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 [[81/80|syntonic commas (81/80)]] with [[Enneadeca|enneadeca ({{Monzo| -14 -19 19 }})]].

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 (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 12.~~0078623975~~..., and temperaments having ''n'' near this value tend to be the most accurate ones ~~– indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is~~.

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.

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-The '''syntonic-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with [[Enneadeca|enneadeca ({{Monzo| -14 -19 19 }})]].
+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 (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 12.0078623975..., and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is.
+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.
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