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{{Infobox regtemp
{{Infobox regtemp
| Title = Delta centauri
| Title = Delta centauri
| Subgroups = 3.5.11
| Subgroups = 3.5.11, 3.5.11.17
| Comma basis = [[1953125/1948617]]
| Comma basis = [[1953125/1948617]] (3.5.11);<br>[[1377/1375]], [[265625/264627]] (3.5.11.17)
| Edo join 1 = b28 | Edo join 2 = b71
| Edo join 1 = b28 | Edo join 2 = b71
| Mapping = 1; 1 9
| Mapping = 1; 1 9
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 ==
One might want to extend this temperament to 3.5.11.13 to complete the 9:11:13:15 [[isoharmonic]] chord, however most extensions 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).
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.


There is a clear extension to 3.5.11.17, by mapping [[17/9]] to 12 generators up.
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 ==
== Interval chain ==
Prime harmonics and subharmonics are bold
<div style="display: inline-grid; margin-right: 25px;">
<div style="display: inline-grid; margin-right: 25px;">
{|class="wikitable"
{|class="wikitable"
Line 31: Line 30:
!Cents*
!Cents*
!Approximate ratios
!Approximate ratios
!Additional ratios<br>in 3.5.11.17 extension
|-
|-
|0
|0
|0.00
|0.00
|[[1/1]]
|[[1/1]]
|
|-
|-
|1
|1
|883.901
|883.901
|'''[[5/3]]'''
|'''[[5/3]]'''
|[[459/275]]
|-
|-
|2
|2
|1767.802
|1767.802
|[[25/9]]
|[[25/9]]
|[[153/55]]
|-
|-
|3
|3
|749.748
|749.748
|[[125/81]]
|[[125/81]]
|[[17/11]]
|-
|-
|4
|4
|1633.649
|1633.649
|[[625/343]]
|[[625/243]]
|[[85/33]]
|-
|-
|5
|5
|615.595
|615.595
|[[891/625]], [[3125/2187]]
|[[891/625]], [[3125/2187]]
|[[425/297]]
|-
|-
|6
|6
|1499.496
|1499.496
|[[297/125]]
|[[297/125]]
|[[289/121]]
|-
|-
|7
|7
|481.442
|481.442
|[[33/25]]
|[[33/25]]
|
|-
|-
|8
|8
|1365.343
|1365.343
|[[11/5]]
|[[11/5]]
|
|-
|-
|9
|9
|347.289
|347.289
|'''[[11/9]]'''
|'''[[11/9]]'''
|[[153/125]]
|-
|-
|10
|10
|1231.190
|1231.190
|[[55/27]]
|[[55/27]]
|[[51/25]]
|-
|-
|11
|11
|213.136
|213.136
|[[275/243]]
|[[275/243]]
|[[17/15]]
|-
|-
|12
|12
|1097.037
|1097.037
|[[1375/729]]
|[[1375/729]]
|'''[[17/9]]'''
|-
|-
|13
|13
|78.983
|78.983
|[[3267/3125]], [[6875/6561]]
|[[3267/3125]], [[6875/6561]]
|[[85/81]]
|-
|-
|14
|14
|962.884
|962.884
|[[1089/625]]
|[[1089/625]]
|[[425/243]]
|-
|-
|15
|15
|1846.785
|1846.785
|[[363/125]]
|[[363/125]]
|[[289/99]]
|}
|}
</div>
</div>
Line 103: Line 119:
!Cents*
!Cents*
!Approximate ratios
!Approximate ratios
!Additional ratios<br>in 3.5.11.17 extension
|-
|-
|0
|0
|0.00
|0.00
|[[1/1]]
|[[1/1]]
|
|-
|-
|1
|1
|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
|1420.513
|1420.513
|[[25/11]]
|[[25/11]]
|
|-
|-
|8
|8
|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]]'''
|-
|-
|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 192: 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 221: 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 236: Line 293:
|  
|  
| 884.630
| 884.630
|
|-
|
| [[17/11]]
| 885.197
|
|
|-
|-
Retrieved from "https://en.xen.wiki/w/Delta_centauri"