Syntonic–31 equivalence continuum: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Tabulate fractional-n temps
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| [[31-comma temperaments|31 & 31c]]
| [[31-comma temperaments|31 & 31c]]
|  
|  
| {{monzo|-49 31}}
| {{monzo| -49 31 }}
|-
|-
| 1
| 1
| 31 & 12c
| 31 & 12c
|  
|  
| {{monzo|-45 27 1}}
| {{monzo| -45 27 1 }}
|-
|-
| 2
| 2
| [[High badness temperaments#Quasimoha|Quasimoha]]
| [[High badness temperaments#Quasimoha|Quasimoha]]
| 2353579470675/2199023255552
| 2353579470675/2199023255552
| {{monzo|-41 23 2}}
| {{monzo| -41 23 2 }}
|-
|-
| 3
| 3
| [[High badness temperaments#Oncle|Oncle]]
| [[High badness temperaments#Oncle|Oncle]]
| 145282683375/137438953472
| 145282683375/137438953472
| {{monzo|-37 19 3}}
| {{monzo| -37 19 3 }}
|-
|-
| 4
| 4
| [[Orwellismic temperaments#Sentinel|Sentinel]]
| [[Orwellismic temperaments#Sentinel|Sentinel]]
| 8968066875/8589934592
| 8968066875/8589934592
| {{monzo|-33 15 4}}
| {{monzo| -33 15 4 }}
|-
|-
| 5
| 5
| [[High badness temperaments#Tritonic|Tritonic]]
| [[High badness 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
| [[Starling temperaments#Myna|Myna]]
| [[Mynic]]
| 10077696/9765625
| 10077696/9765625
| {{monzo|9 9 -10}}
| {{monzo| 9 9 -10 }}
|-
|-
| 11
| 11
| [[Starling temperaments#Nusecond|Nusecond]]
| [[Starling temperaments#Nusecond|Nusecond]]
| 51018336/48828125
| 51018336/48828125
| {{monzo|5 13 -11}}
| {{monzo| 5 13 -11 }}
|-
|-
| 12
| 12
| [[Starling temperaments#Cypress|Cypress]]
| [[Starling temperaments#Cypress|Cypress]]
| 258280326/244140625
| 258280326/244140625
| {{monzo|1 17 -12}}
| {{monzo| 1 17 -12 }}
|-
|-
| 13
| 13
| [[Orwellismic temperaments#Diesic|Diesic]]
| [[Orwellismic 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 }}
|-
|-
| …
| …
Line 96: Line 96:
| [[Meantone family|Meantone]]
| [[Meantone family|Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1}}
| {{monzo| -4 4 -1 }}
|}
|}


Examples of temperaments with fractional values of ''n'':
Examples of temperaments with fractional values of ''n'':
* 31 & 70c (''n'' = 11/2 = 5.5)
 
* [[Marvel temperaments #Slender|Slender]] (''n'' = 13/2 = 6.5)
{| class="wikitable"
* [[Mirkwai clan #Eris|Eris]] (''n'' = 29/4 = 7.25)
|+ Notable temperaments of fractional ''n''
* [[Breedsmic temperaments #Tertiaseptal|Tertiaseptal]] (''n'' = 22/3 = 7.{{overline|3}})
|-
* [[Luna family #Luna|Luna]] (''n'' = 15/2 = 7.5)
! Temperament !! ''n'' !! Comma
* [[Breedsmic temperaments #Quasiorwell|Quasiorwell]] (''n'' = 38/5 = 7.6)
|-
* [[Mirkwai clan #Grendel|Grendel]] (''n'' = 23/3 = 7.{{overline|6}})
| [[Slender]] || 13/2 = 6.5 || {{monzo| -46 10 13 }}
* [[31-comma temperaments #Birds|Birds]] (''n'' = 31/4 = 7.75)
|-
* [[Porwell temperaments #Countermiracle|Countermiracle]] (''n'' = 25/3 = 8.{{overline|3}})
| [[Eris]] || 29/4 = 7.25 || {{monzo| -80 8 29 }}
* [[Hemimean clan #Semisept|Semisept]] (''n'' = 17/2 = 8.5)
|-
* [[Starling temperaments #Casablanca|Casablanca]] (''n'' = 19/2 = 9.5)
| [[Tertiaseptal]] || 22/3 = 7.{{overline|3}} || {{monzo| -59 5 22 }}
|-
| [[Luna]] || 15/2 = 7.5 || {{monzo| 38 -2 -15 }}
|-
| [[Quasiorwell]] || 38/5 = 7.6 || {{monzo| 93 -3 -38 }}
|-
| [[Grendel]] || 23/3 = 7.{{overline|6}} || {{monzo| 55 -1 -23 }}
|-
| [[Birds]] || 31/4 = 7.75 || {{monzo| 72 0 -31 }}
|-
| [[Countermiracle]] || 25/3 = 8.{{overline|3}} || {{monzo| 47 7 -25 }}
|-
| [[Casablanca]] || 19/2 = 9.5 || {{monzo| 22 14 -19 }}
|}


