Schismic–Pythagorean equivalence continuum: Difference between revisions
This m-continuum covers most temps of fractional n. |
No need to put the data of compton, undim, quintaleap, and quindromeda here. If anything a link will suffice |
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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. | 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 | {| class="wikitable center-1" | ||
|+ Temperaments with integer ''n'' | |+ Temperaments with integer ''n'' | ||
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We may invert the continuum by setting ''m'' such that 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… | We may invert the continuum by setting ''m'' such that 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… | ||
{| class="wikitable center-1 | {| class="wikitable center-1" | ||
|+ Temperaments with integer ''m'' | |+ Temperaments with integer ''m'' | ||
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== | == 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. | |||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
Comma list: {{monzo| - | [[Comma list]]: {{monzo| -23 16 -1 }} = 43046721/41943040 | ||
{{Mapping|legend=1| 1 0 -23 | 0 1 16 }} | |||
{{Multival|legend=1| 1 16 23 }} | {{Multival|legend=1| 1 16 23 }} | ||
POTE | [[Optimal tuning]]s: | ||
* [[POTE]]: ~2 = 1\1, ~3/2 = 699.030 | |||
{{Optimal ET sequence|legend=1| 12, …, 79, 91, 103 }} | {{Optimal ET sequence|legend=1| 12, …, 79, 91, 103 }} | ||
[[Badness]]: 0.295079 | |||
Badness | |||
== Sextile | == Sextile == | ||
{{See also| Landscape microtemperaments #Sextile }} | {{See also| Landscape microtemperaments #Sextile }} | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
Comma list: {{monzo| 71 -36 -6 }} | [[Comma list]]: {{monzo| 71 -36 -6 }} | ||
{{Mapping|legend=1| 6 0 71 | 0 1 -6 }}] | |||
POTE | [[Optimal tuning]]s: | ||
* [[POTE]]: ~3/2 = 702.2356 | |||
{{Multival|legend=1| 6 -36 -77 }} | {{Multival|legend=1| 6 -36 -77 }} | ||
| Line 270: | Line 206: | ||
{{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]]: 0.555423 | ||
[[Category:12edo]] | [[Category:12edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Revision as of 10:29, 20 April 2024
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 | World calendar | [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⟩ |
We may invert the continuum by setting m such that 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…
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1 | Python | 43046721/41943040 | [-23 16 -1⟩ |
| 0 | Compton | 531441/524288 | [-19 12⟩ |
| 1 | Schismic | 32805/32768 | [-15 8 1⟩ |
| 2 | Diaschismic | 2048/2025 | [11 -4 -2⟩ |
| 3 | Augmented | 128/125 | [7 0 -3⟩ |
| 4 | Diminished | 648/625 | [3 4 -4⟩ |
| 5 | Ripple | 6561/6250 | [-1 8 -5⟩ |
| 6 | Wronecki | 531441/500000 | [-5 12 -6⟩ |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
| Temperament | n | m |
|---|---|---|
| Passion | 5/3 = 1.6 | 5/2 = 2.5 |
| Quintaleap | 5/2 = 2.5 | 5/3 = 1.6 |
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.
Subgroup: 2.3.5
Comma list: [-23 16 -1⟩ = 43046721/41943040
Mapping: [⟨1 0 -23], ⟨0 1 16]]
Wedgie: ⟨⟨ 1 16 23 ]]
- POTE: ~2 = 1\1, ~3/2 = 699.030
Optimal ET sequence: 12, …, 79, 91, 103
Badness: 0.295079
Sextile
Subgroup: 2.3.5
Comma list: [71 -36 -6⟩
Mapping: [⟨6 0 71], ⟨0 1 -6]]]
- POTE: ~3/2 = 702.2356
Wedgie: ⟨⟨ 6 -36 -77 ]]
Optimal ET sequence: 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc
Badness: 0.555423