Syntonic–Archytas equivalence continuum: Difference between revisions
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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 [[Meantone family|meantone + za]] 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 squares (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 1.267726433120519..., 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 [[Meantone family|meantone + za]] 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 squares (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 1.267726433120519..., and temperaments having ''n'' near this value will be more accurate. | ||
{| class="wikitable center-1 center-2" | |||
|+ Temperament families in the continuum | |||
|- | |||
! rowspan="2" | ''n'' | |||
! rowspan="2" | Temperament family | |||
! colspan="2" | Comma | |||
|- | |||
! Ratio | |||
! Monzo | |||
|- | |||
| 0 | |||
| [[Archytas clan|Archy]] | |||
| [[64/63]] | |||
| {{monzo|6 -2 0 -1}} | |||
|- | |||
| 1 | |||
| [[Hemifamity family|Hemifamity]] | |||
| [[5120/5103]] | |||
| {{monzo|1 5 1 -4}} | |||
|- | |||
| ∞ | |||
| [[Meantone family|Meantone]] | |||
| [[81/80]] | |||
| {{monzo| -4 4 -1 0}} | |||
|} | |||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Revision as of 14:54, 20 June 2023
The breedsmic-syntonic equivalence continuum is a continuum of 7-limit rank-3 temperament families which equate a number of syntonic commas (81/80) with an Archytas comma (64/63). This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by dominant 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 meantone + za 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 squares (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 1.267726433120519..., and temperaments having n near this value will be more accurate.
| n | Temperament family | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Archy | 64/63 | [6 -2 0 -1⟩ |
| 1 | Hemifamity | 5120/5103 | [1 5 1 -4⟩ |
| ∞ | Meantone | 81/80 | [-4 4 -1 0⟩ |