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.
Interval lattice
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-^^{3}Gbb |
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
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
- notJoelTaylorsTreeSpirit-Marvel10-2^{[dead link]}
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