Marvel

Marvel is the rank-3 temperament tempering out 225/224, the marvel comma. It has a canonical 11-limit extension adding 385/384 and 540/539 to the comma list.

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

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, it leaves prime 7 the only variable to be sharp. This indicates that the supermajor second ~8/7 should be flat (towards ~256/225), the subminor third ~7/6 be sharp (towards ~75/64), the supermajor third ~9/7 be flat (towards ~32/25) and the tritone ~7/5 be sharp (towards ~45/32), 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 satify them, but if overtempering is allowed, many smaller edos are possible, such as 31- and 41edo. Interestingly, 72edo is too overtempered to satisfy some of these constraints, whereas 53edo though satisfying these constraints tempers in favor of tuning the intervals closer to the more complex 5-limit interpretations, though because of their comparative simplicity (and thus lesser tuning fidelity), the 7-limit consonances of the 9-odd-limit still clearly work so that 53edo is consistent to distance 2 in the 9-odd-limit if we exclude 7/5 and 10/7 which are the most damaged. 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, 166-, 178-, 197-, and 240edo are all great choices with different intonational characteristics, though note that inconsistencies may often arise in the representation of the 7-limit lattice because 225/224 is itself larger than the size of the step of any of these edos.

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