User:FloraC/Sandbox

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Temperament pages

Databoxes has been canceled, but the cleanup will continue

Note:

  1. Order: subgroup, comma list, (sval) mapping, (sval) mapping generators, gencom mapping, gencom, map to lattice, lattice basis, wedgie, POTE generator(s), minimax tuning, tuning ranges, algebraic generator, vals, 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 tuning are based on tonality diamond, it should explicitly state the odd limit, or a diamond function of ratios.
  5. Replace "EDOs" with Template:Val list.
  6. For subgroup temperaments, "mapping" becomes "sval mapping", add "gencom mapping" and "gencom". If TE is TE is TE (sic), simply show "POTE", otherwise show "POL2" or "POT2" instead of "POTE".

Get a family for:

  • Ripple (3 different 7-limit extensions) done
  • Smate (2 different 7-limit extensions) done
  • Parakleismic (2 different 7-limit extensions) canceled, not strong extensions
  • Superpyth (2 different 7-limit extensions) canceled, not strong extensions

Who's next?

  • Meantone family
  • Archytas clan
  • Father family
  • Trienstonic clan
  • Septisemi temperaments
  • Archytas family
  • Slendro clan
  • Semiphore family
  • Marvel temperaments
  • Marvel family
  • Mint temperaments
  • Mint family
  • Gamelismic clan
  • Gamelismic family
  • Jubilismic clan
  • Jubilismic family
  • Didymus rank three family
  • Kleismic family
  • Kleismic rank three family
  • Shibboleth family
  • Keemic temperaments
  • Keemic family
  • Schismatic family
  • Hemimean clan
  • Hemimean family
  • Luna family
  • Canousmic temperaments
  • Canou family
  • Starling temperaments
  • Starling family
  • Sensipent family
  • Sensamagic clan
  • Sensamagic family
  • Magic family
  • Unicorn family
  • Trisedodge family
  • Sycamore family
  • Rastmic temperaments
  • Semicomma family
  • Orwellismic temperaments
  • Hemimage temperaments
  • Hemimage family
  • Porwell temperaments
  • Porwell family
  • Hemifamity temperaments
  • Hemifamity family
  • Porcupine family
  • Tetracot family
  • Misty family
  • Diaschismic family
  • Amity family
  • Lehmerismic temperaments
  • Kalismic temperaments
  • Ragismic family
  • Landscape family
  • Dimcomp family
  • Mirkwai family
  • Breed family
  • Breedsmic temperaments
  • Ragismic microtemperaments
  • Escapade family
  • Gravity family
  • Cataharry family
  • Cataharry temperaments

As a regular temperament

3-limit

Subgroup: 2.3

Comma list: [-49 31⟩

Mapping: ⟨31 49]

TE octave stretch: +1.63¢

TE absolute error (¢): 1.64

TE relative error (%): 4.22

5-limit

Subgroup: 2.3.5

Comma list: 81/80, 393216/390625

Mapping: ⟨31 49 72]

TE octave stretch: +0.98¢

TE absolute error (¢): 1.63

TE relative error (%): 4.20

7-limit

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125, 1029/1024

Mapping: ⟨31 49 72 87]

TE octave stretch: +0.83¢

TE absolute error (¢): 1.43

TE relative error (%): 3.70

Septimal meantone

Deutsch

Main article: Meantone
See also: Wikipedia: Septimal meantone temperament

The 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, 7/5, C-F#, the tritone, and 21/16, C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas 126/125 and 225/224 and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.

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 ]]

POTE generator: ~3/2 = 696.495

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
Eigenmonzos: 2, 5

Tuning ranges:

  • valid range: [694.737, 700.000] (11\19 to 7\12)
  • nice range: [694.786, 701.955]
  • strict range: [694.786, 700.000]

Algebraic generator: Cybozem, the real root of 15x3 - 10x2 - 18, 503.4257 cents. The recurrence converges quickly.

Template:Val list

Badness: 0.0137

Scales: meantone5, meantone7, meantone12

Archytas

Main article: Archytas

Subgroup: 2.3.5.7

Comma list: 64/63

Mapping: [⟨1 0 0 6], ⟨0 1 0 -2], ⟨0 0 1 0]]

Mapping generators: ~2, ~3, ~5

Map to lattice: [⟨0 1 0 -2], ⟨0 0 1 0]]

Lattice basis:

3/2 length = 1.0508, 5/4 length = 2.3219
Angle (3/2, 5/4) = 90 degrees

POTE generators: ~3/2 = 709.3213, ~5/4 = 393.3747

Minimax tuning:

[[1 0 0 0⟩, [2 1/3 0 -1/3⟩, [2 -2/3 1 -1/3⟩, [2 -2/3 0 2/3⟩]
Eigenmonzos: 2, 6/5, 7/5
[[1 0 0 0⟩, [3/2 1/2 0 -1/4⟩, [3/2 -1/2 1 -1/4⟩, [3 -1 0 1/2⟩]
Eigenmonzos: 2, 6/5, 9/7

Template:Val list

Scales: archytas12, archytas12synch

Scale tree

(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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