User:FloraC/Sandbox: Difference between revisions

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* <s>Sengic family</s>
* <s>Sengic family</s>
* [[Schismatic family]]
* [[Schismatic family]]
* [[Kleismic family]]
* <s>Kleismic family</s>
* [[Kleismic rank three family]]
* <s>Kleismic rank three family</s>
* [[Hemimean clan]]
* [[Hemimean clan]]
* [[Hemimean family]]
* [[Hemimean family]]

Revision as of 05:48, 4 September 2023

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 4 10 4 13 12 ]]

Optimal tuning (CTE): ~3/2 = 696.9521

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness: 0.0137

5-limit rank-2 temperaments by TE simple badness

Breed's simple badness.

Junk temperaments

High-badness temperaments

Temperament Complexity Error (¢) Badness (moct) Mapping Comma list
Yo .367 26.0 7.95 [⟨1 0 -1], ⟨0 1 -2]] 10/9
Antitonic .395 27.0 8.89 [⟨2 3 0], ⟨0 0 1]] 9/8
Father .443 13.2 4.87 [⟨1 0 4], ⟨0 1 -1]] 16/15
Bug .602 11.6 5.81 [⟨1 0 0], ⟨0 2 3]] 27/25
Supersharp 1.06 7.94 7.01 [⟨2 0 -5], ⟨0 1 3]] 800/729
Laconic 1.24 6.52 6.74 [⟨1 1 1], ⟨0 3 7]] 2187/2000
Lafayette 1.30 7.00 7.60 [⟨1 1 2], ⟨0 5 3]] 3456/3125
Symbolic 1.31 6.09 6.66 [⟨1 3 4], ⟨0 -5 -6]] 2048/1875
Sixix 1.37 4.57 5.22 [⟨1 3 2], ⟨0 -4 1]] 3125/2916
Uncle 1.40 7.53 8.80 [⟨1 0 12], ⟨0 1 -6]] 4096/3645
Whitewood 1.42 4.20 4.95 [⟨7 11 0], ⟨0 0 1]] 2187/2048
1.66 4.13 5.72 [⟨1 0 5], ⟨0 3 -5]] 32768/30375

