225/224: Difference between revisions
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{{Wikipedia|Septimal kleisma}} | {{Wikipedia|Septimal kleisma}} | ||
The interval of '''225/224''', the '''septimal kleisma''' or '''marvel comma''' is a [[7-limit]] [[superparticular]] ratio. It pops up as the difference between pairs of 7-limit ratios, for example as ([[15/14]])/([[16/15]]) or ([[45/32]])/([[7/5]]) | The interval of '''225/224''', the '''septimal kleisma''' or '''marvel comma''' is a [[7-limit]] [[superparticular]] ratio. It pops up as the difference between pairs of 7-limit ratios, for example as ([[15/14]])/([[16/15]]) or ([[45/32]])/([[7/5]]). | ||
Another useful relation is as the difference between the [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone. Hence, it is also the difference between [[32/25]] and [[9/7]], and between [[75/64]] and [[7/6]]. | |||
In terms of commas, it is the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit, it factors neatly into ([[385/384]])([[540/539]]). | |||
== Temperaments == | == Temperaments == | ||
Revision as of 09:03, 28 November 2022
| Interval information |
marvel comma
Ruyoyo comma
reduced
S25 × S26 × S27
The interval of 225/224, the septimal kleisma or marvel comma is a 7-limit superparticular ratio. It pops up as the difference between pairs of 7-limit ratios, for example as (15/14)/(16/15) or (45/32)/(7/5).
Another useful relation is as the difference between the 25/24, the classical chromatic semitone, and 28/27, the septimal third-tone. Hence, it is also the difference between 32/25 and 9/7, and between 75/64 and 7/6.
In terms of commas, it is the difference between 81/80 and 126/125 and is tempered out alongside these two commas in septimal meantone. In the 11-limit, it factors neatly into (385/384)(540/539).
Temperaments
Tempering out this comma alone in the 7-limit leads to the marvel temperament, which enables marvel chords. See marvel family for the family of rank-3 temperaments where it is tempered out. See marvel temperaments for a collection of rank-2 temperaments where it is tempered out.
Approximation
If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we get 311edo, where it is equal to 2 steps, meaning 311edo is a circle of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.