Syntonic–kleismic equivalence continuum: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,044 edits
No edit summary
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
No edit summary
Tags: Mobile edit Mobile web edit
 
(21 intermediate revisions by 6 users not shown)
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–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 (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, (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'' − 19 }}.


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 16: Line 17:
|-
|-
| 0
| 0
| 19 &amp; 19c
| [[Graywood]]
| [[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]]
| [[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
| [[Xenial]]
| [[129140163/125000000]]
| [[129140163/125000000]]
| {{monzo|-6 17 -9}}
| {{Monzo| -6 17 -9 }}
|-
|-
| …
| …
Line 73: Line 74:
| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1}}
| {{Monzo| -4 4 -1 }}
|}
|}


Line 79: Line 80:


{| class="wikitable"
{| class="wikitable"
|+ Notable temperaments of fractional ''n''
|+ style="font-size: 105%;" | Notable temperaments of fractional ''n''
|-
|-
! Temperament !! ''n'' !! Comma
! Temperament !! ''n'' !! Comma
Line 102: Line 103:
|}
|}


== Lalasepyo (8c &amp; 11) ==
== Graywood ==
Named by [[CompactStar]] in 2024, graywood tempers out the [[19-comma]], corresponding to {{nowrap| ''n'' {{=}} 0 }}. It takes [[19edo]]'s closed [[circle of fifths]], but adds an independent generator for [[prime interval|prime]] [[5/1|5]]. 19 is the only equal temperament that makes it to the optimal ET sequence as all the small edo tunings, e.g. [[38edo|38c-edo]] or [[57edo|57c-edo]], are not nearly as accurate as 19 itself.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 1162261467/1073741824
 
{{Mapping|legend=1| 19 30 0 | 0 0 1 }}
: mapping generators: ~2187/2048, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2187/2048 = 63.2773{{c}}, ~5/4 = 381.7568{{c}}
: [[error map]]: {{val| +2.268 -3.637 -0.020 }}
* [[CWE]]: ~2187/2048 = 63.1579{{c}}, ~5/4 = 382.7889{{c}}
: error map: {{val| 0.000 -7.218 -3.525 }}
 
{{Optimal ET sequence|legend=1| 19 }}
 
[[Badness]] (Sintel): 32.4
 
== Hogzilla ==
: ''For extensions, see [[Semaphoresmic clan #Helayo]].''
 
Hogzilla is similar to [[godzilla]] in that it is generated by a [[semitwelfth]]. It corresponds to {{nowrap| ''n'' {{=}} 2 }}.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 4428675/4194304
 
{{Mapping|legend=1| 1 0 11 | 0 2 -11 }}
: mapping generators: ~2, ~2048/1215
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.5490{{c}}, ~2048/1215 = 949.3637{{c}}
: [[error map]]: {{val| +2.549 -3.228 -1.275 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2048/1215 = 947.2462{{c}}
: error map: {{val| 0.000 -7.463 -6.022 }}
 
{{Optimal ET sequence|legend=1| 14, 19 }}
 
[[Badness]] (Sintel): 9.96
 
== Stump ==
: ''For extensions, see [[Marvel temperaments #Triton]] and [[Sensamagic clan #Pycnic]].''
 
Stump splits the [[3/1|3rd]] [[harmonic]] into three equal parts, each for [[~]][[64/45]]. It corresponds to {{nowrap| ''n'' {{=}} 3 }}.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 273375/262144
 
{{Mapping|legend=1| 1 0 6 | 0 3 -7 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.6288{{c}}, ~64/45 = 633.1214{{c}}
: [[error map]]: {{val| +2.629 -2.591 -2.391 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/45 = 631.6779{{c}}
: error map: {{val| 0.000 -6.921 -8.059 }}
 
{{Optimal ET sequence|legend=1| 17, 19, 207bbccc }}
 
[[Badness]] (Sintel): 4.71
 
== Negri (5-limit) ==
{{Main| 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 {{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
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -32 10 7 }} = 4613203125/4294967296
[[Comma list]]: 16875/16384


[[Mapping]]: [{{val| 1 -1 6 }}, {{val| 0 7 -10 }}]
{{Mapping|legend=1| 1 2 2 | 0 -4 3 }}
: mapping generators: ~2, ~16/15


[[POTE generator]]: ~675/512 = 442.2674 cents
[[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
 
== Xenial ==
: ''For extensions, see [[Starling temperaments #Xenial]] and [[Sensamagic clan #Xenia]].''
 