Main sequence

Temperament Complexity Error (¢) Badness (moct) Mapping Comma list
Dicot .521 7.09 3.08 [⟨1 1 2], ⟨0 2 1]] 25/24
Meantone .711 1.58 .937 [⟨1 0 -4], ⟨0 1 4]] 81/80
Mavila .795 6.06 4.02 [⟨1 0 7], ⟨0 1 -3]] 135/128
Augmented .894 2.40 1.79 [⟨3 0 7], ⟨0 1 0]] 128/125
Porcupine .960 2.68 2.14 [⟨1 2 3], ⟨0 -3 -5]] 250/243
Blackwood 1.02 4.63 3.93 [⟨5 8 0], ⟨0 0 1]] 256/243
Diminished 1.05 3.10 2.73 [⟨4 0 3], ⟨0 1 1]] 648/625
Srutal 1.22 .835 .852 [⟨2 0 11], ⟨0 1 -2]] 2048/2025
Magic 1.40 1.11 1.29 [⟨1 0 2], ⟨0 5 1]] 3125/3072
Hanson 1.55 .274 .353 [⟨1 0 1], ⟨0 6 5]] 15625/15552
Ripple 1.56 2.82 3.66 [⟨1 2 3], ⟨0 -5 -8]] 6561/6250
Negri 1.58 1.69 2.23 [⟨1 2 2], ⟨0 -4 3]] 16875/16384
Tetracot 1.61 .900 1.21 [⟨1 1 1], ⟨0 4 9]] 20000/19683
Superpyth 1.70 2.11 2.99 [⟨1 0 -12], ⟨0 1 9]] 20480/19683
Helmholtz 1.79 .0570 .0851 [⟨1 0 15], ⟨0 1 -8]] 32805/32768
Wesley 1.91 2.75 4.37 [⟨1 4 3], ⟨0 -7 -2]] 78125/73728
Sensipent 1.97 .356 .584 [⟨1 6 8], ⟨0 7 9]] 78732/78125
Stump 2.02 1.88 3.16 [⟨1 0 6], ⟨0 3 -7]] 273375/262144
Passion 2.02 1.57 2.64 [⟨1 2 2], ⟨0 -5 4]] 262144/253125
Doublewide 2.06 2.00 3.43 [⟨2 1 3], ⟨0 4 3]] 390625/373248
Würschmidt 2.29 .262 .499 [⟨1 7 3], ⟨0 8 1]] 393216/390625
Amity 2.29 .140 .268 [⟨1 3 6], ⟨0 -5 -13]] 1600000/1594323
Valentine 2.34 .736 1.44 [⟨1 1 2], ⟨0 9 5]] 1990656/1953125
Immunity 2.40 1.03 2.06 [⟨1 0 -8], ⟨0 2 13]] 1638400/1594323
Shibboleth 2.42 1.24 2.50 [⟨1 4 5], ⟨0 -9 -10]] 1953125/1889568
Compton 2.44 .504 1.02 [⟨12 19 0], ⟨0 0 1]] 531441/524288
Orson 2.44 .215 .438 [⟨1 0 3], ⟨0 7 -3]] 2109375/2097152
Unicorn 2.52 .725 1.52 [⟨1 2 3], ⟨0 -8 -13]] 1594323/1562500
Mynic 2.60 1.10 2.37 [⟨1 9 9], ⟨0 -10 -9]] 10077696/9765625
Ampersand 2.78 .594 1.38 [⟨1 1 3], ⟨0 6 -7]] 34171875/33554432
Fifive 2.91 .643 1.56 [⟨2 2 3], ⟨0 5 7]] 9765625/9565938
Misty 2.99 .308 .767 [⟨3 0 26], ⟨0 1 -4]] 67108864/66430125
Gravity 2.99 .269 .669 [⟨1 5 12], ⟨0 -6 -17]] 129140163/128000000
Rodan 3.10 .433 1.12 [⟨1 1 -1], ⟨0 3 17]] [20 -17 3⟩
3.11 .757 1.96 [⟨1 0 -23], ⟨0 1 16]] [-23 16 -1⟩
Mabila 3.32 .488 1.35 [⟨1 6 1], ⟨0 -10 3]] [28 -3 -10⟩
Parakleismic 3.47 .0798 .231 [⟨1 5 6], ⟨0 -13 -14]] [8 14 -13⟩
Quartonic 3.48 .214 .621 [⟨1 2 3], ⟨0 -11 -18]] [3 -18 11⟩
Escapade 3.60 .138 .414 [⟨1 2 2], ⟨0 -9 7]] [32 -7 -9⟩
Ditonic 3.68 .258 .792 [⟨1 6 3], ⟨0 -13 -2]] [-27 -2 13⟩
Vishnuzmic 3.71 .0471 .145 [⟨2 4 5], ⟨0 -7 -3]] [23 6 -14⟩
Vulture 3.81 .0576 .183 [⟨1 0 -6], ⟨0 4 21]] [24 -21 4⟩
Trisedodge 3.87 .336 1.08 [⟨5 1 7], ⟨0 3 2]] [19 10 -15⟩
4.00 .157 .524 [⟨1 4 -1], ⟨0 -8 11]] [-36 11 8⟩

