Xenharmonic Wiki

Deneb: Difference between revisions

← Older edit

CompactStar (talk | contribs)

1 user group
5,789 edits
VisualWikitext
CompactStar (talk | contribs)
autopatrolled
5,789 edits
CompactStar (talk | contribs)
autopatrolled
5,789 edits
 
(20 intermediate revisions by the same user not shown)
Line 1: Line 1:
{{Todo|inline=1|complete page|comment=Add infobox. Rewrite, polish interval table. Add tunings section. }}
{{Infobox regtemp
| Title = Deneb
| Subgroups = 3.5.11
| Comma basis = [[6655/6561]]
| Edo join 1 = b28 | Edo join 2 = b39
| Mapping = 1; -3 1
| Generators = 11/9
| Generators tuning = 340.052
| Optimization method = CWE
| MOS scales = {{mos scalesig|1L 4s<3/1>|link=1}}, {{mos scalesig|5L 1s<3/1>|link=1}}, {{mos scalesig|6L 5s<3/1>|link=1}}, {{mos scalesig|11L 6s<3/1>|link=1}}
}}
 
{{Todo|inline=1|complete page|comment=Add minimax error and target scale size. }}
'''Deneb''' is a [[nonoctave]], [[tritave]]-repeating [[rank-2 temperament]] that tempers out the comma [[6655/6561]] in the 3.5.11 subgroup. It is generated by a neutral third that represents a flat [[11/9]], and three generators stacked represent [[9/5]] (and due to tritave-equivalence, -3 generators stacked represents [[5/3]]). It is one of the simplest temperaments in the 3.5.11 subgroup with decent accuracy.  
'''Deneb''' is a [[nonoctave]], [[tritave]]-repeating [[rank-2 temperament]] that tempers out the comma [[6655/6561]] in the 3.5.11 subgroup. It is generated by a neutral third that represents a flat [[11/9]], and three generators stacked represent [[9/5]] (and due to tritave-equivalence, -3 generators stacked represents [[5/3]]). It is one of the simplest temperaments in the 3.5.11 subgroup with decent accuracy.  


