5L 2s/Temperaments: Difference between revisions

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Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L 2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.  
Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L 2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.  


== 5-limit meantone ==
== Meantone ==
{{main| Meantone }}
{{main| Meantone }}
Subgroup: 2.3.5
Subgroup: 2.3.5
Line 64: Line 64:
[[Tuning ranges]]:  
[[Tuning ranges]]:  


* [[diamond monotone]] range: [685.714, 720.000] (7 to 5)
* [[Diamond monotone]] range: [685.714, 720.000] (7 to 5)
* [[diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
* [[Diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
* diamond monotone and tradeoff range: [694.786, 701.955]
* Diamond monotone and tradeoff: [694.786, 701.955]


{{Vals|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
{{Vals|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
Line 152: Line 152:


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/5
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/5
* [[9-odd-limit]]
* [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }}, {{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }}, {{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}]
Line 162: Line 160:


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[diamond monotone]] range: [692.308, 694.737] (26 to 19)
* [[Diamond monotone]] range: [692.308, 694.737] (26 to 19)
* [[diamond tradeoff]] range: [692.353, 701.955]
* [[Diamond tradeoff]] range: [692.353, 701.955]
* diamond monotone and tradeoff range: [692.353, 694.737]
* Diamond monotone and tradeoff: [692.353, 694.737]


Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Line 256: Line 254:


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[diamond monotone]] range: [694.737, 700.000] (19 to 12)
* [[Diamond monotone]] range: [694.737, 700.000] (19 to 12)
* [[diamond tradeoff]] range: [694.786, 701.955]
* [[Diamond tradeoff]] range: [694.786, 701.955]
* diamond monotone and tradeoff range: [694.786, 700.000]
* Diamond monotone and tradeoff: [694.786, 700.000]


Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
Line 289: Line 287:


Tuning ranges:  
Tuning ranges:  
* [[diamond monotone]] range: [694.737, 696.774] (19 to 31)
* [[Diamond monotone]] range: [694.737, 696.774] (19 to 31)
* [[diamond tradeoff]] range: [691.202, 701.955]
* [[Diamond tradeoff]] range: [691.202, 701.955]
* diamond monotone and tradeoff range: [694.737, 696.774]
* Diamond monotone and tradeoff: [694.737, 696.774]


Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Line 310: Line 308:
Mapping: Same as septimal meantone, plus 18 gens = 11/8
Mapping: Same as septimal meantone, plus 18 gens = 11/8


Comma list: 81/80, 126/125, 99/98
Comma list: 81/80, 99/98, 126/125


Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]
Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]
Line 322: Line 320:


Tuning ranges:  
Tuning ranges:  
* [[diamond monotone]] range: [696.774, 700.000] (31 to 12)
* [[Diamond monotone]] range: [696.774, 700.000] (31 to 12)
* [[diamond tradeoff]] range: [691.202, 701.955]
* [[Diamond tradeoff]] range: [691.202, 701.955]
* diamond monotone and tradeoff range: [696.774, 700.000]
* Diamond monotone and tradeoff: [696.774, 700.000]


[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
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== Schismic ==
== Schismic ==
Subgroup: 2.3.5.7.9.11.13.19
Subgroup: 2.3.5.7.11.13.19


Period: 1\1
Period: 1\1
Line 476: Line 474:
Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


{{Val list|legend=1| 41, 53, 94}}
{{Val list|legend=1| 41, 53, 94 }}


== Parapyth ==
== Parapyth ==
Line 568: Line 566:
[[Mapping|Period-generator mapping]]: [&lt;2: 1, 3: 0, 7: -21, 11: -14, 13: -9|, &lt;2: 0, 3: 1, 7: 15, 11: 11, 13: 8|]
[[Mapping|Period-generator mapping]]: [&lt;2: 1, 3: 0, 7: -21, 11: -14, 13: -9|, &lt;2: 0, 3: 1, 7: 15, 11: 11, 13: 8|]


Commas: 169/168, 352/351, 364/363
Comma list: 169/168, 352/351, 364/363


Gencom: [2 3/2; 169/169 352/351 364/363]
Gencom: [2 3/2; 169/169 352/351 364/363]
Line 636: Line 634:
Gencom mapping: [&lt;2: 1, 3: 1, 7: 4|, &lt;2: 0, 3: 1, 7: -2|]
Gencom mapping: [&lt;2: 1, 3: 1, 7: 4|, &lt;2: 0, 3: 1, 7: -2|]


