Syntonic–31 equivalence continuum: Difference between revisions

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


All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ {{monzo| -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 both commas and thus tempers 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.
All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ {{monzo| -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.


{| class="wikitable center-1 center-2"
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments in the continuum
|-
|-
Line 14: Line 14:
|-
|-
| 0
| 0
| [[31st-octave temperaments|31 & 31c]]
| [[31st-octave temperaments|31-commatic]]
|  
|  
| {{monzo| -49 31 }}
| {{Monzo| -49 31 }}
|-
|-
| 1
| 1
| 31 & 12c
| 31 & 12c
|  
|  
| {{monzo| -45 27 1 }}
| {{Monzo| -45 27 1 }}
|-
|-
| 2
| 2
| [[Miscellaneous 5-limit temperaments #Quasimoha|Quasimoha]]
| [[Miscellaneous 5-limit temperaments #Quasimoha|Quasimoha]]
| 2353579470675/2199023255552
| 2353579470675/2199023255552
| {{monzo| -41 23 2 }}
| {{Monzo| -41 23 2 }}
|-
|-
| 3
| 3
| [[Miscellaneous 5-limit temperaments #Oncle|Oncle]]
| [[Miscellaneous 5-limit temperaments #Oncle|Oncle]]
| 145282683375/137438953472
| 145282683375/137438953472
| {{monzo| -37 19 3 }}
| {{Monzo| -37 19 3 }}
|-
|-
| 4
| 4
| [[Miscellaneous 5-limit temperaments #Sentinel|Sentinel]]
| [[Miscellaneous 5-limit temperaments #Sentinel|Sentinel]]
| 8968066875/8589934592
| 8968066875/8589934592
| {{monzo| -33 15 4 }}
| {{Monzo| -33 15 4 }}
|-
|-
| 5
| 5
| [[Miscellaneous 5-limit temperaments #Tritonic|Tritonic]]
| [[Miscellaneous 5-limit temperaments #Tritonic|Tritonic]]
| 553584375/536870912
| 553584375/536870912
| {{monzo| -29 11 5 }}
| {{Monzo| -29 11 5 }}
|-
|-
| 6
| 6
| [[Ampersand]]
| [[Ampersand]]
| 34171875/33554432
| 34171875/33554432
| {{monzo| -25 7 6 }}
| {{Monzo| -25 7 6 }}
|-
|-
| 7
| 7
| [[Orson]]
| [[Orson]]
| 2109375/2097152
| 2109375/2097152
| {{monzo| -21 3 7 }}
| {{Monzo| -21 3 7 }}
|-
|-
| 8
| 8
| [[Würschmidt]]
| [[Würschmidt]]
| 393216/390625
| 393216/390625
| {{monzo| 17 1 -8 }}
| {{Monzo| 17 1 -8 }}
|-
|-
| 9
| 9
| [[Valentine]]
| [[Valentine]]
| 1990656/1953125
| 1990656/1953125
| {{monzo| 13 5 -9 }}
| {{Monzo| 13 5 -9 }}
|-
|-
| 10
| 10
| [[Mynic]]
| [[Mynic]]
| 10077696/9765625
| 10077696/9765625
| {{monzo| 9 9 -10 }}
| {{Monzo| 9 9 -10 }}
|-
|-
| 11
| 11
| [[Miscellaneous 5-limit temperaments #Nusecond|Nusecond]]
| [[Miscellaneous 5-limit temperaments #Nusecond|Nusecond]]
| 51018336/48828125
| 51018336/48828125
| {{monzo| 5 13 -11 }}
| {{Monzo| 5 13 -11 }}
|-
|-
| 12
| 12
| [[Miscellaneous 5-limit temperaments #Cypress|Cypress]]
| [[Miscellaneous 5-limit temperaments #Cypress|Cypress]]
| 258280326/244140625
| 258280326/244140625
| {{monzo| 1 17 -12 }}
| {{Monzo| 1 17 -12 }}
|-
|-
| 13
| 13
| [[Miscellaneous 5-limit temperaments #Diesic|Diesic]]
| [[Miscellaneous 5-limit temperaments #Diesic|Diesic]]
| 10460353203/9765625000
| 10460353203/9765625000
| {{monzo| -3 21 -13 }}
| {{Monzo| -3 21 -13 }}
|-
|-
| 14
| 14
| 31 & 13c
| 31 & 13c
| 847288609443/781250000000
| 847288609443/781250000000
| {{monzo| -7 25 -14 }}
| {{Monzo| -7 25 -14 }}
|-
|-
| …
| …
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| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{Monzo| -4 4 -1 }}
|}
|}


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


[[Comma list]]: {{monzo| -45 27 1 }} = 38127987424935/35184372088832
[[Comma list]]: {{monzo| -45 27 1 }


{{Mapping|legend=1| 1 0 45 | 0 1 -27 }}
{{Mapping|legend=1| 1 0 45 | 0 1 -27 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 696.950
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 696.950{{c}}


{{Optimal ET sequence|legend=1| 12c, 19c, 31, 43c, 50c }}
{{Optimal ET sequence|legend=1| 12c, 19c, 31, 43c, 50c }}
Line 145: Line 145:
: ''For extensions, see [[Gamelismic clan #Miracle]].''
: ''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 [[16/15|classical diatonic semitones]]. It can be described as the 31 & 41 temperament, corresponding to ''n'' = 6.  
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 [[16/15|classical diatonic semitones]]. It can be described as the {{nowrap| 31 & 41 }} temperament, corresponding to {{nowrap| ''n'' {{=}} 6 }}.  


