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Schismic–commatic equivalence continuum: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''schismic–Pythagorean 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]].
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 {{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.
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.
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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–Pythagorean 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"
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== Sextile (5-limit) ==
== Sextile (5-limit) ==
{{See also| Landscape microtemperaments #Sextile }}
{{See also| Garischismic clan #Sextile }}


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.  
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.  
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[[Badness]] (Sintel): 8.02
[[Badness]] (Sintel): 8.02


== Heptacot ==
== Heptacot (5-limit) ==
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 [[Akselai]] in 2024, for it splits the perfect fifth into seven.  
: ''For extensions, see [[Garischismic clan #Heptacot]].''
 
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.  


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