Syntonic–kleismic equivalence continuum: Difference between revisions
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The '''syntonic-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[Enneadeca| | The '''syntonic-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[Enneadeca|enneadeca ({{Monzo| -14 -19 19 }})]]. | ||
All temperaments in the continuum satisfy ( | All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ {{monzo|-14 -19 19}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[19edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 0.1309..., and temperaments having ''n'' near this value tend to be the most accurate ones. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
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|- | |- | ||
| 0 | | 0 | ||
| [[ | | [[Enneadecal]] | ||
| | | | ||
| {{monzo| - | | {{monzo|-14 -19 19}} | ||
|- | |- | ||
| … | | … | ||
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|- | |- | ||
| ∞ | | ∞ | ||
| [[ | | [[Meantone]] | ||
| | | [[81/80]] | ||
| {{monzo|- | | {{monzo| -4 4 -1}} | ||
|} | |} | ||
Revision as of 09:11, 14 March 2021
The syntonic-enneadecal equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the enneadeca ([-14 -19 19⟩).
All temperaments in the continuum satisfy (81/80)n ~ [-14 -19 19⟩. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 19edo (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of n is approximately 0.1309..., and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Enneadecal | [-14 -19 19⟩ | |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |