Glacier: Difference between revisions

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-'''Glacier''' is a [[rank-2 temperament]] in the 2.3.13 subgroup that tempers out the comma [[373248/371293]], the amount by which 5 [[13/12]]'s exceed [[3/2]]. In this temperament, 5 generators make ~3/2 and 6 generators make ~13/8. It possesses [[MOS scale]]s of the families [[1L 4s]], [[1L 5s]], [[1L 6s]], [[1L 7s]], [[8L 1s]], and [[9L 8s]], although the 1L 4s and 1L 5s scales usually end up as extremely lopsided. [[17edo]] and [[26edo]] are good tunings for this temperament, but [[94edo]] achieves a much better effect with practically perfect fifths and ~13/8 2 cents off. Glacier has a generator nearly identical to [[Bleu]] despite it tempering out different commas, of which [[17edo]] also offers a good generator.
+'''Glacier''' is a [[rank-2 temperament]] in the 2.3.13 subgroup that tempers out the comma [[373248/371293]], the amount by which 5 [[13/12]]'s exceed [[3/2]]. In this temperament, 5 generators make ~3/2 and 6 generators make ~13/8. It possesses [[MOS scale]]s of the families [[1L 4s]], [[1L 5s]], [[1L 6s]], [[1L 7s]], [[8L 1s]], and [[9L 8s]], although the 1L 4s and 1L 5s scales usually end up as extremely lopsided. [[17edo]] and [[26edo]] are good tunings for this temperament, but [[94edo]] achieves a much better effect with practically perfect fifths and ~13/8 2 cents off. The optimal patent val in the 2.3.13 subgroup is [[171edo]].
-== Interval chain (CTE tuning) ==
+It has extensions to the full [[13-limit]], but they are contrived. Glacier works much better as a no-5s temperament, whose best subgroup is 2.3.7.11.13.23.29, finding each prime only in positive generators. For technical data on this extension, see [[No-fives subgroup temperaments#Glaishur]].
+The best 2.3.5.13 extension is arguably [[meantone]], adding [[65/64]] and [[81/80]] to the list of tempered commas, of which good edos are [[43edo|43]] and 26. 43 is best in the 2.3.5.13 meantone subgroup. However, adding the schisma is a possibility as 77, 94 and 171edo support schismic, albeit 5/4 will be found at -40 generators.
+Extensions with 7 and 11 are possible. The generator can be close to a pure 13/12, in which case 7/4 will be extremely accurately tuned +7 gens up, tempering out [[62748517/62705664]]. However, the fifths become quite flat as a result. Thusly, [[26edo]] is an optimal tuning for this extension, which also includes [[flattone]] in the 13-limit. This is [[Bleu]].
+The best extension is found by tempering out [[352/351]] and [[729/728]], which is much more complex than Bleu, but much more accurate, called [https://mysingingmonsters.fandom.com/wiki/Glaishur Glaishur.]
+== Interval chain (CWE tuning) ==
 {|class="wikitable"
 |-
 ! Generators up
 ! Cents
+!Mapping
 |-
 | 0
 | 0.0
+|1
 |-
 | 1
-| 140.3
+| 140.384
+|13/12
 |-
 | 2
-| 280.6
+| 280.768
+|27/23
 |-
 | 3
-| 420.9
+| 421.152
+|14/11
 |-
 | 4
-| 561.2
+| 561.536
+|18/13
 |-
 | 5
-| 701.5
+| 701.92
+|'''3/2'''
 |-
 | 6
-| 841.8
+| 842.304
+|'''13/8'''
 |-
 | 7
-| 982.1
+| 982.688
+| 81/46
 |-
 | 8
-| 1122.4
+| 1123.072
+| 44/23
 |-
 | 9
-| 62.7
+| 63.456
+|27/26
 |-
 | 10
-| 203.0
+| 203.84
+|'''9/8'''
 |-
 | 11
-| 343.3
+| 344.224
+|11/9
+|-
+|12
+|484.608
+|81/46
+|-
+|13
+|624.992
+|'''23/16'''
+|-
+|14
+|765.376
+|
+|-
+|15
+|905.76
+|'''27/16'''
+|-
+|16
+|1046.144
+|
+|-
+|17
+|1186.528
+|
+|-
+|18
+|126.912
+|14/13
+|-
+|19
+|267.296
+|
+|-
+|20
+|407.68
+|81/64
+|-
+|21
+|548.064
+|'''11/8'''
+|-
+|22
+|688.448
+|
+|-
+|23
+|828.832
+|
+|-
+|24
+|969.216
+|'''7/4'''
+|-
+|25
+|1109.6
+|243/128
+|-
+|26
+|49.984
+|
+|-
+|27
+|190.368
+|
+|-
+|28
+|330.752
+|
+|-
+|29
+|471.136
+|
+|-
+|30
+|611.52
+|729/512
+|-
+|31
+|751.904
+|
+|-
+|32
+|892.288
+|
+|-
+|33
+|1032.672
+|'''29/16'''
 |}