Delta centauri: Difference between revisions
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This temperament was seemingly first discovered by {{u|MisterShafXen}} in 2025<ref>[https://en.xen.wiki/index.php?title=4-cent_comma&diff=prev&oldid=183697 Xenharmonic Wiki | ''4-cent comma'' (Revision as of 11:56, 28 February 2025 by MisterShafXen)]</ref>, and was named by {{u|CompactStar}} in 2026 after the star {{w|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). | This temperament was seemingly first discovered by {{u|MisterShafXen}} in 2025<ref>[https://en.xen.wiki/index.php?title=4-cent_comma&diff=prev&oldid=183697 Xenharmonic Wiki | ''4-cent comma'' (Revision as of 11:56, 28 February 2025 by MisterShafXen)]</ref>, and was named by {{u|CompactStar}} in 2026 after the star {{w|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). | ||
{{Todo|inline=1|complete page|cleanup|comment= | For technical data, see [[No-twos subgroup temperaments #Delta centauri]]. | ||
{{Todo|inline=1|complete page|cleanup|comment=Complete the infobox. Format the intervals section. }} | |||
== Extensions == | == Extensions == | ||
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== Interval chain == | == Interval chain == | ||
In the following tables, prime harmonics are labeled in '''bold'''. | In the following tables, prime harmonics and subharmonics are labeled in '''bold'''. | ||
<div style="display: inline-grid; margin-right: 25px;"> | <div style="display: inline-grid; margin-right: 25px;"> | ||
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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>]] | ||
== Tunings == | |||
=== Tuning spectrum === | |||
== Tunings == | == Tunings == | ||
=== Tuning spectrum === | === Tuning spectrum === | ||
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! Comments | ! Comments | ||
|- | |- | ||
| [[ | | '''[[9edt|4\9]]''' | ||
| | | | ||
| 877.823 | | '''845.313''' | ||
| | | '''Lower bound of 3.5.11 tonality diamond monotone''' | ||
|- | |||
| '''[[13edt|6\13]]''' | |||
| | |||
| '''877.823''' | |||
| '''Lower bound of 3.5.11.17 tonality diamond monotone''' | |||
|- | |- | ||
| [[41edt|19\41]] | | [[41edt|19\41]] | ||
| Line 305: | Line 314: | ||
| | | | ||
|- | |- | ||
| [[15edt|7\15]] | | '''[[15edt|7\15]]''' | ||
| | |||
| '''887.579''' | |||
| '''Upper bound of 3.5.11.17 tonality diamond monotone''' | |||
|- | |||
| '''[[17edt|8\17]]''' | |||
| | | | ||
| | | '''895.038''' | ||
| | | '''Upper bound of 3.5.11 tonality diamond monotone''' | ||
|} | |} | ||
Latest revision as of 21:33, 21 October 2026
| Delta centauri |
1377/1375, 265625/264627 (3.5.11.17)
13L 2s<3/1>, 15L 13s<3/1>
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).
For technical data, see No-twos subgroup temperaments #Delta centauri.
|
Todo: complete page, cleanup
Complete the infobox. Format the intervals 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. 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 weak extension b15 & b71 (which splits 5/1 into two 739/325s).
Interval chain
In the following tables, prime harmonics and subharmonics are labeled in 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 | 459/275 |
| 2 | 1767.802 | 25/9 | 153/55 |
| 3 | 749.748 | 125/81 | 17/11 |
| 4 | 1633.649 | 625/243 | 85/33 |
| 5 | 615.595 | 891/625, 3125/2187 | 425/297, 867/605 |
| 6 | 1499.496 | 297/125 | 121/51, 289/121 |
| 7 | 481.442 | 33/25 | |
| 8 | 1365.343 | 11/5 | |
| 9 | 347.289 | 11/9 | 153/125 |
| 10 | 1231.190 | 55/27 | 51/25 |
| 11 | 213.136 | 275/243 | 17/15 |
| 12 | 1097.037 | 1375/729 | 17/9 |
| 13 | 78.983 | 3267/3125, 6875/6561 | 85/81, 289/275 |
| 14 | 962.884 | 1089/625 | 425/243 |
| 15 | 1846.785 | 363/125 | 289/99 |
| Number of generators | Cents* | Approximate ratios | Additional ratios in 3.5.11.17 extension |
|---|---|---|---|
| 0 | 0.00 | 1/1 | |
| -1 | 1018.054 | 9/5 | 275/153 |
| -2 | 134.153 | 27/25 | 55/51 |
| -3 | 1152.207 | 243/125 | 33/17 |
| -4 | 268.306 | 729/625 | 99/85 |
| -5 | 1286.360 | 625/297, 6561/3125 | 605/289, 891/425 |
| -6 | 402.459 | 125/99 | 153/121, 363/289 |
| -7 | 1420.513 | 25/11 | |
| -8 | 536.612 | 15/11 | |
| -9 | 1554.547 | 27/11 | 125/51 |
| -10 | 670.765 | 81/55 | 25/17 |
| -11 | 1688.819 | 729/275 | 45/17 |
| -12 | 804.918 | 2187/1375 | 27/17 |
| -13 | 1822.972 | 3125/1089, 19683/6875 | 243/85, 825/289 |
| -14 | 939.071 | 625/363 | 729/425 |
| -15 | 55.170 | 125/121 | 297/289 |
* In 3.5.11 CWE tuning, tritave-reduced
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>
Tunings
Tuning spectrum
Tunings
Tuning spectrum
| Edt generator |
Unchanged interval (eigenmonzo) |
Generator (¢) | Comments |
|---|---|---|---|
| 4\9 | 845.313 | Lower bound of 3.5.11 tonality diamond monotone | |
| 6\13 | 877.823 | Lower bound of 3.5.11.17 tonality diamond monotone | |
| 19\41 | 881.393 | ||
| 13\28 | 883.051 | Smallest EDT with a reasonable tuning of this temperament | |
| 33/17 | 883.328 | ||
| 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 | ||
| 25/17 | 884.210 | ||
| 17/15 | 884.224 | ||
| 17/9 | 884.235 | ||
| 53\114 | 884.242 | ||
| 5/3 | 884.359 | Trithagorean/3.5 subgroup | |
| 20\43 | 884.630 | ||
| 17/11 | 885.197 | ||
| 27\58 | 885.393 | ||
| 7\15 | 887.579 | Upper bound of 3.5.11.17 tonality diamond monotone | |
| 8\17 | 895.038 | Upper bound of 3.5.11 tonality diamond monotone |