Schismic–commatic equivalence continuum: Difference between revisions
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| 0 | | 0 | ||
| [[ | | [[Compton family|Compton]] | ||
| 531441/524288 | | 531441/524288 | ||
| {{monzo|-19 12}} | | {{monzo|-19 12}} | ||
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|- | |- | ||
| 4 | | 4 | ||
| Undim | | [[Hemifamity temperaments|Undim]] | ||
| | | | ||
| {{monzo| 41 -20 -4}} | | {{monzo| 41 -20 -4}} | ||
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== Compton temperament (12&72) == | == Compton temperament (12&72) == | ||
{{see also| Pythagorean comma }} ''and [[ | {{see also| Pythagorean comma }} ''and [[Compton family]]'' | ||
Comma list: {{monzo| -19 12 }} = 531441/524288 | Comma list: {{monzo| -19 12 }} = 531441/524288 | ||
Mapping: [{{val| 12 19 28 }}, {{val| 0 0 -1 }}] | Mapping: [{{val| 12 19 28 }}, {{val| 0 0 -1 }}] | ||
| Line 112: | Line 110: | ||
Wedgie: {{wedgie| 0 12 19 }} | Wedgie: {{wedgie| 0 12 19 }} | ||
{{Val list | POTE generator: ~5/4 = 384.882 | ||
Vals: {{Val list| 12, 48, 60, 72, 84 }} | |||
Badness: 0.094494 | Badness: 0.094494 | ||
| Line 118: | Line 118: | ||
== Lalagu (12&79) == | == Lalagu (12&79) == | ||
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 }}] | ||
| Line 125: | Line 123: | ||
Wedgie: {{wedgie| 1 16 23 }} | Wedgie: {{wedgie| 1 16 23 }} | ||
{{Val list | POTE generator: ~4/3 = 500.970 | ||
Vals: {{Val list| 12, 79, 91, 103 }} | |||
Badness: 0.295079 | Badness: 0.295079 | ||
| Line 133: | Line 133: | ||
Comma list: {{monzo| 37 -16 -5 }} = 137438953472/134521003125 | Comma list: {{monzo| 37 -16 -5 }} = 137438953472/134521003125 | ||
Mapping: [{{val| 1 2 1 }}, {{val| 0 -5 16 }}] | Mapping: [{{val| 1 2 1 }}, {{val| 0 -5 16 }}] | ||
| Line 140: | Line 138: | ||
Wedgie: {{wedgie| 5 -16 -37 }} | Wedgie: {{wedgie| 5 -16 -37 }} | ||
{{Val list | POTE generator: ~135/128 = 99.267 | ||
Vals: {{Val list| 12, 109, 121, 133 }} | |||
Badness: 0.444506 | Badness: 0.444506 | ||
== Undim (12&152) == | == Undim (12&152) == | ||
{{see also| Hemifamity temperaments #Undim }} | |||
Comma list: {{monzo| 41 -20 -4 }} | Comma list: {{monzo| 41 -20 -4 }} | ||
Mapping: [{{val| 4 6 11 }}, {{val| 0 1 -5 }}] | Mapping: [{{val| 4 6 11 }}, {{val| 0 1 -5 }}] | ||
| Line 153: | Line 153: | ||
Wedgie: {{wedgie| 4 -20 -41 }} | Wedgie: {{wedgie| 4 -20 -41 }} | ||
{{Val list | POTE generator: ~3/2 = 702.6054 | ||
Vals: {{Val list| 12, 140, 152, 164, 1076bc, 1228bc }} | |||
Badness: 0.241703 | Badness: 0.241703 | ||
| Line 168: | Line 170: | ||
Wedgie: {{wedgie| 5 -28 -56 }} | Wedgie: {{wedgie| 5 -28 -56 }} | ||
{{Val list | Vals: {{Val list| 12, 193, 205, 217, 422 }} | ||
Badness: 0.399849 | Badness: 0.399849 | ||
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Wedgie: {{wedgie| 6 -36 -77 }} | Wedgie: {{wedgie| 6 -36 -77 }} | ||
{{Val list | Vals: {{Val list| 12, 258, 270, 1878 }} | ||
Badness: 0.555423 | Badness: 0.555423 | ||
Revision as of 00:08, 29 April 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 | ||
| 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 | Quinsa-quingu (12&205) | [56 -28 -5⟩ | |
| 6 | Sextile (12&270) | [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)
- 12 & 121 (n = 5/2 = 2.5)
Compton temperament (12&72)
- and Compton family
Comma list: [-19 12⟩ = 531441/524288
Mapping: [⟨12 19 28], ⟨0 0 -1]]
Wedgie: ⟨⟨0 12 19]]
POTE generator: ~5/4 = 384.882
Vals: Template:Val list
Badness: 0.094494
Lalagu (12&79)
Comma list: [-23 16 -1⟩ = 43046721/41943040
Mapping: [⟨1 2 9], ⟨0 -1 -16]]
Wedgie: ⟨⟨1 16 23]]
POTE generator: ~4/3 = 500.970
Vals: Template:Val list
Badness: 0.295079
Trisa-quingu (12&121)
Comma list: [37 -16 -5⟩ = 137438953472/134521003125
Mapping: [⟨1 2 1], ⟨0 -5 16]]
Wedgie: ⟨⟨5 -16 -37]]
POTE generator: ~135/128 = 99.267
Vals: Template:Val list
Badness: 0.444506
Undim (12&152)
Comma list: [41 -20 -4⟩
Mapping: [⟨4 6 11], ⟨0 1 -5]]
Wedgie: ⟨⟨4 -20 -41]]
POTE generator: ~3/2 = 702.6054
Vals: Template:Val list
Badness: 0.241703
Quinsa-quingu (12&205)
Comma list: [56 -28 -5⟩
POTE generator: ~4428675/4194304 = 99.526
Mapping: [⟨1 2 0], ⟨0 -5 28]]
Wedgie: ⟨⟨5 -28 -56]]
Vals: Template:Val list
Badness: 0.399849
Sextile (12&270)
Comma list: [71 -36 -6⟩
POTE generator: ~3/2 = 702.2356
Mapping: [⟨6 10 11], ⟨0 -1 6]]
Wedgie: ⟨⟨6 -36 -77]]
Vals: Template:Val list
Badness: 0.555423