Kleismic: Difference between revisions

Revision as of 15:13, 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.

For technical data, see Kleismic family #Hanson.

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

* in 2.3.5.13-subgroup CTE tuning

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
	f⁶ + 2f⁵ - 8 = 0	317.1496	1–3–5 equal-beating tuning, DR 3:4:5
	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
	f⁶ - 2f⁵ + 2 = 0	317.9593	1–3–5 equal-beating tuning, DR 4:5:6, close to 2/7-kleisma
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

@@ Line 162: / Line 162: @@
 | 316.9925
 | 1/6-kleisma
-|-
-|
-| 3''g''<sup>3</sup> + 4''g'' - 10 = 0
-| 317.0010
-| [[Delta-rational chord|DR]] 13:15:18
 |-
 |
@@ Line 199: / Line 194: @@
 |-
 |
-| ''g''<sup>6</sup> + 2''g''<sup>5</sup> - 8 = 0
+| ''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
@@ Line 239: / Line 234: @@
 |-
 |
-| [[26/15]]
+| [[15/13]]
 | 317.4197
 | 1/3-marveltwin comma
-|-
-|
-| 3''g''<sup>3</sup> - ''g'' - 4 = 0
-| 317.5679
-| [[Delta-rational chord|DR]] 9:13:15
 |-
 | [[34edo|9\34]]
@@ Line 259: / Line 249: @@
 |-
 |
-| ''g''<sup>6</sup> - 2''g''<sup>5</sup> + 2 = 0
+| ''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
@@ Line 307: / Line 297: @@
 === Other tunings ===
 * [[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 ==