Schismic–commatic equivalence continuum: Difference between revisions

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The '''schismic-Pythagorean equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Pythagorean comma|Pythagorean comma ({{monzo| -19 12 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[12edo]].
{{Technical data page}}
The '''schismic–commatic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Pythagorean comma|Pythagorean comma ({{monzo| -19 12 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[12edo]].


All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo| -19 12 }}. 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 [[12edo]] 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 12.0078623975…, and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is.
All temperaments in the continuum satisfy {{nowrap|(32805/32768)<sup>''n''</sup> ~ {{monzo| -19 12 }}}}. 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 12edo 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 12.0078623975…, and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is.


{| class="wikitable center-1 center-2"
The [[Pythagorean comma]] is the characteristic 3-limit comma tempered out in 12edo, 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 an ''n'' that is not coprime with 12, however, the corresponding temperament splits the [[octave]] into {{nowrap| gcd(''n'', 12) }} parts, and splits the interval class of 3 into {{nowrap| ''n''/gcd(''n'', 12) }}. For example:
|+ Temperaments with integer ''n''
* [[Meantone]] ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth with an unsplit octave;
* [[Diaschismic]] ({{nowrap| ''n'' {{=}} 2 }}) splits the octave in two, as 2 divides 12;
* [[Misty]] ({{nowrap| ''n'' {{=}} 3 }}) splits the octave in three, as 3 divides 12;
* [[Undim]]  ({{nowrap| ''n'' {{=}} 4 }}) splits the octave in four, as 4 divides 12;
* [[Quindromeda]]  ({{nowrap| ''n'' {{=}} 5 }}) does not split the octave but splits the fourth in five, as 5 is coprime with 12.
 
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 12: Line 20:
! Ratio
! Ratio
! Monzo
! Monzo
|-
| -3
| [[Triscordial]]
| (40 digits)
| {{Monzo| -64 36 3 }}
|-
| -2
| [[Biscordial]]
| (30 digits)
| {{Monzo| -49 28 2 }}
|-
|-
| -1
| -1
| [[Marvel temperaments #Gracecordial|Gracecordial]]
| [[Gracecordial]]
| 17433922005/17179869184
| (22 digits)
| {{monzo| -34 20 1 }}
| {{Monzo| -34 20 1 }}
|-
|-
| 0
| 0
| [[Compton family|Compton]]
| [[Compton]]
| [[531441/524288]]
| [[531441/524288]]
| {{monzo| -19 12 }}
| {{Monzo| -19 12 }}
|-
|-
| 1
| 1
| [[Meantone family|Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{Monzo| -4 4 -1 }}
|-
|-
| 2
| 2
| [[Diaschismic family|Diaschismic]]
| [[Diaschismic]]
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| {{Monzo| 11 -4 -2 }}
|-
|-
| 3
| 3
| [[Misty family|Misty]]
| [[Misty]]
| [[67108864/66430125]]
| [[67108864/66430125]]
| {{monzo| 26 -12 -3 }}
| {{Monzo| 26 -12 -3 }}
|-
|-
| 4
| 4
| [[Undim family|Undim]]
| [[Undim]]
|  
| (26 digits)
| {{monzo| 41 -20 -4 }}
| {{Monzo| 41 -20 -4 }}
|-
|-
| 5
| 5
| [[Quindromeda family|Quindromeda]]
| [[Quindromeda]]
|  
| (34 digits)
| {{monzo| 56 -28 -5 }}
| {{Monzo| 56 -28 -5 }}
|-
|-
| 6
| 6
| [[Sextile]]
| [[Sextile]]
|  
| (44 digits)
| {{monzo| 71 -36 -6 }}
| {{Monzo| 71 -36 -6 }}
|-
|-
| 7
| 7
| Sepsa-sepgu (12&amp;323)
| [[Heptacot]]
|
| (52 digits)
| {{monzo| 86 -44 -7 }}
| {{Monzo| 86 -44 -7 }}
|-
|-
| 8
| 8
| [[World calendar]]
| [[World calendar]] restriction
|  
| (62 digits)
| {{monzo| 101 -52 -8 }}
| {{Monzo| 101 -52 -8 }}
|-
|-
| 9
| 9
| Quinbisa-tritrigu (12&amp;441)
| Quinbisa-tritrigu (12 & 441)
|  
| (70 digits)
| {{monzo| 116 -60 -9 }}
| {{Monzo| 116 -60 -9 }}
|-
|-
| 10
| 10
| Lesa-quinbigu (12&amp;494)
| Lesa-quinbigu (12 & 494)
|  
| (80 digits)
| {{monzo| 131 -68 -10 }}
| {{Monzo| 131 -68 -10 }}
|-
|-
| 11
| 11
| Quadtrisa-legu (12&amp;559)
| Quadtrisa-legu (12 & 559)
|  
| (88 digits)
| {{monzo| 146 -76 -11 }}
| {{Monzo| 146 -76 -11 }}
|-
|-
| 12
| 12
| [[Very high accuracy temperaments #Atomic|Atomic]]
| [[Atomic]]
|  
| (98 digits)
| [[Kirnberger's atom|{{monzo| 161 -84 -12 }}]]
| [[Kirnberger's atom|{{Monzo| 161 -84 -12 }}]]
|-
|-
| 13
| 13
| Quintrila-theyo (12&amp;677)
| Quintrila-theyo (12 & 677)
|  
| (106 digits)
| {{monzo| -176 92 13 }}
| {{Monzo| -176 92 13 }}
|-
|-
| …
| …
Line 96: Line 114:
| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| {{Monzo| -15 8 1 }}
|}
|}