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


[[Comma list]]: {{monzo| -25 7 6 }} = 34171875/33554432
[[Comma list]]: 34171875/33554432


{{Mapping|legend=1| 1 1 3 | 0 6 -7 }}
{{Mapping|legend=1| 1 1 3 | 0 6 -7 }}


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


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: ''For extensions, see [[Gamelismic clan #Valentine]].''
: ''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 [[25/24|classical chromatic semitones]]. It can be described as the 31 & 46 temperament, corresponding to ''n'' = 9.  
The 5-limit version of valentine tempers out the [[valentine comma]], which is the difference between a perfect fifth and a stack of nine [[25/24|classical chromatic semitones]]. It can be described as the {{nowrap| 31 & 46 }} temperament, corresponding to {{nowrap| ''n'' {{=}} 9 }}.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 177: Line 177:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~25/24 = 78.039
* [[CTE]]: ~2 = 1200.000{{c}}, ~25/24 = 78.039{{c}}
: [[error map]]: {{val| 0.000 +0.397 +3.882 }}
: [[error map]]: {{val| 0.000 +0.397 +3.882 }}
* [[POTE]]: ~2 = 1200.000, ~25/24 = 78.039
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/24 = 78.039{{c}}
: error map: {{val| 0.000 -0.829 +3.201 }}
: error map: {{val| 0.000 -0.829 +3.201 }}


Line 186: Line 186:
[[Badness]] (Smith): 0.122765
[[Badness]] (Smith): 0.122765


== Quadlaleyo (31 &amp; 70c) ==
== Quadlaleyo (31 & 70c) ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -54 18 11 }} = 18917016064453125/18014398509481984
[[Comma list]]: {{monzo| -54 18 11 }}


{{Mapping|legend=1| 1 3 0 | 0 -11 18 }}
{{Mapping|legend=1| 1 3 0 | 0 -11 18 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~32768/30375 = 154.597
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~32768/30375 = 154.597{{c}}


{{Optimal ET sequence|legend=1| 8c, 23c, 31, 39c, 132, 163 }}
{{Optimal ET sequence|legend=1| 8c, 23c, 31, 39c, 132, 163 }}
Line 201: Line 201:
[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]


== Lalasepbigu (31 &amp; 13c) ==
== Lalasepbigu (31 & 13c) ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -7 25 -14 }} = 847288609443/781250000000
[[Comma list]]: 847288609443/781250000000


{{Mapping|legend=1| 1 7 12 | 0 -14 -25 }}
{{Mapping|legend=1| 1 7 12 | 0 -14 -25 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~25000/19683 = 464.423
[[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, 44c, 49bc, 75c, 80bc }}
Line 224: Line 224:


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~5/4 = 386.8710
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~5/4 = 386.8710{{c}}


{{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 }}

Revision as of 15:03, 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: {{monzo| -45 27 1 }

Mapping: [⟨1 0 45], ⟨0 1 -27]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 696.950 ¢

Optimal ET sequence: 12c, 19c, 31, 43c, 50c

Badness (Smith): 2.993628

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~16/15 = 116.701 ¢
error map: ⟨0.000 -1.750 -3.219]
  • CWE: ~2 = 1200.000 ¢, ~16/15 = 116.680 ¢
error map: ⟨0.000 -1.874 -3.075]

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

Badness:

  • Smith: 0.165755
  • Dirichlet: 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~25/24 = 78.039 ¢
error map: ⟨0.000 +0.397 +3.882]
  • POTE: ~2 = 1200.000 ¢, ~25/24 = 78.039 ¢
error map: ⟨0.000 -0.829 +3.201]

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

Badness (Smith): 0.122765

Quadlaleyo (31 & 70c)

Subgroup: 2.3.5

Comma list: [-54 18 11⟩

Mapping: [⟨1 3 0], ⟨0 -11 18]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~32768/30375 = 154.597 ¢

Optimal ET sequence: 8c, 23c, 31, 39c, 132, 163

Badness (Smith): 2.067160

The temperament finder - 5-limit 31 & 70c

Lalasepbigu (31 & 13c)

Subgroup: 2.3.5

Comma list: 847288609443/781250000000

Mapping: [⟨1 7 12], ⟨0 -14 -25]]

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

Optimal ET sequence: 13c, 18bc, 31, 44c, 49bc, 75c, 80bc

Badness (Smith): 2.094918

The temperament finder - 5-limit 31 & 13c

Counterwürschmidt

See also: Mirkwai clan § Grendel

Subgroup: 2.3.5

Comma list: [55 -1 -23⟩

Mapping: [⟨1 9 2], ⟨0 -23 1]]

mapping generators: ~2, ~5/4

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~5/4 = 386.8710 ¢

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

Badness (Smith): 0.420

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