225/224
| Interval information |
marvel comma
Ruyoyo comma
reduced
S25 × S26 × S27
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.
Canonical extensions of note
The marvel extension hecate has the no-17's 19-limit as its subgroup, and undecimal marvel (aka unimarv), the extension chosen by Gene Ward Smith, can be extended to the 13-limit. They merge in the rank 2 temperament catakleismic (which can be conceptualized as accepting both rank 3 marvel structures simultaneously), for which the smallest reasonable edo tuning for the full no-17's 19-limit is 53edo followed by 72edo, though in 53edo the 11 and 19 are a little off and in 72edo the 13 and 19 are a little off instead; 72edo is positioned better as a full 17-limit marvel system while 53edo is positioned better as a (potentially no-11's) 13-limit marvel system. If we focus on the 11-limit of undecimal marvel (discarding the mapping of 13), 31edo and 41edo are the smallest to clearly succeed, though many accept 41edo's mapping of ~13/8 to the neutral sixth and some accept that mapping for 31edo as contextually usable too.
53edo and 84edo tunings of 7-limit marvel
53edo and 84edo are the smallest edos to tune the supermajor second ~8/7 flat (towards ~256/225), the subminor third ~7/6 sharp (towards ~75/64), the supermajor third ~9/7 flat (towards ~32/25) and the tritone ~7/5 sharp (towards ~45/32), such that every 7-limit 9-odd-limit interval is tuned between itself and the 5-limit interpretation it's separated from by 225/224, though even if you allow overtempering, the only smaller edo to satisfy all of these constraints is 12edo, which is a trivial tuning of it (meaning it is very high-damage owing to conflating many intervals so that the lattice is oversimplified). TE, CTE, CEE and CWE as well as the idea of tempering between pairs of 5- and 7-limit intervals separated by 225/224 all implicate these tuning tendencies of these 7-limit LCJI intervals for optimized 7-limit marvel tunings. Interestingly, 72edo fails some of these constraints and is less optimized for others, in the sense that 53edo tunes closer to the more complex 5-limit interpretations (which arguably need more tuning fidelity), which is something not taken into account by these tuning optimization schemes (so that they generally tune closer to LCJI). By contrast, 84edo, an overlooked superset of 12edo, has the benefit of being a high-limit performer in odd-limits 23 through 51 (inclusive). In fact, 53edo and 84edo are the only edos to satisfy all these constraints consistently when we include not overtempering to overshoot the 5-limit interval, and if we also require 28/27 to be sharp and 25/24 to be flat, 53edo is the only one, making it a uniquely optimized 7-limit marvel tuning; as far as the 9-odd-limit is concerned, the only intervals which are more than 25% off in 53edo are 7/5 and 10/7], so that it is almost consistent to distance 2, and many more complex intervals of the 7-limit are consistent as well (barring the stacking of prime 7 more than once, so that 5 * 7 = 35 is fine but not 7 * 7 = 49, which causes inconsistencies in the 7-limited tonality diamond).
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.