We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. The just value of ''m'' is 1.0908441588…
We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''syntonic–commatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588…. The [[syntonic comma]] is way larger but much simpler than the schisma. As such, this continuum does not contain as many [[microtemperament]]s, but has more useful lower-complexity temperaments.
 
Alternatively, because the the 5-limit otonal detemperament of 12edo is a 4×3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as {{nowrap| ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]) }}. This corresponds to these commas' structural significance via 128/125 being entirely in the [[2.5 subgroup]] while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates is a flip of the ''m''-continuum such that microtemperaments converging to atomic are found as successive mediants towards the [[JIP]]. Specifically, its JIP is at 1.90915584…, which is approximated very closely by the microtemperament atomic at {{nowrap| 21/11 {{=}} 1.90909… }} so that the main ''n''-continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic). It is noted for its nontrivial relation to the other better-motivated (in terms of mapping) coordinates discussed.


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ Temperaments with integer ''m''
|+ style="font-size: 105%;" | Temperaments with integer ''m'' (and thus ''k'')
|-
|-
! rowspan="2" | ''m''
! rowspan="2" | ''m''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 110: Line 131:
! Ratio
! Ratio
! Monzo
! Monzo
|-
| -1
| 4
| [[Python]]
| [[43046721/41943040]]
| {{Monzo| -23 16 -1 }}
|-
|-
| 0
| 0
| [[Compton family|Compton]]
| 3
| [[Compton]]
| [[531441/524288]]
| [[531441/524288]]
| {{monzo| -19 12 }}
| {{Monzo| -19 12 }}
|-
|-
| 1
| 1
| 2
| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| {{Monzo| -15 8 1 }}
|-
|-
| 2
| 2
| [[Diaschismic family|Diaschismic]]
| 1
| [[Diaschismic]]
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| {{Monzo| 11 -4 -2 }}
|-
|-
| 3
| 3
| [[Augmented]]
| 0
| [[Augmented (temperament)|Augmented]]
| [[128/125]]
| [[128/125]]
| {{monzo| 7 0 -3 }}
| {{Monzo| 7 0 -3 }}
|-
|-
| 4
| 4
| [[Diminished]]
| -1
| [[Diminished (temperament)|Diminished]]
| [[648/625]]
| [[648/625]]
| {{monzo| 3 4 -4 }}
| {{Monzo| 3 4 -4 }}
|-
|-
| 5
| 5
| -2
| [[Ripple]]
| [[Ripple]]
| [[6561/6250]]
| [[6561/6250]]
| {{monzo| -1 8 -5 }}
| {{Monzo| -1 8 -5 }}
|-
|-
| 6
| 6
| -3
| [[Wronecki]]
| [[Wronecki]]
| [[531441/500000]]
| [[531441/500000]]
| {{monzo| -5 12 -6 }}
| {{Monzo| -5 12 -6 }}
|-
|-
| …
| …
| …
| …
| …
Line 151: Line 186:
| …
| …
|-
|-
| ∞
| ∞
| ∞
| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{Monzo| -4 4 -1 }}
|}
|}


