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 the 19-comma ({{~~Monzo~~|-30 19}}).

The '''syntonic-kleismic equivalence continuum''' (or '''syntonic-enneadecal equivalence continuum''') is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the 19-comma ({{monzo| -30 19 }}).

All temperaments in the continuum satisfy (81/80)''k'' ~ {{monzo|-30 19}}. Varying ''k'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''k'' approaches infinity. If we allow non-integer and infinite ''k'', 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 ''k'' is approximately 6.~~376...~~, and temperaments having ''k'' near this value tend to be the most accurate ones.

All temperaments in the continuum satisfy (81/80)''n'' ~ {{monzo|-30 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 6.376…, and temperaments having ''n'' near this value tend to be the most accurate ones.

This continuum can be expressed as the relationship between 81/80 and the [[enneadeca]] ({{~~Monzo~~|-14 -19 19}}). That is, (81/80)''n'' ~ {{monzo|-14 -19 19}}. In this case, ''n'' = 3''k'' - 19.

This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, (81/80)''k'' ~ {{monzo| -14 -19 19 }}. In this case, ''k'' = 3''n'' - 19.

{| class="wikitable center-1 center-2"

|+ Temperaments in the continuum

|-

! rowspan="2" | ''k''

! rowspan="2" | ''n''

! rowspan="2" | Temperament

! colspan="2" | Comma

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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 the 19-comma ({{Monzo|-30 19}}).
+The '''syntonic-kleismic equivalence continuum''' (or '''syntonic-enneadecal equivalence continuum''') is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the 19-comma ({{monzo| -30 19 }}).
-All temperaments in the continuum satisfy (81/80)<sup>''k''</sup> ~ {{monzo|-30 19}}. Varying ''k'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''k'' approaches infinity. If we allow non-integer and infinite ''k'', 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 ''k'' is approximately 6.376..., and temperaments having ''k'' near this value tend to be the most accurate ones.
+All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ {{monzo|-30 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 6.376…, and temperaments having ''n'' near this value tend to be the most accurate ones.
-This continuum can be expressed as the relationship between 81/80 and the [[enneadeca]] ({{Monzo|-14 -19 19}}). That is, (81/80)<sup>''n''</sup> ~ {{monzo|-14 -19 19}}. In this case, ''n'' = 3''k'' - 19.
+This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, (81/80)<sup>''k''</sup> ~ {{monzo| -14 -19 19 }}. In this case, ''k'' = 3''n'' - 19.
 {| class="wikitable center-1 center-2"
 |+ Temperaments in the continuum
 |-
-! rowspan="2" | ''k''
+! rowspan="2" | ''n''
 ! rowspan="2" | Temperament
 ! colspan="2" | Comma