Schismic–Mercator equivalence continuum: Difference between revisions

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The '''syntonic-chromatic equivalence continuum''' is a continuum of temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo|-84 53}})]].
{{Technical data page}}
The '''schismic–Mercator equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo| -84 53 }})]]. This continuum is theoretically interesting in that the temperaments associated with its various commas are all 5-limit microtemperaments.


All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo|-84 53}}. 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 [[53edo]] (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 1.8503390493..., 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|-84 53}}}}. 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 [[53edo]] (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 1.8503390493…, 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]].
For a similar but perhaps more intuitive and practical concept, see [[Syntonic–chromatic equivalence continuum]].


{| class="wikitable center-1 center-2"
{| class="wikitable center-1"
|+ Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 16: Line 17:
|-
|-
| 0
| 0
| [[Mercator's comma|Mercator]]
| [[Mercator]]
|  
| (52 digits)
| {{monzo|-84 53}}
| {{Monzo| -84 53 }}
|-
|-
| 1
| 1
| Counterschismic
| [[Counterschismic]]
|  
| (44 digits)
| {{monzo|-69 45 -1}}
| [[Counterschisma|{{Monzo| -69 45 -1 }}]]
|-
|-
| 2
| 2
| [[Very high accuracy temperaments #Monzismic|Monzismic]]
| [[Very high accuracy temperaments #Monzismic|Monzismic]]
|  
| (36 digits)
| [[Monzisma|{{monzo|54 -37 2}}]]
| [[Monzisma|{{Monzo| 54 -37 2 }}]]
|-
|-
| 3
| 3
| [[Tricot]]
| [[Alphatricot]]
|  
| (28 digits)
| {{monzo| 39 -29 3}}
| [[Alphatricot comma|{{Monzo| 39 -29 3 }}]]
|-
|-
| 4
| 4
| [[Vulture]]
| [[Vulture]]
|  
| (22 digits)
| {{monzo| 24 -21 4 }}
| [[Vulture comma|{{Monzo| 24 -21 4 }}]]
|-
|-
| 5
| 5
| [[Amity]]
| [[Amity]]
| [[1600000/1594323]]
| [[1600000/1594323]]
| {{monzo| 9 -13 5 }}
| {{Monzo| 9 -13 5 }}
|-
|-
| 6
| 6
| [[Kleismic]]
| [[Kleismic]]
| [[15625/15552]]
| [[15625/15552]]
| {{monzo|-6 -5 6}}
| {{Monzo|-6 -5 6}}
|-
|-
| 7
| 7
| [[Orson]]
| [[Orson]]
| [[Semicomma|2109375/2097152]]
| [[Semicomma|2109375/2097152]]
| {{monzo|-21 3 7 }}
| {{Monzo|-21 3 7 }}
|-
| 8
| [[Buzzardsmic clan #Demibuzzard|Demibuzzard]]
| (22 digits)
| {{Monzo| -36 11 8 }}
|-
| 9
| [[Miscellaneous 5-limit temperaments #Untriton|Untriton]]
| (32 digits)
| {{Monzo| -51 19 9 }}
|-
|-
| …
| …
Line 63: Line 74:
| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1}}
| {{Monzo| -15 8 1 }}
|}
 
We may invert the continuum by setting ''m'' such that {{nowrap|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''counterschismic–Mercator equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 2.17600… While the counterschisma is of comparable size as the schisma, it is way more complex, so this continuum does not contain as many useful temperaments at integer points.
 
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Mercator]]
| (52 digits)
| {{Monzo| -84 53 }}
|-
| 1
| [[Schismic]]
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
|-
| 2
| [[Monzismic]]
| (36 digits)
| [[Monzisma|{{Monzo| 54 -37 2 }}]]
|-
| …
| …
| …
| …
|-
| ∞
| [[Counterschismic]]
| (44 digits)
| [[Counterschisma|{{Monzo| -69 45 -1 }}]]
|}
|}


