Schismic–countercommatic equivalence continuum: Difference between revisions

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The '''schismic-Mercator 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]] [[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 {{nowrap|(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 [[Syntonic-chromatic equivalence continuum]].
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 [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For example:
* [[Cotoneum]] ({{nowrap|''n'' {{=}} 1}}) is generated by a fifth;
* [[Newt]] ({{nowrap|''n'' {{=}} 2}}) splits its fifth in two;
* Etc.


{| class="wikitable center-1 center-2"
For a similar but perhaps more intuitive and practical concept, see [[Schismic–commatic equivalence continuum]].
|+ Temperaments in the continuum
 
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments of integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 15: Line 20:
! Monzo
! Monzo
|-
|-
| -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]]
| [[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
| [[41-comma|Counterpyth]]
| [[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 & 282
|  
| (68 digits)
| {{monzo|110 -65 -3}}
| {{Monzo| 110 -65 -3 }}
|-
|-
| 4
| 4
| 41&amp;335
| 41 & 335
|  
| (76 digits)
| {{monzo|125 -73 -4}}
| {{Monzo| 125 -73 -4 }}
|-
|-
| 5
| 5
| 41&amp;388
| 41 & 388
|  
| (86 digits)
| {{monzo|140 -81 -5}}
| {{Monzo| 140 -81 -5 }}
|-
|-
| 6
| 6
| 41&amp;441
| 41 & 441
|  
| (94 digits)
| {{monzo|155 -89 -6}}
| {{Monzo| 155 -89 -6 }}
|-
|-
| 7
| 7
| 41&amp;453
| 41 & 453
|  
| (104 digits)
| {{monzo|170 -97 -7}}
| {{Monzo| 170 -97 -7 }}
|-
|-
| 8
| 8
| 41&amp;506
| 41 & 506
|  
| (112 digits)
| {{monzo|185 -105 -8}}
| {{Monzo| 185 -105 -8 }}
|-
|-
| 9
| 9
| 41&amp;559
| 41 & 559
|  
| (122 digits)
| {{monzo|200 -113 -9}}
| {{Monzo| 200 -113 -9 }}
|-
|-
| 10
| 10
| 41&amp;571
| 41 & 571
|  
| (130 digits)
| {{monzo|215 -121 -10}}
| {{Monzo| 215 -121 -10 }}
|-
|-
| 11
| 11
| 41&amp;624
| 41 & 624
|  
| (140 digits)
| {{monzo|-230 129 11}}
| {{Monzo| -230 129 11 }}
|-
|-
| 12
| 12
| 41&amp;677
| 41 & 677
|  
| (148 digits)
| {{monzo|-245 137 12}}
| {{Monzo| -245 137 12 }}
|-
|-
| 13
| 13
| 41&amp;730
| 41 & 730
|  
| (158 digits)
| {{monzo|-260 145 13}}
| {{Monzo| -260 145 13 }}
|-
|-
| …
| …
Line 128: Line 133:
| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1}}
| {{monzo| -15 8 1 }}
|}
|}


Examples of temperaments with fractional values of ''n'':  
Examples of temperaments with fractional values of ''n'':  
* [[Marvel temperaments #Septimin|Septimin]] (''n'' = -11/2 = -5.5)
* [[Septimin]] ({{nowrap|''n'' {{=}} −11/2}})
* [[Shibboleth family|Shibboleth]] (''n'' = -9/2 = -4.5)
* [[Shibboleth]] ({{nowrap|''n'' {{=}} −9/2}})
* [[Mirkwai clan #Pluto|Pluto]] (''n'' = -7/2 = -3.5)
* [[Pluto]] ({{nowrap|''n'' {{=}} −7/2}})
* 3737 &amp; 5585 (''n'' = 31/3 = 10.{{overline|3}})
* 3737 & 5585 ({{nowrap|''n'' {{=}} 31/3 {{=}} 10.{{overline|3}}}})
* 1277 &amp; 2513 (''n'' = 21/2 = 10.5)
* 1277 & 2513 ({{nowrap|''n'' {{=}} 21/2}})
 
== Kwai (5-limit) ==
: ''For extensions, see [[Hemifamity temperaments #Kwai]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 50 -33 1 }}
 
{{Mapping|legend=1| 1 0 -50 | 0 1 33 }}
: mapping generators: ~2, ~3
 
[[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, 111, 152, 2017bbc, 2169bbc }}
 
[[Badness]] (Sintel): 14.9
 
== Cotoneum (5-limit) ==
: ''For extensions, see [[Garischismic clan #Cotoneum]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 80 -49 -1 }}
 
{{Mapping|legend=1| 1 0 80 | 0 1 -49 }}
: mapping generators: ~2, ~3
 
[[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, 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|legend=1| 1 1 -5 | 0 2 25 }}
: mapping generators: ~2, ~655360/531441
 
[[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| 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]]: {{monzo| 95 -57 -2 }}
 
{{Mapping|legend=1| 1 1 19 | 0 2 -57 }}
: mapping generators: ~2, ~{{monzo| 47 -28 -1 }}
 
[[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 }}
 
[[Badness]] (Sintel): 35.9


[[Category:41edo]]
[[Category:41edo]]
[[Category:Theory]]
[[Category:Temperament]]
[[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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