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Schismic–Pythagorean 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 [[duodene|5-limit otonal detemper]] of 12edo is a 4x3 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 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.
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 center-2"
{| class="wikitable center-1 center-2"
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| 9
| 9
| 15/8
| 15/8
| Quinbisa-tritrigu (12&amp;441)
| Quinbisa-tritrigu (12 & 441)
| (70 digits)
| (70 digits)
| {{monzo| 116 -60 -9 }}
| {{monzo| 116 -60 -9 }}
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| 10
| 10
| 17/9
| 17/9
| Lesa-quinbigu (12&amp;494)
| Lesa-quinbigu (12 & 494)
| (80 digits)
| (80 digits)
| {{monzo| 131 -68 -10 }}
| {{monzo| 131 -68 -10 }}
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| 11
| 11
| 19/10
| 19/10
| Quadtrisa-legu (12&amp;559)
| Quadtrisa-legu (12 & 559)
| (88 digits)
| (88 digits)
| {{monzo| 146 -76 -11 }}
| {{monzo| 146 -76 -11 }}
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| 13
| 13
| 23/12
| 23/12
| Quintrila-theyo (12&amp;677)
| Quintrila-theyo (12 & 677)
| (106 digits)
| (106 digits)
| {{monzo| -176 92 13 }}
| {{monzo| -176 92 13 }}
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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–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.  


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
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| 0
| 0
| 3
| 3
| [[Compton family|Compton]]
| [[Compton]]
| [[531441/524288]]
| [[531441/524288]]
| {{monzo| -19 12 }}
| {{monzo| -19 12 }}
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| 2
| 2
| 1
| 1
| [[Diaschismic family|Diaschismic]]
| [[Diaschismic]]
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
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| 3
| 3
| 0
| 0
| [[Augmented]]
| [[Augmented (temperament)|Augmented]]
| [[128/125]]
| [[128/125]]
| {{monzo| 7 0 -3 }}
| {{monzo| 7 0 -3 }}
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| 4
| 4
| -1
| -1
| [[Diminished]]
| [[Diminished (temperament)|Diminished]]
| [[648/625]]
| [[648/625]]
| {{monzo| 3 4 -4 }}
| {{monzo| 3 4 -4 }}
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== Python ==
== Python ==
Python is generated by a fifth, which is typically flatter than 7\12. The ~5/4 is reached by sixteen fifths octave-reduced, which is a double augmented second (C-Dx). It can be described as 12 & 91, and 103edo is a good tuning.  
Python is generated by a fifth, which is typically flatter than 7\12. The ~5/4 is reached by sixteen 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.  


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


[[Comma list]]: {{monzo| -23 16 -1 }} = 43046721/41943040
[[Comma list]]: 43046721/41943040


{{Mapping|legend=1| 1 0 -23 | 0 1 16 }}
{{Mapping|legend=1| 1 0 -23 | 0 1 16 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 699.1680
* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.1680{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 699.0789
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.0789{{c}}


{{Optimal ET sequence|legend=1| 12, …, 79, 91, 103 }}
{{Optimal ET sequence|legend=1| 12, …, 79, 91, 103 }}


[[Badness]]: 0.295079
[[Badness]] (Smith): 0.295079


== Sextile ==
== Sextile ==
{{See also| Landscape microtemperaments #Sextile }}
{{See also| Landscape microtemperaments #Sextile }}


The 5-limit version of sextile reaches the interval class of 5 by &minus;6 perfect fifths minus a period of 1/6-octave.  
The 5-limit version of sextile reaches the interval class of 5 by −6 perfect fifths minus a period of 1/6-octave.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~4096/3645 = 1\6, ~3/2 = 702.2627 (~4428675/4194304 = 97.7373)
* [[CTE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2627{{c}} (~4428675/4194304 = 97.7373{{c}})
* [[CWE]]: ~4096/3645 = 1\6, ~3/2 = 702.2434 (~4428675/4194304 = 97.7566)
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2434{{c}} (~4428675/4194304 = 97.7566{{c}})


{{Optimal ET sequence|legend=1| 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc }}
{{Optimal ET sequence|legend=1| 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc }}


[[Badness]]: 0.555423
[[Badness]] (Smith): 0.555423


== Heptacot ==
== Heptacot ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| -37 19 3 }} = 100.3094
* [[CTE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -37 19 3 }} = 100.3094{{c}}
* [[CWE]]: ~2 = 1\1, ~{{monzo| -37 19 3 }} = 100.3076
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -37 19 3 }} = 100.3076{{c}}


{{Optimal ET sequence|legend=1| 12, …, 287, 299, 311, 323, 981, 1304, 5539bc, 6843bbcc }}
{{Optimal ET sequence|legend=1| 12, …, 287, 299, 311, 323, 981, 1304, 5539bc, 6843bbcc }}


[[Badness]]: 0.682871
[[Badness]] (Smith): 0.682871


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