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 out both commas and thus tempers out 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.

The Pythagorean comma is the characteristic 3-limit comma tempered out in 12edo, and has many advantages as a target. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of harmonic 3 in the generator chain. For an n that is not coprime with 12, however, the corresponding temperament splits the octave into gcd (n, 12) parts, and splits the interval class of 3 into n/gcd(n, 12). For example:

• Meantone (n = 1) is generated by a fifth with an unsplit octave;
• Diaschismic (n = 2) splits the octave in two, as 2 divides 12;
• Misty (n = 3) splits the octave in three, as 3 divides 12;
• Undim (n = 4) splits the octave in four, as 4 divides 12;
• Quindromeda (n = 5) does not split the octave but splits the fourth in five, as 5 is coprime with 12.

Alternatively, because the 5-limit otonal detemper of 12edo is a 4x3 rectangle (known as the duodene), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as (81/80)^k ~ (128/125). This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive way and is noted for its nontrivial relation to the other better-motivated coordinates discussed; specifically, it's related to the inverted continuum by a translation followed by a flip. Its JIP is at 1.90915584... which is approximated closely by the microtemperament atomic at 21/11 = 1.90909... so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic).

Temperaments with integer n

n	k	Temperament	Comma
			Ratio	Monzo
-1	5/2	Gracecordial	(22 digits)	[-34 20 1⟩
0	3/1	Compton	531441/524288	[-19 12⟩
1	∞	Meantone	81/80	[-4 4 -1⟩
2	1	Diaschismic	2048/2025	[11 -4 -2⟩
3	3/2	Misty	67108864/66430125	[26 -12 -3⟩
4	5/3	Undim	(26 digits)	[41 -20 -4⟩
5	7/4	Quindromeda	(34 digits)	[56 -28 -5⟩
6	9/5	Sextile	(44 digits)	[71 -36 -6⟩
7	11/6	Heptacot	(52 digits)	[86 -44 -7⟩
8	13/7	World calendar	(62 digits)	[101 -52 -8⟩
9	15/8	Quinbisa-tritrigu (12&441)	(70 digits)	[116 -60 -9⟩
10	17/9	Lesa-quinbigu (12&494)	(80 digits)	[131 -68 -10⟩
11	19/10	Quadtrisa-legu (12&559)	(88 digits)	[146 -76 -11⟩
12	21/11	Atomic	(98 digits)	[161 -84 -12⟩
13	23/12	Quintrila-theyo (12&677)	(106 digits)	[-176 92 13⟩
…	…	…	…
∞	2	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… The syntonic comma is way larger but much simpler than the schisma. As such, this continuum does not contain as many microtemperaments, but has more useful lower-complexity temperaments.

Temperaments with integer m

m	k	Temperament	Comma
			Ratio	Monzo
-1	4	Python	43046721/41943040	[-23 16 -1⟩
0	3	Compton	531441/524288	[-19 12⟩
1	2	Schismic	32805/32768	[-15 8 1⟩
2	1	Diaschismic	2048/2025	[11 -4 -2⟩
3	0	Augmented	128/125	[7 0 -3⟩
4	-1	Diminished	648/625	[3 4 -4⟩
5	-2	Ripple	6561/6250	[-1 8 -5⟩
6	-3	Wronecki	531441/500000	[-5 12 -6⟩
…	…	…	…
∞	∞	Meantone	81/80	[-4 4 -1⟩

Temperaments with fractional n and m

n	m	k	Temperament	Comma
5/3 = 1.6	5/2 = 2.5	1/2	Passion	[18 -4 -5⟩
5/2 = 2.5	5/3 = 1.6	4/3	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