Named by [[User:Xenllium|Xenllium]] in 2026, xenial splits the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. It corresponds to {{nowrap| ''n'' {{=}} 9 }}. Its [[ploidacot]] is zeta-enneacot, and from this it derives its name.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 129140163/125000000
 
{{Mapping|legend=1| 1 -6 -12 | 0 9 17 }}
: mapping generators: ~2, ~9/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2802{{c}}, ~9/5 = 1011.2914{{c}}
: [[error map]]: {{val| +0.280 -2.013 +2.278 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.0762{{c}}
: error map: {{val| 0.000 -2.269 +1.982 }}
 
{{Optimal ET sequence|legend=1| 19, 70, 89, 108, 127 }}
 
[[Badness]] (Sintel): 8.84
 
== Lalasepyo (8c & 11) ==
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 4613203125/4294967296
 
{{Mapping|legend=1| 1 -1 6 | 0 7 -10 }}
: mapping generators: ~2, ~675/512
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.5641{{c}}, ~675/512 = 443.2124{{c}}
: [[error map]]: {{val| +2.564 -2.033 -3.053 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~675/512 = 442.2692{{c}}
: error map: {{val| 0.000 -6.071 -9.006 }}


{{Optimal ET sequence|legend=1| 8c, 11, 19 }}
{{Optimal ET sequence|legend=1| 8c, 11, 19 }}


[[Badness]]: 1.061630
[[Badness]] (Sintel): 24.9


[http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c]
[http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c]


== Counterhanson ==
== Unsmate ==
{{See also| Ragismic microtemperaments #Counterkleismic }}
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 17578125/16777216
 
{{Mapping|legend=1| 1 -6 4 | 0 9 -2 }}
: mapping generators: ~2, ~1875/1024
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.8892{{c}}, ~1875/1024 = 1012.5428{{c}}
: [[error map]]: {{val| +1.889 -0.405 -3.843 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1875/1024 = 1011.0348{{c}}
: error map: {{val| 0.000 -2.642 -8.383 }}
 
{{Optimal ET sequence|legend=1| 6b, 13, 19, 89c, 108c, 127c, 146cc }}
 
[[Badness]] (Sintel): 10.8
 
== Parakleismic ==
{{Main| Parakleismic }}
: ''For extensions, see [[Ragismic microtemperaments #Parakleismic]] and [[Starling temperaments #Paraguay]].''
 
The 5-limit version of parakleismic tempers out the [[parakleisma]]. It corresponds to {{nowrap| ''n'' {{=}} 13/2 }}, and 13 generator steps give the interval class of [[3/1|3]].
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 1224440064/1220703125
 
{{Mapping|legend=1| 1 -8 -8 | 0 13 14 }}
: mapping generators: ~2, ~5/3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.971{{c}}, ~5/3 = 884.7383{{c}}
: [[error map]]: {{val| -0.029 -0.127 +0.253 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/3 = 884.8576{{c}}
: error map: {{val| 0.000 -0.106 +0.293 }}
 
{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
 
[[Badness]] (Sintel): 1.02
 
== Mowgli ==
: ''For extensions, see [[Hemimean clan #Mowglic]].''
 
[[TE]], [[CTE]] and [[POTE]] coincide at 126.7237{{c}} with pure octaves since prime 2 is not involved in the comma to begin with.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 31381059609/30517578125
 
{{Mapping|legend=1| 1 0 0 | 0 15 22 }}
: mapping generators: ~2, ~27/25
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9478{{c}}, ~27/25 = 126.7236{{c}}
: [[error map]]: {{val| -0.001 -1.100 +1.606 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 126.7237{{c}}
: error map: {{val| 0.000 -1.100 +1.607 }}
 
{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161, 303 }}
 
[[Badness]] (Sintel): 15.3
 
== Enneadecal (5-limit) ==
: ''For extensions, see [[Ragismic microtemperaments #Enneadecal]].''
 