== Quadlayo (31 & 12c) ==
== Quadlayo (31 & 12c) ==
In fifths notation, 5/4 is mapped to the quadruple-diminished fifth.
In fifths notation, 5/4 is mapped to the quadruple-diminished fifth.
Subgroup: 2.3.5


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


Mapping: [{{val| 1 2 -9 }}, {{val| 0 -1 27 }}]
Mapping: {{mapping| 1 2 -9 | 0 -1 27 }}


POTE generator: ~4/3 = 503.050
Optimal tuning (POTE): ~2 = 1\1, ~4/3 = 503.050


{{Optimal ET sequence|legend=1| 12c, 19c, 31, 43c, 50c }}
{{Optimal ET sequence|legend=1| 12c, 19c, 31, 43c, 50c }}
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== Quadlaleyo (31 & 70c) ==
== Quadlaleyo (31 & 70c) ==
Subgroup: 2.3.5
Comma list: {{monzo| -54 18 11 }} = 18917016064453125/18014398509481984
Comma list: {{monzo| -54 18 11 }} = 18917016064453125/18014398509481984


Mapping: [{{val| 1 3 0 }}, {{val| 0 -11 18 }}]
Mapping: {{mapping| 1 3 0 | 0 -11 18 }}


POTE generator: ~32768/30375 = 154.597
Optimal tuning (POTE): ~2 = 1\1, ~32768/30375 = 154.597


{{Optimal ET sequence|legend=1| 8c, 23c, 31, 39c, 132, 163 }}
{{Optimal ET sequence|legend=1| 8c, 23c, 31, 39c, 132, 163 }}
Line 142: Line 159:
== Ampersand (31 & 41) ==
== Ampersand (31 & 41) ==
{{See also|Gamelismic clan #Miracle}}
{{See also|Gamelismic clan #Miracle}}
Subgroup: 2.3.5


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


Mapping: [{{val| 1 1 3 }}, {{val| 0 6 -7 }}]
Mapping: {{mapping| 1 1 3 | 0 6 -7 }}


POTE generator: ~16/15 = 116.673
Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 116.673


{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
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== Lalasepbigu (31 & 13c) ==
== Lalasepbigu (31 & 13c) ==
Subgroup: 2.3.5
Comma list: {{monzo| -7 25 -14 }} = 847288609443/781250000000
Comma list: {{monzo| -7 25 -14 }} = 847288609443/781250000000


Mapping: [{{val| 1 7 12 }}, {{val| 0 -14 -25 }}]
Mapping: {{mapping| 1 7 12 | 0 -14 -25 }}


POTE generator: ~25000/19683 = 464.423
Optimal tuning (POTE): ~2 = 1\1, ~25000/19683 = 464.423


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

Revision as of 09:17, 15 September 2023

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 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.

Temperaments in the continuum
n Temperament Comma
Ratio Monzo
0 31 & 31c [-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⟩
Grendel 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 fifths notation, 5/4 is mapped to the quadruple-diminished fifth.

Subgroup: 2.3.5

Comma list: [-45 27 1⟩ = 38127987424935/35184372088832

Mapping: [⟨1 2 -9], ⟨0 -1 27]]

Optimal tuning (POTE): ~2 = 1\1, ~4/3 = 503.050

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

Badness: 2.993628

The temperament finder - 5-limit 31 & 12c

Quadlaleyo (31 & 70c)

Subgroup: 2.3.5

Comma list: [-54 18 11⟩ = 18917016064453125/18014398509481984

Mapping: [⟨1 3 0], ⟨0 -11 18]]

Optimal tuning (POTE): ~2 = 1\1, ~32768/30375 = 154.597

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

Badness: 2.067160

The temperament finder - 5-limit 31 & 70c

Ampersand (31 & 41)

See also: Gamelismic clan § Miracle

Subgroup: 2.3.5

Comma list: [-25 7 6⟩ = 34171875/33554432

Mapping: [⟨1 1 3], ⟨0 6 -7]]

Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 116.673

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

Badness: 0.165755

Lalasepbigu (31 & 13c)

Subgroup: 2.3.5

Comma list: [-7 25 -14⟩ = 847288609443/781250000000

Mapping: [⟨1 7 12], ⟨0 -14 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~25000/19683 = 464.423

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

Badness: 2.094918

The temperament finder - 5-limit 31 & 13c

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