Syntonic–31 equivalence continuum: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Normalize a mapping
m Style
Line 1: Line 1:
The '''syntonic-31 equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with a [[31-comma 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 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 (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 (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.


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
Line 14: Line 14:
|-
|-
| 0
| 0
| [[31-comma temperaments|31 &amp; 31c]]
| [[31st-octave temperaments|31 &amp; 31c]]
|  
|  
| {{monzo| -49 31 }}
| {{monzo| -49 31 }}
Line 24: Line 24:
|-
|-
| 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 }}
Line 69: Line 69:
|-
|-
| 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 }}

Revision as of 12:08, 3 October 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 the circle-of-fifths notation, 5/4 is mapped to the quadruple-diminished fifth (C-Gbbbb).

Subgroup: 2.3.5

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

Mapping: [⟨1 0 45], ⟨0 1 -27]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.950

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

Retrieved from "https://en.xen.wiki/index.php?title=Syntonic–31_equivalence_continuum&oldid=125037"