The 5-limit version of enneadecal tempers out the [[enneadeca]], which simply equates a stack of nineteen [[6/5]] minor thirds with five [[2/1|octaves]]. It corresponds to {{nowrap| ''n'' {{=}} 19/3 }}, with a 19th-octave period and a generator of a [[3/2|perfect fifth]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -20 -24 25 }} = 298023223876953125/296148833645101056
[[Comma list]]: 19073486328125/19042491875328


[[Mapping]]: [{{val| 1 -5 -4 }}, {{val| 0 25 2 4}}]
{{Mapping|legend=1| 19 0 14 | 0 1 1 }}
: mapping generators: ~648/625, ~3


[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.081
[[Optimal tuning]]s:
* [[WE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9861{{c}}
: [[error map]]: {{val| +0.013 +0.044 -0.095 }}
* [[CWE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9900{{c}}
: error map: {{val| 0.000 -0.035 -0.113 }}


{{Optimal ET sequence|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}
{{Optimal ET sequence|legend=1| 19, 95, 114, 133, 152, 171, 323, 494, 665, 1159, 1824, 2983, 7125c }}


[[Badness]]: 0.317551
[[Badness]] (Sintel): 1.12


== Countermeantone ==
== Countermeantone ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 10 23 -20 }} = 96402615118848/95367431640625
[[Comma list]]: {{monzo| 10 23 -20 }}
 
{{Mapping|legend=1| 1 -10 -11 | 0 20 23 }}
: mapping generators: ~2, ~78125/52488


[[Mapping]]: [{{val| 1 10 12 }}, {{val| 0 -20 -23 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9478{{c}}, ~78125/52488 = 695.0566{{c}}
: [[error map]]: {{val| -0.052 -0.301 +0.562 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~78125/52488 = 695.0846{{c}}
: error map: {{val| 0.000 -0.264 +0.631 }}


[[Optimal tuning]] ([[POTE]]): ~104976/78125 = 504.913
{{Optimal ET sequence|legend=1| 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c }}


{{Optimal ET sequence|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }}
[[Badness]] (Sintel): 8.76


[[Badness]]: 0.373477
== Counterhanson ==
: ''For extensions, see [[Ragismic microtemperaments #Counterkleismic]].''


== Mowgli ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 0 22 -15 }}
[[Comma list]]: {{monzo| -20 -24 25 }}


[[Mapping]]: [{{val| 1 0 0 }}, {{val| 0 15 22 }}]
{{Mapping|legend=1| 1 -5 -4 | 0 25 24 }}
: mapping generators: ~2, ~6/5


[[Optimal tuning]] ([[POTE]]): ~27/25 = 126.7237
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0419{{c}}, ~6/5 = 316.0916{{c}}
: [[error map]]: {{val| +0.042 +0.126 -0.282 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0021{{c}}
: error map: {{val| 0.000 +0.097 -0.344 }}


{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161 }}
{{Optimal ET sequence|legend=1| 19, …, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}


[[Badness]]: 0.653871
[[Badness]] (Sintel): 7.45


== Oviminor ==
== Oviminor ==
Line 167: Line 363:
[[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 -134 -134 | 0 184 185 }}
: mapping generators: ~2, ~5/3


[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.7501
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0094{{c}}, ~5/3 = 884.2568{{c}}
: [[error map]]: {{val| +0.009 +0.033 -0.069 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2499{{c}}
: error map: {{val| 0.000 +0.026 -0.083 }}


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


[[Badness]]: 32.0
[[Badness]] (Sintel): 751


[[Category:19edo]]
[[Category:19edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Latest revision as of 09:57, 14 May 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 Graywood 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 Xenial 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⟩

Graywood

Named by CompactStar in 2024, graywood tempers out the 19-comma, corresponding to n = 0. It takes 19edo's closed circle of fifths, but adds an independent generator for prime 5. 19 is the only equal temperament that makes it to the optimal ET sequence as all the small edo tunings, e.g. 38c-edo or 57c-edo, are not nearly as accurate as 19 itself.