Examples of temperaments with fractional values of ''n'':
{| class="wikitable"
* Python (''n'' = 1/2 = 0.5)
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''
* [[Ripple family|Ripple]] (''n'' = 5/4 = 1.25)
|-
* [[Dimipent family|Diminished]] (''n'' = 4/3 = 1.{{overline|3}})
! ''n'' !! ''m'' !! ''k'' !! Temperament !! Comma
* [[Augmented family|Augmented]] (''n'' = 3/2 = 1.5)
|-
* [[Passion family|Passion]] (''n'' = 5/3 = 1.{{overline|6}})
| 5/3 = 1.{{overline|6}} || 5/2 = 2.5 || 1/2 || [[Passion]] || {{monzo| 18 -4 -5 }}
* [[Quintaleap family|Quintaleap]] (''n'' = 5/2 = 2.5)
|-
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || 4/3 || [[Quintaleap]] || {{monzo| 37 -16 -5 }}
|}
 
== Python ==
Python is generated by a fifth, which is typically flatter than 7\12. The ~5/4 is reached by +16 fifths octave reduced, which is a double-augmented second (C–Dx). It can be described as {{nowrap| 12 & 91 }}, and [[103edo]] is a good tuning. It corresponds to {{nowrap| ''m'' {{=}} -1 }} and {{nowrap| ''n'' {{=}} 1/2 }}.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 43046721/41943040
 
{{Mapping|legend=1| 1 0 -23 | 0 1 16 }}
: Mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8769{{c}}, ~3/2 = 699.5409{{c}}
: [[error map]]: {{val| +0.876 -1.537 +0.203 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.0789{{c}}
: error map: {{val| 0.000 -2.876 -1.051 }}


== Compton (12&amp;72) ==
{{Optimal ET sequence|legend=1| 12, …, 79, 91, 103, 218b }}
{{See also| Pythagorean comma }} ''and [[Compton family]]''


Subgroup: 2.3.5
[[Badness]] (Sintel): 6.92


Comma list: {{monzo| -19 12 }} = 531441/524288
== Gracecordial (5-limit) ==
: ''For extensions, see [[Marvel temperaments #Gracecordial]].''


Mapping: [{{val| 12 19 28 }}, {{val| 0 0 -1 }}]
The 5-limit version of gracecordial is generated by a fifth, which is typically sharp of 7\12 but flat of just. The ~5/4 is reached by -20 fifths octave reduced, which is a triple-diminished fifth (C–Gbbb). It can be described as {{nowrap| 12 & 125 }}, and [[137edo]] is a good tuning. It corresponds to {{nowrap| ''n'' {{=}} -1 }} and {{nowrap| ''m'' {{=}} 1/2 }}.


{{Multival|legend=1| 0 12 19 }}
[[Subgroup]]: 2.3.5


POTE generator: ~5/4 = 384.882
[[Comma list]]: 17433922005/17179869184


{{Optimal ET sequence|legend=1| 12, 48, 60, 72, 84 }}
{{Mapping|legend=1| 1 0 34 | 0 1 -20 }}
: Mapping generators: ~2, ~3


Badness: 0.094494
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3986{{c}}, ~3/2 = 700.9665{{c}}
: [[error map]]: {{val| +0.399 -0.590 -0.064 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.7202{{c}}
: error map: {{val| 0.000 -1.235 -0.718 }}


== Python (12&amp;79) ==
{{Optimal ET sequence|legend=1| 12, 113, 125, 137, 1221bbcc }}
Subgroup: 2.3.5


Comma list: {{monzo| -23 16 -1 }} = 43046721/41943040
[[Badness]] (Sintel): 7.20


Mapping: [{{val| 1 0 -23 }}, {{val| 0 -1 -16 }}]
== Biscordial ==
Named by [[Xenllium]] in 2026, biscordial has a period of half-octave, and tempers out the biscordial comma, {{monzo| -49 28 2 }}. The ~5/4 is reached by 17 periods minus 14 fifths. It corresponds to {{nowrap| ''n'' {{=}} -2 }}.


