Kleismic: Difference between revisions

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== Tunings ==
== Tunings ==
=== Optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Prime-Optimized Tunings
|-
! Weight-skew\Order !! Euclidean
|-
| Tenney || (2.3.5) CTE: ~6/5 = 317.0308¢
|-
| Tenney || (2.3.5) POTE: ~6/5 = 317.007¢
|-
| Tenney || (2.3.5.13) CTE: ~6/5 = 317.1110¢
|-
| Tenney || (2.3.5.13) POTE: ~6/5 = 317.0756¢
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|-
! Optimized chord !! Generator value !! Polynomial !! Further notes
|-
| 3:4:5 (+1 +1) || ~6/5 = 317.1496 || ''g''<sup>6</sup> + 2''g''<sup>5</sup> - 8 = 0 || 1-3-5 equal-beating tuning
|-
| 4:5:6 (+1 +1) || ~6/5 = 317.9593 || ''g''<sup>6</sup> - 2''g''<sup>5</sup> + 2 = 0 || 1-3-5 equal-beating tuning, close to 2/7-kleisma
|-
| 10:12:15 (+2 +3) || ~6/5 = 317.6675 || ''g''<sup>6</sup> - 5''g'' + 3 = 0 || Close to 1/4-kleisma
|-
| 9:13:15 (+2 +1) || ~6/5 = 317.5679 || 3''g''<sup>3</sup> + 4''g'' - 10 = 0 || Close to 13/36-marveltwin comma
|-
| 13:15:18 (+2 +3) || ~6/5 = 317.0010 || 3''g''<sup>3</sup> - ''g'' - 4 = 0 || Close to 13/51-marveltwin comma
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
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| 317.1429
| 317.1429
|  
|  
|-
|
| ''f''<sup>6</sup> + 2''f''<sup>5</sup> - 8 = 0
| 317.1496
| 1–3–5 equal-beating tuning, [[Delta-rational chord|DR]] 3:4:5
|-
|-
|  
|  
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| 317.6681
| 317.6681
| 1/4-kleisma, virtually [[Delta-rational chord|DR]] 10:12:15
| 1/4-kleisma, virtually [[Delta-rational chord|DR]] 10:12:15
|-
|
| ''f''<sup>6</sup> - 2''f''<sup>5</sup> + 2 = 0
| 317.9593
| 1–3–5 equal-beating tuning, [[Delta-rational chord|DR]] 4:5:6, close to 2/7-kleisma
|-
|-
| [[83edo|22\83]]
| [[83edo|22\83]]
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=== Other tunings ===
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983
* [[Delta-rational chord|DR]] 9:13:15: ~2 = 1\1, ~6/5 = 317.0010 (real root of 3''f''<sup>3</sup> - ''f'' - 4 = 0)
* [[Delta-rational chord|DR]] 13:15:18: ~2 = 1\1, ~6/5 = 317.5679 (real root of 3''f''<sup>3</sup> + 4''f'' - 10 = 0)


== Scales ==
== Scales ==

Revision as of 17:10, 17 September 2024

Hanson is a rank-2 temperament of the kleismic family, characterized by the vanishing of the kleisma. It is generated by a classical minor third (6/5), six of which make a twelfth (3/1). This naturally gives us hemitwelfths at only 3 generator steps, which can be interpreted as 26/15 (and thus hemifourths as 15/13), resulting in a low-complexity but high-accuracy extension to the 2.3.5.13 subgroup, sometimes known as cata.

7-limit extensions include keemun, catalan, catakleismic, countercata, and metakleismic.

For technical data, see Kleismic family #Hanson.

Interval chain

In the following table, odd harmonics 1–15 are labeled in bold.

# Cents* Approximate Ratios
0 0.0 1/1
1 317.1 6/5
2 634.2 13/9
3 950.3 26/15
4 68.4 25/24, 26/25, 27/26
5 385.6 5/4
6 702.7 3/2
7 1019.8 9/5
8 136.9 13/12, 27/25
9 454.0 13/10
10 771.1 25/16
11 1088.2 15/8
12 205.3 9/8
13 522.4 27/20
14 839.6 13/8
15 1156.7 39/20
16 273.8 75/64
17 590.9 45/32
18 908.0 27/16
19 25.1 65/64, 81/80

* in 2.3.5.13-subgroup CTE tuning

Tunings

Optimized tunings

Prime-Optimized Tunings
Weight-skew\Order Euclidean
Tenney (2.3.5) CTE: ~6/5 = 317.0308¢
Tenney (2.3.5) POTE: ~6/5 = 317.007¢
Tenney (2.3.5.13) CTE: ~6/5 = 317.1110¢
Tenney (2.3.5.13) POTE: ~6/5 = 317.0756¢
DR and equal-beating tunings
Optimized chord Generator value Polynomial Further notes
3:4:5 (+1 +1) ~6/5 = 317.1496 g6 + 2g5 - 8 = 0 1-3-5 equal-beating tuning
4:5:6 (+1 +1) ~6/5 = 317.9593 g6 - 2g5 + 2 = 0 1-3-5 equal-beating tuning, close to 2/7-kleisma
10:12:15 (+2 +3) ~6/5 = 317.6675 g6 - 5g + 3 = 0 Close to 1/4-kleisma
9:13:15 (+2 +1) ~6/5 = 317.5679 3g3 + 4g - 10 = 0 Close to 13/36-marveltwin comma
13:15:18 (+2 +3) ~6/5 = 317.0010 3g3 - g - 4 = 0 Close to 13/51-marveltwin comma

Tuning spectrum

Edo
Generator 		Eigenmonzo
(Unchanged-interval)* 		Generator (¢) Comments
6/5 315.6413 Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
5\19 315.7895 Lower bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone
27/26 316.3343 1/4-tunbarsma
29\110 316.3636 110ff val
24\91 316.4835 91f val
27/25 316.6547 1/8-kleisma
19\72 316.6667
9/5 316.7995 1/7-kleisma
33\125 316.8000 125f val
26/25 316.9750 1/4-marveltwin comma
14\53 316.9811
3/2 316.9925 1/6-kleisma
75/52 317.0274 1/2-tunbarsma
51\193 317.0984
15/8 317.1153 2/11-kleisma
88\333 317.1171
13/10 317.1349
37\140 317.1429
13/8 317.1805
60\227 317.1807
23\87 317.2414
5/4 317.2627 1/5-kleisma, upper bound of 5-odd-limit diamond tradeoff
13/12 317.3216
32\121 317.3554
41\155 317.4194
15/13 317.4197 1/3-marveltwin comma
9\34 317.6471
25/24 317.6681 1/4-kleisma, virtually DR 10:12:15
22\83 318.0723 83f val
13/9 318.3088 1/2-marveltwin comma, upper bound of 2.3.5.13-subgroup 15-odd-limit diamond tradeoff
125/72 318.3437 1/3-kleisma
13\49 318.3673 49f val
125/104 318.4135 Full tunbarsma
625/432 319.6949 1/2-kleisma
4\15 320.0000 Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone
65/54 320.9764 Full marveltwin comma

* besides the octave

Other tunings

  • DKW (2.3.5): ~2 = 1\1, ~6/5 = 317.1983

Scales

Music

Petr Pařízek
Chris Vaisvil

External links

Retrieved from "https://en.xen.wiki/index.php?title=Kleismic&oldid=155230"