Delta centauri: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
CompactStar (talk | contribs)
autopatrolled
5,358 edits
Re-organize a bit. Update todo
 
(20 intermediate revisions by 2 users not shown)
Line 8: Line 8:
| Generators tuning = 883.9
| Generators tuning = 883.9
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[2L 9s (3/1-equivalent)|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)|2L 11s<3/1>]], <br>[[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.


This temperament was seemingly first discovered by [[User:MisterShafXen|MisterShafXen]] in 2025<ref>https://en.xen.wiki/index.php?title=4-cent_comma&diff=prev&oldid=183697</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).


== Scales ==
{{Todo|inline=1|complete page|cleanup|comment= Document the technical data in [[No-twos subgroup temperaments]]. Complete the infobox. Format the intervals section. }}
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>]], [[28L 15s (3/1-equivalent)|28L 15s<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. }}


== 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 errors of around 1-2 cents on 11th and 17th harmonics if properly tuned, but it's still decent accuracy.
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).
Line 57: Line 54:
|4
|4
|1633.649
|1633.649
|[[625/343]]
|[[625/243]]
|[[85/33]]
|[[85/33]]
|-
|-
Line 68: Line 65:
|1499.496
|1499.496
|[[297/125]]
|[[297/125]]
|
|[[289/121]]
|-
|-
|7
|7
Line 98: Line 95:
|1097.037
|1097.037
|[[1375/729]]
|[[1375/729]]
|[[17/9]]
|'''[[17/9]]'''
|-
|-
|13
|13
Line 113: Line 110:
|1846.785
|1846.785
|[[363/125]]
|[[363/125]]
|
|[[289/99]]
|}
|}
</div>
</div>
Line 132: Line 129:
|1018.054
|1018.054
|'''[[9/5]]'''
|'''[[9/5]]'''
|
|[[275/153]]
|-
|-
|2
|2
|134.153
|134.153
|[[27/25]]
|[[27/25]]
|
|[[55/51]]
|-
|-
|3
|3
|1152.207
|1152.207
|[[243/125]]
|[[243/125]]
|
|[[33/17]]
|-
|-
|4
|4
|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 164:
|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
|804.918
|804.918
|[[2187/1375]]
|[[2187/1375]]
|[[27/17]]
|'''[[27/17]]'''
|-
|-
|13
|13
|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 ==
== 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>]]
 
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edt<br>generator
! Edt<br>generator
Line 227: Line 229:
| 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]]
Line 256: Line 263:
|  
|  
| 884.007
| 884.007
|
|-
|  
|  
| [[25/17]]
| 884.210
|
|-
|
| [[17/15]]
| 884.224
|
|-
|
| [[17/9]]
| 884.235
|
|-
|-
| [[114edt|53\114]]
| [[114edt|53\114]]
Line 271: Line 293:
|  
|  
| 884.630
| 884.630
|
|-
|
| [[17/11]]
| 885.197
|
|
|-
|-

Latest revision as of 08:59, 9 October 2026

Delta centauri
Subgroups 3.5.11, 3.5.11.17
Comma basis 1953125/1948617 (3.5.11);
1377/1375, 265625/264627 (3.5.11.17)
Reduced mapping ⟨1; 1 9]
ET join b28 & b71
Generators (CWE) ~5/3 = 883.9 ¢
MOS scales 2L 9s<3/1>, 2L 11s<3/1>,
13L 2s<3/1>, 15L 13s<3/1>
Ploidacot mono5cleft
Minimax error -throdd-limit:  ¢
Target scale size -throdd-limit: notes

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).

Todo: complete page, cleanup

Document the technical data in No-twos subgroup temperaments. 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 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 5/3 459/275
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
6 1499.496 297/125 289/121
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
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 891/425
6 402.459 125/99 363/289
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
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

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
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 Sharpest generator of 13L 2s<3/1> MOS
This is an exo tuning as 11/9 is 380.4 cents

References

  1. ↑ Xenharmonic Wiki | 4-cent comma (Revision as of 11:56, 28 February 2025 by MisterShafXen)
Retrieved from "https://en.xen.wiki/index.php?title=Delta_centauri&oldid=239001"