{{Multival|legend=1| 1 16 23 }}
[[Subgroup]]: 2.3.5


POTE generator: ~4/3 = 500.970
[[Comma list]]: 571919811374025/562949953421312


{{Optimal ET sequence|legend=1| 12, …, 79, 91, 103 }}
{{Mapping|legend=1| 2 0 49 | 0 1 -14 }}
: Mapping generators: ~23914845/16777216, ~3


Badness: 0.295079
[[Optimal tuning]]s:  
* [[WE]]: ~23914845/16777216 = 600.153{{c}}, ~3/2 = 701.211{{c}}
* [[CWE]]: ~23914845/16777216 = 600.000{{c}}, ~3/2 = 701.019{{c}}


== Quintaleap (12&amp;121) ==
{{Optimal ET sequence|legend=1| 12, 166, 178, 190, 392, 582 }}
{{See also| Quintaleap family }}


Subgroup: 2.3.5
[[Badness]] (Sintel): 15.7


Comma list: {{monzo| 37 -16 -5 }} = 137438953472/134521003125
== Triscordial ==
Named by [[Xenllium]] in 2026, triscordial has a period of 1/3-octave, and tempers out the triscordial comma, {{monzo| -64 36 3 }}. The ~5/4 is reached by 22 periods minus 12 fifths. It corresponds to {{nowrap| ''n'' {{=}} -3 }}.


Mapping: [{{val| 1 2 1 }}, {{val| 0 -5 16 }}]
[[Subgroup]]: 2.3.5


{{Multival|legend=1| 5 -16 -37 }}
[[Comma list]]: {{Monzo| -64 36 3 }}


POTE generator: ~135/128 = 99.267
{{Mapping|legend=1| 3 0 64 | 0 1 -12 }}


{{Optimal ET sequence|legend=1| 12, …, 85, 97, 109, 121, 133, 278c, 411bc, 544bc }}
: Mapping generators: ~2657205/2097152, ~3


Badness: 0.444506
[[Optimal tuning]]s:  
* [[WE]]: ~2657205/2097152 = 400.084{{c}}, ~3/2 = 701.343{{c}}
* [[CWE]]: ~2657205/2097152 = 400.000{{c}}, ~3/2 = 701.182{{c}}


== Undim (12&amp;152) ==
{{Optimal ET sequence|legend=1| 12, 231, 243, 255, 498, 753 }}
{{See also| Undim family }}


Subgroup: 2.3.5
[[Badness]] (Sintel): 28.4


Comma list: {{monzo| 41 -20 -4 }}
== Sextile (5-limit) ==
{{See also| Garischismic clan #Sextile }}


Mapping: [{{val| 4 0 41 }}, {{val| 0 1 -5 }}]
The 5-limit version of sextile reaches the [[interval class]] of [[5/1|5]] by −6 [[3/2|perfect fifths]] (i.e. a diminished fifth) minus a period of 1/6-octave. It corresponds to {{nowrap| ''n'' {{=}} 6 }}, meaning the Pythagorean comma is equated with a stack of six schismas.


{{Multival|legend=1| 4 -20 -41 }}
[[Subgroup]]: 2.3.5


POTE generator: ~3/2 = 702.6054
[[Comma list]]: {{monzo| 71 -36 -6 }}


{{Optimal ET sequence|legend=1| 12, …, 104, 116, 128, 140, 152, 610, 772, 924c, 1076bc, 1228bc }}
{{Mapping|legend=1| 6 0 71 | 0 1 -6 }}]
: Mapping generators: ~4096/3645, ~3


Badness: 0.241703
[[Optimal tuning]]s:  
* [[WE]]: ~4096/3645 = 199.9836{{c}}, ~3/2 = 702.1782{{c}} (~4428675/4194304 = 97.7564{{c}})
: [[error map]]: {{val| -0.098 +0.125 +0.045 }}
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2434{{c}} (~4428675/4194304 = 97.7566{{c}})
: error map: {{val| 0.000 +0.288 +0.226 }}


== Quindromeda (12&amp;205) ==
{{Optimal ET sequence|legend=1| 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc }}
{{See also| Quindromeda family }}


Subgroup: 2.3.5
[[Badness]] (Sintel): 13.0


Comma list: {{monzo| 56 -28 -5 }}
== Wronecki ==
Wronecki equates a stack of six ~[[10/9]]'s with the octave. It reaches the interval class of 5 by +2 [[3/2|perfect fifths]] (i.e. a major second) plus a period of 1/6-octave. It corresponds to {{nowrap| ''m'' {{=}} 6 }}, meaning the Pythagorean comma is equated with a stack of six syntonic commas.


