# 225/224

(Redirected from Septimal kleisma)
 Ratio 225/224 Factorization 2-5 × 32 × 52 × 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 $\text{d}{-2}^{5,5}_{7}$ Special properties square superparticular,reduced Tenney height (log2 nd) 15.6211 Weil height (log2 max(n, d)) 15.6276 Wilson height (sopfr (nd)) 33 Harmonic entropy(Shannon, $\sqrt{nd}$) ~1.73575 bits Comma size small S-expressions S15,S25 × S26 × S27 open this interval in xen-calc
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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).

## 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.