225/224

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Interval information
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 [math]\text{d}{-2}^{5,5}_{7}[/math]
Special properties square superparticular,
reduced
Tenney height (log2 n⋅d) 15.6211
Weil height (max(n, d)) 225
Benedetti height (n⋅d) 50400
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~2.64751 bits
Comma size small
S-expressions S15,
S25 × S26 × S27
open this interval in xen-calc
English Wikipedia has an article on:

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 get 311edo, where it is equal to 2 steps, meaning 311edo is 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.

See also