Syntonic–kleismic equivalence continuum: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled
15,004 edits
m grammar
 
(35 intermediate revisions by 9 users not shown)
Line 1: Line 1:
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 }})]].  
{{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 }}).


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 {{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, {{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"
{| class="wikitable center-1 center-2"
|+ Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments in the continuum
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 14: Line 17:
|-
|-
| 0
| 0
| [[Enneadecal]]
| 19 &amp; 19c
|  
| [[19-comma|1162261467/1073741824]]
| {{monzo|-14 -19 19}}
| {{monzo|-30 19}}
|-
| 1
| 7c & 12c
| [[71744535/67108864]]
| {{monzo|-26 15 1}}
|-
| 2
| [[High badness temperaments #Hogzilla|Hogzilla]]
| [[4428675/4194304]]
| {{monzo|-22 11 2}}
|-
| 3
| [[High badness temperaments #Stump|Stump]]
| [[273375/262144]]
| {{monzo|-18 7 3}}
|-
| 4
| [[Negri]]
| [[16875/16384]]
| {{monzo|-14 3 4}}
|-
| 5
| [[Magic]]
| [[3125/3072]]
| {{monzo|-10 -1 5}}
|-
| 6
| [[Hanson]]
| [[15625/15552]]
| {{monzo|-6 -5 6}}
|-
| 7
| [[Sensipent family#Sensipent|Sensipent]]
| [[78732/78125]]
| {{monzo|2 9 -7}}
|-
| 8
| [[Unicorn]]
| [[1594323/1562500]]
| {{monzo|-2 13 -8}}
|-
| 9
| 19 &amp; 51c
| [[129140163/125000000]]
| {{monzo|-6 17 -9}}
|-
|-
| …
| …
Line 28: Line 76:
| {{monzo| -4 4 -1}}
| {{monzo| -4 4 -1}}
|}
|}
Examples of temperaments with fractional values of ''k'':
{| class="wikitable"
|+ style="font-size: 105%;" | Notable temperaments of fractional ''n''
|-
! Temperament !! ''n'' !! Comma
|-
| [[Unsmate]] || 9/2 = 4.5 || {{monzo| -24 2 9 }}
|-
| [[Sycamore]] || 11/2 = 5.5 || {{monzo| -16 -6 11 }}
|-
| [[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) ==
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -32 10 7 }} = 4613203125/4294967296
[[Mapping]]: [{{val| 1 -1 6 }}, {{val| 0 7 -10 }}]
[[POTE generator]]: ~675/512 = 442.2674 cents
{{Optimal ET sequence|legend=1| 8c, 11, 19 }}
[[Badness]]: 1.061630
[http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c]
== Counterhanson ==
{{See also| Ragismic microtemperaments #Counterkleismic }}
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -20 -24 25 }} = 298023223876953125/296148833645101056
[[Mapping]]: [{{val| 1 -5 -4 }}, {{val| 0 25 2 4}}]
[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.081
{{Optimal ET sequence|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}
[[Badness]]: 0.317551
== Countermeantone ==
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 10 23 -20 }} = 96402615118848/95367431640625
[[Mapping]]: [{{val| 1 10 12 }}, {{val| 0 -20 -23 }}]
[[Optimal tuning]] ([[POTE]]): ~104976/78125 = 504.913
{{Optimal ET sequence|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }}
[[Badness]]: 0.373477
== Mowgli ==
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 0 22 -15 }}
[[Mapping]]: [{{val| 1 0 0 }}, {{val| 0 15 22 }}]
[[Optimal tuning]] ([[POTE]]): ~27/25 = 126.7237
{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161 }}
[[Badness]]: 0.653871
== 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]]: {{monzo| -134 -185 184 }}
[[Mapping]]: [{{val| 1 50 51 }}, {{val| 0 -184 -185 }}]
[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.7501
{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}
[[Badness]]: 32.0


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

Latest revision as of 09:18, 14 July 2025

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 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 ([-14 -19 19⟩). That is, (81/80)k ~ [-14 -19 19⟩. In this case, k = 3n − 19.

Temperaments in the continuum
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⟩

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 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

Lalasepyo (8c & 11)

Subgroup: 2.3.5

Comma list: [-32 10 7⟩ = 4613203125/4294967296

Mapping: [⟨1 -1 6], ⟨0 7 -10]]

POTE generator: ~675/512 = 442.2674 cents

Optimal ET sequence: 8c, 11, 19

Badness: 1.061630

The temperament finder - 5-limit 19 & 8c

Counterhanson

See also: Ragismic microtemperaments § Counterkleismic

Subgroup: 2.3.5

Comma list: [-20 -24 25⟩ = 298023223876953125/296148833645101056

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: 0.317551

Countermeantone

Subgroup: 2.3.5

Comma list: [10 23 -20⟩ = 96402615118848/95367431640625

Mapping: [⟨1 10 12], ⟨0 -20 -23]]

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

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

Badness: 0.373477

Mowgli

Subgroup: 2.3.5

Comma list: [0 22 -15⟩

Mapping: [⟨1 0 0], ⟨0 15 22]]

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

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

Badness: 0.653871

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): ~6/5 = 315.7501

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

Badness: 32.0

Retrieved from "https://en.xen.wiki/index.php?title=Syntonic–kleismic_equivalence_continuum&oldid=204512"