Mapping: [{{val| 1 2 0 }}, {{val| 0 -5 28 }}]
[[Subgroup]]: 2.3.5


POTE generator: ~4428675/4194304 = 99.526
[[Comma list]]: 531441/500000


{{Multival|legend=1| 5 -28 -56 }}
{{Mapping|legend=1| 6 0 -5 | 0 1 2 }}


{{Optimal ET sequence|legend=1| 12, …, 181, 193, 205, 217, 422 }}
: Mapping generators: ~10/9, ~3


Badness: 0.399849
[[Optimal tuning]]s:  
* [[WE]]: ~10/9 = 200.1488{{c}}, ~3/2 = 695.5574{{c}}
: [[error map]]: {{val| +0.893 -5.505 +5.843 }}
* [[CWE]]: ~10/9 = 1200.0000{{c}}, ~3/2 = 695.6109{{c}}
: error map: {{val| 0.000 -6.344 +4.908 }}


== Sextile (12&amp;270) ==
{{Optimal ET sequence|legend=1| 12, 66b, 78b, 90b, 102b }}
{{See also| Landscape microtemperaments #Sextile }}


Subgroup: 2.3.5
[[Badness]] (Sintel): 8.02


Comma list: {{monzo| 71 -36 -6 }}
== Heptacot (5-limit) ==
: ''For extensions, see [[Garischismic clan #Heptacot]].''


Mapping: [{{val| 6 0 71 }}, {{val| 0 1 -6 }}]
The 5-limit version of heptacot tempers out the [[heptacot comma]] and divides the [[3/2|perfect fifth]] into seven equal parts, the most notable example being [[12edo]] (7\12). It corresponds to {{nowrap| ''n'' {{=}} 7 }}, meaning the Pythagorean comma is equated with a stack of seven schismas. It was named by [[Tristan Bay]] in 2024, for it splits the perfect fifth into seven.


POTE generator: ~3/2 = 702.2356
[[Subgroup]]: 2.3.5


{{Multival|legend=1| 6 -36 -77 }}
[[Comma list]]: {{monzo| 86 -44 -7 }}


{{Optimal ET sequence|legend=1| 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc }}
{{Mapping|legend=1| 1 1 6 | 0 7 -44 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9328{{c}}, ~{{monzo| -37 19 3 }} = 100.3012{{c}}
: [[error map]]: {{val| -0.067 +0.086 +0.029 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -37 19 3 }} = 100.3076{{c}}
: error map: {{val| 0.000 +0.198 +0.153 }}
 
{{Optimal ET sequence|legend=1| 12, …, 287, 299, 311, 323, 981, 1304, 5539bc, 6843bbcc }}


Badness: 0.555423
[[Badness]] (Sintel): 16.0


[[Category:12edo]]
[[Category:12edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Latest revision as of 11:03, 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–commatic equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with Pythagorean comma ([-19 12⟩). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 12edo.

All temperaments in the continuum satisfy (32805/32768)n ~ [-19 12⟩. 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 12edo 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 12.0078623975…, and temperaments having n near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small Kirnberger's atom (the difference between 12 schismas and the Pythagorean comma) is.

The Pythagorean comma is the characteristic 3-limit comma tempered out in 12edo, 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 an n that is not coprime with 12, however, the corresponding temperament splits the octave into gcd(n, 12) parts, and splits the interval class of 3 into n/gcd(n, 12). For example:

  • Meantone (n = 1) is generated by a fifth with an unsplit octave;
  • Diaschismic (n = 2) splits the octave in two, as 2 divides 12;
  • Misty (n = 3) splits the octave in three, as 3 divides 12;
  • Undim (n = 4) splits the octave in four, as 4 divides 12;
  • Quindromeda (n = 5) does not split the octave but splits the fourth in five, as 5 is coprime with 12.
Temperaments with integer n
n Temperament Comma
Ratio Monzo
-3 Triscordial (40 digits) [-64 36 3⟩
-2 Biscordial (30 digits) [-49 28 2⟩
-1 Gracecordial (22 digits) [-34 20 1⟩
0 Compton 531441/524288 [-19 12⟩
1 Meantone 81/80 [-4 4 -1⟩
2 Diaschismic 2048/2025 [11 -4 -2⟩
3 Misty 67108864/66430125 [26 -12 -3⟩
4 Undim (26 digits) [41 -20 -4⟩
5 Quindromeda (34 digits) [56 -28 -5⟩
6 Sextile (44 digits) [71 -36 -6⟩
7 Heptacot (52 digits) [86 -44 -7⟩
8 World calendar restriction (62 digits) [101 -52 -8⟩
9 Quinbisa-tritrigu (12 & 441) (70 digits) [116 -60 -9⟩
10 Lesa-quinbigu (12 & 494) (80 digits) [131 -68 -10⟩
11 Quadtrisa-legu (12 & 559) (88 digits) [146 -76 -11⟩
12 Atomic (98 digits) [161 -84 -12⟩
13 Quintrila-theyo (12 & 677) (106 digits) [-176 92 13⟩
… … … …
∞ 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 syntonic–commatic equivalence continuum, which is essentially the same thing. The just value of m is 1.0908441588…. The syntonic comma is way larger but much simpler than the schisma. As such, this continuum does not contain as many microtemperaments, but has more useful lower-complexity temperaments.

Alternatively, because the the 5-limit otonal detemperament of 12edo is a 4×3 rectangle (known as the duodene), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as (81/80)k ~ (128/125). This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates is a flip of the m-continuum such that microtemperaments converging to atomic are found as successive mediants towards the JIP. Specifically, its JIP is at 1.90915584…, which is approximated very closely by the microtemperament atomic at 21/11 = 1.90909… so that the main n-continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic). It is noted for its nontrivial relation to the other better-motivated (in terms of mapping) coordinates discussed.

Temperaments with integer m (and thus k)
m k Temperament Comma
Ratio Monzo
-1 4 Python 43046721/41943040 [-23 16 -1⟩
0 3 Compton 531441/524288 [-19 12⟩
1 2 Schismic 32805/32768 [-15 8 1⟩
2 1 Diaschismic 2048/2025 [11 -4 -2⟩
3 0 Augmented 128/125 [7 0 -3⟩
4 -1 Diminished 648/625 [3 4 -4⟩
5 -2 Ripple 6561/6250 [-1 8 -5⟩
6 -3 Wronecki 531441/500000 [-5 12 -6⟩
… … … … …
∞ ∞ Meantone 81/80 [-4 4 -1⟩
Temperaments with fractional n and m
n m k Temperament Comma
5/3 = 1.6 5/2 = 2.5 1/2 Passion [18 -4 -5⟩
5/2 = 2.5 5/3 = 1.6 4/3 Quintaleap [37 -16 -5⟩

Python

Python is generated by a fifth, which is typically flatter than 7\12. The ~5/4 is reached by +16 fifths octave reduced, which is a double-augmented second (C–Dx). It can be described as 12 & 91, and 103edo is a good tuning. It corresponds to m = -1 and n = 1/2.

Subgroup: 2.3.5

Comma list: 43046721/41943040

Mapping: [⟨1 0 -23], ⟨0 1 16]]

Mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.8769 ¢, ~3/2 = 699.5409 ¢
error map: ⟨+0.876 -1.537 +0.203]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.0789 ¢
error map: ⟨0.000 -2.876 -1.051]

Optimal ET sequence: 12, …, 79, 91, 103, 218b

Badness (Sintel): 6.92

Gracecordial (5-limit)

For extensions, see Marvel temperaments #Gracecordial.

The 5-limit version of gracecordial is generated by a fifth, which is typically sharp of 7\12 but flat of just. The ~5/4 is reached by -20 fifths octave reduced, which is a triple-diminished fifth (C–Gbbb). It can be described as 12 & 125, and 137edo is a good tuning. It corresponds to n = -1 and m = 1/2.

Subgroup: 2.3.5

Comma list: 17433922005/17179869184

Mapping: [⟨1 0 34], ⟨0 1 -20]]

Mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.3986 ¢, ~3/2 = 700.9665 ¢
error map: ⟨+0.399 -0.590 -0.064]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.7202 ¢
error map: ⟨0.000 -1.235 -0.718]

Optimal ET sequence: 12, 113, 125, 137, 1221bbcc

Badness (Sintel): 7.20

Biscordial

Named by Xenllium in 2026, biscordial has a period of half-octave, and tempers out the biscordial comma, [-49 28 2⟩. The ~5/4 is reached by 17 periods minus 14 fifths. It corresponds to n = -2.

