Schismic–commatic equivalence continuum: Difference between revisions
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| [[Marvel temperaments #Gracecordial|Gracecordial]] | | [[Marvel temperaments #Gracecordial|Gracecordial]] | ||
| 17433922005/17179869184 | | 17433922005/17179869184 | ||
| {{monzo|-34 20 1}} | | {{monzo| -34 20 1 }} | ||
|- | |- | ||
| 0 | | 0 | ||
| [[Compton family|Compton]] | | [[Compton family|Compton]] | ||
| 531441/524288 | | 531441/524288 | ||
| {{monzo|-19 12}} | | {{monzo| -19 12 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| [[Meantone family|Meantone]] | | [[Meantone family|Meantone]] | ||
| 81/80 | | 81/80 | ||
| {{monzo|-4 4 -1}} | | {{monzo|-4 4 -1 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| [[Diaschismic family|Diaschismic]] | | [[Diaschismic family|Diaschismic]] | ||
| 2048/2025 | | 2048/2025 | ||
| {{monzo| 11 -4 -2}} | | {{monzo| 11 -4 -2 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| [[Misty family|Misty]] | | [[Misty family|Misty]] | ||
| 67108864/66430125 | | 67108864/66430125 | ||
| {{monzo| 26 -12 -3}} | | {{monzo| 26 -12 -3 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| [[Hemifamity temperaments|Undim]] | | [[Hemifamity temperaments|Undim]] | ||
| | | | ||
| {{monzo| 41 -20 -4}} | | {{monzo| 41 -20 -4 }} | ||
|- | |- | ||
| 5 | | 5 | ||
| [[ | | [[Quindromeda family|Quindromeda]] | ||
| | | | ||
| {{monzo| 56 -28 -5}} | | {{monzo| 56 -28 -5 }} | ||
|- | |- | ||
| 6 | | 6 | ||
| Sextile | | [[Sextile]] | ||
| | | | ||
| {{monzo| 71 -36 -6}} | | {{monzo| 71 -36 -6 }} | ||
|- | |- | ||
| 7 | | 7 | ||
| Sepsa-sepgu (12&323) | | Sepsa-sepgu (12&323) | ||
| | | | ||
| {{monzo| 86 -44 -7}} | | {{monzo| 86 -44 -7 }} | ||
|- | |- | ||
| 8 | | 8 | ||
| Tritrisa-quadbigu (12&388) | | Tritrisa-quadbigu (12&388) | ||
| | | | ||
| {{monzo| 101 -52 -8}} | | {{monzo| 101 -52 -8 }} | ||
|- | |- | ||
| 9 | | 9 | ||
| Quinbisa-tritrigu (12&441) | | Quinbisa-tritrigu (12&441) | ||
| | | | ||
| {{monzo| 116 -60 -9}} | | {{monzo| 116 -60 -9 }} | ||
|- | |- | ||
| 10 | | 10 | ||
| Lesa-quinbigu (12&494) | | Lesa-quinbigu (12&494) | ||
| | | | ||
| {{monzo| 131 -68 -10}} | | {{monzo| 131 -68 -10 }} | ||
|- | |- | ||
| 11 | | 11 | ||
| Quadtrisa-legu (12&559) | | Quadtrisa-legu (12&559) | ||
| | | | ||
| {{monzo| 146 -76 -11}} | | {{monzo| 146 -76 -11 }} | ||
|- | |- | ||
| 12 | | 12 | ||
| [[Very high accuracy temperaments #Atomic|Atomic]] | | [[Very high accuracy temperaments #Atomic|Atomic]] | ||
| | | | ||
| [[Kirnberger's atom|{{monzo|161 -84 -12}}]] | | [[Kirnberger's atom|{{monzo| 161 -84 -12 }}]] | ||
|- | |- | ||
| 13 | | 13 | ||
| Quintrila-theyo (12&677) | | Quintrila-theyo (12&677) | ||
| | | | ||
| {{monzo|-176 92 13}} | | {{monzo| -176 92 13 }} | ||
|- | |- | ||
| … | | … | ||
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| [[Schismic]] | | [[Schismic]] | ||
| [[32805/32768]] | | [[32805/32768]] | ||
| {{monzo| -15 8 1}} | | {{monzo| -15 8 1 }} | ||
|} | |} | ||
