Syntonic–31 equivalence continuum: Difference between revisions

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Line 14: Line 14:
|-
|-
| 0
| 0
| [[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}}
Line 56: Line 56:
| [[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
Line 84: Line 84:
|-
|-
| 14
| 14
| 31 & 13c
| 31 & 13c
| 847288609443/781250000000
| 847288609443/781250000000
| {{monzo|-7 25 -14}}
| {{monzo|-7 25 -14}}
Line 100: Line 100:


Examples of temperaments with fractional values of ''n'':
Examples of temperaments with fractional values of ''n'':
* 31 & 70c (''n'' = 11/2 = 5.5)
* 31 & 70c (''n'' = 11/2 = 5.5)
* [[Marvel temperaments#Slender|Slender]] (''n'' = 13/2 = 6.5)
* [[Marvel temperaments#Slender|Slender]] (''n'' = 13/2 = 6.5)
* [[Breedsmic temperaments#Tertiaseptal|Tertiaseptal]] (''n'' = 22/3 = 7.{{overline|3}})
* [[Luna family#Luna|Luna]] (''n'' = 15/2 = 7.5)
* [[Luna family#Luna|Luna]] (''n'' = 15/2 = 7.5)
* [[Mirkwai clan #Grendel|Grendel]] (''n'' = 23/3 = 7.{{overline|6}})
* [[31-comma temperaments#Birds|Birds]] (''n'' = 31/4 = 7.75)
* [[Porwell temperaments|Countermiracle]] (''n'' = 25/3 = 8.{{overline|3}})
* [[Hemimean clan#Semisept|Semisept]] (''n'' = 17/2 = 8.5)
* [[Hemimean clan#Semisept|Semisept]] (''n'' = 17/2 = 8.5)
* [[Casablanca node|Casablanca]] (''n'' = 19/2 = 9.5)
* [[Casablanca node|Casablanca]] (''n'' = 19/2 = 9.5)


== 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.


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 }}]


POTE generator: ~4/3 = 503.0504
POTE generator: ~4/3 = 503.0504


Mapping: [{{val| 1 2 -9 }}, {{val| 0 -1 27 }}]
Vals: {{Val list| 12c, 19c, 31, 43c, 50c }}


{{Val list|legend=1| 12c, 19c, 31, 43c, 50c }}
Badness: 2.993628


[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]


== 31 & 13c ==
== Lalasepbigu (31 & 13c) ==
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 }}]


POTE generator: 464.4231
POTE generator: ~25000/19683 = 464.4231


Mapping: [{{val| 1 7 12 }}, {{val| 0 -14 -25 }}]
Vals: {{Val list| 13c, 18bc, 31, 44c, 49bc, 75c, 80bc }}


{{Val list|legend=1| 13c, 18bc, 31, 44c, 49bc, 75c, 80bc }}
Badness: 2.094918


[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]


== 31 & 70c ==
== Quadlaleyo (31 & 70c) ==
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 }}]


POTE generator: 154.5972
POTE generator: ~32768/30375 = 154.5972


Mapping: [{{val| 1 3 0 }}, {{val| 0 -11 18 }}]
Vals: {{Val list| 8c, 23c, 31, 39c, 132, 163 }}


{{Val list|legend=1| 8c, 23c, 31, 39c, 132, 163 }}
Badness: 2.067160


[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]

Revision as of 11:11, 22 May 2021

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 Myna 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:

Quadlayo (31 & 12c)

In fifths notation, 5/4 is mapped to the quadruple-diminished fifth.

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

Mapping: [⟨1 2 -9], ⟨0 -1 27]]

POTE generator: ~4/3 = 503.0504

Vals: Template:Val list

Badness: 2.993628

The temperament finder - 5-limit 31 & 12c

Lalasepbigu (31 & 13c)

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

Mapping: [⟨1 7 12], ⟨0 -14 -25]]

POTE generator: ~25000/19683 = 464.4231

Vals: Template:Val list

Badness: 2.094918

The temperament finder - 5-limit 31 & 13c

Quadlaleyo (31 & 70c)

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

Mapping: [⟨1 3 0], ⟨0 -11 18]]

POTE generator: ~32768/30375 = 154.5972

Vals: Template:Val list

Badness: 2.067160

The temperament finder - 5-limit 31 & 70c

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