Delta centauri: Difference between revisions
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| Generators tuning = 883.9 | | Generators tuning = 883.9 | ||
| Optimization method = CWE | | Optimization method = CWE | ||
| MOS scales = [[2L 1s (3/1-equivalent)|2L 1s<3/1>]],[[2L 5s (3/1-equivalent))|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)|15L 13s<3/1>]] | |||
}} | }} | ||
'''Delta centauri''' is the [[non-octave]] [[regular temperament|temperament]] of the [[3.5.11 subgroup]] that [[tempering out|tempers out]] the 4-cent comma, [[1953125/1948617]]. This temperament is generated by an almost-just [[5/3]] and stacking it nine times and tritave-reducing reaches the representation of [[11/9]]. Basically it is a slightly-tempered [[3.5 subgroup]]. The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. It is a [[microtemperament]] with sub-cent error on the 5th and 11th harmonics in optimal tunings, and fairly low [[complexity]] as far as microtemperaments go. | '''Delta centauri''' is the [[non-octave]] [[regular temperament|temperament]] of the [[3.5.11 subgroup]] that [[tempering out|tempers out]] the 4-cent comma, [[1953125/1948617]]. This temperament is generated by an almost-just [[5/3]] and stacking it nine times and tritave-reducing reaches the representation of [[11/9]]. Basically it is a slightly-tempered [[3.5 subgroup]]. The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. It is a [[microtemperament]] with sub-cent error on the 5th and 11th harmonics in optimal tunings, and fairly low [[complexity]] as far as microtemperaments go. | ||
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== Scales == | == Scales == | ||
The same small [[MOS scale]]s as the 3.5 subgroup are produced: [[2L 1s (3/1-equivalent)|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)|15L 13s<3/1>]] | The same small [[MOS scale]]s as the 3.5 subgroup are produced: [[2L 1s (3/1-equivalent)|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)|15L 13s<3/1>]] | ||
{{Todo|inline=1|complete page|comment= Document the technical data in [[No-twos subgroup temperaments]]. Complete the infobox. Make the intervals and tunings section. }} | {{Todo|inline=1|complete page|comment= Document the technical data in [[No-twos subgroup temperaments]]. Complete the infobox. Make the intervals and tunings section. }} | ||
== Extensions == | == Extensions == | ||
There is a clear strong extension to 3.5.11.17, by tempering out [[1377/1375]] and mapping [[17/9]] to 12 generators up. This provides a simpler interpretation of several intervals in the MOS scales. This extension sacrifices a little bit of accuracy and has | There is a clear strong extension to 3.5.11.17, by tempering out [[1377/1375]] and mapping [[17/9]] to 12 generators up. This provides a simpler interpretation of several intervals in the MOS scales. This extension sacrifices a little bit of accuracy and has [[damage]] of around 1-2 cents on 11/9 and 17/9 in CWE tuning, but it's still decent accuracy. | ||
One might want to extend this temperament to 3.5.11.13 to complete the 9:11:13:15 [[isoharmonic]] chord, however most extensions in that subgroup aren't very good because they seem to sacrifice either the high accuracy or low complexity that the 3.5.11 version has. Probably the best strong extension is b15 & b28, which maps [[13/9]] to 5 generators up, but this will make 13/9 very flat (18 cents of [[damage]] in CWE tuning). There is also the high-complexity weak extension b15 & b99 (which has a period of [[13/9]] and tempers out [[2197/2187]]) and the high-complexity strong extension b15 & b71 (which splits [[5/1]] into two 739/325s). | One might want to extend this temperament to 3.5.11.13 to complete the 9:11:13:15 [[isoharmonic]] chord, however most extensions in that subgroup aren't very good because they seem to sacrifice either the high accuracy or low complexity that the 3.5.11 version has. Probably the best strong extension is b15 & b28, which maps [[13/9]] to 5 generators up, but this will make 13/9 very flat (18 cents of [[damage]] in CWE tuning). There is also the high-complexity weak extension b15 & b99 (which has a period of [[13/9]] and tempers out [[2197/2187]]) and the high-complexity strong extension b15 & b71 (which splits [[5/1]] into two 739/325s). | ||
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|268.306 | |268.306 | ||
|[[729/625]] | |[[729/625]] | ||
| | |[[99/85]] | ||
|- | |- | ||
|5 | |5 | ||
|1286.360 | |1286.360 | ||
|[[625/297]], [[6561/3125]] | |[[625/297]], [[6561/3125]] | ||
| | |[[891/425]] | ||
|- | |- | ||
|6 | |6 | ||
|402.459 | |402.459 | ||
|[[125/99]] | |[[125/99]] | ||
| | |[[363/289]] | ||
|- | |- | ||
|7 | |7 | ||
| Line 167: | Line 168: | ||
|536.612 | |536.612 | ||
|[[15/11]] | |[[15/11]] | ||
| | |||
|- | |- | ||
|9 | |9 | ||
|1554.547 | |1554.547 | ||
|'''[[27/11]]''' | |'''[[27/11]]''' | ||
| | |[[125/51]] | ||
|- | |- | ||
|10 | |10 | ||
|670.765 | |670.765 | ||
|[[81/55]] | |[[81/55]] | ||
| | |[[25/17]] | ||
|- | |- | ||
|11 | |11 | ||
|1688.819 | |1688.819 | ||
|[[729/275]] | |[[729/275]] | ||
| | |[[45/17]] | ||
|- | |- | ||
|12 | |12 | ||
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|1822.972 | |1822.972 | ||
|[[3125/1089]], [[19683/6875]] | |[[3125/1089]], [[19683/6875]] | ||
| | |[[243/85]] | ||
|- | |- | ||
|14 | |14 | ||
|939.071 | |939.071 | ||
|[[625/363]] | |[[625/363]] | ||
| | |[[729/425]] | ||
|- | |- | ||
|15 | |15 | ||
|55.170 | |55.170 | ||
|[[125/121]] | |[[125/121]] | ||
| | |[[297/289]] | ||
|} | |} | ||
</div> | </div> | ||
<nowiki/>* In 3.5.11 CWE tuning | <nowiki/>* In 3.5.11 CWE tuning, tritave-reduced | ||
== Tuning spectrum == | == Tuning spectrum == | ||
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| 883.051 | | 883.051 | ||
| Smallest EDT with a reasonable tuning of this temperament | | Smallest EDT with a reasonable tuning of this temperament | ||
|- | |||
| | |||
| [[33/17]] | |||
| 883.328 | |||
| | |||
|- | |- | ||
|[[99edt|46\99]] | |[[99edt|46\99]] | ||
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| | | | ||
| 884.007 | | 884.007 | ||
| | |||
|- | |||
| | | | ||
| [[25/17]] | |||
| 884.210 | |||
| | |||
|- | |||
| | |||
| [[17/15]] | |||
| 884.224 | |||
| | |||
|- | |||
| | |||
| [[17/9]] | |||
| 884.235 | |||
| | |||
|- | |- | ||
| [[114edt|53\114]] | | [[114edt|53\114]] | ||
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| | | | ||
| 884.630 | | 884.630 | ||
| | |||
|- | |||
| | |||
| [[17/11]] | |||
| 885.197 | |||
| | | | ||
|- | |- | ||