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* [[Augmented family|Augmented]] (''n'' = 3/2 = 1.5) | * [[Augmented family|Augmented]] (''n'' = 3/2 = 1.5) | ||
* [[Passion family|Passion]] (''n'' = 5/3 = 1.{{overline|6}}) | * [[Passion family|Passion]] (''n'' = 5/3 = 1.{{overline|6}}) | ||
* [[ | * [[Quintaleap family|Quintaleap]] (''n'' = 5/2 = 2.5) | ||
== Compton | == Compton (12&72) == | ||
{{See also| Pythagorean comma }} ''and [[Compton family]]'' | {{See also| Pythagorean comma }} ''and [[Compton family]]'' | ||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| -19 12 }} = 531441/524288 | Comma list: {{monzo| -19 12 }} = 531441/524288 | ||
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Mapping: [{{val| 12 19 28 }}, {{val| 0 0 -1 }}] | Mapping: [{{val| 12 19 28 }}, {{val| 0 0 -1 }}] | ||
{{Multival|legend=1| 0 12 19 }} | |||
POTE generator: ~5/4 = 384.882 | POTE generator: ~5/4 = 384.882 | ||
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== Lalagu (12&79) == | == Lalagu (12&79) == | ||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| -23 16 -1 }} = 43046721/41943040 | Comma list: {{monzo| -23 16 -1 }} = 43046721/41943040 | ||
Mapping: [{{val| 1 2 9 }}, {{val| 0 -1 -16 }}] | Mapping: [{{val| 1 2 9 }}, {{val| 0 -1 -16 }}] | ||
{{Multival|legend=1| 1 16 23 }} | |||
POTE generator: ~4/3 = 500.970 | POTE generator: ~4/3 = 500.970 | ||
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Badness: 0.295079 | Badness: 0.295079 | ||
== Quintaleap (12&121 | == Quintaleap (12&121) == | ||
{{See also| | {{See also| Quintaleap family }} | ||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| 37 -16 -5 }} = 137438953472/134521003125 | Comma list: {{monzo| 37 -16 -5 }} = 137438953472/134521003125 | ||
| Line 141: | Line 147: | ||
Mapping: [{{val| 1 2 1 }}, {{val| 0 -5 16 }}] | Mapping: [{{val| 1 2 1 }}, {{val| 0 -5 16 }}] | ||
{{Multival|legend=1| 5 -16 -37 }} | |||
POTE generator: ~135/128 = 99.267 | POTE generator: ~135/128 = 99.267 | ||
| Line 151: | Line 157: | ||
== Undim (12&152) == | == Undim (12&152) == | ||
{{see also| Hemifamity temperaments #Undim }} | {{see also| Hemifamity temperaments #Undim }} | ||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| 41 -20 -4 }} | Comma list: {{monzo| 41 -20 -4 }} | ||
| Line 156: | Line 164: | ||
Mapping: [{{val| 4 6 11 }}, {{val| 0 1 -5 }}] | Mapping: [{{val| 4 6 11 }}, {{val| 0 1 -5 }}] | ||
{{Multival|legend=1| 4 -20 -41 }} | |||
POTE generator: ~3/2 = 702.6054 | POTE generator: ~3/2 = 702.6054 | ||
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Badness: 0.241703 | Badness: 0.241703 | ||
== Quindromeda (12&205 | == Quindromeda (12&205) == | ||
{{ | {{See also| Quindromeda family }} | ||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| 56 -28 -5 }} | Comma list: {{monzo| 56 -28 -5 }} | ||
Mapping: [{{val| 1 2 0 }}, {{val| 0 -5 28 }}] | |||
POTE generator: ~4428675/4194304 = 99.526 | POTE generator: ~4428675/4194304 = 99.526 | ||
{{Multival|legend=1| 5 -28 -56 }} | |||