Subgroup: 2.3.5

Comma list: 1162261467/1073741824

Mapping: [⟨19 30 0], ⟨0 0 1]]

mapping generators: ~2187/2048, ~5

Optimal tunings:

  • WE: ~2187/2048 = 63.2773 ¢, ~5/4 = 381.7568 ¢
error map: ⟨+2.268 -3.637 -0.020]
  • CWE: ~2187/2048 = 63.1579 ¢, ~5/4 = 382.7889 ¢
error map: ⟨0.000 -7.218 -3.525]

Optimal ET sequence: 19

Badness (Sintel): 32.4

Hogzilla

For extensions, see Semaphoresmic clan #Helayo.

Hogzilla is similar to godzilla in that it is generated by a semitwelfth. It corresponds to n = 2.

Subgroup: 2.3.5

Comma list: 4428675/4194304

Mapping: [⟨1 0 11], ⟨0 2 -11]]

mapping generators: ~2, ~2048/1215

Optimal tunings:

  • WE: ~2 = 1202.5490 ¢, ~2048/1215 = 949.3637 ¢
error map: ⟨+2.549 -3.228 -1.275]
  • CWE: ~2 = 1200.0000 ¢, ~2048/1215 = 947.2462 ¢
error map: ⟨0.000 -7.463 -6.022]

Optimal ET sequence: 14, 19

Badness (Sintel): 9.96

Stump

For extensions, see Marvel temperaments #Triton and Sensamagic clan #Pycnic.

Stump splits the 3rd harmonic into three equal parts, each for ~64/45. It corresponds to n = 3.

Subgroup: 2.3.5

Comma list: 273375/262144

Mapping: [⟨1 0 6], ⟨0 3 -7]]

Optimal tunings:

  • WE: ~2 = 1202.6288 ¢, ~64/45 = 633.1214 ¢
error map: ⟨+2.629 -2.591 -2.391]
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 631.6779 ¢
error map: ⟨0.000 -6.921 -8.059]

Optimal ET sequence: 17, 19, 207bbccc

Badness (Sintel): 4.71

Negri (5-limit)

Main article: 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

Xenial

For extensions, see Starling temperaments #Xenial and Sensamagic clan #Xenia.

Named by Xenllium in 2026, xenial splits the perfect eleventh into nine equal parts, each for ~10/9. It corresponds to n = 9. Its ploidacot is zeta-enneacot, and from this it derives its name.

Subgroup: 2.3.5

Comma list: 129140163/125000000

Mapping: [⟨1 -6 -12], ⟨0 9 17]]

mapping generators: ~2, ~9/5

Optimal tunings:

  • WE: ~2 = 1200.2802 ¢, ~9/5 = 1011.2914 ¢
error map: ⟨+0.280 -2.013 +2.278]
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.0762 ¢
error map: ⟨0.000 -2.269 +1.982]

Optimal ET sequence: 19, 70, 89, 108, 127

Badness (Sintel): 8.84

Lalasepyo (8c & 11)

Subgroup: 2.3.5

Comma list: 4613203125/4294967296

Mapping: [⟨1 -1 6], ⟨0 7 -10]]

mapping generators: ~2, ~675/512

Optimal tunings:

  • WE: ~2 = 1202.5641 ¢, ~675/512 = 443.2124 ¢
error map: ⟨+2.564 -2.033 -3.053]
  • CWE: ~2 = 1200.0000 ¢, ~675/512 = 442.2692 ¢
error map: ⟨0.000 -6.071 -9.006]

Optimal ET sequence: 8c, 11, 19

Badness (Sintel): 24.9

The temperament finder - 5-limit 19 & 8c

Unsmate

Subgroup: 2.3.5

Comma list: 17578125/16777216

Mapping: [⟨1 -6 4], ⟨0 9 -2]]

mapping generators: ~2, ~1875/1024

Optimal tunings:

  • WE: ~2 = 1201.8892 ¢, ~1875/1024 = 1012.5428 ¢
error map: ⟨+1.889 -0.405 -3.843]
  • CWE: ~2 = 1200.0000 ¢, ~1875/1024 = 1011.0348 ¢
error map: ⟨0.000 -2.642 -8.383]

Optimal ET sequence: 6b, 13, 19, 89c, 108c, 127c, 146cc

Badness (Sintel): 10.8

Parakleismic

Main article: Parakleismic
For extensions, see Ragismic microtemperaments #Parakleismic and Starling temperaments #Paraguay.