Tunings to be reviewed

5-limit Prime-Optimized Tunings
Weight-skew\Order Chebyshevian Euclidean Manhattan
Tenney CTC
~3/2 = 696.5784¢
Eigenmonzo basis: 2.5 (1/4-comma tuning)		 CTE
~3/2 = 697.2143¢		 CTOP
~3/2 = 698.0197¢
Tenney-Weil CTWC
~3/2 = 696.5784¢
Eigenmonzo basis: 2.5 (1/4-comma tuning)		 CTWE
~3/2 = 696.6512¢		 CTWOP
~3/2 = 696.5784¢
Eigenmonzo basis: 2.5 (1/4-comma tuning)
Equilateral CEC
~3/2 = 696.5784¢
Eigenmonzo basis: 2.5 (1/4-comma tuning)		 CEE
~3/2 = 696.8947¢
Eigenmonzo basis: 2.1875 (4/17-comma tuning)		 CEOP
~3/2 = 697.6537¢
Eigenmonzo basis: 2.15 (1/5-comma tuning)
Equilateral-Weil CEWC
~3/2 = 696.5784¢
Eigenmonzo basis: 2.5 (1/4-comma tuning)		 CEWE
~3/2 = 696.4534¢
Eigenmonzo basis: 2.48828125/3 (11/43-comma tuning)		 CEWOP
~3/2 = 696.5784¢
Eigenmonzo basis: 2.5 (1/4-comma tuning)
Benedetti CBC
~3/2 = 696.5784¢
Eigenmonzo basis: 2.5 (1/4-comma tuning)		 CBE
~3/2 = 697.3738¢
Eigenmonzo basis: 2.[0 25 36⟩ (36/169-comma tuning)		 CBOP
~3/2 = 698.1598¢
Eigenmonzo basis: 2.30375 (3/17-comma tuning)
Benedetti-Weil CBWC
~3/2 = 696.5784¢
Eigenmonzo basis: 2.5 (1/4-comma tuning)		 CBWE
~3/2 = 696.7868¢
Eigenmonzo basis: 2.[0 5 31⟩ (31/129-comma tuning)		 CBWOP
~3/2 = 696.5784¢
Eigenmonzo basis: 2.5 (1/4-comma tuning)
7-limit Prime-Optimized Tunings
Weight-skew\Order Chebyshevian Euclidean Manhattan
Tenney CTC
~3/2 = 696.8826¢
Eigenmonzo basis: 2.7		 CTE
~3/2 = 696.9521¢		 CTOP
~3/2 = 696.6458¢
Tenney-Weil CTWC
~3/2 = 696.8826¢
Eigenmonzo basis: 2.7		 CTWE
~3/2 = 696.6562¢		 CTWOP
~3/2 = 696.8826¢
Eigenmonzo basis: 2.7
Equilateral CEC
~3/2 = 696.8826¢
Eigenmonzo basis: 2.7		 CEE
~3/2 = 696.8843¢
Eigenmonzo basis: 2.[0 1 4 10⟩		 CEOP
~3/2 = 697.3437¢
Eigenmonzo basis: 2.21
Equilateral-Weil CEWC
~3/2 = 696.8826¢
Eigenmonzo basis: 2.7		 CEWE
~3/2 = 696.7248¢
Eigenmonzo basis: 2.4117715/9		 CEWOP
~3/2 = 696.8826¢
Eigenmonzo basis: 2.7
Benedetti CBC
~3/2 = 696.8826¢
Eigenmonzo basis: 2.7		 CBE
~3/2 = 697.0147¢
Eigenmonzo basis: 2.[0 1225 1764 2250⟩		 CBOP
~3/2 = 697.8422¢
Eigenmonzo basis: 2.750141
Benedetti-Weil CBWC
~3/2 = 696.8826¢
Eigenmonzo basis: 2.7		 CBWE
~3/2 = 696.6306¢
Eigenmonzo basis: 2.[0 -3290 3171 7215⟩		 CBWOP
~3/2 = 696.8826¢
Eigenmonzo basis: 2.7

Temperament pages

Note:

  1. Order: subgroup, comma list, (sval) mapping, (sval) mapping generators, gencom mapping, gencom, lattice basis, wedgie, optimal tuning (CTE), minimax tuning, tuning ranges, algebraic generator, Optimal ET sequence, badness, complexity spectrum, and others.
  2. Comma list should show the simplest commas sufficient to define the temperament, stated in Normal lists #Normal interval list.
  3. Mapping generators should show all the ratios as used in the mapping, including the period.
  4. Since minimax tunings are based on tonality diamond, it should explicitly state the odd limit, or a diamond function of ratios.
  5. For subgroup temperaments, "mapping" becomes "sval mapping", add "gencom mapping" and "gencom". If TE is TE is TE (sic), simply show "CTE", otherwise show "subgroup CTE" or "inharmonic CTE" instead.

Get a family for:

  • Ripple (3 different 7-limit extensions) done
  • Smate (2 different 7-limit extensions) done
  • Passion (4 different 7-limit extensions, 3 strong and 1 weak) done
  • Undim done
  • Quintaleap done
  • Quindromeda done
  • Parakleismic (many reasonable but unnamed 7-limit extensions)
  • Schismatic rank three family (perhaps)

Who's next?

