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 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 | ||
| [[31st-octave temperaments|31 & | | [[31st-octave temperaments|31 & 31c]] | ||
| | | | ||
| {{monzo| -49 31 }} | | {{monzo| -49 31 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| 31 & | | 31 & 12c | ||
| | | | ||
| {{monzo| -45 27 1 }} | | {{monzo| -45 27 1 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| [[ | | [[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]] | ||
| 145282683375/137438953472 | | 145282683375/137438953472 | ||
| {{monzo| -37 19 3 }} | | {{monzo| -37 19 3 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| [[ | | [[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]] | ||
| 553584375/536870912 | | 553584375/536870912 | ||
| {{monzo| -29 11 5 }} | | {{monzo| -29 11 5 }} | ||
| Line 69: | Line 69: | ||
|- | |- | ||
| 11 | | 11 | ||
| [[ | | [[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]] | ||
| 258280326/244140625 | | 258280326/244140625 | ||
| {{monzo| 1 17 -12 }} | | {{monzo| 1 17 -12 }} | ||
|- | |- | ||
| 13 | | 13 | ||
| [[ | | [[Miscellaneous 5-limit temperaments #Diesic|Diesic]] | ||
| 10460353203/9765625000 | | 10460353203/9765625000 | ||
| {{monzo| -3 21 -13 }} | | {{monzo| -3 21 -13 }} | ||
|- | |- | ||
| 14 | | 14 | ||
| 31 & | | 31 & 13c | ||
| 847288609443/781250000000 | | 847288609443/781250000000 | ||
| {{monzo| -7 25 -14 }} | | {{monzo| -7 25 -14 }} | ||
| Line 94: | Line 94: | ||
|- | |- | ||
| ∞ | | ∞ | ||
| [[ | | [[Meantone]] | ||
| [[81/80]] | | [[81/80]] | ||
| {{monzo| -4 4 -1 }} | | {{monzo| -4 4 -1 }} | ||
| Line 126: | Line 126: | ||
== Quadlayo (31 & 12c) == | == Quadlayo (31 & 12c) == | ||
In the [[ | 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 }} = 38127987424935/35184372088832 | [[Comma list]]: {{monzo| -45 27 1 }} = 38127987424935/35184372088832 | ||
{{Mapping|legend=1| 1 0 45 | 0 1 -27 }} | |||
Optimal tuning (POTE): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 696.950 | ||
{{Optimal ET sequence|legend=1| 12c, 19c, 31, 43c, 50c }} | {{Optimal ET sequence|legend=1| 12c, 19c, 31, 43c, 50c }} | ||
Badness: 2.993628 | [[Badness]] (Smith): 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] | ||
== Quadlaleyo (31 & 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 }} = 18917016064453125/18014398509481984 | ||
{{Mapping|legend=1| 1 3 0 | 0 -11 18 }} | |||
Optimal tuning (POTE): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~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 }} | ||
Badness: 2.067160 | [[Badness]] (Smith): 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] | ||
| Line 160: | Line 160: | ||
{{See also| Gamelismic clan #Miracle }} | {{See also| Gamelismic clan #Miracle }} | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
Comma list: {{monzo| -25 7 6 }} = 34171875/33554432 | [[Comma list]]: {{monzo| -25 7 6 }} = 34171875/33554432 | ||
{{Mapping|legend=1| 1 1 3 | 0 6 -7 }} | |||
Optimal tuning (POTE): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16/15 = 116.673 | ||
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }} | {{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }} | ||
Badness: 0.165755 | [[Badness]] (Smith): 0.165755 | ||
== Counterwürschmidt == | == Counterwürschmidt == | ||
{{See also| Mirkwai clan #Grendel }} | {{See also| Mirkwai clan #Grendel }} | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
Comma list: {{monzo| 55 -1 -23 }} | [[Comma list]]: {{monzo| 55 -1 -23 }} | ||
{{Mapping|legend=1| 1 9 2 | 0 -23 1 }} | |||
: mapping generators: ~2, ~5/4 | : mapping generators: ~2, ~5/4 | ||
Optimal tuning (CTE): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~5/4 = 386.8710 | ||
{{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 }} | ||
Badness: 0.420 | [[Badness]] (Smith): 0.420 | ||
== Lalasepbigu (31 & 13c) == | == Lalasepbigu (31 & 13c) == | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
Comma list: {{monzo| -7 25 -14 }} = 847288609443/781250000000 | [[Comma list]]: {{monzo| -7 25 -14 }} = 847288609443/781250000000 | ||
{{Mapping|legend=1| 1 7 12 | 0 -14 -25 }} | |||
Optimal tuning (POTE): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~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 }} | ||
Badness: 2.094918 | [[Badness]] (Smith): 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] | ||
Revision as of 13:16, 6 January 2025
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.
| 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:
| 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⟩ = 38127987424935/35184372088832
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
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 = 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
Ampersand
Subgroup: 2.3.5
Comma list: [-25 7 6⟩ = 34171875/33554432
Mapping: [⟨1 1 3], ⟨0 6 -7]]
Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 116.673
Optimal ET sequence: 10, 21, 31, 41, 72
Badness (Smith): 0.165755
Counterwürschmidt
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 = 1200.000, ~5/4 = 386.8710
Optimal ET sequence: 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c
Badness (Smith): 0.420
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 = 1200.000, ~25000/19683 = 464.423
Optimal ET sequence: 13c, 18bc, 31, 44c, 49bc, 75c, 80bc
Badness (Smith): 2.094918