The 5-limit version of parakleismic tempers out the parakleisma. It corresponds to n = 13/2, and 13 generator steps give the interval class of 3.

Subgroup: 2.3.5

Comma list: 1224440064/1220703125

Mapping: [⟨1 -8 -8], ⟨0 13 14]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1199.971 ¢, ~5/3 = 884.7383 ¢
error map: ⟨-0.029 -0.127 +0.253]
  • CWE: ~2 = 1200.000 ¢, ~5/3 = 884.8576 ¢
error map: ⟨0.000 -0.106 +0.293]

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

Badness (Sintel): 1.02

Mowgli

For extensions, see Hemimean clan #Mowglic.

TE, CTE and POTE coincide at 126.7237 ¢ with pure octaves since prime 2 is not involved in the comma to begin with.

Subgroup: 2.3.5

Comma list: 31381059609/30517578125

Mapping: [⟨1 0 0], ⟨0 15 22]]

mapping generators: ~2, ~27/25

Optimal tunings:

  • WE: ~2 = 1199.9478 ¢, ~27/25 = 126.7236 ¢
error map: ⟨-0.001 -1.100 +1.606]
  • CWE: ~2 = 1200.0000 ¢, ~27/25 = 126.7237 ¢
error map: ⟨0.000 -1.100 +1.607]

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

Badness (Sintel): 15.3

Enneadecal (5-limit)

For extensions, see Ragismic microtemperaments #Enneadecal.

The 5-limit version of enneadecal tempers out the enneadeca, which simply equates a stack of nineteen 6/5 minor thirds with five octaves. It corresponds to n = 19/3, with a 19th-octave period and a generator of a perfect fifth.

Subgroup: 2.3.5

Comma list: 19073486328125/19042491875328

Mapping: [⟨19 0 14], ⟨0 1 1]]

mapping generators: ~648/625, ~3

Optimal tunings:

  • WE: ~648/625 = 63.1579 ¢, ~3/2 = 701.9861 ¢
error map: ⟨+0.013 +0.044 -0.095]
  • CWE: ~648/625 = 63.1579 ¢, ~3/2 = 701.9900 ¢
error map: ⟨0.000 -0.035 -0.113]

Optimal ET sequence: 19, 95, 114, 133, 152, 171, 323, 494, 665, 1159, 1824, 2983, 7125c

Badness (Sintel): 1.12

Countermeantone

Subgroup: 2.3.5

Comma list: [10 23 -20⟩

Mapping: [⟨1 -10 -11], ⟨0 20 23]]

mapping generators: ~2, ~78125/52488

Optimal tunings:

  • WE: ~2 = 1199.9478 ¢, ~78125/52488 = 695.0566 ¢
error map: ⟨-0.052 -0.301 +0.562]
  • CWE: ~2 = 1200.0000 ¢, ~78125/52488 = 695.0846 ¢
error map: ⟨0.000 -0.264 +0.631]

Optimal ET sequence: 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c

Badness (Sintel): 8.76

Counterhanson

For extensions, see Ragismic microtemperaments #Counterkleismic.

Subgroup: 2.3.5

Comma list: [-20 -24 25⟩

Mapping: [⟨1 -5 -4], ⟨0 25 24]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1200.0419 ¢, ~6/5 = 316.0916 ¢
error map: ⟨+0.042 +0.126 -0.282]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0021 ¢
error map: ⟨0.000 +0.097 -0.344]

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

Badness (Sintel): 7.45

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 -134 -134], ⟨0 184 185]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1200.0094 ¢, ~5/3 = 884.2568 ¢
error map: ⟨+0.009 +0.033 -0.069]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.2499 ¢
error map: ⟨0.000 +0.026 -0.083]

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

Badness (Sintel): 751

Retrieved from "https://en.xen.wiki/index.php?title=Syntonic–kleismic_equivalence_continuum&oldid=230326"