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. Gene carried it to the 7-limit restriction in 2004.

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. 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.

Marvel/hecate nomenclature
for selected intervals

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

• Marvel9
• Marvel10
• Marvel11
• Marvel12
• Marvel19
• Marvel22

Undecimal marvel hobbit scales

• Marvel22 11
• Unimarv19
• Unimarv22

Other marvel scales

• Pump12 1
• Pump12 2
• Pump13
• Pump14
• Pump15
• Pump16
• Pump17
• Pump18
• Diamond9plus-marvel

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

Billy Stiltner

• notJoelTaylorsTreeSpirit-Marvel10-2^{[dead link]}

Chris Vaisvil

• 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