Examples of temperaments with fractional values of ''n'':
{| class="wikitable"
* 3684 & 11105 (''n'' = 11/6 = 1.8{{overline|3}})
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''
|-
! Temperament !! ''n'' !! ''m'' !! Comma
|-
| 53 & 3684 || 11/6 = 1.8{{overline|3}} || 11/5 = 2.2 || {{Monzo| -339 230 -11 }}
|-
| 53 & 4296 || 13/7 = 1.{{overline|857142}} || 13/6 = 2.1{{overline|6}} || {{Monzo| 393 -267 13 }}
|-
| [[Countritonic]] || 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} || {{Monzo| 33 -34 9 }}
|-
| [[Quartonic]] || 11/2 = 5.5 || 11/9 = 1.{{overline|2}} || {{Monzo| 3 -18 11 }}
|-
| [[Maja]] || 17/3 = 5.{{overline|6}} || 17/14 = 1.2{{overline|142857}} || {{Monzo| -3 -23 17 }}
|-
| [[Ditonic]] || 13/2 = 6.5 || 13/11 = 1.{{overline|18}} || {{Monzo| -27 -2 13 }}
|}
 
== Counterschismic ==
{{See also| Counterschisma }}


== Mercator ==
Counterschismic is generated by a [[3/2|perfect fifth]], like [[schismic]], but the [[5/1|5th]] [[harmonic]] is located at +45 fifths instead of schismic's -8. They unite at [[53edo]], of course. [[730edo]] may be recommended as a tuning.
{{see also| Mercator's comma }}


Comma list: {{monzo| -84 53 }}
[[Subgroup]]: 2.3.5


POTE genrator: ~5/4 = 386.264
[[Comma list]]: {{monzo| -69 45 -1 }}


Mapping: [{{val| 53 84 123 }}, {{val| 0 0 1 }}]
{{Mapping|legend=1| 1 0 -69 | 0 1 45 }}
: mapping generators: ~2, ~3


Wedgie: {{wedgie| 0 53 84 }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0116{{c}}, ~3/2 = 701.9243{{c}}
: [[error map]]: {{val| +0.012 -0.019 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9177{{c}}
: error map: {{val| 0.000 -0.037 -0.017 }}


{{Val list|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}
{{Optimal ET sequence|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056, 16219b }}


Badness: 0.2843
[[Badness]] (Sintel): 2.14


== Counterschismic ==
== Demibuzzard (5-limit) ==
{{see also| Counterschisma }}
: ''For extensions, see [[Buzzardsmic clan #Demibuzzard]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 69198046875/68719476736
 
{{Mapping|legend=1| 1 -4 10 | 0 8 -11 }}
: mapping generators: ~2, ~16384/10125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~16384/10125 = 837.8342{{c}}
: [[error map]]: {{val| +0.222 -0.171 -0.267 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16384/10125 = 837.6794{{c}}
: error map: {{val| 0.000 -0.520 -0.787 }}
 
{{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
 
[[Badness]] (Sintel): 3.06
 
== Countritonic ==
: ''For extensions, see [[Hemifamity temperaments #Countriton]], [[Ragismic microtemperaments #Ragitritonic]], and [[Garischismic clan #Garitritonic]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 33 -34 9 }}
 
{{Mapping|legend=1| 1 -3 -15 | 0 9 34 }}
: mapping generators: ~2, ~20480000/14348907
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9228{{c}}, ~20480000/14348907 = 611.3248{{c}}
: [[error map]]: {{val| -0.077 +0.200 -0.113 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~20480000/14348907 = 611.3614{{c}}
: error map: {{val| 0.000 +0.297 -0.027 }}
 
{{Optimal ET sequence|legend=1| 51c, 53, 263, 316, 369, 422, 475, 528, 2587b, 3115b, 3643b }}
 
[[Badness]] (Sintel): 6.00


Comma list: {{monzo| -69 45 -1 }}
== 53 & 3684 ==
[[Subgroup]]: 2.3.5


POTE generator: ~3/2 = 701.9175
[[Comma list]]: {{monzo| -339 230 -11 }}


Mapping: [{{val| 1 2 21 }}, {{val| 0 -1 -45 }}]
{{Mapping|legend=1| 1 2 11 | 0 -11 -230 }}
: mapping generators: ~2, ~10737418240/10460353203