Line 30: Line 42:
| 340.052
| 340.052
| '''[[11/9]]'''
| '''[[11/9]]'''
|  
| [[40/33]]
|-
|-
| 2
| 2
| 680.104
| 680.104
| [[81/55]], [[121/81]]
| [[81/55]], [[121/81]]
|
| [[40/27]]
|-
|-
| 3
| 3
| 1020.156
| 1020.156
| '''[[9/5]]'''
| '''[[9/5]]'''
|
|
|-
|-
| 4
| 4
| 1360.208
| 1360.208
| [[11/5]]
| [[11/5]]
|
| [[24/11]]
|-
|-
| 5
| 5
Line 55: Line 67:
| 138.357
| 138.357
| [[27/25]]
| [[27/25]]
|
| [[88/81]]
|-
|-
| 7
| 7
| 478.409
| 478.409
| [[33/25]]
| [[33/25]]
|
| [[72/55]], [[320/243]]
|-
|-
| 8
| 8
| 818.461
| 818.461
| [[121/75]]
| [[121/75]]
|
| [[8/5]]
|-
|-
| 9
| 9
| 1158.513
| 1158.513
| [[243/125]]
| [[243/125]]
|
| [[64/33]], [[88/45]]
|-
|-
| 10
| 10
Line 80: Line 92:
| 1838.617
| 1838.617
| [[363/125]]
| [[363/125]]
|
| [[72/25]]
|-
|-
| 12
| 12
| 276.714
| 276.714
| [[729/625]]
| [[729/625]]
|
| [[64/55]]
|-
|-
| 13
| 13
| 616.766
| 616.766
| [[891/625]]
| [[891/625]]
|
| [[64/45]]
|-
|-
| 14
| 14
| 956.818
| 956.818
| [[1089/625]]
| [[1089/625]]
|
| [[512/297]]
|-
|-
| 15
| 15
| 1296.870
| 1296.870
| [[1331/625]], [[6561/3125]]
| [[1331/625]], [[6561/3125]]
| [[512/243]]
| [[512/243]], [[576/275]]
|-
|-
| 16
| 16
| 1636.922
| 1636.922
| [[8019/3125]], [[14641/5625]]
| [[8019/3125]], [[14641/5625]]
|
| [[64/25]]
|}
|}
</div>
</div>
Line 123: Line 135:
| 0.000
| 0.000
| [[1/1]]
| [[1/1]]
|
|  
|-
|-
| -1
| -1
| 1561.903
| 1561.903
| '''[[27/11]]'''
| '''[[27/11]]'''
|
| [[99/40]]
|-
|-
| -2
| -2
| 1221.851
| 1221.851
| [[55/27]], [[243/121]]
| [[55/27]], [[243/121]]
|
| [[81/40]]
|-
|-
| -3
| -3
Line 143: Line 155:
| 541.747
| 541.747
| [[15/11]]
| [[15/11]]
|
| [[11/8]]
|-
|-
| -5
| -5
| 201.695
| 201.695
| [[135/121]]
| [[135/121]]
|
| [[9/8]]
|-
|-
| -6
| -6
| 1763.598
| 1763.598
| [[25/9]]
| [[25/9]]
|
| [[243/88]]
|-
|-
| -7
| -7
| 1423.546
| 1423.546
| [[25/11]]
| [[25/11]]
|
| [[55/24]], [[729/320]]
|-
|-
| -8
| -8
| 1083.494
| 1083.494
| [[225/121]]
| [[225/121]]
|
| [[15/8]]
|-
|-
| -9
| -9
| 743.442
| 743.442
| [[125/81]]
| [[125/81]]
|
| [[99/64]], [[135/88]]
|-
|-
| -10
| -10
| 403.390
| 403.390
| [[125/99]]
| [[125/99]]
|
| [[81/64]]
|-
|-
| -11
| -11
| 63.338
| 63.338
| [[125/121]]
| [[125/121]]
|
| [[25/24]]
|-
|-
| -12
| -12
| 1625.241
| 1625.241
| [[625/243]]
| [[625/243]]
|
| [[165/64]]
|-
|-
| -13
| -13
| 1285.189
| 1285.189
| [[625/297]]
| [[625/297]]
|
| [[135/64]]
|-
|-
| -14
| -14
| 945.137
| 945.137
| [[625/363]]
| [[625/363]]
|
| [[891/512]]
|-
|-
| -15
| -15
| 605.085
| 605.085
| [[1875/1331]], [[3125/2187]]
| [[1875/1331]], [[3125/2187]]
|
| [[275/192]], [[729/512]]
|-
|-
| -16
| -16
| 265.033
| 265.033
| [[3125/2673]], [[16875/14641]]
| [[3125/2673]], [[16875/14641]]
|
| [[75/64]]
|}
|}
</div>
</div>
Line 219: Line 231:
== Extensions ==
== Extensions ==
=== Add 13 ===
=== Add 13 ===
There are multiple possible extensions to 3.5.11.13 and higher no-2s no-7s limits. The only practical [[strong extension]] of Deneb to 3.5.11.13, b28 & b39, maps [[13/9]] to +13 generators up. This only really works for generators sharp of 5\28edt. 3.5.11 Deneb can suffice using the diatonic-like 11-note MOS {{mos scalesig|6L 5s<3/1>|link=1}}, but this extension requires at least the 17-note MOS {{mos scalesig|11L 6s<3/1>|link=1}} to properly use the ratios of 13.
There are several possible extensions to 3.5.11.13 and higher no-2s no-7s limits, but it's mainly weak extensions. The only practical [[strong extension]] of Deneb to 3.5.11.13, b28 & b39, maps [[13/9]] to +13 generators up. This only really works for generators sharp of 5\28edt. 3.5.11 Deneb can suffice using the diatonic-like 11-note MOS {{mos scalesig|6L 5s<3/1>|link=1}}, but this extension requires at least the 17-note MOS {{mos scalesig|11L 6s<3/1>|link=1}} to properly use the ratios of 13.


