 [[1216/1215|S16/S18]] we equate [[225/224|S15]] with [[400/399|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|legend=1| 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|legend=1| 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 microtemperament-accuracy mapping for prime 41 via tempering out [[6561/6560|S81]] = (81/80)/(82/81) (the second of the aforementioned commas) s.t any accurate schismic tuning (one with a very slightly flat 81/80) will have a good tuning for an otonal supermajor third [[41/32]] and a flat supermajor second (41/32)/(9/8) = [[41/36]]. | |||
== History and etymology == | |||
''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio 887/886 {{nowrap|(2<sup>-1</sup>⋅443<sup>-1</sup>⋅887)}}, it is used interchangably with this interval in some of Helmholtz' writing. | |||
== Trivia == | == Trivia == | ||
The schisma explains how the greatly composite numbers 1048576 (2<sup>20</sup>) and 104976 (18<sup>4</sup>) look alike in decimal. The largest common power of two between these numbers is 2<sup>5</sup>, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768. | The schisma explains how the greatly composite numbers 1048576 (2<sup>20</sup>) and 104976 (18<sup>4</sup>) look alike in decimal. The largest common power of two between these numbers is 2<sup>5</sup>, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768. | ||
It | It is also very close in size—about 0.0013{{c}} off—from the difference between 3/2 and 7\12, which is about 1.9550009{{c}}. Tempering out this difference instead results in [[atomic]], an extremely high accuracy temperament. | ||
== See also == | == See also == | ||
Latest revision as of 17:12, 6 June 2026
| Interval information |
reduced harmonic
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).
Other intervals
Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the Pythagorean comma and septimal comma is called the septimal schisma. Other examples are undevicesimal schisma, Alpharabian schisma and tridecaschisma.
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). This corresponds to making 19/16 the minor third and 24/19~19/15 the major third; a good tuning for this is 65edo, or if you prefer a more accurate 19/16, 77edo.
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 that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)6 / (2/1). This is derived as 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 microtemperament-accuracy mapping for prime 41 via tempering out S81 = (81/80)/(82/81) (the second of the aforementioned commas) s.t any accurate schismic tuning (one with a very slightly flat 81/80) will have a good tuning for an otonal supermajor third 41/32 and a flat supermajor second (41/32)/(9/8) = 41/36.
History and etymology
Schisma is a borrowing of Ancient Greek, meaning "split". The term was first used by Boethius (6th century), in his De institutione musica, using it to refer to half of the Pythagorean comma. The modern sense was introduced by Helmholtz' On the Sensations of Tone, in particular the translation by Alexander Ellis, where it is spelled skhisma. Since it is extremely close to the superparticular ratio 887/886 (2-1⋅443-1⋅887), it is used interchangably with this interval in some of Helmholtz' writing.
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.
It is also very close in size—about 0.0013 ¢ off—from the difference between 3/2 and 7\12, which is about 1.9550009 ¢. Tempering out this difference instead results in atomic, an extremely high accuracy temperament.