Schismic–countercommatic equivalence continuum: Difference between revisions

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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.
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]] [[microtemperament]]s [[support]]ed by [[41edo]].


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 [[tempering out|tempers out]] both commas and thus tempers out 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]].


{| class="wikitable center-1 center-2"
{| class="wikitable center-1"
|+ Temperaments in the continuum
|+ Temperaments of integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 16: Line 16:
|-
|-
| -7
| -7
| [[Marvel temperaments #Merman|Merman]]
| [[Merman]]
| 1121008359375 / 1099511627776
| 1121008359375 / 1099511627776
| {{monzo|-40 15 7}}
| {{monzo| -40 15 7 }}
|-
|-
| -6
| -6
| [[Syntonic-31 equivalence continuum|Ampersand]]
| [[Syntonic-31 equivalence continuum|Ampersand]]
| 34171875 / 33554432
| 34171875 / 33554432
| {{monzo|-25 7 6}}
| {{monzo| -25 7 6 }}
|-
|-
| -5
| -5
| [[Magic family|Magic]]
| [[Magic]]
| 3125 / 3072
| 3125 / 3072
| {{monzo|-10 -1 5}}
| {{monzo| -10 -1 5 }}
|-
|-
| -4
| -4
| [[Tetracot family|Tetracot]]
| [[Tetracot]]
| 20000 / 19683
| 20000 / 19683
| {{monzo|5 -9 4}}
| {{monzo| 5 -9 4 }}
|-
|-
| -3
| -3
| [[Gamelismic clan #Rodan|Rodan]]
| [[Rodan]]
| 131072000 / 129140163
| 131072000 / 129140163
| {{monzo|20 -17 3}}
| {{monzo| 20 -17 3 }}
|-
|-
| -2
| -2
| [[Breedsmic temperaments #Hemififths|Hemififths]]
| [[Hemififths]]
| 858993459200 / 847288609443
| 858993459200 / 847288609443
| {{monzo|35 -25 2}}
| {{monzo| 35 -25 2 }}
|-
|-
| -1
| -1
| [[Mirkwai clan #Kwai|Kwai]]
| [[Kwai]]
|  
| (32 digits)
| {{monzo|50 -33 1}}
| {{monzo| 50 -33 1 }}
|-
|-
| 0
| 0
| [[Countercomp family #Countercomp|Countercomp]]
| [[Countercomp]]
|  
| (40 digits)
| {{monzo|65 -41}}
| {{monzo| 65 -41 }}
|-
|-
| 1
| 1
| [[Hemimage temperaments #Cotoneum|Cotoneum]]
| [[Cotoneum]]
|  
| (50 digits)
| {{monzo|80 -49 -1}}
| {{monzo| 80 -49 -1 }}
|-
|-
| 2
| 2
| [[Breedsmic temperaments #Newt|Newt]]
| [[Newt]]
|  
| (58 digits)
| {{monzo|95 -57 -2}}
| {{monzo| 95 -57 -2 }}
|-
|-
| 3
| 3
| 41&amp;282
| 41 &amp; 282
|  
| (68 digits)
| {{monzo|110 -65 -3}}
| {{monzo| 110 -65 -3 }}
|-
|-
| 4
| 4
| 41&amp;335
| 41 &amp; 335
|  
| (76 digits)
| {{monzo|125 -73 -4}}
| {{monzo| 125 -73 -4 }}
|-
|-
| 5
| 5
| 41&amp;388
| 41 &amp; 388
|  
| (86 digits)
| {{monzo|140 -81 -5}}
| {{monzo| 140 -81 -5 }}
|-
|-
| 6
| 6
| 41&amp;441
| 41 &amp; 441
|  
| (94 digits)
| {{monzo|155 -89 -6}}
| {{monzo| 155 -89 -6 }}
|-
|-
| 7
| 7
| 41&amp;453
| 41 &amp; 453
|  
| (104 digits)
| {{monzo|170 -97 -7}}
| {{monzo| 170 -97 -7 }}
|-
|-
| 8
| 8
| 41&amp;506
| 41 &amp; 506
|  
| (112 digits)
| {{monzo|185 -105 -8}}
| {{monzo| 185 -105 -8 }}
|-
|-
| 9
| 9
| 41&amp;559
| 41 &amp; 559
|  
| (122 digits)
| {{monzo|200 -113 -9}}
| {{monzo| 200 -113 -9 }}
|-
|-
| 10
| 10
| 41&amp;571
| 41 &amp; 571
|  
| (130 digits)
| {{monzo|215 -121 -10}}
| {{monzo| 215 -121 -10 }}
|-
|-
| 11
| 11
| 41&amp;624
| 41 &amp; 624
|  
| (140 digits)
| {{monzo|-230 129 11}}
| {{monzo| -230 129 11 }}
|-
|-
| 12
| 12
| 41&amp;677
| 41 &amp; 677
|  
| (148 digits)
| {{monzo|-245 137 12}}
| {{monzo| -245 137 12 }}
|-
|-
| 13
| 13
| 41&amp;730
| 41 &amp; 730
|  
| (158 digits)
| {{monzo|-260 145 13}}
| {{monzo| -260 145 13 }}
|-
|-
| …
| …
Line 128: Line 128:
| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1}}
| {{monzo| -15 8 1 }}
|}
|}


