225/224: Difference between revisions
Undo revision 176831 by Godtone (talk). Obscure? It's right there in the opening sentence of the temperaments section, which is where ppl should be looking for Tag: Undo |
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{{Wikipedia|Septimal kleisma}} | {{Wikipedia|Septimal kleisma}} | ||
The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between | The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between a 7-limit ratio and a 5-limit ratio. For example, it's the difference between [[16/15]] and [[15/14]], and between [[7/5]] and [[45/32]]. Moreover, it can be seen as the amount by which [[8/7]] exceeds a stack of two {{nowrap|[[16/15]]'s}}, or as the amount by which a stack of two {{nowrap|[[5/4]]'s}} exceeds [[14/9]]. It's also the difference between [[75/64]] and [[7/6]], and between [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone. | ||
As a comma with a single power of 7 in it, it is tremendously useful in terms of bringing prime 7 into the framework of [[5-limit]] [[just intonation|JI]]; tempering it out maps [[7/4]] to the classic augmented sixth, [[225/128]] and enables all of the aforementioned equivalences. | |||
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]]), and in the 13-limit, ([[351/350]])([[625/624]]) or ([[325/324]])([[729/728]]). | 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]]), and in the 13-limit, ([[351/350]])([[625/624]]) or ([[325/324]])([[729/728]]). | ||
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== See also == | == See also == | ||
* [[Marvel]] | |||
* [[Small comma]] | * [[Small comma]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
Latest revision as of 11:31, 6 June 2025
| Interval information |
septimal kleisma
Ruyoyo comma
reduced
S25 × S26 × S27
The interval of 225/224, the marvel comma, otherwise known as the septimal kleisma, is a 7-limit superparticular comma. It pops up as the difference between a 7-limit ratio and a 5-limit ratio. For example, it's the difference between 16/15 and 15/14, and between 7/5 and 45/32. Moreover, it can be seen as the amount by which 8/7 exceeds a stack of two 16/15's, or as the amount by which a stack of two 5/4's exceeds 14/9. It's also the difference between 75/64 and 7/6, and between 25/24, the classical chromatic semitone, and 28/27, the septimal third-tone.
As a comma with a single power of 7 in it, it is tremendously useful in terms of bringing prime 7 into the framework of 5-limit JI; tempering it out maps 7/4 to the classic augmented sixth, 225/128 and enables all of the aforementioned equivalences.
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), and in the 13-limit, (351/350)(625/624) or (325/324)(729/728).
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 almost return to the starting point at the 311th step, meaning 311edo is a consistent 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.
Etymology
Marvel comma was named after the corresponding temperament, marvel, which was named by Gene Ward Smith in 2002–2003.