Delta centauri: Difference between revisions
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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. | 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. | ||
One might want to extend this temperament to 3.5.11.13 to complete the 9:11:13:15 [[isoharmonic]] chord, however most extensions 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 b51 & 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 b51 & 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). | ||
== Interval chain == | == Interval chain == | ||
Revision as of 00:34, 9 October 2026
| Delta centauri |
1377/1375, 265625/264627 (3.5.11.17)
Delta centauri is the non-octave temperament of the 3.5.11 subgroup that 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.
This temperament was seemingly first discovered by MisterShafXen in 2025[1], and was named by CompactStar in 2026 after the star Delta Centauri, following a series of non-octave temperaments that are named after stars, and as a reference to the 4-cent comma (delta being the 4th letter of the Greek alphabet).
Scales
The same small MOS scales as the 3.5 subgroup are produced: 2L 1s<3/1>, 2L 3s<3/1>, 2L 5s<3/1>, 2L 7s<3/1>, 2L 9s<3/1>, 2L 11s<3/1>, 13L 2s<3/1>, 15L 13s<3/1>, 28L 15s<3/1>
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Todo: complete page
Document the technical data in No-twos subgroup temperaments. Complete the infobox. Make the intervals and tunings section. |
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.
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 b51 & 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).
Interval chain
Prime harmonics and subharmonics are bold
| Number of generators | Cents* | Approximate ratios | Additional ratios in 3.5.11.17 extension |
|---|---|---|---|
| 0 | 0.00 | 1/1 | |
| 1 | 883.901 | 5/3 | |
| 2 | 1767.802 | 25/9 | |
| 3 | 749.748 | 125/81 | |
| 4 | 1633.649 | 625/343 | |
| 5 | 615.595 | 891/625, 3125/2187 | |
| 6 | 1499.496 | 297/125 | |
| 7 | 481.442 | 33/25 | |
| 8 | 1365.343 | 11/5 | |
| 9 | 347.289 | 11/9 | |
| 10 | 1231.190 | 55/27 | |
| 11 | 213.136 | 275/243 | |
| 12 | 1097.037 | 1375/729 | |
| 13 | 78.983 | 3267/3125, 6875/6561 | |
| 14 | 962.884 | 1089/625 | |
| 15 | 1846.785 | 363/125 |
| Number of generators | Cents* | Approximate ratios | Additional ratios in 3.5.11.17 extension |
|---|---|---|---|
| 0 | 0.00 | 1/1 | |
| 1 | 1018.054 | 9/5 | |
| 2 | 134.153 | 27/25 | |
| 3 | 1152.207 | 243/125 | |
| 4 | 268.306 | 729/625 | |
| 5 | 1286.360 | 625/297, 6561/3125 | |
| 6 | 402.459 | 125/99 | |
| 7 | 1420.513 | 25/11 | |
| 8 | 536.612 | 15/11 | |
| 9 | 1554.547 | 27/11 | |
| 10 | 670.765 | 81/55 | |
| 11 | 1688.819 | 729/275 | |
| 12 | 804.918 | 2187/1375 | |
| 13 | 1822.972 | 3125/1089, 19683/6875 | |
| 14 | 939.071 | 625/363 | |
| 15 | 55.170 | 125/121 |
* In 3.5.11 CWE tuning
Tuning spectrum
| Edt generator |
Unchanged interval (eigenmonzo) |
Generator (¢) | Comments |
|---|---|---|---|
| 6\13 | 877.823 | Flattest generator of 13L 2s<3/1> MOS This is an exo tuning as 11/9 is 292.6 cents | |
| 19\41 | 881.393 | ||
| 13\28 | 883.051 | Smallest EDT with a reasonable tuning of this temperament | |
| 46\99 | 883.737 | ||
| 33/25 | 883.787 | ||
| 11/5 | 883.859 | ||
| 11/9 | 883.914 | ||
| 55/27 | 883.959 | Error is equally distributed between 11/9 and 5/3 | |
| 33\71 | 884.007 | ||
| 53\114 | 884.242 | ||
| 5/3 | 884.359 | Trithagorean/3.5 subgroup | |
| 20\43 | 884.630 | ||
| 27\58 | 885.393 | ||
| 7\15 | 887.579 | Sharpest generator of 13L 2s<3/1> MOS This is an exo tuning as 11/9 is 380.4 cents |