Wedgie: {{wedgie| 1 45 69 }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.000272{{c}}, ~10737418240/10460353203 = 45.276910{{c}}
: [[error map]]: {{val| +0.0003 -0.0005 +0.0000 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~10737418240/10460353203 = 45.276898{{c}}
: error map: {{val| 0.0000 -0.0009 -0.0003 }}


{{Val list|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}}
{{Optimal ET sequence|legend=1| 53, …, 3313, 3366, 3419, 3472, 3525, 3578, 3631, 3684, 7421, 11105, 25894, 36999 }}


Badness: 0.09123
[[Badness]] (Sintel): 6.48


== 3684 & 11105 ==
== 53 & 4296 ==
Comma list: {{monzo| -339 230 -11 }}
[[Subgroup]]: 2.3.5


Map: [{{val|1 2 11}}, {{val|0 -11 -230}}]
[[Comma list]]: {{monzo| 393 -267 13 }}


Wedgie: {{wedgie|11 230 339}}
{{Mapping|legend=1| 1 -7 -174 | 0 13 267 }}
: mapping generators: ~2, ~{{monzo| 61 -41 2 }}


POTE generator: 45.2769
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.999891{{c}}, ~{{monzo| 61 -41 2 }} = 792.458032{{c}}
: [[error map]]: {{val| -0.0001 +0.0002 -0.0000 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 61 -41 2 }} = 792.458104{{c}}
: error map: {{val| 0.0000 +0.0004 +0.0002 }}


{{Val list|legend=1| 53, 3684, 11105 }}
{{Optimal ET sequence|legend=1| 53, …, 3872, 3925, 3978, 4031, 4084, 4137, 4190, 4243, 4296, 34315, 38611, 42907, 47203, 51499, 55795, 60091, 64387, 68683 }}


== Unnamed temperament (''n'' = 13/7) ==
[[Badness]] (Sintel): 4.07
Comma list: {{monzo| 393 -267 13 }}


[[Category:Theory]]
[[Category:53edo]]
[[Category:Temperament]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Latest revision as of 11:23, 20 May 2026

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 schismic–Mercator equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with Mercator's comma ([-84 53⟩). This continuum is theoretically interesting in that the temperaments associated with its various commas are all 5-limit microtemperaments.

All temperaments in the continuum satisfy (32805/32768)n ~ [-84 53⟩. 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 53edo (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 1.8503390493…, 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.

Temperaments with integer n
n Temperament Comma
Ratio Monzo
0 Mercator (52 digits) [-84 53⟩
1 Counterschismic (44 digits) [-69 45 -1⟩
2 Monzismic (36 digits) [54 -37 2⟩
3 Alphatricot (28 digits) [39 -29 3⟩
4 Vulture (22 digits) [24 -21 4⟩
5 Amity 1600000/1594323 [9 -13 5⟩
6 Kleismic 15625/15552 [-6 -5 6⟩
7 Orson 2109375/2097152 [-21 3 7⟩
8 Demibuzzard (22 digits) [-36 11 8⟩
9 Untriton (32 digits) [-51 19 9⟩
… … … …
∞ Schismic 32805/32768 [-15 8 1⟩

We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the counterschismic–Mercator equivalence continuum, which is essentially the same thing. The just value of m is 2.17600… While the counterschisma is of comparable size as the schisma, it is way more complex, so this continuum does not contain as many useful temperaments at integer points.

Temperaments with integer m
m Temperament Comma
Ratio Monzo
0 Mercator (52 digits) [-84 53⟩
1 Schismic 32805/32768 [-15 8 1⟩
2 Monzismic (36 digits) [54 -37 2⟩
… … … …
∞ Counterschismic (44 digits) [-69 45 -1⟩
Temperaments with fractional n and m
Temperament n m Comma
53 & 3684 11/6 = 1.83 11/5 = 2.2 [-339 230 -11⟩
53 & 4296 13/7 = 1.857142 13/6 = 2.16 [393 -267 13⟩
Countritonic 9/2 = 4.5 9/7 = 1.285714 [33 -34 9⟩
Quartonic 11/2 = 5.5 11/9 = 1.2 [3 -18 11⟩
Maja 17/3 = 5.6 17/14 = 1.2142857 [-3 -23 17⟩
Ditonic 13/2 = 6.5 13/11 = 1.18 [-27 -2 13⟩

Counterschismic

See also: Counterschisma

Counterschismic is generated by a perfect fifth, like schismic, but the 5th harmonic is located at +45 fifths instead of schismic's -8. They unite at 53edo, of course. 730edo may be recommended as a tuning.