EDOs: 5, 12, 17, 22, 27, 137bc
Vals: 5, 12, 17, 22, 27, 137bc


[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
Line 716: Line 714:
Gencom mapping: [&lt;2: 1, 3: 1, 7: 4, 11: 7|, &lt;2: 0, 3: 1, 7: -2, 11: -6|]
Gencom mapping: [&lt;2: 1, 3: 1, 7: 4, 11: 7|, &lt;2: 0, 3: 1, 7: -2, 11: -6|]


EDOs: 5, 12, 17, 39c, 56c
Vals: 5, 12, 17, 39c, 56c


[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
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|}
|}
<references/></div></div>
<references/></div></div>
[[Mapping|Period-generator mapping]]: [&lt;1 0 -12 6|, &lt;0 1 9 -2|]
[[Mapping|Period-generator mapping]]: [{{val|1 0 -12 6}}, {{val|0 1 9 -2}}]


[[Comma]]s: 64/63, 245/243
[[Comma]]s: 64/63, 245/243
Line 794: Line 792:
[[Wedgie]]: {{wedgie|1 9 -2 12 -6 -30}}
[[Wedgie]]: {{wedgie|1 9 -2 12 -6 -30}}


EDOs: 5, 17, 22, 27, 49
Vals: 5, 17, 22, 27, 49


Badness: 0.0323
Badness: 0.0323

Revision as of 11:16, 25 June 2021

Below are some important rank-2 temperaments with optimal generator size in the diatonic (5L 2s) range (the period is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.

Meantone

Main article: Meantone

Subgroup: 2.3.5

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.239

EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50

Scales (Scala files): Meantone5, Meantone7, Meantone12

Interval table (7-note MOS, 2.3.5.7 POTE tuning)
# Cents[1] Approximate ratios[2]
0 0.00 1/1
1 696.2 3/2
2 192.5 9/8, 10/9
3 888.7 5/3
4 385.0 5/4
5 1081.2 15/8
6 577.4 25/18
  1. ↑ octave-reduced
  2. ↑ 2.3.5, odd limit ≤ 27

Comma list: 81/80

Mapping: [⟨1 0 -4], ⟨0 1 4]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 4]]

Tuning ranges:

  • Diamond monotone range: [685.714, 720.000] (7 to 5)
  • Diamond tradeoff range: [694.786, 701.955] (1/3 comma to Pythagorean)
  • Diamond monotone and tradeoff: [694.786, 701.955]

Optimal ET sequence: 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb

Badness: 0.00736

Flattone

Subgroup: 2.3.5.7.13

Period: 1\1

Optimal (POTE) generator: ~3/2 = 693.7498

EDO generators: 11\19, 15\26, 26\45, 37\64

Scales (Scala files): Flattone12

Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)
#Gens up Cents [1] Approximate ratios[2]
0 0.00 1/1
1 693.7 3/2
2 187.5 9/8, 10/9
3 881.2 5/3
4 375.0 5/4, 16/13
5 1068.7 15/8, 24/13
6 562.5 18/13
7 56.2
8 750.0 20/13
9 243.7 8/7
10 937.5 12/7
11 431.2 9/7
  1. ↑ octave-reduced
  2. ↑ 2.3.5.7.13, odd limit ≤ 27

Comma list: 81/80, 525/512

Mapping: [⟨1 0 -4 17], ⟨0 1 4 -9]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 -9 4 -17 -32]]

Minimax tuning:

[[1 0 0 0⟩, [21/13 0 1/13 -1/13⟩, [32/13 0 4/13 -4/13⟩, [32/13 0 -9/13 9/13⟩]
Eigenmonzos (unchanged intervals): 2, 7/5
[[1 0 0 0⟩, [17/11 2/11 0 -1/11⟩, [24/11 8/11 0 -4/11⟩, [34/11 -18/11 0 9/11⟩]
Eigenmonzos (unchanged intervals): 2, 9/7

Tuning ranges:

Algebraic generator: Squarto, the positive root of 8x2 - 4x - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

