Schismic–Pythagorean equivalence continuum: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
CompactStar (talk | contribs)
5,020 edits
No edit summary
The inverted continuum
Line 1: Line 1:
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 – 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 center-2"
{| class="wikitable center-1 center-2"
|+ Temperaments in the continuum
|+ Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 20: Line 20:
| 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 }}
|-
|-
Line 97: Line 97:
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| {{monzo| -15 8 1 }}
|}
We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. The just value of ''m'' is 1.0908441588…
{| class="wikitable center-1 center-2"
|+ Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Compton family|Compton]]
| [[531441/524288]]
| {{monzo| -19 12 }}
|-
| 1
| [[Schismic]]
| [[32805/32768]]
| {{monzo| -15 8 1 }}
|-
| 2
| [[Diaschismic family|Diaschismic]]
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
|-
| 3
| [[Augmented]]
| [[128/125]]
| {{monzo| 7 0 -3 }}
|-
| 4
| [[Diminished]]
| [[648/625]]
| {{monzo| 3 4 -4 }}
|-
| 5
| [[Ripple]]
| [[6561/6250]]
| {{monzo| -1 8 -5 }}
|-
| 6
| [[Wronecki]]
| [[531441/500000]]
| {{monzo| -5 12 -6 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Meantone]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
|}
|}


Revision as of 09:59, 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.

Temperaments with integer n
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. The just value of m is 1.0908441588…

Temperaments with integer m
m Temperament Comma
Ratio Monzo
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⟩

Examples of temperaments with fractional values of n:

Compton (12&72)

See also: Pythagorean comma
 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

Optimal ET sequence: 12, 48, 60, 72, 84

Badness: 0.094494

Python (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

Optimal ET sequence: 12, …, 79, 91, 103

Badness: 0.295079

Quintaleap (12&121)

See also: Quintaleap family

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

Optimal ET sequence: 12, …, 85, 97, 109, 121, 133, 278c, 411bc, 544bc

Badness: 0.444506

Undim (12&152)

See also: Undim family

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

Optimal ET sequence: 12, …, 104, 116, 128, 140, 152, 610, 772, 924c, 1076bc, 1228bc

Badness: 0.241703

Quindromeda (12&205)

See also: Quindromeda family

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 ]]

Optimal ET sequence: 12, …, 181, 193, 205, 217, 422

Badness: 0.399849

Sextile (12&270)

See also: Landscape microtemperaments § Sextile

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 ]]

Optimal ET sequence: 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc

Badness: 0.555423

Retrieved from "https://en.xen.wiki/index.php?title=Schismic–Pythagorean_equivalence_continuum&oldid=141476"