Schisma: Difference between revisions
all the temps mentioned here except one (which is undocumented AFAIK!) are both non-duplicate (are presented to link and refer to other existing temps and only give an overview) and are here because they are simple and efficient ways of expressing the schisma. i thought it's agreed not to delete info and make the wiki less navigable Tag: Reverted |
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{{Wikipedia| Schisma }} | {{Wikipedia| Schisma }} | ||
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]]) | The '''schisma''', '''32805/32768''', is a small interval about 2 [[cent]]s. It arises as 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]]). | ||
== 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. | ||
== | == 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. | ||
== 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]] and [[Alpharabian schisma]]. | |||
The | |||
== Trivia == | == Trivia == | ||
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[[Category:Schismic]] | [[Category:Schismic]] | ||
[[Category:Commas named for their regular temperament properties]] | |||
Revision as of 01:16, 1 June 2026
| Interval information |
reduced harmonic
The schisma, 32805/32768, is a small interval about 2 cents. It arises as 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.
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.
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 and Alpharabian schisma.
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.