Line 145: Line 145:
[[Comma list]]: 131072000/129140163
[[Comma list]]: 131072000/129140163


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


[[Optimal tuning]] ([[POTE]]): ~729/640 = 234.528
[[Optimal tuning]] ([[POTE]]): ~729/640 = 234.528
Line 160: Line 160:
[[Comma list]]: 858993459200/847288609443
[[Comma list]]: 858993459200/847288609443


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


[[Optimal tuning]] ([[POTE]]): ~655360/531441 = 351.476
[[Optimal tuning]] ([[POTE]]): ~655360/531441 = 351.476
Line 175: Line 175:
[[Comma list]]: {{monzo| 50 -33 1 }} = 5629499534213120/5559060566555523
[[Comma list]]: {{monzo| 50 -33 1 }} = 5629499534213120/5559060566555523


[[Mapping]]: [{{val| 1 0 -50 }}, {{val| 0 1 33 }}]
{{Mapping|legend=1| 1 0 -50 | 0 1 33 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.630
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.630
Line 190: Line 190:
[[Comma list]]: {{monzo| 65 -41 }}
[[Comma list]]: {{monzo| 65 -41 }}


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


[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.668
[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.668
Line 199: Line 199:


== Cotoneum (5-limit) ==
== Cotoneum (5-limit) ==
{{See also| Hemimage temperaments #Cotoneum }}
{{See also| Garischismic clan #Cotoneum }}


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 205: Line 205:
[[Comma list]]: {{monzo| 80 -49 -1 }}
[[Comma list]]: {{monzo| 80 -49 -1 }}


[[Mapping]]: [{{val| 1 0 80 }}, {{val| 0 1 -49 }}]
{{Mapping|legend=1| 1 0 80 | 0 1 -49 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.315
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.315
Line 220: Line 220:
[[Comma list]]: {{monzo| 95 -57 -2 }}
[[Comma list]]: {{monzo| 95 -57 -2 }}


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


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

Revision as of 10:17, 27 July 2024

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 supported by 41edo.

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 out both commas and thus tempers out 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 of integer n
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 (32 digits) [50 -33 1⟩
0 Countercomp (40 digits) [65 -41⟩
1 Cotoneum (50 digits) [80 -49 -1⟩
2 Newt (58 digits) [95 -57 -2⟩
3 41 & 282 (68 digits) [110 -65 -3⟩
4 41 & 335 (76 digits) [125 -73 -4⟩
5 41 & 388 (86 digits) [140 -81 -5⟩
6 41 & 441 (94 digits) [155 -89 -6⟩
7 41 & 453 (104 digits) [170 -97 -7⟩
8 41 & 506 (112 digits) [185 -105 -8⟩
9 41 & 559 (122 digits) [200 -113 -9⟩
10 41 & 571 (130 digits) [215 -121 -10⟩
11 41 & 624 (140 digits) [-230 129 11⟩
12 41 & 677 (148 digits) [-245 137 12⟩
13 41 & 730 (158 digits) [-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

Optimal ET sequence: 5, 31c, 36c, 41, 46, 87, 220, 307

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

Optimal ET sequence: 41, 58, 99, 239, 338, 915b, 1253bc

Badness: 0.372848

Kwai (5-limit)

See also: Mirkwai clan § Kwai

Subgroup: 2.3.5

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

Mapping: [⟨1 0 -50], ⟨0 1 33]]

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

Optimal ET sequence: 41, 111, 152

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

Optimal ET sequence: 41, 123, 164, 205, 369, 574, 779, 2132bc

Badness: 0.934310

Cotoneum (5-limit)

See also: Garischismic clan § Cotoneum

Subgroup: 2.3.5

Comma list: [80 -49 -1⟩

Mapping: [⟨1 0 80], ⟨0 1 -49]]

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

Optimal ET sequence: 41, 135c, 176, 217, 475, 1167, 1642, 2117b

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

Optimal ET sequence: 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc

Badness: 1.528465

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