Optimal ET sequence: 7, 19, 26, 45

Badness: 0.0386

Septimal meantone

Subgroup: 2.3.5.7

Period: 1\1

Optimal (POTE) generator: 696.495

EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50

Scales (Scala files): Meantone5, Meantone7, Meantone12

Interval table (12-note MOS, 2.3.5.7 POTE tuning)
#Gens up Cents [1] Approximate ratios[2]
0 0.00 1/1
1 696.5 3/2
2 193.0 9/8, 10/9
3 889.5 5/3
4 386.0 5/4
5 1082.5 15/8, 28/15
6 579.0 7/5
7 75.5 21/20, 25/24, 28/27
8 772.0 14/9, 25/16
9 268.5 7/6
10 965.0 7/4
11 461.4 21/16
  1. ↑ octave-reduced
  2. ↑ 2.3.5.7, odd limit ≤ 27

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]]

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
Eigenmonzos (unchanged intervals): 2, 5

Tuning ranges:

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

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

Badness: 0.0137

Meanpop

Subgroup: 2.3.5.7.11

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.434

EDO generators: 29\50, 40\69, 47\81

Mapping: Same as septimal meantone, plus -13 gens = 11/8

Comma list: 81/80, 126/125, 385/384

Mapping: [⟨1 0 -4 -13 24], ⟨0 1 4 10 -13]]

Mapping generator: ~2, ~3

Minimax tuning:

[[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [-3 0 5/2 0 0⟩, [11 0 -13/4 0 0⟩]
Eigenmonzos (unchanged intervals): 2, 5

Tuning ranges:

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x3 + 6x - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

Optimal ET sequence: 12e, 19, 31, 81

Badness: 0.0215

Huygens

Subgroup: 2.3.5.7.11

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.967

EDO generators: 25\43, 43\74

Mapping: Same as septimal meantone, plus 18 gens = 11/8

Comma list: 81/80, 99/98, 126/125

Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]

Mapping generators: ~2, ~3

Minimax tuning:

[[1 0 0 0 0⟩, [25/16 -1/8 0 0 1/16⟩, [9/4 -1/2 0 0 1/4⟩, [21/8 -5/4 0 0 5/8⟩, [25/8 -9/4 0 0 9/8⟩]
Eigenmonzos (unchanged intervals): 2, 11/9

Tuning ranges:

Algebraic generator: Traverse, the positive real root of x4 + 2x - 13, or 696.9529 cents.

Optimal ET sequence: 12, 19e, 31, 105, 136b, 167be, 198be

Badness: 0.0170

Schismic

Subgroup: 2.3.5.7.11.13.19

Period: 1\1

Optimal (POTE) generator: ~3/2 = 702.1044

EDO generators: 24\41, 31\53, 55\94

Scales: Garibaldi12, Garibaldi17

Interval table (29-note MOS, 2.3.5.7.11.13.19 POTE tuning)
#Gens up Cents [1] Approximate ratios[2]
0 0.00 1/1
1 702.10 3/2
2 204.21 9/8
3 906.31 27/16, 32/19
4 408.42
5 1110.52
6 612.63 10/7
7 114.73 15/14, 16/15
8 816.84 8/5
9 318.94 6/5
10 1021.04 9/5
11 523.15 27/20
12 25.25 81/80
13 727.36 32/21
14 229.462 8/7
15 931.57 12/7
16 433.67 9/7
17 1135.77 54/28
18 637.88 13/9
19 139.98 13/12
20 842.09 13/8
21 344.19 11/9, 39/32
22 1046.30 11/6
23 548.40 11/8, 26/19
24 50.51 33/32
25 752.61
26 254.714 22/19
27 956.82 26/15
28 458.92 13/10
  1. ↑ octave-reduced
  2. ↑ 2.3.5.7.11.13.19

Comma list: 225/224, 275/273, 325/324, 385/384, 513/512

Mapping: [⟨2: 1, 3: 0, 5: 15, 7: 25, 11: -33, 13: -28, 19: 9], ⟨2: 0, 3: 1, 5: -8, 7: -14, 11: 23, 13: 20, 19: -3]]

