Marvel–syntonic equivalence continuum
The marvel-syntonic equivalence continuum is a continuum of 7-limit rank-3 temperament families which equate a number of marvel commas (225/224) with an syntonic comma (81/80). This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by septimal meantone temperament.
All temperaments in the continuum satisfy (81/80)n ~ 64/63. Varying n results in different temperament families listed in the table below. It converges to marvel as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 7-limit temperament families supported by septimal meantone (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of n is approximately 2.788851128489581..., and temperaments having n near this value will be more accurate.
| n | Temperament family | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| Meantone | 81/80 | [-4 4 -1 0⟩ | |
| 1 | Starling | 126/125 | [1 2 -3 1⟩⟩ |
| 2 | Hemimean | 3136/3125 | [6 0 -5 2⟩ |
| 3 | Meter | 703125/702464 | [-11 2 7 -3⟩ |
| ∞ | Marvel | 225/224 | [-5 2 2 -1⟩ |