Syntonic–31 equivalence continuum: Difference between revisions

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== Quadlayo (31 & 12c) ==
== Quadlayo (31 & 12c) ==
In the [[chain-of-fifths notation]], 5/4 is mapped to the quadruple-diminished fifth (C-Gbbbb).
In the [[chain-of-fifths notation]], 5/4 is mapped to the quadruple-diminished fifth (C-Gbbbb).


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -45 27 1 }
[[Comma list]]: {{monzo| -45 27 1 }}


{{Mapping|legend=1| 1 0 45 | 0 1 -27 }}
{{Mapping|legend=1| 1 0 45 | 0 1 -27 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 696.950{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.6167{{c}}, ~3/2 = 697.8886{{c}}
: [[error map]]: {{val| +1.617 -2.450 -0.204 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 696.9075{{c}}
: error map: {{val| 0.000 -5.048 -2.815 }}


{{Optimal ET sequence|legend=1| 12c, 19c, 31, 43c, 50c }}
{{Optimal ET sequence|legend=1| 12c, 19c, 31, 136bc, 167bc, 198bc, 229bc }}


[[Badness]] (Smith): 2.993628
[[Badness]] (Sintel): 70.2


[http://x31eq.com/cgi-bin/rt.cgi?ets=31_12c&limit=5 The temperament finder - 5-limit 31 & 12c]
[http://x31eq.com/cgi-bin/rt.cgi?ets=31_12c&limit=5 The temperament finder - 5-limit 31 & 12c]
Line 152: Line 157:


{{Mapping|legend=1| 1 1 3 | 0 6 -7 }}
{{Mapping|legend=1| 1 1 3 | 0 6 -7 }}
: mapping generators: ~2, ~16/15


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~16/15 = 116.701{{c}}
* [[WE]]: ~2 = 1200.8367{{c}}, ~16/15 = 116.7546{{c}}
: [[error map]]: {{val| 0.000 -1.750 -3.219 }}
: [[error map]]: {{val| +0.837 -0.591 -1.086 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~16/15 = 116.680{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~16/15 = 116.6802{{c}}
: error map: {{val| 0.000 -1.874 -3.075 }}
: error map: {{val| 0.000 -1.874 -3.075 }}


{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}


[[Badness]]:
[[Badness]] (Sintel): 3.89
* Smith: 0.165755
* Dirichlet: 3.89


== Valentine (5-limit) ==
== Valentine (5-limit) ==
Line 175: Line 179:


{{Mapping|legend=1| 1 1 2 | 0 9 5 }}
{{Mapping|legend=1| 1 1 2 | 0 9 5 }}
: mapping generators: ~2, ~25/24


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~25/24 = 78.039{{c}}
* [[WE]]: ~2 = 1199.3579{{c}}, ~25/24 = 77.9973{{c}}
: [[error map]]: {{val| 0.000 +0.397 +3.882 }}
: [[error map]]: {{val| -0.642 -0.621 +2.389 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/24 = 78.039{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/24 = 77.9807{{c}}
: error map: {{val| 0.000 -0.829 +3.201 }}
: error map: {{val| 0.000 -0.129 +3.590 }}


{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 123 }}
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 123 }}


[[Badness]] (Smith): 0.122765
[[Badness]] (Sintel): 2.88


== Quadlaleyo (31 & 70c) ==
== Quadlaleyo (31 & 70c) ==
Line 191: Line 196:
[[Comma list]]: {{monzo| -54 18 11 }}
[[Comma list]]: {{monzo| -54 18 11 }}


{{Mapping|legend=1| 1 3 0 | 0 -11 18 }}
{{Mapping|legend=1| 1 -8 18 | 0 11 -18 }}
: mapping generators: ~2, ~30375/16384


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~32768/30375 = 154.597{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0416{{c}}, ~32768/30375 = 1046.3102{{c}}
: [[error map]]: {{val| +1.042 -0.876 -1.149 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32768/30375 = 1045.4008{{c}}
: error map: {{val| 0.000 -2.546 -3.529 }}


{{Optimal ET sequence|legend=1| 8c, 23c, 31, 39c, 132, 163 }}
{{Optimal ET sequence|legend=1| 8c, 31, 101c, 132, 163 }}


[[Badness]] (Smith): 2.067160
[[Badness]] (Sintel): 48.5


[http://x31eq.com/cgi-bin/rt.cgi?ets=31_70c&limit=5 The temperament finder - 5-limit 31 & 70c]
[http://x31eq.com/cgi-bin/rt.cgi?ets=31_70c&limit=5 The temperament finder - 5-limit 31 & 70c]
Line 206: Line 216:
[[Comma list]]: 847288609443/781250000000
[[Comma list]]: 847288609443/781250000000


