Syntonic–kleismic equivalence continuum: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
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 }}).
The '''syntonic–kleismic equivalence continuum''' (or '''syntonic–enneadecal equivalence continuum''') is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|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 {{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.
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 [[support]]ed by [[19edo]] (due to it being the unique equal temperament that tempers out both commas and thus tempers out 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, {{nowrap|(81/80)<sup>''k''</sup> ~ {{monzo| -14 -19 19 }}}}. In this case, {{nowrap|''k'' {{=}} 3''n'' &minus; 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'' − 19 }}.


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ style="font-size: 105%;" | Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 17: Line 17:
|-
|-
| 0
| 0
| 19 &amp; 19c
| 19 & 19c
| [[19-comma|1162261467/1073741824]]
| [[19-comma|1162261467/1073741824]]
| {{monzo|-30 19}}
| {{Monzo| -30 19 }}
|-
|-
| 1
| 1
| 7c & 12c
| 7c & 12c
| [[71744535/67108864]]
| [[71744535/67108864]]
| {{monzo|-26 15 1}}
| {{Monzo| -26 15 1 }}
|-
|-
| 2
| 2
| [[High badness temperaments #Hogzilla|Hogzilla]]
| [[Hogzilla]]
| [[4428675/4194304]]
| [[4428675/4194304]]
| {{monzo|-22 11 2}}
| {{monzo|-22 11 2}}
|-
|-
| 3
| 3
| [[High badness temperaments #Stump|Stump]]
| [[Stump]]
| [[273375/262144]]
| [[273375/262144]]
| {{monzo|-18 7 3}}
| {{Monzo| -18 7 3 }}
|-
|-
| 4
| 4
| [[Negri]]
| [[Negri]]
| [[16875/16384]]
| [[16875/16384]]
| {{monzo|-14 3 4}}
| {{Monzo| -14 3 4 }}
|-
|-
| 5
| 5
| [[Magic]]
| [[Magic]]
| [[3125/3072]]
| [[3125/3072]]
| {{monzo|-10 -1 5}}
| {{Monzo| -10 -1 5 }}
|-
|-
| 6
| 6
| [[Hanson]]
| [[Hanson]]
| [[15625/15552]]
| [[15625/15552]]
| {{monzo|-6 -5 6}}
| {{Monzo| -6 -5 6 }}
|-
|-
| 7
| 7
| [[Sensipent family#Sensipent|Sensipent]]
| [[Sensipent]]
| [[78732/78125]]
| [[78732/78125]]
| {{monzo|2 9 -7}}
| {{Monzo| 2 9 -7 }}
|-
|-
| 8
| 8
| [[Unicorn]]
| [[Unicorn]]
| [[1594323/1562500]]
| [[1594323/1562500]]
| {{monzo|-2 13 -8}}
| {{Monzo| -2 13 -8 }}
|-
|-
| 9
| 9
| 19 &amp; 51c
| 19 & 51c
| [[129140163/125000000]]
| [[129140163/125000000]]
| {{monzo|-6 17 -9}}
| {{Monzo| -6 17 -9 }}
|-
|-
| …
| …
Line 74: Line 74:
| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1}}
| {{Monzo| -4 4 -1 }}
|}
|}


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{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
: mapping generators: ~2, ~6/5
: mapping generators: ~2, ~6/5


Line 134: Line 133:


{{Mapping|legend=1| 1 2 2 | 0 -4 3 }}
{{Mapping|legend=1| 1 2 2 | 0 -4 3 }}
: mapping generators: ~2, ~16/15
: mapping generators: ~2, ~16/15


Line 152: Line 150:
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -20 -24 25 }} = 298023223876953125/296148833645101056
[[Comma list]]: {{monzo| -20 -24 25 }}


[[Mapping]]: [{{val| 1 -5 -4 }}, {{val| 0 25 2 4}}]
{{Mapping|legend=1| 1 -5 -4 | 0 25 2 4}}


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


{{Optimal ET sequence|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 }}
Line 165: Line 163:
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 10 23 -20 }} = 96402615118848/95367431640625
[[Comma list]]: {{monzo| 10 23 -20 }}


[[Mapping]]: [{{val| 1 10 12 }}, {{val| 0 -20 -23 }}]
{{Mapping|legend=1| 1 10 12 | 0 -20 -23 }}


[[Optimal tuning]] ([[POTE]]): ~104976/78125 = 504.913
[[Optimal tuning]] ([[POTE]]): 1200.000{{c}}, ~104976/78125 = 504.913{{c}}


{{Optimal ET sequence|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 }}
Line 185: Line 183:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]] and [[POTE]]: ~2 = 1200.000, ~27/25 = 126.724
* [[CTE]] and [[POTE]]: ~2 = 1200.000{{c}}, ~27/25 = 126.724{{c}}


{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161, 303 }}
{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161, 303 }}
Line 191: Line 189:
[[Badness]] (Sintel): 15.3
[[Badness]] (Sintel): 15.3


== Lalasepyo (8c &amp; 11) ==
== Lalasepyo (8c & 11) ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -32 10 7 }} = 4613203125/4294967296
[[Comma list]]: 4613203125/4294967296


[[Mapping]]: [{{val| 1 -1 6 }}, {{val| 0 7 -10 }}]
{{Mapping|legend=1| 1 -1 6 | 0 7 -10 }}


[[POTE generator]]: ~675/512 = 442.2674 cents
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~675/512 = 442.2674{{c}}


{{Optimal ET sequence|legend=1| 8c, 11, 19 }}
{{Optimal ET sequence|legend=1| 8c, 11, 19 }}
Line 215: Line 213:
[[Comma list]]: {{monzo| -134 -185 184 }}
[[Comma list]]: {{monzo| -134 -185 184 }}


[[Mapping]]: [{{val| 1 50 51 }}, {{val| 0 -184 -185 }}]
{{Mapping|legend=1| 1 50 51 | 0 -184 -185 }}


[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.7501
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 315.7501{{c}}


{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}
{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}

Revision as of 08:00, 14 March 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The syntonic–kleismic equivalence continuum (or syntonic–enneadecal equivalence continuum) is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the 19-comma ([-30 19⟩).

