Schismic–countercommatic equivalence continuum: Difference between revisions

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m FloraC moved page Schismic-counterpyth equivalence continuum to Schismic-countercommatic equivalence continuum: Name change following pythagorean -> compton
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The '''schismic-counterpyth equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[41-comma|counterpyth comma ({{monzo|65 -41}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.
The '''schismic-countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with the [[41-comma|Pythagorean countercomma ({{monzo| 65 -41 }})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.


All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo|65 -41}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[41edo]] (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 10.1575233481..., and temperaments having ''n'' near this value tend to be the most accurate ones.  
All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo| 65 -41 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[41edo]] (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 10.1575233481..., and temperaments having ''n'' near this value tend to be the most accurate ones.  


For a similar but perhaps more intuitive and practical concept, see [[Schismic-Pythagorean equivalence continuum]].
For a similar but perhaps more intuitive and practical concept, see [[Schismic-Pythagorean equivalence continuum]].
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|-
|-
| 0
| 0
| [[41-comma|Counterpyth]]
| [[Countercomp family #Countercomp|Countercomp]]
|  
|  
| {{monzo|65 -41}}
| {{monzo|65 -41}}
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== Rodan (5-limit) ==
== Rodan (5-limit) ==
{{See also|Gamelismic clan #Rodan}}
{{See also| Gamelismic clan #Rodan }}


[[Comma]]: 131072000/129140163
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 131072000/129140163


[[Mapping]]: [{{val| 1 1 -1 }}, {{val| 0 3 17 }}]
[[Mapping]]: [{{val| 1 1 -1 }}, {{val| 0 3 17 }}]


[[POTE generator]]: ~729/640 = 234.528
[[Optimal tuning]] ([[POTE]]): ~729/640 = 234.528


{{Val list|legend=1| 5, 31c, 36c, 41, 46, 87, 220, 307 }}
{{Val list|legend=1| 5, 31c, 36c, 41, 46, 87, 220, 307 }}
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{{See also|Breedsmic temperaments #Hemififths}}
{{See also|Breedsmic temperaments #Hemififths}}


[[Comma]]: 858993459200/847288609443
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 858993459200/847288609443


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


[[POTE generator]]: ~655360/531441 = 351.476
[[Optimal tuning]] ([[POTE]]): ~655360/531441 = 351.476


{{Val list|legend=1| 41, 58, 99, 239, 338, 915b, 1253bc }}
{{Val list|legend=1| 41, 58, 99, 239, 338, 915b, 1253bc }}
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== Kwai (5-limit) ==
== Kwai (5-limit) ==
{{See also|Mirkwai clan #Kwai}}
{{See also| Mirkwai clan #Kwai }}
 
[[Subgroup]]: 2.3.5


[[Comma]]: {{Monzo|50 -33 1}} = 5629499534213120/5559060566555523
[[Comma list]]: {{monzo| 50 -33 1 }} = 5629499534213120/5559060566555523


[[Mapping]]: [{{val| 1 2 16 }}, {{val| 0 -1 -33 }}]
[[Mapping]]: [{{val| 1 2 16 }}, {{val| 0 -1 -33 }}]


[[POTE generator]]: ~4/3 = 497.370 (or ~3/2 = 702.630)
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.630


{{Val list|legend=1| 41, 111, 152 }}
{{Val list|legend=1| 41, 111, 152 }}
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[[Badness]]: 0.636715
[[Badness]]: 0.636715


== Counterpyth ==
== Countercomp ==
:''See also: [[Counterpyth family]] and [[41-comma]]''
:''See also: [[Countercomp family]] and [[41-comma]]''
 
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo|65 -41}}
[[Comma list]]: {{monzo| 65 -41 }}


[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}]
[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}]


[[POTE generator]]: ~5/4 = 386.668
[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.668


{{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}
{{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}
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== Cotoneum (5-limit) ==
== Cotoneum (5-limit) ==
{{See also|Hemimage temperaments #Cotoneum}}
{{See also| Hemimage temperaments #Cotoneum }}


[[Comma]]: {{Monzo|80 -49 -1}}
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 80 -49 -1 }}


[[Mapping]]: [{{val| 1 2 -18 }}, {{val| 0 -1 49 }}]
[[Mapping]]: [{{val| 1 2 -18 }}, {{val| 0 -1 49 }}]


[[POTE generator]]: ~4/3 = 497.685 (or ~3/2 = 702.315)
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.315


{{Val list|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b }}
{{Val list|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b }}
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== Newt (5-limit) ==
== Newt (5-limit) ==
{{See also|Breedsmic temperaments #Newt}}
{{See also| Breedsmic temperaments #Newt }}
 
