Marvel
Marvel is a rank-3 temperament with the same lattice structure as 5-limit JI, while identifying the harmonic seventh (7/4) as a stack of two classical major sevenths (15/8) octave-reduced, tempering out 225/224. It is the head of the marvel family, and the canonical 11-limit extension adding 385/384 and 540/539 to the comma list makes it a member of both keenansimic and swetismic temperaments.
The temperament was named by Gene Ward Smith in 2002–2003, when the 11-limit version was found first[1][2]. Gene carried it to the 7-limit restriction in 2004[3].
Extending marvel to the 13-limit is not as obvious. While Gene has chosen 351/350 as the canonical extension, hecate, tempering out 325/324 and 729/728, arguably makes more sense as it is closer in tuning[4]. Hecate has a natural extension to the no-17 19-limit, by tempering out 400/399 and 513/512.
See Marvel family #Marvel for technical data.
Interval lattice
-
11-limit marvel -
13-limit marvel/hecate -
2.3.5.7.11.13.19 subgroup marvel/hecate
Notation
Marvel can be notated the same as 5-limit just intonation since they share the same lattice structure. One way to do this is to take the conventional circle-of-fifths notation with an additional module of accidentals for the 81/80 comma. In this system, 5/4 is a major third, 7/4 an augmented sixth, and 11/8 a double diminished 5th.
| Ratio | Nominal | Example |
|---|---|---|
| 3/2 | Perfect fifth | C-G |
| 5/4 | Down major third | C-vE |
| 7/4 | Dudaugmented sixth | C-vvA# |
| 11/8 | Trup double-diminished fifth | C-^3Gbb |
| 13/8 | Dup minor sixth | C-^^Ab |
| 19/16 | Minor third | C-Eb |
Alternatively, it can be notated the same as full prime-limit just intonation, with a distinct accidental pair for each prime. That makes some intervals more intuitive, at the cost of hiding the structure of marvel tempering. For example, it is customary of the 5/4 to be a major third, and 7/4 to be a minor seventh. As a result, the fact that the 14/9 is a stack of two 5/4's is not revealed, and the related chords can be less convenient.
Chords
Marvel enables essentially tempered chords of marvel, keenanismic, swetismic, and undecimal marvel. Extending the temperament to the 13-limit through 325/324, resulting in hecate, enables chords of marveltwin and squbemic. Hecate hexad is a chord peculiar to this temperament.
Alternative 11-limit extensions give different sets of chords. One notable example, tempering out 441/440, enables prodigy chords.
Scales
Marvel hobbit scales
Undecimal marvel hobbit scales
Other marvel scales
Tunings
In the 7-limit, the optimal way such as that taken by TE and derivatives to close out the comma 225/224 is to tune primes 3 and 5 flat, and 2 and 7 sharp. If we tune the octave pure, the other inclinations remain. This indicates that the diminished third ~256/225 should be sharp (towards ~8/7), the augmented second ~75/64 be flat (towards ~7/6), the diminished fourth ~32/25 be sharp (towards ~9/7) and the tritone ~45/32 be flat (towards ~7/5), such that every 7-limit 9-odd-limit interval is tuned between itself and the 5-limit interpretation it is separated from by 225/224. If we take these as hard constraints, then 53edo and 84edo are the smallest edo tunings to satisfy them, but if overtempering is allowed, many smaller edos are possible, such as 31- and 41edo. Interestingly, 72edo, though very performant as a 7- and 11-limit tuning, is overtempered for some of these constraints, whereas 53edo, though satisfying these constraints, tempers the intervals closer to the more complex 5-limit interpretations, though the 7-limit concordances of the 9-odd-limit still clearly work. 84edo, another superset of 12edo, is an interesting edo to look at for its high performance in large odd-limits. Going up to larger edos, 125-, 166-, 178-, 197-, and 240edo are all great choices with different intonational characteristics.
The marvel extension hecate has the no-17's 19-limit as its subgroup, and tridecimal marvel, the extension chosen by Gene Ward Smith, is in the 13-limit. They merge in the rank-2 temperament catakleismic, which can be conceptualized as accepting both rank-3 marvel structures simultaneously. One such tuning is excellently given by 125edo. If we are looking for a small edo tuning instead, 53edo and 72edo are also reasonable edo tunings for the full no-17's 19-limit catakleismic, 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.
Tuning spectrum
This spectrum assumes pure 2 and 7.
| Eigenmonzo (Unchanged-interval) |
Fifth (¢) |
Major Third (¢) |
Comments |
|---|---|---|---|
| 5/4 | 698.099 | 386.314 | |
| 6/5 | 700.027 | 384.386 | 7-odd-limit minimax |
| 15/11 | 700.351 | 384.062 | |
| 10/9 | 700.670 | 383.743 | 9-odd-limit minimax |
| 11/10 | 700.885 | 383.528 | |
| 15/13 | 700.916 | 383.497 | 15-odd-limit hecate minimax |
| 13/10 | 701.065 | 383.348 | 13-odd-limit hecate minimax |
| 13/11 | 701.199 | 383.214 | |
| 18/13 | 701.361 | 383.052 | |
| 13/12 | 701.480 | 382.933 | |
| 16/13 | 701.559 | 382.854 | |
| 4/3 | 701.955 | 382.458 | |
| 14/11 | 702.278 | 382.135 | |
| 11/8 | 702.278 | 382.135 | |
| 12/11 | 702.602 | 381.811 |
Music
- Pump1 – in pump12 1, 197edo tuning
- Semimarvelous Blue Drawf (2010) – in Dwarf17marv, equal-beating tuning
See also
- Marvel temperaments, the collection of rank-2 temperaments that temper out the marvel comma
Notes
- ↑ Yahoo! Tuning Group | Relative complexity and scale construction – first mention of marvel.
- ↑ Yahoo! Tuning Group | Top 135 11-limit planar temperaments – establishment as an 11-limit temperament.
- ↑ Yahoo! Tuning Group | Marvel
- ↑ Yahoo! Tuning Group | 13-limit marvel