Vals: {{Val list| 12, 193, 205, 217, 422 }} | Vals: {{Val list| 12, 193, 205, 217, 422 }} | ||
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== Sextile (12&270) == | == Sextile (12&270) == | ||
{{ | {{See also| Landscape microtemperaments #Sextile }} | ||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| 71 -36 -6 }} | Comma list: {{monzo| 71 -36 -6 }} | ||
Mapping: [{{val| 6 10 11 }}, {{val| 0 -1 6 }}] | |||
POTE generator: ~3/2 = 702.2356 | POTE generator: ~3/2 = 702.2356 | ||
{{Multival|legend=1| 6 -36 -77 }} | |||
Vals: {{Val list| 12, 258, 270, 1878 }} | Vals: {{Val list| 12, 258, 270, 1878 }} | ||
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[[Category:12edo]] | [[Category:12edo]] | ||
[[Category: | [[Category:Regular temperament theory]] | ||
[[Category:Temperament]] | [[Category:Temperament collection]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Revision as of 16:05, 21 November 2021
The Schismic-Pythagorean 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 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.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1 | Gracecordial | 17433922005/17179869184 | [-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 | [41 -20 -4⟩ | |
| 5 | Quindromeda | [56 -28 -5⟩ | |
| 6 | Sextile | [71 -36 -6⟩ | |
| 7 | Sepsa-sepgu (12&323) | [86 -44 -7⟩ | |
| 8 | Tritrisa-quadbigu (12&388) | [101 -52 -8⟩ | |
| 9 | Quinbisa-tritrigu (12&441) | [116 -60 -9⟩ | |
| 10 | Lesa-quinbigu (12&494) | [131 -68 -10⟩ | |
| 11 | Quadtrisa-legu (12&559) | [146 -76 -11⟩ | |
| 12 | Atomic | [161 -84 -12⟩ | |
| 13 | Quintrila-theyo (12&677) | [-176 92 13⟩ | |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |
Examples of temperaments with fractional values of n:
- 12 & 79 (n = 1/2 = 0.5)
- Diminished (n = 4/3 = 1.3)
- Augmented (n = 3/2 = 1.5)
- Passion (n = 5/3 = 1.6)
- Quintaleap (n = 5/2 = 2.5)
Compton (12&72)
- and Compton family
Subgroup: 2.3.5
Comma list: [-19 12⟩ = 531441/524288
Mapping: [⟨12 19 28], ⟨0 0 -1]]
⟨⟨0 12 19]]
POTE generator: ~5/4 = 384.882
Vals: Template:Val list
Badness: 0.094494
Lalagu (12&79)
Subgroup: 2.3.5
Comma list: [-23 16 -1⟩ = 43046721/41943040
Mapping: [⟨1 2 9], ⟨0 -1 -16]]
⟨⟨1 16 23]]
POTE generator: ~4/3 = 500.970
Vals: Template:Val list
Badness: 0.295079
Quintaleap (12&121)
Subgroup: 2.3.5
Comma list: [37 -16 -5⟩ = 137438953472/134521003125
Mapping: [⟨1 2 1], ⟨0 -5 16]]
⟨⟨5 -16 -37]]
POTE generator: ~135/128 = 99.267
Vals: Template:Val list
Badness: 0.444506
Undim (12&152)
Subgroup: 2.3.5
Comma list: [41 -20 -4⟩
Mapping: [⟨4 6 11], ⟨0 1 -5]]
⟨⟨4 -20 -41]]
POTE generator: ~3/2 = 702.6054
Vals: Template:Val list
Badness: 0.241703
Quindromeda (12&205)
Subgroup: 2.3.5
Comma list: [56 -28 -5⟩
Mapping: [⟨1 2 0], ⟨0 -5 28]]
POTE generator: ~4428675/4194304 = 99.526
⟨⟨5 -28 -56]]
Vals: Template:Val list
Badness: 0.399849
Sextile (12&270)
Subgroup: 2.3.5
Comma list: [71 -36 -6⟩
Mapping: [⟨6 10 11], ⟨0 -1 6]]
POTE generator: ~3/2 = 702.2356
⟨⟨6 -36 -77]]
Vals: Template:Val list
Badness: 0.555423