[[Subgroup]]: 2.3.5


[[Comma]]: {{Monzo|95 -57 -2}}
[[Comma list]]: {{monzo| 95 -57 -2 }}


[[Mapping]]: [{{val| 1 1 19 }}, {{val| 0 2 -57 }}]
[[Mapping]]: [{{val| 1 1 19 }}, {{val| 0 2 -57 }}]


[[POTE generator]]: ~{{Monzo|47 -28 -1}} = 351.114
[[Optimal tuning]] ([[POTE]]): ~{{monzo| 47 -28 -1 }} = 351.114


{{Val list|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc }}
{{Val list|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc }}

Revision as of 06:01, 22 December 2022

The schismic-countercommatic equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with the Pythagorean countercomma ([65 -41⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments.

All temperaments in the continuum satisfy (32805/32768)n ~ [65 -41⟩. Varying n results in different temperaments listed in the table below. It converges to schismic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 41edo (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 10.1575233481..., and temperaments having n near this value tend to be the most accurate ones.

For a similar but perhaps more intuitive and practical concept, see Schismic-Pythagorean equivalence continuum.

Temperaments in the continuum
n Temperament Comma
Ratio Monzo
-7 Merman 1121008359375 / 1099511627776 [-40 15 7⟩
-6 Ampersand 34171875 / 33554432 [-25 7 6⟩
-5 Magic 3125 / 3072 [-10 -1 5⟩
-4 Tetracot 20000 / 19683 [5 -9 4⟩
-3 Rodan 131072000 / 129140163 [20 -17 3⟩
-2 Hemififths 858993459200 / 847288609443 [35 -25 2⟩
-1 Kwai [50 -33 1⟩
0 Countercomp [65 -41⟩
1 Cotoneum [80 -49 -1⟩
2 Newt [95 -57 -2⟩
3 41&282 [110 -65 -3⟩
4 41&335 [125 -73 -4⟩
5 41&388 [140 -81 -5⟩
6 41&441 [155 -89 -6⟩
7 41&453 [170 -97 -7⟩
8 41&506 [185 -105 -8⟩
9 41&559 [200 -113 -9⟩
10 41&571 [215 -121 -10⟩
11 41&624 [-230 129 11⟩
12 41&677 [-245 137 12⟩
13 41&730 [-260 145 13⟩
… … … …
∞ Schismic 32805/32768 [-15 8 1⟩

Examples of temperaments with fractional values of n:

  • Septimin (n = -11/2 = -5.5)
  • Shibboleth (n = -9/2 = -4.5)
  • Pluto (n = -7/2 = -3.5)
  • 3737 & 5585 (n = 31/3 = 10.3)
  • 1277 & 2513 (n = 21/2 = 10.5)

Rodan (5-limit)

See also: Gamelismic clan § Rodan

Subgroup: 2.3.5

Comma list: 131072000/129140163

Mapping: [⟨1 1 -1], ⟨0 3 17]]

Optimal tuning (POTE): ~729/640 = 234.528

Template:Val list

Badness: 0.168264

Hemififths (5-limit)

See also: Breedsmic temperaments § Hemififths

Subgroup: 2.3.5

Comma list: 858993459200/847288609443

Mapping: [⟨1 1 -5], ⟨0 2 25]]

Optimal tuning (POTE): ~655360/531441 = 351.476

Template:Val list

Badness: 0.372848

Kwai (5-limit)

See also: Mirkwai clan § Kwai

Subgroup: 2.3.5

Comma list: [50 -33 1⟩ = 5629499534213120/5559060566555523

Mapping: [⟨1 2 16], ⟨0 -1 -33]]

Optimal tuning (POTE): ~3/2 = 702.630

Template:Val list

Badness: 0.636715

Countercomp

See also: Countercomp family and 41-comma

Subgroup: 2.3.5

Comma list: [65 -41⟩

Mapping: [⟨41 65 0], ⟨0 0 1]]

Optimal tuning (POTE): ~5/4 = 386.668

Template:Val list

Badness: 0.934310

Cotoneum (5-limit)

See also: Hemimage temperaments § Cotoneum

Subgroup: 2.3.5

Comma list: [80 -49 -1⟩

Mapping: [⟨1 2 -18], ⟨0 -1 49]]

Optimal tuning (POTE): ~3/2 = 702.315

Template:Val list

Badness: 1.240078

Newt (5-limit)

See also: Breedsmic temperaments § Newt

Subgroup: 2.3.5

Comma list: [95 -57 -2⟩

Mapping: [⟨1 1 19], ⟨0 2 -57]]

Optimal tuning (POTE): ~[47 -28 -1⟩ = 351.114

Template:Val list

Badness: 1.528465

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