Syntonic–kleismic equivalence continuum: Difference between revisions

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The '''syntonic-kleismic equivalence continuum''' (or '''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 }}).

The '''syntonic–kleismic equivalence continuum''' (or '''syntonic–enneadecal equivalence continuum''') is a [[equivalence continuum|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)''n'' ~ {{monzo|-30 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 6.376…, and temperaments having ''n'' near this value tend to be the most accurate ones.

All temperaments in the continuum satisfy {{nowrap|(81/80)''n'' ~ {{monzo|-30 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 6.376…, and temperaments having ''n'' near this value tend to be the most accurate ones.

This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, (81/80)''k'' ~ {{monzo| -14 -19 19 }}. In this case, ''k'' = 3''n'' - 19.

This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, {{nowrap|(81/80)''k'' ~ {{monzo| -14 -19 19 }}}}. In this case, {{nowrap|''k'' {{=}} 3''n'' − 19}}.

{| class="wikitable center-1 center-2"

|+ Temperaments in the continuum

|+ style="font-size: 105%;" | Temperaments in the continuum

|-

! rowspan="2" | ''n''

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|-

| 7

| [[~~Sensi~~]]

| [[Sensipent family#Sensipent|Sensipent]]

| [[78732/78125]]

| {{monzo|2 9 -7}}

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Examples of temperaments with fractional values of ''k'':

* 8c & ~~11 (~~''n'' ~~= 3.5)~~

{| class="wikitable"

* [[High badness temperaments#Unsmate|~~Unsmate]] (~~''n'' = 4.5)

|+ style="font-size: 105%;" | Notable temperaments of fractional ''n''

* [[~~Sycamore family#Sycamore|~~Sycamore]] ~~(''n''~~ = 5.5)

|-

* [[Counterhanson]] ~~(''n'' =~~ 25/4 = 6.25)

! Temperament !! ''n'' !! Comma

* [[Enneadecal]] ~~(''n'' =~~ 19/3 = 6.{{overline|3}})

|-

* [[~~Very high accuracy temperaments#Egads|~~Egads]] ~~(''n'' =~~ 51/8 = 6.375)

| [[Unsmate]] || 9/2 = 4.5 || {{monzo| -24 2 9 }}

* [[Acrokleismic]] ~~(''n'' =~~ 32/5 = 6.4)

|-

* [[Parakleismic]] ~~(''n''~~ = 6.5)

| [[Sycamore]] || 11/2 = 5.5 || {{monzo| -16 -6 11 }}

* [[Countermeantone]] ~~(''n'' =~~ 20/3 = 6.{{overline|6}})

|-

* [[Mowgli]] (''n'' = 7.5)

| [[Counterhanson]] || 25/4 = 6.25 || {{monzo| -20 -24 25 }}

|-

| [[Enneadecal]] || 19/3 = 6.{{overline|3}} || {{monzo| -14 -19 19 }}

|-

| [[Egads]] || 51/8 = 6.375 || {{monzo| -36 -52 51 }}

|-

| [[Acrokleismic]] || 32/5 = 6.4 || {{monzo| 22 33 -32 }}

|-

| [[Parakleismic]] || 13/2 = 6.5 || {{monzo| 8 14 -13 }}

|-

| [[Countermeantone]] || 20/3 = 6.{{overline|6}} || {{monzo| 10 23 -20 }}

|-

| [[Mowgli]] || 15/2 = 7.5 || {{monzo| 0 22 -15 }}

|}

== Negri (5-limit) ==

: ''For extensions, see [[Semaphoresmic clan #Negri]].''

The 5-limit version of negri tempers out the [[negri comma]], spliting a perfect fourth into four ~16/15 generators. It corresponds to {{nowrap| ''n'' {{=}} 4 }}. The only 7-limit extension that make any sense to use is to map the hemifourth to 7/6~8/7.

[[Subgroup]]: 2.3.5

[[Comma list]]: 16875/16384

: mapping generators: ~2, ~16/15

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1202.3403{{c}}, ~16/15 = 126.0002{{c}}

: [[error map]]: {{val| +2.340 -1.275 -3.633 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 125.6610{{c}}

: error map: {{val| 0.000 -4.599 -9.331 }}

[[Badness]] (Sintel): 2.04

== Lalasepyo (8c & 11) ==

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[[POTE generator]]: ~675/512 = 442.2674 cents

[[Badness]]: 1.061630

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[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.081

{{~~Val list~~|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}

{{Optimal ET sequence|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}

[[Badness]]: 0.317551

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[[Optimal tuning]] ([[POTE]]): ~104976/78125 = 504.913

{{~~Val list~~|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }}

{{Optimal ET sequence|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }}

[[Badness]]: 0.373477

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[[Optimal tuning]] ([[POTE]]): ~27/25 = 126.7237

[[Badness]]: 0.653871

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[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.7501