{{Mapping|legend=1| 1 7 12 | 0 -14 -25 }}
{{Mapping|legend=1| 1 -7 -13 | 0 14 25 }}
: mapping generators: ~2, ~19683/12500
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3614{{c}}, ~19683/12500 = 735.7984{{c}}
: [[error map]]: {{val| +0.361 -3.307 +3.498 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~19683/12500 = 735.5950{{c}}
: error map: {{val| 0.000 -3.625 -3.560 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25000/19683 = 464.423{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25000/19683 = 464.423{{c}}


{{Optimal ET sequence|legend=1| 13c, 18bc, 31, 44c, 49bc, 75c, 80bc }}
{{Optimal ET sequence|legend=1| 13c, 18bc, 31 }}


[[Badness]] (Smith): 2.094918
[[Badness]] (Sintel): 49.1


[http://x31eq.com/cgi-bin/rt.cgi?ets=31_13c&limit=5 The temperament finder - 5-limit 31 & 13c]
[http://x31eq.com/cgi-bin/rt.cgi?ets=31_13c&limit=5 The temperament finder - 5-limit 31 & 13c]


== Counterwürschmidt ==
== Counterwürschmidt ==
{{See also| Mirkwai clan #Grendel }}
: ''For extensions, see [[Mirkwai clan #Grendel]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Comma list]]: {{monzo| 55 -1 -23 }}
[[Comma list]]: {{monzo| 55 -1 -23 }}


{{Mapping|legend=1| 1 9 2 | 0 -23 1 }}
{{Mapping|legend=1| 1 -14 3 | 0 23 -1 }}
: mapping generators: ~2, ~5/4
: mapping generators: ~2, ~8/5


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~5/4 = 386.8710{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~8/5 = 813.0556{{c}}
: [[error map]]: {{val| -0.120 +0.005 +0.271 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 813.1344{{c}}
: error map: {{val| 0.000 +0.135 +0.552 }}


{{Optimal ET sequence|legend=1| 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c }}
{{Optimal ET sequence|legend=1| 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c, 2649cc }}


[[Badness]] (Smith): 0.420
[[Badness]] (Sintel): 9.86


[[Category:31edo]]
[[Category:31edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Latest revision as of 15:18, 25 February 2026

The syntonic–31 equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with a 31-comma ([-49 31⟩). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 31edo.

All temperaments in the continuum satisfy (81/80)n ~ [-49 31⟩. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 31edo due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them. The just value of n is approximately 7.46781…, and temperaments having n near this value tend to be the most accurate ones.

Temperaments in the continuum
n Temperament Comma
Ratio Monzo
0 31-commatic [-49 31⟩
1 31 & 12c [-45 27 1⟩
2 Quasimoha 2353579470675/2199023255552 [-41 23 2⟩
3 Oncle 145282683375/137438953472 [-37 19 3⟩
4 Sentinel 8968066875/8589934592 [-33 15 4⟩
5 Tritonic 553584375/536870912 [-29 11 5⟩
6 Ampersand 34171875/33554432 [-25 7 6⟩
7 Orson 2109375/2097152 [-21 3 7⟩
8 Würschmidt 393216/390625 [17 1 -8⟩
9 Valentine 1990656/1953125 [13 5 -9⟩
10 Mynic 10077696/9765625 [9 9 -10⟩
11 Nusecond 51018336/48828125 [5 13 -11⟩
12 Cypress 258280326/244140625 [1 17 -12⟩
13 Diesic 10460353203/9765625000 [-3 21 -13⟩
14 31 & 13c 847288609443/781250000000 [-7 25 -14⟩
… … … …
∞ Meantone 81/80 [-4 4 -1⟩

Examples of temperaments with fractional values of n:

Notable temperaments of fractional n
Temperament n Comma
Slender 13/2 = 6.5 [-46 10 13⟩
Eris 29/4 = 7.25 [-80 8 29⟩
Tertiaseptal 22/3 = 7.3 [-59 5 22⟩
Luna 15/2 = 7.5 [38 -2 -15⟩
Quasiorwell 38/5 = 7.6 [93 -3 -38⟩
Counterwürschmidt 23/3 = 7.6 [55 -1 -23⟩
Birds 31/4 = 7.75 [72 0 -31⟩
Countermiracle 25/3 = 8.3 [47 7 -25⟩
Casablanca 19/2 = 9.5 [22 14 -19⟩

Quadlayo (31 & 12c)

In the chain-of-fifths notation, 5/4 is mapped to the quadruple-diminished fifth (C-Gbbbb).

