225/224
| Interval information |
septimal kleisma
Ruyoyo comma
reduced
(Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])
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.