Marvel–syntonic equivalence continuum: Difference between revisions
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The '''marvel-syntonic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[225/224|marvel commas (225/224)]] with | The '''marvel-syntonic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[225/224|marvel commas (225/224)]] with a [[81/80|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)<sup>''n''</sup> ~ 64/63. Varying ''n'' results in different temperament families listed in the table below. It converges to [[marvel family|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. | All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 64/63. Varying ''n'' results in different temperament families listed in the table below. It converges to [[marvel family|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. | ||
Revision as of 15:10, 20 June 2023
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 a 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 | ||
| 0 | 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⟩ |