Subgroup: 2.3.5

Comma list: [-45 27 1⟩

Mapping: [⟨1 0 45], ⟨0 1 -27]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1201.6167 ¢, ~3/2 = 697.8886 ¢
error map: ⟨+1.617 -2.450 -0.204]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 696.9075 ¢
error map: ⟨0.000 -5.048 -2.815]

Optimal ET sequence: 12c, 19c, 31, 136bc, 167bc, 198bc, 229bc

Badness (Sintel): 70.2

The temperament finder - 5-limit 31 & 12c

Ampersand

For extensions, see Gamelismic clan #Miracle.

Ampersand is the 5-limit version of miracle, tempering out the ampersand comma, which is the difference between a perfect fifth and a stack of six classical diatonic semitones. It can be described as the 31 & 41 temperament, corresponding to n = 6.

Subgroup: 2.3.5

Comma list: 34171875/33554432

Mapping: [⟨1 1 3], ⟨0 6 -7]]

mapping generators: ~2, ~16/15

Optimal tunings:

  • WE: ~2 = 1200.8367 ¢, ~16/15 = 116.7546 ¢
error map: ⟨+0.837 -0.591 -1.086]
  • CWE: ~2 = 1200.000 ¢, ~16/15 = 116.6802 ¢
error map: ⟨0.000 -1.874 -3.075]

Optimal ET sequence: 10, 21, 31, 41, 72

Badness (Sintel): 3.89

Valentine (5-limit)

For extensions, see Gamelismic clan #Valentine.

The 5-limit version of valentine tempers out the valentine comma, which is the difference between a perfect fifth and a stack of nine classical chromatic semitones. It can be described as the 31 & 46 temperament, corresponding to n = 9.

Subgroup: 2.3.5

Comma list: 1990656/1953125

Mapping: [⟨1 1 2], ⟨0 9 5]]

mapping generators: ~2, ~25/24

Optimal tunings:

  • WE: ~2 = 1199.3579 ¢, ~25/24 = 77.9973 ¢
error map: ⟨-0.642 -0.621 +2.389]
  • CWE: ~2 = 1200.0000 ¢, ~25/24 = 77.9807 ¢
error map: ⟨0.000 -0.129 +3.590]

Optimal ET sequence: 15, 31, 46, 77, 123

Badness (Sintel): 2.88

Quadlaleyo (31 & 70c)

Subgroup: 2.3.5

Comma list: [-54 18 11⟩

Mapping: [⟨1 -8 18], ⟨0 11 -18]]

mapping generators: ~2, ~30375/16384

Optimal tunings:

  • WE: ~2 = 1201.0416 ¢, ~32768/30375 = 1046.3102 ¢
error map: ⟨+1.042 -0.876 -1.149]
  • CWE: ~2 = 1200.0000 ¢, ~32768/30375 = 1045.4008 ¢
error map: ⟨0.000 -2.546 -3.529]

Optimal ET sequence: 8c, 31, 101c, 132, 163

Badness (Sintel): 48.5

The temperament finder - 5-limit 31 & 70c

Lalasepbigu (31 & 13c)

Subgroup: 2.3.5

Comma list: 847288609443/781250000000

Mapping: [⟨1 -7 -13], ⟨0 14 25]]

mapping generators: ~2, ~19683/12500

Optimal tunings:

  • WE: ~2 = 1200.3614 ¢, ~19683/12500 = 735.7984 ¢
error map: ⟨+0.361 -3.307 +3.498]
  • CWE: ~2 = 1200.0000 ¢, ~19683/12500 = 735.5950 ¢
error map: ⟨0.000 -3.625 -3.560]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25000/19683 = 464.423 ¢

Optimal ET sequence: 13c, 18bc, 31

Badness (Sintel): 49.1

The temperament finder - 5-limit 31 & 13c

Counterwürschmidt

For extensions, see Mirkwai clan #Grendel.

Subgroup: 2.3.5

Comma list: [55 -1 -23⟩

Mapping: [⟨1 -14 3], ⟨0 23 -1]]

mapping generators: ~2, ~8/5

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~8/5 = 813.0556 ¢
error map: ⟨-0.120 +0.005 +0.271]
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1344 ¢
error map: ⟨0.000 +0.135 +0.552]

Optimal ET sequence: 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c, 2649cc

Badness (Sintel): 9.86

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