User:FloraC/Sandbox: Difference between revisions

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== Temperament pages ==
Databoxes has been canceled, but the cleanup will continue
Note:
# Order: subgroup, [[comma list]], [[mapping]], mapping generators, gencom mapping, [[gencom]], map to lattice, lattice basis, [[wedgie]], [[minimax tuning]], [[tuning ranges]], [[algebraic generator]], vals, [[badness]], [[complexity spectrum]].
# Comma list shows the simplest commas sufficient to define the temperament, stated in [[Normal lists #Normal interval list]].
# Mapping generators should show all the ratios as used in the mapping, including the period.
# Minimax tuning are based on tonality diamond, so it should explicitly state the ''odd'' limit, or a diamond function of ratios.
# Use [[Template:Val list]].
# 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:
* <s>Ripple (3 different 7-limit extensions)</s>
* <s>Smate (2 different 7-limit extensions)</s>
* Maybe [[parakleismic]] (2 different 7-limit extensions)
* Maybe [[superpyth]] (2 different 7-limit extensions)
Who's next?
* <s>Meantone family</s>
* <s>Archytas clan</s>
* <s>Father family</s>
* <s>Trienstonic clan</s>
* <s>Septisemi temperaments</s>
* <s>Archytas family</s>
* <s>Slendro clan</s>
* <s>Semiphore family</s>
* <s>Marvel temperaments</s>
* <s>Marvel family</s>
* <s>Mint temperaments</s>
* <s>Mint family</s>
* [[Gamelismic clan]]
* <s>Gamelismic family</s>
* <s>Jubilismic clan</s>
* <s>Jubilismic family</s>
* <s>Didymus rank three family</s>
* <s>Kleismic family</s>
* [[Kleismic rank three family]]
* <s>Shibboleth family</s>
* <s>Keemic temperaments</s>
* <s>Keemic family</s>
* <s>Schismatic family</s>
* <s>Hemimean clan</s>
* <s>Hemimean family</s>
* <s>Luna family</s>
* <s>Canousmic temperaments</s>
* <s>Canou family</s>
* [[Starling temperaments]]
* [[Starling family]]
* <s>Sensipent family</s>
* [[Sensamagic clan]]
* [[Sensamagic family]]
* [[Magic family]]
* [[Unicorn family]]
* [[Trisedodge family]]
== Septimal meantone ==
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
{{main| 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]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]
Mapping generators: ~2, ~3
{{Multival|legend=1| 1 4 10 4 13 12 }}
[[POTE generator]]: ~3/2 = 696.495
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo]]s: 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 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.
{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
[[Badness]]: 0.0137
Scales: [[meantone5]], [[meantone7]], [[meantone12]]
== Archytas ==
{{main| Archytas }}
Subgroup: 2.3.5.7
[[Comma list]]: 64/63
[[Mapping]]: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
Mapping generators: ~2, ~3, ~5
Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
Lattice basis:
: 3/2 length = 1.0508, 5/4 length = 2.3219
: Angle (3/2, 5/4) = 90 degrees
[[POTE generator]]s: ~3/2 = 709.3213, ~5/4 = 393.3747
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 1/3 0 -1/3 }}, {{monzo| 2 -2/3 1 -1/3 }}, {{monzo| 2 -2/3 0 2/3 }}]
: [[Eigenmonzo]]s: 2, 6/5, 7/5
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 }}, {{monzo| 3/2 -1/2 1 -1/4 }}, {{monzo| 3 -1 0 1/2 }}]
: [[Eigenmonzo]]s: 2, 6/5, 9/7
{{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 49 }}
Scales: [[archytas12]], [[archytas12synch]]
== Scale tree ==
== Scale tree ==
Version 1 (bounded by branch depth = 7)
(Bounded by branch depth = 7)


