Superkleismic: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,044 edits
m minimax
Squib (talk | contribs)
363 edits
subtemperament of portent
Tags: Mobile edit Mobile web edit Advanced mobile edit
 
(19 intermediate revisions by 3 users not shown)
Line 1: Line 1:
'''Superkleismic''' is a [[regular temperament]] defined in the [[7-limit]] such that three [[6/5]] generators reach [[7/4]] (tempering out [[square superparticular|S5/S6]] = [[875/864]], the keema) and such that three [[8/7]] intervals reach [[3/2]] (tempering out S7/S8 = [[1029/1024]], the gamelisma), making it a member of the [[gamelismic clan]] and a [[keemic temperaments|keemic temperament]]. It extends extremely easily to the [[11-limit]] as well, by tempering out S10 = [[100/99]] (as well as [[385/384]] and [[441/440]]) so that two generators reach [[16/11]], which serves to [[extension|extend]] the structure of [[orgone]] in the 2.7.11 subgroup. Since in superkleismic, the interval [[21/20]] stands for half [[10/9]] = [[20/19]] × [[19/18]], we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out S19 = [[361/360]] and S20 = [[400/399]]. Superkleismic can also be defined in the [[13-limit]], where two generators are identified with [[13/9]] alongside 16/11, tempering out [[144/143]] and [[325/324]].
{{Infobox regtemp
| Title = Superkleismic
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.19
| Comma basis = [[875/864]], [[1029/1024]] (7-limit); <br>[[100/99]], [[245/242]], [[385/384]] (11-limit); <br>[[100/99]], [[133/132]], [[190/189]], [[385/384]] (L11.19)
| Mapping = 1; 9 10 -3 2 14
| Edo join 1 = 15 | Edo join 2 = 26
| Generators = 5/3 | Generators tuning = 878.2 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[11L 4s]], [[15L 11s]]
| Pergen = (P8, ccP4/9)
| Odd limit 1 = 7 | Mistuning 1 = 6.09 | Complexity 1 = 15
| Odd limit 2 = 2.3.5.7.11.19 21 | Mistuning 2 = 8.85 | Complexity 2 = 26
}}
'''Superkleismic''' is a [[regular temperament]] defined in the [[7-limit]] such that three [[6/5]] generators reach [[7/4]] (tempering out [[875/864]] ([[S-expression|S5/S6]]), the keema) and such that three [[8/7]] intervals reach [[3/2]] (tempering out [[1029/1024]] ([[S-expression|S7/S8]]), the gamelisma), making it a member of the [[gamelismic clan]] and a [[keemic temperaments|keemic temperament]]; its [[5-limit]] comma is [[1953125/1889568]], the shibboleth comma. It [[extension|extends]] extremely easily to the [[11-limit]] as well, by tempering out [[100/99]] ({{S|10}}) so that two generators reach [[16/11]], which also serves to extend the structure of [[orgone]] in the 2.7.11 subgroup. This implies [[385/384]] and [[441/440]] are tempered out as well, making it a subtemperament of [[portent]]. Furthermore, since in superkleismic, the interval [[21/20]] stands for half [[10/9]] = ([[19/18]])⋅([[20/19]]), we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out [[361/360]] ({{S|19}}) and [[400/399]] ({{S|20}}). Superkleismic can also be defined in the [[13-limit]], where two generators are identified with [[13/9]] alongside 16/11, tempering out [[144/143]] and [[325/324]], and extended to 17 to reach the full [[19-limit]], based on the equivalence (8/7)<sup>2</sup> ~ [[17/13]] (natural in slendric) and tempering out [[273/272]] and [[833/832]], in addition to [[120/119]] and [[170/169]].


