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 in the continuum

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⟩

Examples of temperaments with fractional values of n:

• Python (n = 1/2 = 0.5)
• Ripple (n = 5/4 = 1.25)
• Diminished (n = 4/3 = 1.3)
• Augmented (n = 3/2 = 1.5)
• Passion (n = 5/3 = 1.6)
• Quintaleap (n = 5/2 = 2.5)

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