[[Fomalhaut]] is a [[weak extension]] of Deneb to higher limits that splits the interval of [[11/3]] in three. Fomalhaut was considered in the no-2s no-7s 23-limit (3.5.11.13.17.19.23) from the start. It is considerably more complex than Deneb, and requires at least an 18-note MOS to properly use (technically, all prime harmonics 3-23 are accessible in the 13-note MOS, but the 13-note MOS is very [[hard]], and said prime harmonics are in different directions).
[[Fomalhaut]] is a [[weak extension]] of Deneb to higher limits that splits the interval of [[11/3]] in three. Fomalhaut was considered in the no-2s no-7s 23-limit (3.5.11.13.17.19.23) from the start. It is considerably more complex than Deneb, and requires at least an 18-note MOS to properly use (technically, all prime harmonics 3-23 are accessible in the 13-note MOS, but the 13-note MOS is very [[hard]], and said prime harmonics are in different directions).


Also, the 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient 3.5.7.11 temperament. The way it is a weak extension of Deneb is that [[27/11]] (the tritave-complement of 11/9) is split into two [[11/7]]s. Mintra can then be further extended to 3.5.7.11.13.
Another possible extension of Deneb to 3.5.11.13 is [[No-twos subgroup temperaments#Threneb|Threneb]], which temper out the threedie, [[2197/2187]]. This splits the tritave into 3 [[13/9]]s, and 13/9 becomes the new period while the generator remains the same. The EDT join of this extension is b39&b45 (only EDTs that are a multiple of 3 are supported). Unfortunately, the smaller MOS scales generated by this are once again very [[hard]].
 
Also, the 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament. The way it is a weak extension of Deneb is that [[27/11]] (the tritave-complement of 11/9) is split into two [[11/7]]s. Mintra can then be further extended to 3.5.7.11.13.


=== Add 8 ===
=== Add 8 ===
5 generators stacked actually is a very good approximation of [[8/3]], so if you're okay with introducing limited factors of 2, Deneb can be easily extended to the 3.5.8.11 subgroup (or using the technically "canonical" ordering, 8.3.5.11, but using [[8/1]] as an equivalence would be rather impractical). This extension tempers out [[121/120]] and [[2200/2187]] and is related to the octave-repeating [[amity]]/[[hitchcock]] temperament. Interestingly, in the 3.5.8.11 extension, ~[[8/1]] = ~[[55/27]]<sup>2</sup> * ~[[243/125]], with ~55/27 being the aforementioned sharp quasi-octave, and ~243/125 being an ultramajor seventh / infraoctave of sorts. Furthermore, ~55/27 and ~243/125 share the same scale degree in Deneb[11].
5 generators stacked actually is a very good approximation of [[8/3]], so if you're okay with introducing limited factors of 2, Deneb can be easily extended to the 3.5.8.11 subgroup (or using the technically "canonical" ordering, 8.3.5.11, but using [[8/1]] as an equivalence would be rather impractical). This extension tempers out [[121/120]] and [[2200/2187]] and is related to the octave-repeating [[amity]]/[[hitchcock]] temperament. Interestingly, in the 3.5.8.11 extension, ~[[8/1]] = ~[[55/27]]<sup>2</sup> * ~[[64/33]], with ~55/27 being the aforementioned sharp quasi-octave, and ~64/33 being the undecimal infraoctave. Furthermore, ~55/27 and ~64/33 share the same scale degree in Deneb[11].


== Tunings ==
== Tunings ==
Retrieved from "https://en.xen.wiki/w/Deneb"