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 96: Line 220:
| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
|-
|-
| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden parapyth
| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden neogothic
|-
|-
| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 ||
| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 ||
Line 140: Line 264:
| 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
| 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
|}
|}
Version 2 (bounded by integer limit = 10)
{| class="wikitable center-all"
! colspan="10" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
| 4\7 || || || || || || || || || || 685.714 || 1 || 1 || 1.000 ||
|-
| || || || || || || || || || 39\68 || 688.235 || 10 || 9 || 1.111 ||
|-
| || || || || || || || || 35\61 || || 688.525 || 9 || 8 || 1.125 ||
|-
| || || || || || || || 31\54 || || || 688.889 || 8 || 7 || 1.143 ||
|-
| || || || || || || 27\47 || || || || 689.362 || 7 || 6 || 1.167 ||
|-
| || || || || || 23\40 || || || || || 690.000 || 6 || 5 || 1.200 ||
|-
| || || || || 19\33 || || || || || || 690.909 || 5 || 4 || 1.250 ||
|-
| || || || || || 34\59 || || || || || 691.525 || 9 || 7 || 1.286 ||
|-
| || || || 15\26 || || || || || || || 692.308 || 4 || 3 || 1.333 ||
|-
| || || || || 26\45 || || || || || || 693.333 || 7 || 5 || 1.400 ||
|-
| || || || || || 37\64 || || || || || 693.750 || 10 || 7 || 1.429 ||
|-
| || || 11\19 || || || || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
|-
| || || || || 29\50 || || || || || || 696.000 || 8 || 5 || 1.600 || Golden meantone
|-
| || || || 18\31 || || || || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region
|-
| || || || || 25\43 || || || || || || 697.674 || 7 || 4 || 1.750 ||
|-
| || || || || ||32\55 || || || || || 698.182 || 9 || 5 || 1.800 ||
|-
| || 7\12 || || || || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)
|-
| || || || || || 31\53 || || || || || 701.887 || 9 || 4 || 2.250 ||
|-
| || || || || 24\41 || || || || || || 702.409 || 7 || 3 || 2.333 ||
|-
| || || || 17\29 || || || || || || || 703.448 || 5 || 2 || 2.500 ||
|-
| || || || || 27\46 || || || || || || 704.348 || 8 || 3 || 2.667 || Golden parapyth
|-
| || || 10\17 || || || || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
|-
| || || || || || 33\56 || || || || || 707.143 || 10 || 3 || 3.333 ||
|-
| || || || || 23\39 || || || || || || 707.692 || 7 || 2 || 3.500 ||
|-
| || || || 13\22 || || || || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region
|-
| || || || || || 29\49 || || || || || 710.204 || 9 || 2 || 4.500 ||
|-
| || || || || 16\27 || || || || || || 711.111 || 5 || 1 || 5.000 ||
|-
| || || || || || 19\32 || || || || || 712.500 || 6 || 1 || 6.000 ||
|-
| || || || || || || 22\37 || || || || 713.514 || 7 || 1 || 7.000 ||
|-
| || || || || || || || 25\42 || || ||714.286 || 8 || 1 || 8.000 ||
|-
| || || || || || || || || 28\47 || ||714.894 || 9 || 1 || 9.000 ||
|-
| || || || || || || || || || 31\52 ||715.385 || 10 || 1 || 10.000 ||
|-
| 3\5 || || || || || || || || || || 720.000 || 1 || 0 || → inf ||
|}
== Temperament pages ==
Databoxes has been canceled, but the cleanup will continue
Note:
# Order: subgroup, [[comma list]], [[mapping]], mapping generators, gencom mapping, [[gencom]], map to lattice, lattice basis, [[wedgie]], [[minimax tuning]], [[tuning ranges]], [[algebraic generator]], vals, [[badness]], [[complexity spectrum]].
# Comma list shows the simplest commas sufficient to define the temperament, stated in [[Normal lists #Normal interval list]].
# Mapping generators should show all the ratios as used in the mapping, including the period.
# Minimax tuning are based on tonality diamond, so it should explicitly state the ''odd'' limit, or a diamond function of ratios.
# Use [[Template:Val list]].
# 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:
* <s>Ripple (3 different 7-limit extensions)</s>
* <s>Smate (2 different 7-limit extensions)</s>
* Maybe [[parakleismic]] (2 different 7-limit extensions)
* Maybe [[superpyth]] (2 different 7-limit extensions)
Who's next?
* <s>Meantone family</s>
* <s>Archytas clan</s>
* <s>Father family</s>
* <s>Trienstonic clan</s>
* <s>Septisemi temperaments</s>
* <s>Archytas family</s>
* <s>Slendro clan</s>
* <s>Semiphore family</s>
* <s>Marvel temperaments</s>
* <s>Marvel family</s>
* <s>Mint temperaments</s>
* <s>Mint family</s>
* [[Gamelismic clan]]
* <s>Gamelismic family</s>
* <s>Jubilismic clan</s>
* <s>Jubilismic family</s>
* <s>Didymus rank three family</s>
* <s>Kleismic family</s>
* [[Kleismic rank three family]]
* <s>Shibboleth family</s>
* <s>Keemic temperaments</s>
* <s>Keemic family</s>
* <s>Schismatic family</s>
* <s>Hemimean clan</s>
* <s>Hemimean family</s>
* <s>Luna family</s>
* <s>Canousmic temperaments</s>
* <s>Canou family</s>
* [[Starling temperaments]]
* [[Starling family]]
* <s>Sensipent family</s>
* [[Sensamagic clan]]
* [[Sensamagic family]]
* [[Magic family]]
== Septimal meantone ==
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
{{main| 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]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]
Mapping generators: ~2, ~3
{{Multival|legend=1| 1 4 10 4 13 12 }}
[[POTE generator]]: ~3/2 = 696.495
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo]]s: 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 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.
{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
[[Badness]]: 0.0137
Scales: [[meantone5]], [[meantone7]], [[meantone12]]
== Archytas ==
{{main| Archytas }}
Subgroup: 2.3.5.7
[[Comma list]]: 64/63
[[Mapping]]: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
Mapping generators: ~2, ~3, ~5
Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
Lattice basis:
: 3/2 length = 1.0508, 5/4 length = 2.3219
: Angle (3/2, 5/4) = 90 degrees
[[POTE generator]]s: ~3/2 = 709.3213, ~5/4 = 393.3747
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 1/3 0 -1/3 }}, {{monzo| 2 -2/3 1 -1/3 }}, {{monzo| 2 -2/3 0 2/3 }}]
: [[Eigenmonzo]]s: 2, 6/5, 7/5
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 }}, {{monzo| 3/2 -1/2 1 -1/4 }}, {{monzo| 3 -1 0 1/2 }}]
: [[Eigenmonzo]]s: 2, 6/5, 9/7
{{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 49 }}
Scales: [[archytas12]], [[archytas12synch]]


