Schismic–commatic 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	Tritrisa-quadbigu (12&388)		[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:

• 12 & 79 (n = 1/2 = 0.5)
• Diminished (n = 4/3 = 1.3)
• Augmented (n = 3/2 = 1.5)
• Passion (n = 5/3 = 1.6)
• Quintaleap / Quintupole (n = 5/2 = 2.5)

Compton temperament (12&72)

See also: Pythagorean comma

and Compton family

Comma list: [-19 12⟩ = 531441/524288

Mapping: [⟨12 19 28], ⟨0 0 -1]]

Wedgie: ⟨⟨0 12 19]]

POTE generator: ~5/4 = 384.882

Vals: Template:Val list

Badness: 0.094494

Lalagu (12&79)

Comma list: [-23 16 -1⟩ = 43046721/41943040

Mapping: [⟨1 2 9], ⟨0 -1 -16]]

Wedgie: ⟨⟨1 16 23]]

POTE generator: ~4/3 = 500.970

Vals: Template:Val list

Badness: 0.295079

Quintaleap (12&121, trisa-quingu)

See also: 16ed5/2 § 16ed5/2 as a generator

Comma list: [37 -16 -5⟩ = 137438953472/134521003125

Mapping: [⟨1 2 1], ⟨0 -5 16]]

Wedgie: ⟨⟨5 -16 -37]]

POTE generator: ~135/128 = 99.267

Vals: Template:Val list

Badness: 0.444506

Undim (12&152)

See also: Hemifamity temperaments § Undim

Comma list: [41 -20 -4⟩

Mapping: [⟨4 6 11], ⟨0 1 -5]]

Wedgie: ⟨⟨4 -20 -41]]

POTE generator: ~3/2 = 702.6054

Vals: Template:Val list

Badness: 0.241703

Quindromeda (12&205, quinsa-quingu)

See also: 28ed5 § 28ed5 as a generator

Comma list: [56 -28 -5⟩

POTE generator: ~4428675/4194304 = 99.526

Mapping: [⟨1 2 0], ⟨0 -5 28]]

Wedgie: ⟨⟨5 -28 -56]]

Vals: Template:Val list

Badness: 0.399849

Sextile (12&270)

See also: Landscape microtemperaments § Sextile

Comma list: [71 -36 -6⟩

POTE generator: ~3/2 = 702.2356

Mapping: [⟨6 10 11], ⟨0 -1 6]]

Wedgie: ⟨⟨6 -36 -77]]

Vals: Template:Val list

Badness: 0.555423