Schismic–countercommatic equivalence continuum: Difference between revisions

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* Etc.
* Etc.


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–commatic equivalence continuum]].


{| class="wikitable center-1"
{| class="wikitable center-1"
Line 22: Line 22:
| −7
| −7
| [[Merman]]
| [[Merman]]
| 1121008359375 / 1099511627776
| 1121008359375/1099511627776
| {{monzo| -40 15 7 }}
| {{Monzo| -40 15 7 }}
|-
|-
| −6
| −6
| [[Syntonic–31 equivalence continuum|Ampersand]]
| [[Ampersand]]
| 34171875 / 33554432
| 34171875/33554432
| {{monzo| -25 7 6 }}
| {{Monzo| -25 7 6 }}
|-
|-
| −5
| −5
| [[Magic]]
| [[Magic]]
| 3125 / 3072
| 3125/3072
| {{monzo| -10 -1 5 }}
| {{Monzo| -10 -1 5 }}
|-
|-
| −4
| −4
| [[Tetracot]]
| [[Tetracot]]
| 20000 / 19683
| 20000/19683
| {{monzo| 5 -9 4 }}
| {{monzo| 5 -9 4 }}
|-
|-
| −3
| −3
| [[Rodan]]
| [[Rodan]]
| 131072000 / 129140163
| 131072000/129140163
| {{monzo| 20 -17 3 }}
| {{Monzo| 20 -17 3 }}
|-
|-
| −2
| −2
| [[Hemififths]]
| [[Hemififths]]
| 858993459200 / 847288609443
| 858993459200/847288609443
| {{monzo| 35 -25 2 }}
| {{monzo| 35 -25 2 }}
|-
|-
Line 53: Line 53:
| [[Kwai]]
| [[Kwai]]
| (32 digits)
| (32 digits)
| {{monzo| 50 -33 1 }}
| {{Monzo| 50 -33 1 }}
|-
|-
| 0
| 0
| [[Countercomp]]
| [[Countercomp]]
| (40 digits)
| (40 digits)
| {{monzo| 65 -41 }}
| {{Monzo| 65 -41 }}
|-
|-
| 1
| 1
| [[Cotoneum]]
| [[Cotoneum]]
| (50 digits)
| (50 digits)
| {{monzo| 80 -49 -1 }}
| {{Monzo| 80 -49 -1 }}
|-
|-
| 2
| 2
| [[Newt]]
| [[Newt]]
| (58 digits)
| (58 digits)
| {{monzo| 95 -57 -2 }}
| {{Monzo| 95 -57 -2 }}
|-
|-
| 3
| 3
| 41 & 282
| 41 & 282
| (68 digits)
| (68 digits)
| {{monzo| 110 -65 -3 }}
| {{Monzo| 110 -65 -3 }}
|-
|-
| 4
| 4
| 41 & 335
| 41 & 335
| (76 digits)
| (76 digits)
| {{monzo| 125 -73 -4 }}
| {{Monzo| 125 -73 -4 }}
|-
|-
| 5
| 5
| 41 & 388
| 41 & 388
| (86 digits)
| (86 digits)
| {{monzo| 140 -81 -5 }}
| {{Monzo| 140 -81 -5 }}
|-
|-
| 6
| 6
| 41 & 441
| 41 & 441
| (94 digits)
| (94 digits)
| {{monzo| 155 -89 -6 }}
| {{Monzo| 155 -89 -6 }}
|-
|-
| 7
| 7
| 41 & 453
| 41 & 453
| (104 digits)
| (104 digits)
| {{monzo| 170 -97 -7 }}
| {{Monzo| 170 -97 -7 }}
|-
|-
| 8
| 8
| 41 & 506
| 41 & 506
| (112 digits)
| (112 digits)
| {{monzo| 185 -105 -8 }}
| {{Monzo| 185 -105 -8 }}
|-
|-
| 9
| 9
| 41 & 559
| 41 & 559
| (122 digits)
| (122 digits)
| {{monzo| 200 -113 -9 }}
| {{Monzo| 200 -113 -9 }}
|-
|-
| 10
| 10
| 41 & 571
| 41 & 571
| (130 digits)
| (130 digits)
| {{monzo| 215 -121 -10 }}
| {{Monzo| 215 -121 -10 }}
|-
|-
| 11
| 11
| 41 & 624
| 41 & 624
| (140 digits)
| (140 digits)
| {{monzo| -230 129 11 }}
| {{Monzo| -230 129 11 }}
|-
|-
| 12
| 12
| 41 & 677
| 41 & 677
| (148 digits)
| (148 digits)
| {{monzo| -245 137 12 }}
| {{Monzo| -245 137 12 }}
|-
|-
| 13
| 13
| 41 & 730
| 41 & 730
| (158 digits)
| (158 digits)
| {{monzo| -260 145 13 }}
| {{Monzo| -260 145 13 }}
|-
|-
| …
| …
Line 137: Line 137:


Examples of temperaments with fractional values of ''n'':  
Examples of temperaments with fractional values of ''n'':  
* [[Marvel temperaments #Septimin|Septimin]] ({{nowrap|''n'' {{=}} −11/2}})
* [[Septimin]] ({{nowrap|''n'' {{=}} −11/2}})
* [[Shibboleth]] ({{nowrap|''n'' {{=}} −9/2}})
* [[Shibboleth]] ({{nowrap|''n'' {{=}} −9/2}})
* [[Mirkwai clan #Pluto|Pluto]] ({{nowrap|''n'' {{=}} −7/2}})
* [[Pluto]] ({{nowrap|''n'' {{=}} −7/2}})
* 3737 & 5585 ({{nowrap|''n'' {{=}} 31/3 {{=}} 10.{{overline|3}}}})
* 3737 & 5585 ({{nowrap|''n'' {{=}} 31/3 {{=}} 10.{{overline|3}}}})
* 1277 & 2513 ({{nowrap|''n'' {{=}} 21/2}})
* 1277 & 2513 ({{nowrap|''n'' {{=}} 21/2}})


== Hemififths (5-limit) ==
== Kwai (5-limit) ==
: ''For extensions, see [[Breedsmic temperaments #Hemififths]].''
: ''For extensions, see [[Hemifamity temperaments #Kwai]].''


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


[[Comma list]]: 858993459200/847288609443
[[Comma list]]: {{monzo| 50 -33 1 }}


{{Mapping|legend=1| 1 1 -5 | 0 2 25 }}
{{Mapping|legend=1| 1 0 -50 | 0 1 33 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~655360/531441 = 351.476
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.792{{c}}, ~3/2 = 702.5077{{c}}
: [[error map]]: {{val| -0.208 +0.345 -0.023 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6243{{c}}
: error map: {{val| 0.000 +0.669 +0.288 }}


{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338, 915b, 1253bc }}
{{Optimal ET sequence|legend=1| 41, 111, 152, 2017bbc, 2169bbc }}


[[Badness]]: 0.372848
[[Badness]] (Sintel): 14.9


== Kwai (5-limit) ==
== Cotoneum (5-limit) ==
: ''For extensions, see [[Mirkwai clan #Kwai]].''
: ''For extensions, see [[Garischismic clan #Cotoneum]].''


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


[[Comma list]]: {{monzo| 50 -33 1 }} = 5629499534213120/5559060566555523
[[Comma list]]: {{monzo| 80 -49 -1 }}


{{Mapping|legend=1| 1 0 -50 | 0 1 33 }}
{{Mapping|legend=1| 1 0 80 | 0 1 -49 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.630
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8849{{c}}, ~3/2 = 702.2471{{c}}
: [[error map]]: {{val| -0.115 +0.177 +0.008 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3162{{c}}
: error map: {{val| 0.000 +0.361 +0.190 }}


{{Optimal ET sequence|legend=1| 41, 111, 152 }}
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b, 3759bbc }}


[[Badness]]: 0.636715
[[Badness]] (Sintel): 29.1


== Cotoneum (5-limit) ==
== Hemififths (5-limit) ==
: ''For extensions, see [[Garischismic clan #Cotoneum]].''
: ''For extensions, see [[Breedsmic temperaments #Hemififths]].''