The minor-third generator of superkleismic is ~6.3 cents sharp of pure 6/5, even wider than the [[kleismic]] minor third (~317 cents), and from this it derives its name. The two mappings unite at [[15edo]]. While not as simple or accurate as kleismic in the 5-limit, it comes into its own as a 7- and 11-limit temperament, approximating both simply and accurately in good tunings. Discarding the harmonics 3 and 5 and concentrating purely on that subgroup gets you orgone. [[41edo]] is a good tuning for superkleismic, with a minor-third generator of 11\41, and [[mos]]ses of 11 ([[4L 7s]]), 15 ([[11L 4s]]), or 26 notes ([[15L 11s]]) are available.
The minor-third generator of superkleismic is ~6.3 cents sharp of pure 6/5, even wider than the [[kleismic]] minor third (~317 cents), and from this it derives its name. The two mappings unite at [[15edo]]. While not as simple or accurate as kleismic in the 5-limit, it comes into its own as a 7- and 11-limit temperament, approximating both simply and accurately in good tunings. Discarding the harmonics 3 and 5 and concentrating purely on that subgroup gets you orgone. [[41edo]] is a good tuning for superkleismic, with a minor-third generator of 11\41, and [[mos]]ses of 11 ([[4L 7s]]), 15 ([[11L 4s]]), or 26 notes ([[15L 11s]]) are available.


See [[Gamelismic clan #Superkleismic]] for more technical data.  
See [[Gamelismic clan #Superkleismic]] for more technical data.


== Interval chain ==
== Interval chain ==
Line 9: Line 21:


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! #
! rowspan="2" | #
! Cents*
! rowspan="2" | Cents*
! Approximate 11-limit add-19 ratios
! colspan="2" | Approximate ratios
! 13-limit extension
|-
! 11-limit add-19
! Full 19-limit extension
|-
|-
| 0
| 0
Line 20: Line 34:
|-
|-
| 1
| 1
| 322.0
| 321.8
| 6/5
| 6/5
|
|
|-
|-
| 2
| 2
| 644.0
| 643.6
| '''16/11''', 36/25
| '''16/11''', 36/25
| 13/9, 19/13
| 13/9, 19/13
|-
|-
| 3
| 3
| 966.0
| 965.4
| '''7/4''', 33/19
| '''7/4''', 33/19
| 26/15
| 26/15, 30/17
|-
|-
| 4
| 4
| 88.0
| 87.3
| 20/19, 19/18, 21/20, 22/21
| 20/19, 19/18, 21/20, 22/21
|
| 18/17
|-
|-
| 5
| 5
| 410.0
| 409.1
| 14/11, 19/15, 24/19
| 14/11, 19/15, 24/19
|
| 34/27
|-
|-
| 6
| 6
| 732.0
| 730.9
| '''32/21''', 38/25
| '''32/21''', 38/25
| 20/13
| 20/13, 26/17
|-
|-
| 7
| 7
| 1053.9
| 1052.7
| 11/6
| 11/6
| 24/13
| 24/13
|-
|-
| 8
| 8
| 175.9
| 174.5
| 10/9, 11/10, 21/19
| 10/9, 11/10, 21/19
|
| 19/17
|-
|-
| 9
| 9
| 497.9
| 496.3
| '''4/3''', 33/25
| '''4/3''', 33/25
|
|
|-
|-
| 10
| 10
| 819.9
| 818.2
| '''8/5'''
| '''8/5'''
|
|
|-
|-
| 11
| 11
| 1141.9
| 1140.0
| 35/18, 48/25, 64/33
| 35/18, 48/25, 64/33
| 52/27
| 52/27
|-
|-
| 12
| 12
| 263.9
| 261.8
| 7/6, 22/19
| 7/6, 22/19
|
| 20/17
|-
|-
| 13
| 13
| 585.9
| 583.6
| 7/5
| 7/5
|
| 24/17
|-
|-
| 14
| 14
| 907.9
| 905.4
| '''32/19''', 42/25, 56/33
| '''32/19''', 42/25, 56/33
| 22/13
| 22/13
|-
|-
| 15
| 15
| 29.9
| 27.2
| 49/48, 55/54, 56/55, 64/63
| 49/48, 55/54, 56/55, 64/63
| 40/39
| 40/39
|-
|-
| 16
| 16
| 351.9
| 349.1
| 11/9
| 11/9
| '''16/13'''
| '''16/13'''
|-
|-
| 17
| 17
| 673.9
| 670.9
| 22/15, 28/19, 40/27
| 22/15, 28/19, 40/27
|
|
|-
|-
| 18
| 18
| 995.9
| 992.7
| '''16/9''', 44/25
| '''16/9''', 44/25
|
|
|-
|-
| 19
| 19
| 117.9
| 114.5
| '''16/15'''
| '''16/15'''
| 14/13
| 14/13
|-
|-
| 20
| 20
| 439.9
| 436.3
| 32/25
| 32/25
|
| 22/17
|-
|-
| 21
| 21
| 761.8
| 768.1
| 14/9
| 14/9
|
| 80/51
|-
|-
| 22
| 22
| 1083.8
| 1080.0
| 28/15
| 28/15
|
| '''32/17'''
|-
|-
| 23
| 23
| 205.8
| 201.8
| 28/25
| 28/25
| 44/39
| 44/39
|-
|-
| 24
| 24
| 527.8
| 523.6
| 49/36
| 49/36
|
|
|-
|-
| 25
| 25
| 849.8
| 845.4
| 44/27
| 44/27
| 64/39
| 28/17, 64/39
|-
|-
| 26
| 26
| 1171.8
| 1167.2
| 49/25, 88/45, 160/81
| 49/25, 88/45, 160/81
| 128/65
| 128/65
|}
|}
<nowiki>*</nowiki> in 13-limit CWE tuning
<nowiki>*</nowiki> In 11-limit add-19 [[CWE]] tuning, octave reduced


