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Syntonic–kleismic equivalence continuum: Difference between revisions

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The '''syntonic-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[Enneadeca|enneadeca ({{Monzo| -14 -19 19 }})]].  
The '''syntonic-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the 19-comma ({{Monzo| -30 19}}).


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>''k''</sup> ~ {{monzo|-30 19}}. Varying ''k'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''k'' approaches infinity. If we allow non-integer and infinite ''k'', 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 ''k'' is approximately 6.376..., and temperaments having ''k'' near this value tend to be the most accurate ones.


Alternatively, we can express the continuum in terms of (81/80)<sup>''k''</sup> ~ {{monzo|-30 19}}; in this case, ''k'' = (''n'' + 19) / 3.
This continuum used to be expressed as the relationship between 81/80 and the [[enneadeca]] ({{Monzo|-14 -19 19}}). That is, (81/80)<sup>''n''</sup> ~ {{monzo|-14 -19 19}}. In this case, ''n'' = 3''k'' - 19.


{| class="wikitable center-1 center-2 center-3"
{| class="wikitable center-1 center-2"
|+ Temperaments in the continuum
|+ Temperaments in the continuum
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''k''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
Line 16: Line 15:
! Monzo
! Monzo
|-
|-
| -19
| 0
| 0
| 19edo
| 19edo
Line 22: Line 20:
| {{monzo|-30 19}}
| {{monzo|-30 19}}
|-
|-
| -16
| 1
| 1
| Lalayo
| Lalayo
Line 28: Line 25:
| {{monzo|-26 15 1}}
| {{monzo|-26 15 1}}
|-
|-
| -13
| 2
| 2
| Lala-Yoyo
| Lala-Yoyo
Line 34: Line 30:
| {{monzo|-22 11 2}}
| {{monzo|-22 11 2}}
|-
|-
| -12
| 7/3
| 19 & 8ccc
|
| {{monzo|-62 29 7}}
|-
| -11
| 8/3
| 19 & 1cc
|
| {{monzo|-58 25 8}}
|-
| -10
| 3
| 3
| Latriyo
| Latriyo
Line 52: Line 35:
| {{monzo|-18 7 3}}
| {{monzo|-18 7 3}}
|-
|-
| -9
| 10/3
| 19 & 6c
|
| {{monzo|-50 17 10}}
|-
| -8
| 11/3
| 19 & 11c
|
| {{monzo|-46 13 11}}
|-
| -7
| 4
| 4
| [[Negri]]
| [[Negri]]
Line 70: Line 40:
| {{monzo|-14 3 4}}
| {{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
| 5
| [[Magic]]
| [[Magic]]
Line 88: Line 45:
| {{monzo|-10 -1 5}}
| {{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
| 6
| [[Hanson]]
| [[Hanson]]
Line 106: Line 50:
| {{monzo|-6 -5 6}}
| {{monzo|-6 -5 6}}
|-
|-
| 0
| 19/3
| [[Enneadecal]]
|
| {{monzo|-14 -19 19}}
|-
| 1
| 20/3
| [[Countermeantone]]
|
| {{monzo|-10 -23 20}}
|-
| 2
| 7
| 7
| [[Sensi]]
| [[Sensi]]
Line 124: Line 55:
| {{monzo|2 9 -7}}
| {{monzo|2 9 -7}}
|-
|-
| 3
| 22/3
| 19 & 169c
|
| {{monzo|2 31 -22}}
|-
| 4
| 23/3
| 19 & 162c
|
| {{monzo|-2 35 -23}}
|-
| 5
| 8
| 8
| [[Unicorn]]
| [[Unicorn]]
| [[1594323/1562500]]
| [[1594323/1562500]]
| {{monzo|-2 13 -8}}
| {{monzo|-2 13 -8}}
|-
| 9
| 19 & 51c
| [[129140163/125000000]]
| {{monzo|-6 17 -9}}
|-
|-
| …
| …
Line 147: Line 70:
| …
| …
|-
|-
| ∞
| ∞
| ∞
| [[Meantone]]
| [[Meantone]]
Line 156: Line 78:
Examples of temperaments with fractional values of ''n'':
Examples of temperaments with fractional values of ''n'':


* [[Parakleismic]] (''n'' = 0.5)
* [[Enneadecal]] (''k'' = 19/3 = 6.{{overline|3}})
* 19 & 506 (''n'' = 1/3 = 0.{{overline|3}})
* 19 & 506 (''k'' = 58/9 = 6.{{overline|4}})
* [[Parakleismic]] (''k'' = 6.5)
* [[Countermeantone]] (''k'' = 20/3 = 6.{{overline|6}})


== 19 & 506 ==
== 19 & 506 ==
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