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.
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.
|5/4||Down major third||C-vE|
|11/8||Trup double-diminished fifth||C-^3Gbb|
|13/8||Dup minor sixth||C-^^Ab|
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.
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.
Marvel hobbit scales
Undecimal marvel hobbit scales
Other marvel scales
This spectrum assumes pure 2 and 7.
|15/13||700.916||383.497||15-odd-limit hecate minimax|
|13/10||701.065||383.348||13-odd-limit hecate minimax|
- 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