== Tunings ==
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~5/3 = 878.2017{{c}}
| CWE: ~5/3 = 878.1077{{c}}
| POTE: ~5/3 = 878.0699{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~5/3 = 878.1854{{c}}
| CWE: ~5/3 = 878.1606{{c}}
| POTE: ~5/3 = 878.1534{{c}}
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>Generators
! Edo<br>generators
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 168: Line 214:
|  
|  
| '''320.000'''
| '''320.000'''
| '''Lower bound of 7- through 21-odd-limit diamond monotone'''
| '''Lower bound of 7- through (L11.19) 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 278: Line 324:
|  
|  
| '''321.951'''
| '''321.951'''
| '''Upper bound of 15- through 21-odd-limit diamond monotone'''
| '''Upper bound of (L11.19) 15- through 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 339: Line 385:
[[Category:Superkleismic| ]] <!-- main article -->
[[Category:Superkleismic| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Shibboleth family]]
[[Category:Gamelismic clan]]
[[Category:Gamelismic clan]]
[[Category:Keemic temperaments]]
[[Category:Keemic temperaments]]
[[Category:Octagar temperaments]]

Latest revision as of 19:43, 2 March 2026

Superkleismic
Subgroups 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.19
Comma basis 875/864, 1029/1024 (7-limit);
100/99, 245/242, 385/384 (11-limit);
100/99, 133/132, 190/189, 385/384 (L11.19)
Reduced mapping ⟨1; 9 10 -3 2 14]
ET join 15 & 26
Generators (CWE) ~5/3 = 878.2 ¢
MOS scales 3L 1s, 4L 3s, 4L 7s, 11L 4s, 15L 11s
Ploidacot wau-enneacot
Pergen (P8, ccP4/9)
Minimax error 7-odd-limit: 6.09 ¢;
2.3.5.7.11.19 21-odd-limit: 8.85 ¢
Target scale size 7-odd-limit: 15 notes;
2.3.5.7.11.19 21-odd-limit: 26 notes

Superkleismic is a regular temperament defined in the 7-limit such that three 6/5 generators reach 7/4 (tempering out 875/864 (S5/S6), the keema) and such that three 8/7 intervals reach 3/2 (tempering out 1029/1024 (S7/S8), the gamelisma), making it a member of the gamelismic clan and a keemic temperament; its 5-limit comma is 1953125/1889568, the shibboleth comma. It extends extremely easily to the 11-limit as well, by tempering out 100/99 (S10) so that two generators reach 16/11, which also serves to extend the structure of orgone in the 2.7.11 subgroup. This implies 385/384 and 441/440 are tempered out as well, making it a subtemperament of portent. Furthermore, since in superkleismic, the interval 21/20 stands for half 10/9 = (19/18)⋅(20/19), we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out 361/360 (S19) and 400/399 (S20). Superkleismic can also be defined in the 13-limit, where two generators are identified with 13/9 alongside 16/11, tempering out 144/143 and 325/324, and extended to 17 to reach the full 19-limit, based on the equivalence (8/7)2 ~ 17/13 (natural in slendric) and tempering out 273/272 and 833/832, in addition to 120/119 and 170/169.