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-49 31⟩ [⟨31 49]] +1.63 1.64 4.22
2.3.5 81/80, 393216/390625 [⟨31 49 72]] +0.98 1.63 4.20
2.3.5.7 81/80, 126/125, 1029/1024 [⟨31 49 72 87]] +0.83 1.43 3.70
2.3.5.7.11 81/80, 99/98, 121/120, 126/125 [⟨31 49 72 87 107]] +1.21 1.49 3.84

Scale tree

6-tone
1L 5s, 2L 4s, 3L 3s, 4L 2s, 5L 1s
7-tone
1L 6s, 2L 5s, 3L 4s, 4L 3s, 5L 2s, 6L 1s
8-tone
1L 7s, 2L 6s, 3L 5s, 4L 4s, 5L 3s, 6L 2s, 7L 1s
9-tone
1L 8s, 2L 7s, 3L 6s, 4L 5s, 5L 4s, 6L 3s, 7L 2s, 8L 1s
10-tone
1L 9s, 2L 8s, 3L 7s, 4L 6s, 5L 5s, 6L 4s, 7L 3s, 8L 2s, 9L 1s
11-tone
1L 10s, 2L 9s, 3L 8s, 4L 7s, 5L 6s, 6L 5s, 7L 4s, 8L 3s, 9L 2s, 10L 1s
12-tone
1L 11s, 2L 10s, 3L 9s, 4L 8s, 5L 7s, 6L 6s, 7L 5s, 8L 4s, 9L 3s, 10L 2s, 11L 1s

(Bounded by branch depth = 7)

Generator Cents L s L/s Comments
4\7 685.714 1 1 1.000
27\47 689.362 7 6 1.167
23\40 690.000 6 5 1.200
42\73 690.411 11 9 1.222
19\33 690.909 5 4 1.250
53\92 691.304 14 11 1.273
34\59 691.525 9 7 1.286
49\85 691.765 13 10 1.300
15\26 692.308 4 3 1.333
56\97 692.784 15 11 1.364
41\71 692.958 11 8 1.375
67\116 693.103 18 13 1.385
26\45 693.333 7 5 1.400
63\109 693.578 17 12 1.417
37\64 693.750 10 7 1.429
48\83 693.976 13 9 1.444
11\19 694.737 3 2 1.500 L/s = 3/2
51\88 695.455 14 9 1.556
40\69 695.652 11 7 1.571
69\119 695.798 19 12 1.583
29\50 696.000 8 5 1.600
66\131 696.183 21 13 1.615 Golden meantone
47\81 696.296 13 8 1.625
65\112 696.429 18 11 1.636
18\31 696.774 5 3 1.667 Meantone is in this region
61\105 697.143 17 10 1.700
43\74 697.297 12 7 1.714
68\117 697.436 19 11 1.727
25\43 697.674 7 4 1.750
57\98 697.959 16 9 1.778
32\55 698.182 9 5 1.800
39\67 698.507 11 6 1.833
7\12 700.000 2 1 2.000 Basic diatonic
(Generators smaller than this are proper)
38\65 701.539 11 5 2.200
31\53 701.887 9 4 2.250
55\94 702.128 16 7 2.286
24\41 702.409 7 3 2.333
65\111 702.703 19 8 2.375
41\70 702.857 12 5 2.400
58\99 703.030 17 7 2.428
17\29 703.448 5 2 2.500
61\104 703.846 18 7 2.571
44\75 704.000 13 5 2.600
71\121 704.132 21 8 2.625 Golden neogothic
27\46 704.348 8 3 2.667
64\109 704.587 19 7 2.714
37\63 704.762 11 4 2.750
47\80 705.000 14 5 2.800
10\17 705.882 3 1 3.000 L/s = 3/1
43\73 706.849 13 4 3.250
33\56 707.143 10 3 3.333
56\95 707.368 17 5 3.400
23\39 707.692 7 2 3.500
59\100 708.000 18 5 3.600
36\61 708.197 11 3 3.667
49\83 708.434 15 4 3.750
13\22 709.091 4 1 4.000 Archy is in this region
42\71 709.859 13 3 4.333
29\49 710.204 9 2 4.500
45\76 710.526 14 3 4.667
16\27 711.111 5 1 5.000
35\59 711.864 11 2 5.500
19\32 712.500 6 1 6.000
22\37 713.514 7 1 7.000
3\5 720.000 1 0 → inf
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