Schisma: Difference between revisions
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{{Interwiki | |||
| en = 32805/32768 | |||
| de = 32805/32768 | |||
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The | }} | ||
{{Infobox Interval | |||
| Ratio = 32805/32768 | |||
| Name = schisma | |||
[[ | | Color name = LyM, layoma | ||
< | | Comma = yes | ||
}} | |||
{{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]]). | |||
== 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 [[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 == | |||
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 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 == | |||
* [[Pythagorean tuning]] | |||
* [[Unnoticeable comma]] | |||
[[Category:Schismic]] | |||
[[Category:Commas named for their regular temperament properties]] | |||