Mapping generators: ~2, ~3

Template:Val list

Parapyth

Subgroup: 2.3.7.11.13

Period: 1\1

Optimal (POTE) generator: ~3/2 = 704.745

EDO generators: 10\17, 17\29, 27\46

Interval table (17-note MOS, 2.3.7.11.13 POTE tuning)
#Gens up Cents [1] Approximate ratios[2]
0 0.00 1/1
1 704.7 3/2
2 209.5 9/8
3 914.2 22/13
4 419.0 14/11
5 1123.7
6 628.5 13/9, (23/16)
7 133.2 13/12, 14/13
8 838.0 13/8
9 342.7 11/9
10 1047.5 11/6
11 552.2 11/8
12 56.9 28/27
13 761.7 14/9
14 266.4 7/6
15 971.2 7/4
16 475.9 21/16
  1. ↑ octave-reduced
  2. ↑ 2.3.7.11.13, odd limit ≤ 27

Period-generator mapping: [<2: 1, 3: 0, 7: -21, 11: -14, 13: -9|, <2: 0, 3: 1, 7: 15, 11: 11, 13: 8|]

Comma list: 169/168, 352/351, 364/363

Gencom: [2 3/2; 169/169 352/351 364/363]

Gencom mapping: [<2: 1, 3: 1, 7: -6, 11: -3, 13: -1|, <2: 0, 3: 1, 7: 15, 11: 11, 13: 8|]

EDOs: 17, 46, 63

RMS error: 0.7541 cents

Archy

Subgroup: 2.3.7

Period: 1\1

Optimal (POTE) generator: ~3/2 = 709.321

EDO generators: 10\17, 13\22, 16\27

Scales: Archy5, Archy7, Archy12

Interval table (7-note MOS, 2.3.7 POTE tuning)
#Gens up Cents [1] Approximate ratios[2]
0 0.00 1/1
1 709.3 3/2
2 218.6 9/8, 8/7
3 927.8 12/7
4 437.3 9/7
5 1146.6 27/14
6 655.9
  1. ↑ octave-reduced
  2. ↑ 2.3.7, odd limit ≤ 27

Period-generator mapping: [<2: 1 3: 2, 7: 2|, <2: 0, 3: -1, 7: 2|]

Comma: 64/63

Gencom: [2 3/2; 64/63]

Gencom mapping: [<2: 1, 3: 1, 7: 4|, <2: 0, 3: 1, 7: -2|]

Vals: 5, 12, 17, 22, 27, 137bc

RMS error: 1.856 cents

Supra

Subgroup: 2.3.7.11

Period: 1\1

Optimal (POTE) generator: ~3/2 = 707.192

EDO generators: 10\17, 13\22, 23\39

Scales: Supra7, Supra12

Interval table (12-note MOS, 2.3.7.11 POTE tuning)
#Gens up Cents [1] Approximate ratios[2]
0 0.00 1/1
1 707.2 3/2
2 214.4 9/8, 8/7
3 921.6 12/7
4 428.8 9/7, 14/11
5 1136.0 27/14
6 643.2 16/11
7 150.3 12/11
8 857.5 18/11
9 364.7
10 1071.9
11 579.1
  1. ↑ octave-reduced
  2. ↑ 2.3.7.11, odd limit ≤ 27

Period-generator mapping: [<2: 1, 3: 0, 7: 6, 11: 13|, <2: 0, 3: 1, 7: -2, 11: -6|]

Commas: 64/63, 99/98

Gencom: [2 3/2; 64/63 99/98]

Gencom mapping: [<2: 1, 3: 1, 7: 4, 11: 7|, <2: 0, 3: 1, 7: -2, 11: -6|]

Vals: 5, 12, 17, 39c, 56c

RMS error: 1.977 cents

Superpyth

Subgroup: 2.3.5.7

Period: 1\1

Optimal (POTE) generator: ~3/2 = 710.291

EDO generators: 13\22, 18\27, 31\49

Interval table (12-note MOS, 2.3.5.7 POTE tuning)
#Gens up Cents [1] Approximate ratios[2]
0 0.00 1/1
1 710.3 3/2
2 220.6 9/8, 8/7
3 930.9 12/7
4 441.2 9/7
5 1151.5
6 661.7 40/27
7 172.0 10/9
8 882.3 5/3
9 392.6 5/4
10 1102.9 15/8
11 613.2 10/7
  1. ↑ octave-reduced
  2. ↑ 2.3.5.7, odd limit ≤ 27

Period-generator mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Commas: 64/63, 245/243

Wedgie: ⟨⟨1 9 -2 12 -6 -30]]

Vals: 5, 17, 22, 27, 49

Badness: 0.0323

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