Subgroup: 2.3.5

Comma list: 571919811374025/562949953421312

Mapping: [⟨2 0 49], ⟨0 1 -14]]

Mapping generators: ~23914845/16777216, ~3

Optimal tunings:

  • WE: ~23914845/16777216 = 600.153 ¢, ~3/2 = 701.211 ¢
  • CWE: ~23914845/16777216 = 600.000 ¢, ~3/2 = 701.019 ¢

Optimal ET sequence: 12, 166, 178, 190, 392, 582

Badness (Sintel): 15.7

Triscordial

Named by Xenllium in 2026, triscordial has a period of 1/3-octave, and tempers out the triscordial comma, [-64 36 3⟩. The ~5/4 is reached by 22 periods minus 12 fifths. It corresponds to n = -3.

Subgroup: 2.3.5

Comma list: [-64 36 3⟩

Mapping: [⟨3 0 64], ⟨0 1 -12]]

Mapping generators: ~2657205/2097152, ~3

Optimal tunings:

  • WE: ~2657205/2097152 = 400.084 ¢, ~3/2 = 701.343 ¢
  • CWE: ~2657205/2097152 = 400.000 ¢, ~3/2 = 701.182 ¢

Optimal ET sequence: 12, 231, 243, 255, 498, 753

Badness (Sintel): 28.4

Sextile (5-limit)

See also: Garischismic clan § Sextile

The 5-limit version of sextile reaches the interval class of 5 by −6 perfect fifths (i.e. a diminished fifth) minus a period of 1/6-octave. It corresponds to n = 6, meaning the Pythagorean comma is equated with a stack of six schismas.

Subgroup: 2.3.5

Comma list: [71 -36 -6⟩

Mapping: [⟨6 0 71], ⟨0 1 -6]]]

Mapping generators: ~4096/3645, ~3

Optimal tunings:

  • WE: ~4096/3645 = 199.9836 ¢, ~3/2 = 702.1782 ¢ (~4428675/4194304 = 97.7564 ¢)
error map: ⟨-0.098 +0.125 +0.045]
  • CWE: ~4096/3645 = 200.0000 ¢, ~3/2 = 702.2434 ¢ (~4428675/4194304 = 97.7566 ¢)
error map: ⟨0.000 +0.288 +0.226]

Optimal ET sequence: 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc

Badness (Sintel): 13.0

Wronecki

Wronecki equates a stack of six ~10/9's with the octave. It reaches the interval class of 5 by +2 perfect fifths (i.e. a major second) plus a period of 1/6-octave. It corresponds to m = 6, meaning the Pythagorean comma is equated with a stack of six syntonic commas.

Subgroup: 2.3.5

Comma list: 531441/500000

Mapping: [⟨6 0 -5], ⟨0 1 2]]

Mapping generators: ~10/9, ~3

Optimal tunings:

  • WE: ~10/9 = 200.1488 ¢, ~3/2 = 695.5574 ¢
error map: ⟨+0.893 -5.505 +5.843]
  • CWE: ~10/9 = 1200.0000 ¢, ~3/2 = 695.6109 ¢
error map: ⟨0.000 -6.344 +4.908]

Optimal ET sequence: 12, 66b, 78b, 90b, 102b

Badness (Sintel): 8.02

Heptacot (5-limit)

For extensions, see Garischismic clan #Heptacot.

The 5-limit version of heptacot tempers out the heptacot comma and divides the perfect fifth into seven equal parts, the most notable example being 12edo (7\12). It corresponds to n = 7, meaning the Pythagorean comma is equated with a stack of seven schismas. It was named by Tristan Bay in 2024, for it splits the perfect fifth into seven.

Subgroup: 2.3.5

Comma list: [86 -44 -7⟩

Mapping: [⟨1 1 6], ⟨0 7 -44]]

Optimal tunings:

  • WE: ~2 = 1199.9328 ¢, ~[-37 19 3⟩ = 100.3012 ¢
error map: ⟨-0.067 +0.086 +0.029]
  • CWE: ~2 = 1200.0000 ¢, ~[-37 19 3⟩ = 100.3076 ¢
error map: ⟨0.000 +0.198 +0.153]

Optimal ET sequence: 12, …, 287, 299, 311, 323, 981, 1304, 5539bc, 6843bbcc

Badness (Sintel): 16.0

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