# 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[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.

## 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.

## Tuning spectrum

This spectrum assumes pure 2 and 7.

Eigenmonzo
(Unchanged-interval)
Fifth
(¢)
Major
Third (¢)
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

Billy Stiltner
Chris Vaisvil