Syntonic–kleismic equivalence continuum: Difference between revisions

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All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ {{monzo|-14 -19 19}}. 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 [[19edo]] (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 0.1309..., 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|-14 -19 19}}. 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 [[19edo]] (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 0.1309..., and temperaments having ''n'' near this value tend to be the most accurate ones.


{| class="wikitable center-1 center-2"
Alternatively, we can express the continuum in terms of (81/80)<sup>''k''</sup> ~ {{monzo|-30 19}}; in this case, ''k'' = (''n'' + 19) / 3.
 
{| class="wikitable center-1 center-2 center-3"
|+ Temperaments in the continuum
|+ Temperaments in the continuum
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 13: Line 16:
! Monzo
! Monzo
|-
|-
| -19
| 0
| 0
| 19edo
| [[1162261467/1073741824]]
| {{monzo|-30 19}}
|-
| -16
| 1
| Lalayo
| [[71744535/67108864]]
| {{monzo|-26 15 1}}
|-
| -13
| 2
| Lala-Yoyo
| [[4428675/4194304]]
| {{monzo|-22 11 2}}
|-
| -12
| 7/3
| 19 & 8ccc
|
| {{monzo|-62 29 7}}
|-
| -11
| 8/3
| 19 & 1cc
|
| {{monzo|-58 25 8}}
|-
| -10
| 3
| Latriyo
| [[273375/262144]]
| {{monzo|-18 7 3}}
|-
| -9
| 10/3
| 19 & 6c
|
| {{monzo|-50 17 10}}
|-
| -8
| 11/3
| 19 & 11c
|
| {{monzo|-46 13 11}}
|-
| -7
| 4
| [[Negri]]
| [[16875/16384]]
| {{monzo|-14 3 4}}
|-
| -6
| 13/3
| 19 & 23b
| [[296630859375/274877906944]]
| {{monzo|-38 5 13}}
|-
| -5
| 14/3
| 19 & 35b
| [[18310546875/17179869184]]
| {{monzo|-34 1 14}}
|-
| -4
| 5
| [[Magic]]
| [[3125/3072]]
| {{monzo|-10 -1 5}}
|-
| -3
| 16/3
| 19 & 78
| [[152587890625/146767085568]]
| {{monzo|-26 -7 16}}
|-
| -2
| 17/3
| 19 & 109
| [[762939453125/743008370688]]
| {{monzo|-22 -11 17}}
|-
| -1
| 6
| [[Hanson]]
| [[15625/15552]]
| {{monzo|-6 -5 6}}
|-
| 0
| 19/3
| [[Enneadecal]]
| [[Enneadecal]]
|  
|  
Line 19: Line 113:
|-
|-
| 1
| 1
| 20/3
| [[Countermeantone]]
| [[Countermeantone]]
|  
|  
Line 24: Line 119:
|-
|-
| 2
| 2
| 7
| [[Sensi]]
| [[Sensi]]
| [[78732/78125]]
| [[78732/78125]]
Line 29: Line 125:
|-
|-
| 3
| 3
| 22/3
| 19 & 169c
| 19 & 169c
|  
|  
Line 34: Line 131:
|-
|-
| 4
| 4
| 23/3
| 19 & 162c
| 19 & 162c
|  
|  
Line 39: Line 137:
|-
|-
| 5
| 5
| 8
| [[Unicorn]]
| [[Unicorn]]
| [[1594323/1562500]]
| [[1594323/1562500]]
Line 48: Line 147:
| …
| …
|-
|-
| ∞
| ∞
| ∞
| [[Meantone]]
| [[Meantone]]

Revision as of 12:07, 14 March 2021

The syntonic-enneadecal equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the enneadeca ([-14 -19 19⟩).

All temperaments in the continuum satisfy (81/80)n ~ [-14 -19 19⟩. 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 19edo (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 0.1309..., and temperaments having n near this value tend to be the most accurate ones.

Alternatively, we can express the continuum in terms of (81/80)k ~ [-30 19⟩; in this case, k = (n + 19) / 3.

Temperaments in the continuum
n k Temperament Comma
Ratio Monzo
-19 0 19edo 1162261467/1073741824 [-30 19⟩
-16 1 Lalayo 71744535/67108864 [-26 15 1⟩
-13 2 Lala-Yoyo 4428675/4194304 [-22 11 2⟩
-12 7/3 19 & 8ccc [-62 29 7⟩
-11 8/3 19 & 1cc [-58 25 8⟩
-10 3 Latriyo 273375/262144 [-18 7 3⟩
-9 10/3 19 & 6c [-50 17 10⟩
-8 11/3 19 & 11c [-46 13 11⟩
-7 4 Negri 16875/16384 [-14 3 4⟩
-6 13/3 19 & 23b 296630859375/274877906944 [-38 5 13⟩
-5 14/3 19 & 35b 18310546875/17179869184 [-34 1 14⟩
-4 5 Magic 3125/3072 [-10 -1 5⟩
-3 16/3 19 & 78 152587890625/146767085568 [-26 -7 16⟩
-2 17/3 19 & 109 762939453125/743008370688 [-22 -11 17⟩
-1 6 Hanson 15625/15552 [-6 -5 6⟩
0 19/3 Enneadecal [-14 -19 19⟩
1 20/3 Countermeantone [-10 -23 20⟩
2 7 Sensi 78732/78125 [2 9 -7⟩
3 22/3 19 & 169c [2 31 -22⟩
4 23/3 19 & 162c [-2 35 -23⟩
5 8 Unicorn 1594323/1562500 [-2 13 -8⟩
… … … …
∞ ∞ Meantone 81/80 [-4 4 -1⟩

Examples of temperaments with fractional values of n:

19 & 506

Commas: [38 61 -58⟩

POTE generator: 505.1394 cents

Map: [<1 26 28|, <0 -58 -61|]

EDOs: 19, 38, 57, 468, 487, 506, 525, 544, 1012, 1031

The temperament finder - 5-limit 19 & 506

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