User:FilterNashi/Temperaments and tuning methods

Interval information
Ratio	7378945280/7341576489
Factorization	28 × 3-2 × 5 × 78 × 13-8
Monzo	[8 -2 1 8 0 -8⟩
Size in cents	8.78967¢
Name	peninsula comma
FJS name	[math]\displaystyle{ \text{AA3}^{5,7,7,7,7,7,7,7,7}_{13,13,13,13,13,13,13,13} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	65.5542
Weil norm (log2 max(n, d))	65.5615
Wilson norm (sopfr(nd))	187
Comma size	small
	Open this interval in xen-calc

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peninsula and bandao

7378945280/7341576489 is called the peninsula comma, it's the difference between a stack of 8 14/13s plus a 10/9 and 2/1.

Tempering out this comma in 2.3.5.7.13 subgroup leads to the rank-4 peninsula temperament.

Tempering out this comma also leads to a 1L 8s scale with a period of 2/1, which is called the bandao scale. We can explore the peninsula clan with identifying the intervals within this scale as other intervals.

One of the obvious options here is tempering out 81/80, or 28812/28561 which leads to the same clan. It identifies a small step as a quarter of a 4/3, a large step as 9/8, so four small steps is 4/3 and L+4s is 3/2. We can also tempering out 10985/10976, identifies three small steps as 5/4. Tempering out 196/195 indentifies two small steps as 15/13. Tempering out 105/104 identifies one small step as 16/15. Only 19edo tempering all of them out at 2.3.5.7.13 subgroup.

Dianeutral

2.3.5.11 subgroup

clans: diaschismic, neutral

comma basis: 243/242, 2048/2025

edos: 34, 24, 10, 58, 14, 92[+11], 44[+11], 82[+5], 126[+11], 150[+5, +11], 106[+5], 140[+5, +11], 160[+11], 208[+5, +11]

edo join: 34 & 24

mapping: [<2 0 11 -1], <0 2 -4 5]]

generators: ~45/32, ~96/55

WE tuning: 599.208 950.822

WE errors: -1.584, -0.311, 1.688, 3.583

CWE tuning: 600.000 952.030

CWE errors: 0.000, 2.105, 5.566, 8.832

badness(sintel): 1.110

all the patent val: 10 14 24 34 58

2.3.5.11.13 subgroup

clans: island

comma basis: 144/143, 352/351, 832/825

edos: 34, 24, 10, 58, 92[+11, +13], 14, 44[+11], 82[+5, +13], 126[+11, +13], 150[+5, +11, +13], 106[+5, +13], 140[+5, +11, +13], 160[+11, +13], 208[+5, +11, +13]

edo join: 34 & 24

mapping: [<2 0 11 -1 9], <0 2 -4 5 -1]]

generators: ~45/32, ~26/15

WE tuning: 599.165 950.779

WE errors: -1.671, -0.397, 1.383, 3.412, 1.176

CWE tuning: 600.000 952.074

CWE errors: 0.000, 2.194, 5.389, 9.054, 7.398

badness(sintel): 0.591

all the patent val: 10 14 24 34 58

2.3.5.11.13.17 subgroup

clans: semitonic, srutal archagall

comma basis: 144/143, 136/135, 352/351, 832/825

edos: 34, 24, 10, 58, 14, 92[+11, +13], 44[+11], 82[+5, +13]

edo join: 34 & 24

mapping: [<2 0 11 -1 9 5], <0 2 -4 5 -1 2]]

generators: ~45/32, ~26/15

WE tuning: 599.309 951.088

WE errors: -1.381, 0.221, 1.737, 4.812, 2.169, -6.233

CWE tuning: 600.000 952.149

CWE errors: 0.000, 2.343, 5.091, 9.426, 7.324, -0.658

badness(sintel): 0.570

all the patent val: 10 14 24 34 58

Restriction

Semineutral

subgroup: 2.3.11.17

clans: neutral, semitonic

comma basis: 243/242, 289/288

edos: 24, 10, 14, 34, 58, 38, 82, 62, 106, 6[+11], 130, 86[-11, -17], 44[+11], 154

edo join: 24 & 10

mapping: [<2 0 -1 5], <0 2 5 2]]

generators: ~17/12, ~153/88

WE tuning: 600.224 950.678

WE errors: 0.447, -0.600, 1.847, -2.482

CWE tuning: 600.000 950.476

CWE errors: 0.000, -1.003, 1.061, -4.004

badness(sintel): 0.771

all the patent val: 10 14 24 34 38 48 58 62 72 82 96 106 120 130 154

(Dianeutral)

subgroup: 2.3.5.11.17

clans: diaschismic, srutal archagall

comma basis: 136/135, 256/255, 243/242

edos: 34, 24, 10, 58, 14, 92[+11], 44[+11], 82[+5]

edo join: 34 & 24

mapping: [<2 0 11 -1 5], <0 2 -4 5 2]]

generators: ~45/32, ~55/32

WE tuning: 599.376 951.142

WE errors: -1.247, 0.330, 2.256, 5.018, -5.789

CWE tuning: 600.000 952.085

CWE errors: 0.000, 2.215, 5.346, 9.107, -0.785

badness(sintel): 0.771

all the patent val: 10 14 24 34 58