225/224: Difference between revisions
m Make "marvel comma" the main common name since it has a correponding temp name |
m it's strange that the main page for tempering out 225/224 is not under "see also" but instead only appears as a singular obscure link under the "Temperaments" section |
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== See also == | == See also == | ||
* [[Marvel]] | |||
* [[Small comma]] | * [[Small comma]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
Revision as of 16:52, 17 January 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).
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.