All temperaments in the continuum satisfy (81/80)n ~ [-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 out both commas and thus tempers out 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 ([-14 -19 19⟩). That is, (81/80)k ~ [-14 -19 19⟩. In this case, k = 3n − 19.

Temperaments with integer n
n Temperament Comma
Ratio Monzo
0 19 & 19c 1162261467/1073741824 [-30 19⟩
1 7c & 12c 71744535/67108864 [-26 15 1⟩
2 Hogzilla 4428675/4194304 [-22 11 2⟩
3 Stump 273375/262144 [-18 7 3⟩
4 Negri 16875/16384 [-14 3 4⟩
5 Magic 3125/3072 [-10 -1 5⟩
6 Hanson 15625/15552 [-6 -5 6⟩
7 Sensipent 78732/78125 [2 9 -7⟩
8 Unicorn 1594323/1562500 [-2 13 -8⟩
9 19 & 51c 129140163/125000000 [-6 17 -9⟩
… … … …
∞ Meantone 81/80 [-4 4 -1⟩

Examples of temperaments with fractional values of k:

Notable temperaments of fractional n
Temperament n Comma
Unsmate 9/2 = 4.5 [-24 2 9⟩
Sycamore 11/2 = 5.5 [-16 -6 11⟩
Counterhanson 25/4 = 6.25 [-20 -24 25⟩
Enneadecal 19/3 = 6.3 [-14 -19 19⟩
Egads 51/8 = 6.375 [-36 -52 51⟩
Acrokleismic 32/5 = 6.4 [22 33 -32⟩
Parakleismic 13/2 = 6.5 [8 14 -13⟩
Countermeantone 20/3 = 6.6 [10 23 -20⟩
Mowgli 15/2 = 7.5 [0 22 -15⟩

Parakleismic

For extensions, see Ragismic microtemperaments #Parakleismic.

Subgroup: 2.3.5

Comma list: 1224440064/1220703125

Mapping: [⟨1 5 6], ⟨0 -13 -14]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1199.971 ¢, ~6/5 = 315.233 ¢
error map: ⟨-0.029 -0.127 +0.253]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 315.242 ¢
error map: ⟨0.000 -0.106 +0.293]

Optimal ET sequence: 19, 61, 80, 99, 118, 453, 571, 689, 1496

Badness (Sintel): 1.02

Negri

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 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: [⟨1 2 2], ⟨0 -4 3]]

mapping generators: ~2, ~16/15

Optimal tunings:

  • WE: ~2 = 1202.3403 ¢, ~16/15 = 126.0002 ¢
error map: ⟨+2.340 -1.275 -3.633]
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 125.6610 ¢
error map: ⟨0.000 -4.599 -9.331]

Optimal ET sequence: 9, 10, 19, 67c, 86c, 105c

Badness (Sintel): 2.04

Counterhanson

See also: Ragismic microtemperaments § Counterkleismic

Subgroup: 2.3.5

Comma list: [-20 -24 25⟩

Mapping: [⟨1 -5 -4], ⟨0 25 2 4]]

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

Optimal ET sequence: 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c

Badness (Sintel): 7.45

Countermeantone

Subgroup: 2.3.5

Comma list: [10 23 -20⟩

Mapping: [⟨1 10 12], ⟨0 -20 -23]]

Optimal tuning (POTE): 1200.000 ¢, ~104976/78125 = 504.913 ¢

Optimal ET sequence: 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c

Badness (Sintel): 8.76

Mowgli

For extensions, see Hemimean clan #Mowglic.

Subgroup: 2.3.5

Comma list: 31381059609/30517578125

Mapping: [⟨1 0 0], ⟨0 15 22]]

Optimal tunings:

  • CTE and POTE: ~2 = 1200.000 ¢, ~27/25 = 126.724 ¢

Optimal ET sequence: 19, 85c, 104c, 123, 142, 161, 303

Badness (Sintel): 15.3

Lalasepyo (8c & 11)

Subgroup: 2.3.5

Comma list: 4613203125/4294967296

Mapping: [⟨1 -1 6], ⟨0 7 -10]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~675/512 = 442.2674 ¢

Optimal ET sequence: 8c, 11, 19

Badness (Sintel): 24.9

The temperament finder - 5-limit 19 & 8c

Oviminor

See also: Ragismic microtemperaments § Oviminor

Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past egads, though it is less accurate.

Subgroup: 2.3.5

Comma list: [-134 -185 184⟩

Mapping: [⟨1 50 51], ⟨0 -184 -185]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 315.7501 ¢

Optimal ET sequence: 19, …, 1600, 3219, 4819

Badness (Sintel): 750.8

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