[[Badness]]: 32.0

@@ Line 1: / Line 1: @@
-The '''syntonic-kleismic equivalence continuum''' (or '''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 }}).
+{{Technical data page}}
+The '''syntonic&ndash;kleismic equivalence continuum''' (or '''syntonic&ndash;enneadecal equivalence continuum''') is a [[equivalence continuum|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|-30 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 6.376…, 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|-30 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 6.376…, and temperaments having ''n'' near this value tend to be the most accurate ones.
-This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, (81/80)<sup>''k''</sup> ~ {{monzo| -14 -19 19 }}. In this case, ''k'' = 3''n'' - 19.
+This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, {{nowrap|(81/80)<sup>''k''</sup> ~ {{monzo| -14 -19 19 }}}}. In this case, {{nowrap|''k'' {{=}} 3''n'' &minus; 19}}.
 {| class="wikitable center-1 center-2"
-|+ Temperaments in the continuum
+|+ style="font-size: 105%;" | Temperaments in the continuum
 |-
 ! rowspan="2" | ''n''
@@ Line 51: / Line 52: @@
 |-
 | 7
-| [[Sensi]]
+| [[Sensipent family#Sensipent|Sensipent]]
 | [[78732/78125]]
 | {{monzo|2 9 -7}}
@@ Line 78: / Line 79: @@
 Examples of temperaments with fractional values of ''k'':
-* 8c &amp; 11 (''n'' = 3.5)
+{| class="wikitable"
-* [[High badness temperaments#Unsmate|Unsmate]] (''n'' = 4.5)
+|+ style="font-size: 105%;" | Notable temperaments of fractional ''n''
-* [[Sycamore family#Sycamore|Sycamore]] (''n'' = 5.5)
+|-
-* [[Counterhanson]] (''n'' = 25/4 = 6.25)
+! Temperament !! ''n'' !! Comma
-* [[Enneadecal]] (''n'' = 19/3 = 6.{{overline|3}})
+|-
-* [[Very high accuracy temperaments#Egads|Egads]] (''n'' = 51/8 = 6.375)
+| [[Unsmate]] || 9/2 = 4.5 || {{monzo| -24 2 9 }}
-* [[Acrokleismic]] (''n'' = 32/5 = 6.4)
+|-
-* [[Parakleismic]] (''n'' = 6.5)
+| [[Sycamore]] || 11/2 = 5.5 || {{monzo| -16 -6 11 }}
-* [[Countermeantone]] (''n'' = 20/3 = 6.{{overline|6}})
+|-
-* [[Mowgli]] (''n'' = 7.5)
+| [[Counterhanson]] || 25/4 = 6.25 || {{monzo| -20 -24 25 }}
+|-
+| [[Enneadecal]] || 19/3 = 6.{{overline|3}} || {{monzo| -14 -19 19 }}
+|-
+| [[Egads]] || 51/8 = 6.375 || {{monzo| -36 -52 51 }}
+|-
+| [[Acrokleismic]] || 32/5 = 6.4 || {{monzo| 22 33 -32 }}
+|-
+| [[Parakleismic]] || 13/2 = 6.5 || {{monzo| 8 14 -13 }}
+|-
+| [[Countermeantone]] || 20/3 = 6.{{overline|6}} || {{monzo| 10 23 -20 }}
+|-
+| [[Mowgli]] || 15/2 = 7.5 || {{monzo| 0 22 -15 }}
+|}
+== Negri (5-limit) ==
+: ''For extensions, see [[Semaphoresmic clan #Negri]].''
+The 5-limit version of negri tempers out the [[negri comma]], spliting a perfect fourth into four ~16/15 generators. It corresponds to {{nowrap| ''n'' {{=}} 4 }}. The only 7-limit extension that make any sense to use is to map the hemifourth to 7/6~8/7.
+[[Subgroup]]: 2.3.5
+[[Comma list]]: 16875/16384
+{{Mapping|legend=1| 1 2 2 | 0 -4 3 }}
+: mapping generators: ~2, ~16/15
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1202.3403{{c}}, ~16/15 = 126.0002{{c}}
+: [[error map]]: {{val| +2.340 -1.275 -3.633 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 125.6610{{c}}
+: error map: {{val| 0.000 -4.599 -9.331 }}
+{{Optimal ET sequence|legend=1| 9, 10, 19, 67c, 86c, 105c }}
+[[Badness]] (Sintel): 2.04
 == Lalasepyo (8c &amp; 11) ==
@@ Line 98: / Line 135: @@
 [[POTE generator]]: ~675/512 = 442.2674 cents
-{{Val list|legend=1| 8c, 11, 19 }}
+{{Optimal ET sequence|legend=1| 8c, 11, 19 }}
 [[Badness]]: 1.061630
@@ Line 115: / Line 152: @@
 [[Optimal tuning]] ([[POTE]]): ~6/5 = 316.081
-{{Val list|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}
+{{Optimal ET sequence|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}
 [[Badness]]: 0.317551
@@ Line 128: / Line 165: @@
 [[Optimal tuning]] ([[POTE]]): ~104976/78125 = 504.913
-{{Val list|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }}
+{{Optimal ET sequence|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }}
 [[Badness]]: 0.373477
@@ Line 141: / Line 178: @@
 [[Optimal tuning]] ([[POTE]]): ~27/25 = 126.7237
-{{Val list|legend=1| 19, 85c, 104c, 123, 142, 161 }}
+{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161 }}
 [[Badness]]: 0.653871
@@ Line 158: / Line 195: @@
 [[Optimal tuning]] ([[CTE]]): ~6/5 = 315.7501
-{{Val list|legend=1| 19, …, 1600, 3219, 4819 }}
+{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}
 [[Badness]]: 32.0