Glacier: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Eufalesio (talk | contribs)
autopatrolled
634 edits
mNo edit summary
Eufalesio (talk | contribs)
autopatrolled
634 edits
Added some things and revamped the interval table
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
'''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]]. 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]].


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. [[No-fives subgroup temperaments#Glaceir|See here for technical information.]]
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.
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.
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]].


By tempering the [[144/143]], a sharp ~[[11/8]] can be found +4 gens up.
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.]


The CTE generator of 140.3 reaches a very accurate 39/32 +11 gens up, which can be equated with 11/9 by tempering [[352/351]], becoming a [[majorminth temperament]]. This means that 11/8 is found here +21 gens up, which is quite complex.
== Interval chain (CWE tuning) ==
 
== Interval chain (CTE tuning) ==
{|class="wikitable"
{|class="wikitable"
|-
|-
! Generators up
! Generators up
! Cents
! Cents
!2.3.7.13 mapping
!Mapping
!Majorminthic
!Grossmic
|-
|-
| 0
| 0
| 0.0
| 0.0
|
|1
|
|
|-
|-
| 1
| 1
| 140.3
| 140.384
|13/12
|13/12
|88/81
|
|-
|-
| 2
| 2
| 280.6
| 280.768
|7/6
|27/23
|
|
|-
|-
| 3
| 3
| 420.9
| 421.152
|9/7
|14/11
|
|
|-
|-
| 4
| 4
| 561.2
| 561.536
|18/13
|18/13
|-
| 5
| 701.92
|'''3/2'''
|-
| 6
| 842.304
|'''13/8'''
|-
| 7
| 982.688
| 81/46
|-
| 8
| 1123.072
| 44/23
|-
| 9
| 63.456
|27/26
|-
| 10
| 203.84
|'''9/8'''
|-
| 11
| 344.224
|11/9
|-
|12
|484.608
|81/46
|-
|13
|624.992
|'''23/16'''
|-
|14
|765.376
|
|
|11/8
|-
|-
| 5
|15
| 701.5
|905.76
|3/2
|'''27/16'''
|-
|16
|1046.144
|
|
|-
|17
|1186.528
|
|
|-
|-
| 6
|18
| 841.8
|126.912
|13/8
|14/13
|44/27
|-
|19
|267.296
|
|
|-
|-
| 7
|20
| 982.1
|407.68
| 7/4
|81/64
|-
|21
|548.064
|'''11/8'''
|-
|22
|688.448
|
|
|-
|23
|828.832
|
|
|-
|-
| 8
|24
| 1122.4
|969.216
| 27/14
|'''7/4'''
|-
|25
|1109.6
|243/128
|-
|26
|49.984
|
|
|-
|27
|190.368
|
|
|-
|-
| 9
|28
| 62.7
|330.752
|27/26
|
|
|33/32
|-
|-
| 10
|29
| 203.0
|471.136
|9/8
|
|
|-
|30
|611.52
|729/512
|-
|31
|751.904
|
|
|-
|-
| 11
|32
| 343.3
|892.288
|39/32
|11/9
|
|
|-
|33
|1032.672
|'''29/16'''
|}
|}


Latest revision as of 01:33, 23 January 2026

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 scales 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.

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 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 Glaishur.

Interval chain (CWE tuning)

Generators up Cents Mapping
0 0.0 1
1 140.384 13/12
2 280.768 27/23
3 421.152 14/11
4 561.536 18/13
5 701.92 3/2
6 842.304 13/8
7 982.688 81/46
8 1123.072 44/23
9 63.456 27/26
10 203.84 9/8
11 344.224 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
Retrieved from "https://en.xen.wiki/index.php?title=Glacier&oldid=222224"