Schismic–commatic equivalence continuum: Difference between revisions

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* [[Quindromeda]]  ({{nowrap| ''n'' {{=}} 5 }}) does not split the octave but splits the fourth in five, as 5 is coprime with 12.  
* [[Quindromeda]]  ({{nowrap| ''n'' {{=}} 5 }}) does not split the octave but splits the fourth in five, as 5 is coprime with 12.  


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 has the advantage of finding all temperaments discussed in a relatively intuitive and simple way so that less accurate but structurally simpler temperaments are found at integer points while 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 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"
 
{| class="wikitable center-1 center-2"
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
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|-
|-
| -3
| -3
| 9/4
| [[Triscordial]]
| [[Triscordial]]
| (40 digits)
| (40 digits)
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|-
|-
| -2
| -2
| 7/3
| [[Biscordial]]
| [[Biscordial]]
| (30 digits)
| (30 digits)
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|-
|-
| -1
| -1
| 5/2
| [[Gracecordial]]
| [[Gracecordial]]
| (22 digits)
| (22 digits)
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|-
|-
| 0
| 0
| 3
| [[Compton]]
| [[Compton]]
| [[531441/524288]]
| [[531441/524288]]
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|-
|-
| 1
| 1
| ∞
| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
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|-
|-
| 2
| 2
| 1
| [[Diaschismic]]
| [[Diaschismic]]
| [[2048/2025]]
| [[2048/2025]]
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|-
|-
| 3
| 3
| 3/2
| [[Misty]]
| [[Misty]]
| [[67108864/66430125]]
| [[67108864/66430125]]
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|-
|-
| 4
| 4
| 5/3
| [[Undim]]
| [[Undim]]
| (26 digits)
| (26 digits)
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|-
|-
| 5
| 5
| 7/4
| [[Quindromeda]]
| [[Quindromeda]]
| (34 digits)
| (34 digits)
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|-
|-
| 6
| 6
| 9/5
| [[Sextile]]
| [[Sextile]]
| (44 digits)
| (44 digits)
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|-
|-
| 7
| 7
| 11/6
| [[Heptacot]]
| [[Heptacot]]
| (52 digits)
| (52 digits)
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|-
|-
| 8
| 8
| 13/7
| [[World calendar]] restriction
| [[World calendar]] restriction
| (62 digits)
| (62 digits)
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|-
|-
| 9
| 9
| 15/8
| Quinbisa-tritrigu (12 & 441)
| Quinbisa-tritrigu (12 & 441)
| (70 digits)
| (70 digits)
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|-
|-
| 10
| 10
| 17/9
| Lesa-quinbigu (12 & 494)
| Lesa-quinbigu (12 & 494)
| (80 digits)
| (80 digits)
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|-
|-
| 11
| 11
| 19/10
| Quadtrisa-legu (12 & 559)
| Quadtrisa-legu (12 & 559)
| (88 digits)
| (88 digits)
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|-
|-
| 12
| 12
| 21/11
| [[Atomic]]
| [[Atomic]]
| (98 digits)
| (98 digits)
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|-
|-
| 13
| 13
| 23/12
| Quintrila-theyo (12 & 677)
| Quintrila-theyo (12 & 677)
| (106 digits)
| (106 digits)
| {{Monzo| -176 92 13 }}
| {{Monzo| -176 92 13 }}
|-
|-
| …
| …
| …
| …
| …
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|-
|-
| ∞
| ∞
| 2
| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
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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.  
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"

Revision as of 12:04, 24 April 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: Landscape microtemperaments § 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

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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