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{{Todo|inline=1|complete page|comment=Add infobox. Rewrite, polish interval table. Add tunings section. }}
{{Infobox regtemp
'''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 possesses [[MOS scale]]s of the families {{mos scalesig|1L 4s<3/1>|link=1}}, {{mos scalesig|5L 1s<3/1>|link=1}}, {{mos scalesig|6L 5s<3/1>|link=1}}, and {{mos scalesig|11L 6s<3/1>|link=1}}.
| 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}}
}}


This temperament was first discovered and named by {{u|CompactStar}} in 2023 after the star {{w|Deneb}}, following a series of non-octave temperaments that are named after stars.  
{{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.
 
This temperament was first discovered and named by {{u|CompactStar}} in 2023 after the star {{w|Deneb}}, following a series of non-octave temperaments that are named after stars.


For technical data, see [[No-twos subgroup temperaments #Deneb]].
For technical data, see [[No-twos subgroup temperaments #Deneb]].


== Extensions ==
=== Add 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).
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 ===
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 3.8.5.11 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].


== Interval chain ==
== Interval chain ==
One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121; in this temperament, it is equated to [[55/27]] and placed 2 generators down. Unless you are using an usually sharp generator, this pseudo-octave will be very smeary, being almost 22 cents sharp in optimal tuning, but it is probably still meriting of special treatment in terms of consonance and dissonance. (Incidentally, an extension equating the pseudo-octave to [[2/1]] would be [[dicot]])
One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121; in this temperament, it is equated to [[55/27]] and placed 2 generators down. Unless you are using an unusually sharp generator, this quasi-octave will be very smeary, being almost 22 cents sharp in the CWE tuning, but it is probably still meriting of special treatment in terms of consonance and dissonance. (Incidentally, extending Deneb to actually equate the quasi-octave to [[2/1]] would result in the 2.3.5.11 version of [[dicot]])


In the following table, prime harmonics and subharmonics are labeled in '''bold'''.  
In the following table, prime harmonics and subharmonics are labeled in '''bold'''.  
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{| class="wikitable sortable center-1 center-2 right-3"
{| class="wikitable sortable center-1 center-2 right-3"
|-
|-
! #
! rowspan="2"| #
! Cents*
! rowspan="2"| Cents*
! class="unsortable" | Approximate ratios
! colspan="2" class="unsortable" | Approximate ratios
|-
! 3.5.11 subgroup
! 3.8.5.11 extension
|-
|-
| 0
| 0
| 0.000
| 0.000
| [[1/1]]
| [[1/1]]
|
|-
|-
| 1
| 1
| 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
| 1700.260
| 1700.260
| [[121/45]]
| [[121/45]]
| [[8/3]]
|-
|-
| 6
| 6
| 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
| 1498.565
| 1498.565
| [[297/125]]
| [[297/125]]
| [[64/27]]
|-
| 11
| 1838.617
| [[363/125]]
| [[72/25]]
|-
| 12
| 276.714
| [[729/625]]
| [[64/55]]
|-
| 13
| 616.766
| [[891/625]]
| [[64/45]]
|-
| 14
| 956.818
| [[1089/625]]
| [[512/297]]
|-
| 15
| 1296.870
| [[1331/625]], [[6561/3125]]
| [[512/243]], [[576/275]]
|-
| 16
| 1636.922
| [[8019/3125]], [[14641/5625]]
| [[64/25]]
|}
|}
</div>
</div>
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{| class="wikitable sortable center-1 center-2 right-3"
{| class="wikitable sortable center-1 center-2 right-3"
|-
|-
! #
! rowspan="2"| #
! Cents*
! rowspan="2"| Cents*
! class="unsortable" | Approximate ratios
! colspan="2" class="unsortable" | Approximate ratios
|-
! 3.5.11 subgroup
! 3.8.5.11 extension
|-
|-
|-
| 0
| 0
| 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
| 881.799
| 881.799
| '''[[5/3]]'''
| '''[[5/3]]'''
|
|-
|-
| -4
| -4
| 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
| 63.338
| [[125/121]]
| [[25/24]]
|-
| -12
| 1625.241
| [[625/243]]
| [[165/64]]
|-
| -13
| 1285.189
| [[625/297]]
| [[135/64]]
|-
| -14
| 945.137
| [[625/363]]
| [[891/512]]
|-
| -15
| 605.085
| [[1875/1331]], [[3125/2187]]
| [[275/192]], [[729/512]]
|-
| -16
| 265.033
| [[3125/2673]], [[16875/14641]]
| [[75/64]]
|}
|}
</div>
</div>
<nowiki>*</nowiki> In 3.5.11-subgroup [[CWE tuning]]
<nowiki>*</nowiki> In 3.5.11-subgroup [[CWE tuning]]
== Scales ==
Reasonable tunings of Deneb possess [[MOS scale]]s of the families {{mos scalesig|1L 4s<3/1>|link=1}}, {{mos scalesig|5L 1s<3/1>|link=1}}, {{mos scalesig|6L 5s<3/1>|link=1}}, and {{mos scalesig|11L 6s<3/1>|link=1}}. Interestingly, the {{mos scalesig|6L 5s<3/1>}} Deneb[11] scale is melodically similar to softer diatonic scales because both approximate [[7edo]]/[[11edt]].
== Chords and harmony ==
9:11:15 could serve as the fundamental consonant chord for Deneb analogous to 4:5:6 for [[meantone]] and 3:5:7 for [[BPS]]. 9:11:15 is somewhat close to a first-inversion minor triad, so it may sound very vaguely familiar to some. 9:11:15 also has a simpler (but very wide) tritave-equivalent inversion, 5:9:11. The 3.5.8.11 extension has 3:5:8 and 9:11:15:24. The 3.5.11.13 extensions allow for the [[isoharmonic chord]] 9:11:13:15 (although, for high-complexity extensions, it will appear infrequently).
Due to the presence of a quasi-octave and quasi-fifth, there are Mavila-esque neutral triads splitting the quasi-fifth in half (1-[[11/9]]-[[81/55]]). Technically, Deneb makes 1-[[11/9]]-[[81/55]] an [[essentially tempered dyadic chord]].
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
! Edt<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
| '''[[6edt|1\6]]'''
|
| 316.993
| '''Lower bound of (3, 5, 11) tonality diamond monotone'''
|-
| [[17edt|3\17]]
|
| 335.639
| Lower bound to generate {{mos scalesig|11L 6s<3/1>|link=1}} MOS scale
|-
| [[45edt|8\45]]
|
| 338.125
|
|-
|
| [[5/3]]
| 339.199
|
|-
| [[28edt|5\28]]
|
| 339.635
|
|-
|
| [[33/25]]
| 340.372
|
|-
| [[67edt|12\67]]
|
| 340.649
|
|-
|
| [[11/5]]
| 341.251
| Equal error on 11/9 and 5/3
|-
| [[39edt|7\39]]
|
| 341.377
|
|-
| [[50edt|9\50]]
|
| 342.352
|
|-
| [[11edt|2\11]]
|
| 345.810
| Upper bound to generate {{mos scalesig|11L 6s<3/1>|link=1}} MOS scale
|-
|
| [[11/9]]
| 347.408
| Untempered tuning
|-
| '''[[5edt|1\5]]'''
|
| 380.391
| '''Upper bound of (3, 5, 11) tonality diamond monotone'''
|}
[[Category:Deneb| ]] <!-- main article -->
[[Category:Deneb| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]
[[Category:Non-octave temperaments]]
Retrieved from "https://en.xen.wiki/w/Deneb"