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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 pairs of 7-limit ratios, for example as ([[15/14]])/([[16/15]]) or ([[45/32]])/([[7/5]]). Maybe most simply, it can be seen as the amount by which {{nowrap| [[15/8]] * [[15/8]] exceeds [[7/2]] }}, or equivalently, the amount by which {{nowrap| (15/8)/(16/15) = [[225/128]] }} exceeds 7/4. | 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 pairs of 7-limit ratios, for example as {{nowrap| ([[15/14]])/([[16/15]]) }} or {{nowrap| ([[45/32]])/([[7/5]]) }}. Maybe most simply, it can be seen as the amount by which {{nowrap| [[15/8]] * [[15/8]] exceeds [[7/2]] }}, or equivalently, the amount by which {{nowrap| (15/8)/(16/15) = [[225/128]] }} exceeds 7/4. | ||
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]]. | 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]]. | ||
Revision as of 03:01, 18 March 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 pairs of 7-limit ratios, for example as (15/14)/(16/15) or (45/32)/(7/5). Maybe most simply, it can be seen as the amount by which 15/8 * 15/8 exceeds 7/2, or equivalently, the amount by which 225/128 exceeds 7/4.
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), 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.