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


[[Comma list]]: {{monzo| 80 -49 -1 }}
[[Comma list]]: 858993459200/847288609443


{{Mapping|legend=1| 1 0 80 | 0 1 -49 }}
{{Mapping|legend=1| 1 1 -5 | 0 2 25 }}
: mapping generators: ~2, ~655360/531441


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.315
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7047{{c}}, ~655360/531441 = 351.3898{{c}}
: [[error map]]: {{val| -0.295 +0.529 -0.091 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~655360/531441 = 351.4654{{c}}
: error map: {{val| 0.000 +0.976 +0.322 }}


{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b }}
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338, 915b, 1253bc }}


[[Badness]]: 1.240078
[[Badness]] (Sintel): 8.75


== Newt (5-limit) ==
== Newt (5-limit) ==
: ''For extensions, see [[Breedsmic temperaments #Newt]].''
: ''For extensions, see [[Garischismic clan #Newt]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 196: Line 211:


{{Mapping|legend=1| 1 1 19 | 0 2 -57 }}
{{Mapping|legend=1| 1 1 19 | 0 2 -57 }}
: mapping generators: ~2, ~{{monzo| 47 -28 -1 }}


[[Optimal tuning]] ([[POTE]]): ~{{monzo| 47 -28 -1 }} = 351.114
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9120{{c}}, ~{{monzo| 47 -28 -1 }} = 351.0878{{c}}
: [[error map]]: {{val| -0.088 +0.133 +0.010 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 47 -28 -1 }} = 351.1146{{c}}
: error map: {{val| 0.000 +0.274 +0.152 }}


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


[[Badness]]: 1.528465
[[Badness]] (Sintel): 35.9


[[Category:41edo]]
[[Category:41edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Latest revision as of 10:07, 20 May 2026

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.

The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, and has many advantages as a target. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of harmonic 3 in the generator chain. For example:

  • Cotoneum (n = 1) is generated by a fifth;
  • Newt (n = 2) splits its fifth in two;
  • Etc.

For a similar but perhaps more intuitive and practical concept, see Schismic–commatic 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)
  • Shibboleth (n = −9/2)
  • Pluto (n = −7/2)
  • 3737 & 5585 (n = 31/3 = 10.3)
  • 1277 & 2513 (n = 21/2)

Kwai (5-limit)

For extensions, see Hemifamity temperaments #Kwai.

Subgroup: 2.3.5

Comma list: [50 -33 1⟩

Mapping: [⟨1 0 -50], ⟨0 1 33]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.792 ¢, ~3/2 = 702.5077 ¢
error map: ⟨-0.208 +0.345 -0.023]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6243 ¢
error map: ⟨0.000 +0.669 +0.288]

Optimal ET sequence: 41, 111, 152, 2017bbc, 2169bbc

Badness (Sintel): 14.9

Cotoneum (5-limit)

For extensions, see Garischismic clan #Cotoneum.

Subgroup: 2.3.5

Comma list: [80 -49 -1⟩

Mapping: [⟨1 0 80], ⟨0 1 -49]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.8849 ¢, ~3/2 = 702.2471 ¢
error map: ⟨-0.115 +0.177 +0.008]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3162 ¢
error map: ⟨0.000 +0.361 +0.190]

Optimal ET sequence: 41, 135c, 176, 217, 475, 1167, 1642, 2117b, 3759bbc

Badness (Sintel): 29.1

Hemififths (5-limit)

For extensions, see Breedsmic temperaments #Hemififths.

Subgroup: 2.3.5

Comma list: 858993459200/847288609443

Mapping: [⟨1 1 -5], ⟨0 2 25]]

mapping generators: ~2, ~655360/531441

Optimal tunings:

  • WE: ~2 = 1199.7047 ¢, ~655360/531441 = 351.3898 ¢
error map: ⟨-0.295 +0.529 -0.091]
  • CWE: ~2 = 1200.0000 ¢, ~655360/531441 = 351.4654 ¢
error map: ⟨0.000 +0.976 +0.322]

Optimal ET sequence: 17c, 41, 58, 99, 239, 338, 915b, 1253bc

Badness (Sintel): 8.75

Newt (5-limit)

For extensions, see Garischismic clan #Newt.

Subgroup: 2.3.5

Comma list: [95 -57 -2⟩

Mapping: [⟨1 1 19], ⟨0 2 -57]]

mapping generators: ~2, ~[47 -28 -1⟩

Optimal tunings:

  • WE: ~2 = 1199.9120 ¢, ~[47 -28 -1⟩ = 351.0878 ¢
error map: ⟨-0.088 +0.133 +0.010]
  • CWE: ~2 = 1200.0000 ¢, ~[47 -28 -1⟩ = 351.1146 ¢
error map: ⟨0.000 +0.274 +0.152]

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

Badness (Sintel): 35.9

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