225/224: Difference between revisions
Added content about importance in Septimal meantone. It is naturally important in Marvel as well, but that is evident from the name 'marvel comma'. And Miracle, where it is also tempered, is a pretty natural extension of Marvel. Tags: Mobile edit Mobile web edit |
Marvelma as a product of schismas and garischismas |
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{{Infobox Interval | {{Infobox Interval | ||
| Name = marvel comma, septimal kleisma | |||
| Color name = ryy-2, ruyoyo negative 2nd,<br>ryyM, ruyoyoma | |||
| Comma = yes | |||
| Name = septimal kleisma | |||
| Color name = ryy-2, ruyoyo negative 2nd | |||
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}} | }} | ||
{{Wikipedia|Septimal kleisma}} | |||
The interval of '''225/224''', the ''' | 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]], is tempered out alongside these two commas in [[septimal meantone]]; and the product of two [[Schisma|schismas]] and one [[garischisma]], equivalently of a [[schisma]] and an [[aberschisma]]. 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 == | == Temperaments == | ||
Tempering out this comma leads to the [[marvel family]] of temperaments, which | 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. | |||
== See also == | == See also == | ||
* [[Marvel | * [[Marvel]] | ||
* [[Small comma]] | * [[Small comma]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
[[Category:Marvel]] | [[Category:Marvel]] | ||
[[Category:Commas with unknown etymology]] | |||
Latest revision as of 09:05, 23 June 2026
| Interval information |
septimal kleisma
ryyM, ruyoyoma
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, is tempered out alongside these two commas in septimal meantone; and the product of two schismas and one garischisma, equivalently of a schisma and an aberschisma. 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.