== Commas ==
== Commas ==

Revision as of 10:13, 13 April 2021

Temperament pages

Databoxes has been canceled, but the cleanup will continue

Note:

  1. Order: subgroup, comma list, mapping, mapping generators, gencom mapping, gencom, map to lattice, lattice basis, wedgie, minimax tuning, tuning ranges, algebraic generator, vals, badness, complexity spectrum.
  2. Comma list shows 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. Minimax tuning are based on tonality diamond, so it should explicitly state the odd limit, or a diamond function of ratios.
  5. Use 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)
  • Smate (2 different 7-limit extensions)
  • Maybe parakleismic (2 different 7-limit extensions)
  • Maybe superpyth (2 different 7-limit extensions)

Who's next?

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

Commas

41edo tempers out the following commas using its patent val, ⟨41 65 95 115 142 152 168 174 185 199 203].

Prime
limit 		Ratio[1] Name(s)
3 [65 -41⟩ 41-comma
5 [-5 -10 9⟩ Shibboleth
5 [-25 7 6⟩ Ampersand
5 3125/3072 Magic comma
5 [5 -9 4⟩ Tetracot comma
5 [20 -17 3⟩ Roda
5 [-15 8 1⟩ Schisma
7 [0 -7 6 -1⟩ Great BP diesis
7 [-10 7 8 -7⟩ Blackjackisma
7 875/864 Keema
7 3125/3087 Gariboh
7 [10 -11 2 1⟩ Tolerma
7 [-15 3 2 2⟩ Mirwomo comma
7 245/243 Sensamagic
7 4000/3969 Octagar
7 [-15 0 -2 7⟩ Quince
7 1029/1024 Gamelisma
7 225/224 Marvel comma
7 [0 3 4 -5⟩ Mirkwai
7 [5 -7 -1 3⟩ Hemimage
7 5120/5103 Hemifamity
7 [25 -14 0 -1⟩ Garischisma
7 2401/2400 Breedsma
11 [15 0 1 0 -5⟩ Thuja comma
11 245/242 Cassacot
11 100/99 Ptolemisma
11 1344/1331 Hemimin
11 896/891 Pentacircle
11 [16 0 0 -2 -3⟩ Orgonisma
11 243/242 Rastma
11 385/384 Keenanisma
11 441/440 Werckisma
11 1375/1372 Moctdel
11 540/539 Swetisma
11 3025/3024 Lehmerisma
11 [-1 2 -4 5 -2⟩ Odiheim
13 343/338
13 105/104 Animist comma
13 [12 -7 0 1 0 -1⟩ Secorian
13 275/273 Gassorma
13 144/143 Grossma
13 196/195 Mynucuma
13 640/637 Huntma
13 1188/1183 Kestrel comma
13 325/324 Marveltwin
13 352/351 Minthma
13 364/363 Gentle comma
13 847/845 Cuthbert
13 729/728 Squbema
13 4096/4095 Schismina
13 [3 -2 0 -1 3 -2⟩ Harmonisma
17 2187/2176 Septendecimal schisma
17 256/255 Septendecimal kleisma
17 715/714 Septendecimal bridge comma
19 210/209 Spleen comma
19 361/360 Go comma
19 513/512 Undevicesimal comma
19 1216/1215 Eratosthenes' comma
23 736/729 Vicesimotertial comma
29 145/144 29th-partial chroma
  1. ↑ Ratios with more than 9 digits are presented in monzos
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