Schismic–commatic equivalence continuum: Difference between revisions
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The '''Schismic-Pythagorean equivalence continuum''' is a continuum of 5-limit 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 supported by [[12edo]]. | The '''Schismic-Pythagorean equivalence continuum''' is a continuum of 5-limit 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 supported by [[12edo]]. | ||
All temperaments in the continuum satisfy (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]] | All temperaments in the continuum satisfy (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 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. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
| Line 39: | Line 39: | ||
|- | |- | ||
| 4 | | 4 | ||
| [[ | | [[Undim family|Undim]] | ||
| | | | ||
| {{monzo| 41 -20 -4 }} | | {{monzo| 41 -20 -4 }} | ||
| Line 135: | Line 135: | ||
POTE generator: ~4/3 = 500.970 | POTE generator: ~4/3 = 500.970 | ||
Vals: {{Val list| 12, 79, 91, 103 }} | Vals: {{Val list| 12, …, 79, 91, 103 }} | ||
Badness: 0.295079 | Badness: 0.295079 | ||
| Line 152: | Line 152: | ||
POTE generator: ~135/128 = 99.267 | POTE generator: ~135/128 = 99.267 | ||
Vals: {{Val list| 12, 109, 121, 133 }} | Vals: {{Val list| 12, …, 85, 97, 109, 121, 133, 278c, 411bc, 544bc }} | ||
Badness: 0.444506 | Badness: 0.444506 | ||
== Undim (12&152) == | == Undim (12&152) == | ||
{{ | {{See also| Undim family }} | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
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POTE generator: ~3/2 = 702.6054 | POTE generator: ~3/2 = 702.6054 | ||
Vals: {{Val list| 12, 140, 152, | Vals: {{Val list| 12, …, 104, 116, 128, 140, 152, 610, 772, 924c, 1076bc, 1228bc }} | ||
Badness: 0.241703 | Badness: 0.241703 | ||
| Line 186: | Line 186: | ||
{{Multival|legend=1| 5 -28 -56 }} | {{Multival|legend=1| 5 -28 -56 }} | ||
Vals: {{Val list| 12, 193, 205, 217, 422 }} | Vals: {{Val list| 12, …, 181, 193, 205, 217, 422 }} | ||
Badness: 0.399849 | Badness: 0.399849 | ||
| Line 203: | Line 203: | ||
{{Multival|legend=1| 6 -36 -77 }} | {{Multival|legend=1| 6 -36 -77 }} | ||
Vals: {{Val list| 12, 258, 270, 1878 }} | Vals: {{Val list| 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc }} | ||
Badness: 0.555423 | Badness: 0.555423 | ||
Revision as of 19:48, 11 December 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)
- Ripple (n = 5/4 = 1.25)
- 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]]
Wedgie: ⟨⟨ 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 0 -23], ⟨0 -1 -16]]
Wedgie: ⟨⟨ 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]]
Wedgie: ⟨⟨ 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 0 41], ⟨0 1 -5]]
Wedgie: ⟨⟨ 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
Wedgie: ⟨⟨ 5 -28 -56 ]]
Vals: Template:Val list
Badness: 0.399849
Sextile (12&270)
Subgroup: 2.3.5
Comma list: [71 -36 -6⟩
Mapping: [⟨6 0 71], ⟨0 1 -6]]
POTE generator: ~3/2 = 702.2356
Wedgie: ⟨⟨ 6 -36 -77 ]]
Vals: Template:Val list
Badness: 0.555423