Syntonic–kleismic equivalence continuum: Difference between revisions
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The '''syntonic | {{Technical data page}} | ||
The '''syntonic–kleismic equivalence continuum''' (or '''syntonic–enneadecal equivalence continuum''') is a [[equivalence continuum|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>''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. | All temperaments in the continuum satisfy {{nowrap|(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 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'' | This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, {{nowrap|(81/80)<sup>''k''</sup> ~ {{monzo| -14 -19 19 }}}}. In this case, {{nowrap|''k'' {{=}} 3''n'' − 19}}. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments in the continuum | |+ style="font-size: 105%;" | Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ Notable temperaments of fractional ''n'' | |+ style="font-size: 105%;" | Notable temperaments of fractional ''n'' | ||
|- | |- | ||
! Temperament !! ''n'' !! Comma | ! Temperament !! ''n'' !! Comma | ||
| Line 101: | Line 102: | ||
| [[Mowgli]] || 15/2 = 7.5 || {{monzo| 0 22 -15 }} | | [[Mowgli]] || 15/2 = 7.5 || {{monzo| 0 22 -15 }} | ||
|} | |} | ||
== Negri (5-limit) == | |||
: ''For extensions, see [[Semaphoresmic clan #Negri]].'' | |||
The 5-limit version of negri tempers out the [[negri comma]], spliting a perfect fourth into four ~16/15 generators. It corresponds to {{nowrap| ''n'' {{=}} 4 }}. The only 7-limit extension that make any sense to use is to map the hemifourth to 7/6~8/7. | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: 16875/16384 | |||
{{Mapping|legend=1| 1 2 2 | 0 -4 3 }} | |||
: mapping generators: ~2, ~16/15 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1202.3403{{c}}, ~16/15 = 126.0002{{c}} | |||
: [[error map]]: {{val| +2.340 -1.275 -3.633 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 125.6610{{c}} | |||
: error map: {{val| 0.000 -4.599 -9.331 }} | |||
{{Optimal ET sequence|legend=1| 9, 10, 19, 67c, 86c, 105c }} | |||
[[Badness]] (Sintel): 2.04 | |||
== Lalasepyo (8c & 11) == | == Lalasepyo (8c & 11) == | ||