Schismic–Pythagorean equivalence continuum

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	(22 digits)	[-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	(26 digits)	[41 -20 -4⟩
5	Quindromeda	(34 digits)	[56 -28 -5⟩
6	Sextile	(44 digits)	[71 -36 -6⟩
7	Heptacot	(52 digits)	[86 -44 -7⟩
8	World calendar	(62 digits)	[101 -52 -8⟩
9	Quinbisa-tritrigu (12&441)	(70 digits)	[116 -60 -9⟩
10	Lesa-quinbigu (12&494)	(80 digits)	[131 -68 -10⟩
11	Quadtrisa-legu (12&559)	(88 digits)	[146 -76 -11⟩
12	Atomic	(98 digits)	[161 -84 -12⟩
13	Quintrila-theyo (12&677)	(106 digits)	[-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…

Temperaments with integer m

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⟩

Temperaments with fractional n and m

n	m	Temperament	Comma
5/3 = 1.6	5/2 = 2.5	Passion	[18 -4 -5⟩
5/2 = 2.5	5/3 = 1.6	Quintaleap	[37 -16 -5⟩

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]]

mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 16 23 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 699.1680
• CWE: ~2 = 1\1, ~3/2 = 699.0789

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

Badness: 0.295079

Sextile

See also: Landscape microtemperaments § Sextile

The 5-limit version of sextile reaches the interval class of 5 by -6 perfect fifths minus a period of 1/6-octave.

Subgroup: 2.3.5

Comma list: [71 -36 -6⟩

Mapping: [⟨6 0 71], ⟨0 1 -6]]]

mapping generators: ~4096/3645, ~3

Optimal tunings:

• CTE: ~4096/3645 = 1\6, ~3/2 = 702.2627 (~4428675/4194304 = 97.7373)
• CWE: ~4096/3645 = 1\6, ~3/2 = 702.2434 (~4428675/4194304 = 97.7566)

Wedgie: ⟨⟨ 6 -36 -77 ]]

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

Badness: 0.555423

Heptacot

Heptacot tempers out the heptacot comma and divides the fifth into seven equal parts, the most notable example being 12edo (7\12).

Subgroup: 2.3.5

Comma list: [86 -44 -7⟩

Mapping: [⟨1 1 6], ⟨0 7 -44]]]

Optimal tunings:

• CTE: ~2 = 1\1, ~[-37 19 3⟩ = 100.3094
• CWE: ~2 = 1\1, ~[-37 19 3⟩ = 100.3076

Optimal ET sequence: 12, …, 287, 299, 311, 323, 981, 1304, 5539bc, 6843bbcc

Badness: 0.683