Subgroup: 2.3.5

Comma list: [-69 45 -1⟩

Mapping: [⟨1 0 -69], ⟨0 1 45]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.0116 ¢, ~3/2 = 701.9243 ¢
error map: ⟨+0.012 -0.019 +0.001]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9177 ¢
error map: ⟨0.000 -0.037 -0.017]

Optimal ET sequence: 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056, 16219b

Badness (Sintel): 2.14

Demibuzzard (5-limit)

For extensions, see Buzzardsmic clan #Demibuzzard.

Subgroup: 2.3.5

Comma list: 69198046875/68719476736

Mapping: [⟨1 -4 10], ⟨0 8 -11]]

mapping generators: ~2, ~16384/10125

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~16384/10125 = 837.8342 ¢
error map: ⟨+0.222 -0.171 -0.267]
  • CWE: ~2 = 1200.0000 ¢, ~16384/10125 = 837.6794 ¢
error map: ⟨0.000 -0.520 -0.787]

Optimal ET sequence: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc

Badness (Sintel): 3.06

Countritonic

For extensions, see Hemifamity temperaments #Countriton, Ragismic microtemperaments #Ragitritonic, and Garischismic clan #Garitritonic.

Subgroup: 2.3.5

Comma list: [33 -34 9⟩

Mapping: [⟨1 -3 -15], ⟨0 9 34]]

mapping generators: ~2, ~20480000/14348907

Optimal tunings:

  • WE: ~2 = 1199.9228 ¢, ~20480000/14348907 = 611.3248 ¢
error map: ⟨-0.077 +0.200 -0.113]
  • CWE: ~2 = 1200.0000 ¢, ~20480000/14348907 = 611.3614 ¢
error map: ⟨0.000 +0.297 -0.027]

Optimal ET sequence: 51c, 53, 263, 316, 369, 422, 475, 528, 2587b, 3115b, 3643b

Badness (Sintel): 6.00

53 & 3684

Subgroup: 2.3.5

Comma list: [-339 230 -11⟩

Mapping: [⟨1 2 11], ⟨0 -11 -230]]

mapping generators: ~2, ~10737418240/10460353203

Optimal tunings:

  • WE: ~2 = 1200.000272 ¢, ~10737418240/10460353203 = 45.276910 ¢
error map: ⟨+0.0003 -0.0005 +0.0000]
  • CWE: ~2 = 1200.000000 ¢, ~10737418240/10460353203 = 45.276898 ¢
error map: ⟨0.0000 -0.0009 -0.0003]

Optimal ET sequence: 53, …, 3313, 3366, 3419, 3472, 3525, 3578, 3631, 3684, 7421, 11105, 25894, 36999

Badness (Sintel): 6.48

53 & 4296

Subgroup: 2.3.5

Comma list: [393 -267 13⟩

Mapping: [⟨1 -7 -174], ⟨0 13 267]]

mapping generators: ~2, ~[61 -41 2⟩

Optimal tunings:

  • WE: ~2 = 1199.999891 ¢, ~[61 -41 2⟩ = 792.458032 ¢
error map: ⟨-0.0001 +0.0002 -0.0000]
  • CWE: ~2 = 1200.000000 ¢, ~[61 -41 2⟩ = 792.458104 ¢
error map: ⟨0.0000 +0.0004 +0.0002]

Optimal ET sequence: 53, …, 3872, 3925, 3978, 4031, 4084, 4137, 4190, 4243, 4296, 34315, 38611, 42907, 47203, 51499, 55795, 60091, 64387, 68683

Badness (Sintel): 4.07

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