The minor-third generator of superkleismic is ~6.3 cents sharp of pure 6/5, even wider than the kleismic minor third (~317 cents), and from this it derives its name. The two mappings unite at 15edo. While not as simple or accurate as kleismic in the 5-limit, it comes into its own as a 7- and 11-limit temperament, approximating both simply and accurately in good tunings. Discarding the harmonics 3 and 5 and concentrating purely on that subgroup gets you orgone. 41edo is a good tuning for superkleismic, with a minor-third generator of 11\41, and mosses of 11 (4L 7s), 15 (11L 4s), or 26 notes (15L 11s) are available.

See Gamelismic clan #Superkleismic for more technical data.

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are bolded.

# Cents* Approximate ratios
11-limit add-19 Full 19-limit extension
0 0.0 1/1
1 321.8 6/5
2 643.6 16/11, 36/25 13/9, 19/13
3 965.4 7/4, 33/19 26/15, 30/17
4 87.3 20/19, 19/18, 21/20, 22/21 18/17
5 409.1 14/11, 19/15, 24/19 34/27
6 730.9 32/21, 38/25 20/13, 26/17
7 1052.7 11/6 24/13
8 174.5 10/9, 11/10, 21/19 19/17
9 496.3 4/3, 33/25
10 818.2 8/5
11 1140.0 35/18, 48/25, 64/33 52/27
12 261.8 7/6, 22/19 20/17
13 583.6 7/5 24/17
14 905.4 32/19, 42/25, 56/33 22/13
15 27.2 49/48, 55/54, 56/55, 64/63 40/39
16 349.1 11/9 16/13
17 670.9 22/15, 28/19, 40/27
18 992.7 16/9, 44/25
19 114.5 16/15 14/13
20 436.3 32/25 22/17
21 768.1 14/9 80/51
22 1080.0 28/15 32/17
23 201.8 28/25 44/39
24 523.6 49/36
25 845.4 44/27 28/17, 64/39
26 1167.2 49/25, 88/45, 160/81 128/65

* In 11-limit add-19 CWE tuning, octave reduced

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/3 = 878.2017 ¢ CWE: ~5/3 = 878.1077 ¢ POTE: ~5/3 = 878.0699 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/3 = 878.1854 ¢ CWE: ~5/3 = 878.1606 ¢ POTE: ~5/3 = 878.1534 ¢

Tuning spectrum

Edo
generators 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
6/5 315.641 Untempered tuning
4\15 320.000 Lower bound of 7- through (L11.19) 21-odd-limit diamond monotone
22/21 320.134
11/10 320.626
24/19 320.888
21/20 321.117 1/4-keema
19\71 321.127
22/19 321.150
11/6 321.338
22/15 321.356
8/5 321.369 5-odd-limit minimax, 1/10-shibboleth comma
15\56 321.429
32/21 321.537
32/19 321.606
26\97 321.649
21/19 321.658
16/15 321.670 2/19-shibboleth comma
11/9 321.713
7/5 321.732 7- and 11- through (L11.19) 21-odd-limit minimax
37\138 321.739 138e val
28/19 321.842
28/15 321.844
19/15 321.849
11\41 321.951 Upper bound of (L11.19) 15- through 21-odd-limit diamond monotone
4/3 322.005 9-odd-limit minimax, 1/9-shibboleth comma
14/9 322.139
20/19 322.200
7/6 322.239
18\67 322.388 67ch val
10/9 322.800 1/8-shibboleth comma
7/4 322.942 1/3-keema
7\26 323.077 Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
19/18 323.401
14/11 323.502
16/11 324.341

* besides the octave

Retrieved from "https://en.xen.wiki/index.php?title=Superkleismic&oldid=225040"