225/224
Ratio | 225/224 |
Factorization | 2^{-5} × 3^{2} × 5^{2} × 7^{-1} |
Monzo | [-5 2 2 -1⟩ |
Size in cents | 7.711523¢ |
Names | septimal kleisma, marvel comma |
Color name | ryy-2, ruyoyo negative 2nd, Ruyoyo comma |
FJS name | [math]\text{d}{-2}^{5,5}_{7}[/math] |
Special properties | square superparticular, reduced |
Tenney height (log_{2} nd) | 15.6211 |
Weil height (log_{2} max(n, d)) | 15.6276 |
Wilson height (sopfr (nd)) | 33 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~1.73575 bits |
Comma size | small |
S-expressions | S15, S25 × S26 × S27 |
open this interval in xen-calc |
The interval of 225/224, the septimal kleisma or marvel comma is a 7-limit superparticular ratio. 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 practically a 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.