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]] (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 – in fact, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] is. | 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 – in fact, 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" | ||
Revision as of 06:59, 14 March 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 – in fact, 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 | Tribisa-tribigu (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)
- 12 & 4 (n = 4/3 = 1.3)
Compton temperament (12&72)
Comma list: [-19 12⟩ = 531441/524288
POTE generator: ~5/4 = 384.882
Mapping: [⟨12 19 28], ⟨0 0 -1]]
Wedgie: ⟨⟨0 12 19]]
Badness: 0.094494
Undim (12&152)
Comma list: [41 -20 -4⟩
POTE generator: ~3/2 = 702.6054
Mapping: [⟨4 6 11], ⟨0 1 -5]]
Wedgie: ⟨⟨4 -20 -41]]
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]]
Badness: 0.399849
Tribisa-tribigu (12 & 270)
Comma list: [71 -36 -6⟩
POTE generator: ~3/2 = 702.2356
Mapping: [⟨6 10 11], ⟨0 -1 6]]
Wedgie: ⟨⟨6 -36 -77]]
Badness: 0.555423