Syntonic-diatonic equivalence continuum

The syntonic-diatonic equivalence continuum is a continuum of temperaments which equate a number of syntonic commas (81/80) with the limma (256/243).

All temperaments in the continuum satisfy (81/80)ⁿ ~ 256/243. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 5edo 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 4.1952…, and temperaments near this tend to be the most accurate ones.

256/243 is the characteristic 3-limit comma tempered out in 5edo. In each case, we notice that n equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain.

However, if we let k = n + 1 (meaning n = k - 1) so that k = 0 means n = -1, k = 1 means n = 0, etc. then the continuum corresponds to (81/80)^k = 16/15. Some prefer this way of conceptualising it because:

• 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at k = 0, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)⁰ ~ 1/1 ~ 16/15.
• k = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan (k = 4), with the only exception being meantone (n = k = (unsigned) infinity). (Temperaments corresponding to k = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
• 16/15 is the simplest ratio to be tempered in the continuum.

Temperaments with integer n

k	n	Temperament	Comma
			Ratio	Monzo
-3	-4	Laquadgu (5 & 28)	177147/160000	[-8 11 -4⟩
-2	-3	Laconic	2187/2000	[-4 7 -3⟩
-1	-2	Bug	27/25	[0 3 -2⟩
0	-1	Father	16/15	[4 -1 -1⟩
1	0	Blackwood	256/243	[8 -5⟩
2	1	Superpyth	20480/19683	[12 -9 1⟩
3	2	Immunity	1638400/1594323	[16 -13 2⟩
4	3	Rodan	131072000/129140163	[20 -17 3⟩
5	4	Vulture	10485760000/10460353203	[24 -21 4⟩
6	5	Pental		[-28 25 -5⟩
7	6	Hemiseven		[-32 29 -6⟩
…	…	…	…
∞	∞	Meantone	81/80	[-4 4 -1⟩

We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the superpyth-diatonic equivalence continuum, which is essentially the same thing. The just value of m is 1.3130…

Temperaments with integer m

m	Temperament	Comma
		Ratio	Monzo
-1	Ultrapyth	5242880/4782969	[20 -14 1⟩
0	Blackwood	256/243	[8 -5⟩
1	Meantone	81/80	[-4 4 -1⟩
2	Immunity	1638400/1594323	[16 -13 2⟩
3	5 & 56		[28 -22 3⟩
…	…	…	…
∞	Superpyth	20480/19683	[12 -9 1⟩

Temperaments with fractional n and m

Temperament	n	m
University	-3/2 = -1.5	3/5 = 0.6
Uncle	-1/2 = -0.5	1/3 = 0.3
Counterpental	5/2 = 2.5	5/3 = 1.6
Septiquarter	7/2 = 3.5	7/5 = 1.4
559 & 2513	21/5 = 4.2	21/16 = 1.3125
5 & 118	9/2 = 4.5	9/7 = 1.285714
5 & 137	11/2 = 5.5	11/9 = 1.2

Hemiseven

See also: Gamelismic clan #Hemiseven

Subgroup: 2.3.5

Comma list: [32 -29 6⟩

Mapping: [⟨1 4 14], ⟨0 -6 -29]]

Mapping generators: ~2, ~320/243

POTE generator: ~320/243 = 483.2474

Optimal ET sequence: 5, 62c, 67c, 72, 149, 221, 370, 591b, 961bb

Badness: 0.720465

Ultrapyth

See also: Archytas clan #Ultrapyth

Subgroup: 2.3.5

Comma list: 5242880/4782969

Mapping: [⟨1 0 -20], ⟨0 -1 -14]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 713.8287

Optimal ET sequence: 5, 27c, 32, 37, 79bc, 116bbc

Badness: 0.795243

Trisatriyo (5 & 56)

Subgroup: 2.3.5

Comma list: [28 -22 3⟩ = 33554432000/31381059609

Mapping: [⟨1 1 -2], ⟨0 3 22]]

Mapping generators: ~2, ~2560/2187

POTE generator: ~2560/2187 = 235.8673

Optimal ET sequence: 5, 56, 61

Badness: 1.323443

The temperament finder - 5-limit 5 & 56

Counterpental

See also: Orwellismic temperaments #Pentorwell

Subgroup: 2.3.5

Comma list: [36 -30 5⟩

Mapping: [⟨5 0 -36], ⟨0 1 6]]

Mapping generators: ~729/640, ~3

POTE generator: ~3/2 = 704.5722

Optimal ET sequence: 5, 75, 80

Badness: 1.500224

Septiquarter

See also: Hemifamity temperaments #Septiquarter

Subgroup: 2.3.5

Comma list: [44 -38 7⟩

Mapping: [⟨1 3 10], ⟨0 -7 -38]]

Mapping generators: ~2, ~204800/177147

POTE generator: ~204800/177147 = 242.4567

Optimal ET sequence: 5, 89c, 94, 99, 193, 292, 391

Badness: 0.971284

559 & 2513

Subgroup: 2.3.5

Comma list: [-124 109 -21⟩

Mapping: [⟨1 10 46], ⟨0 -21 -109]]

Mapping generators: ~2, ~3355443200000/2541865828329

POTE generator: ~3355443200000/2541865828329 = 480.8595

Optimal ET sequence: 5, 267c, 272c, 277, 559, 1395, 1954, 2513, 40767, 43280, 45793, 48306, 50819, 53332, 55845, 58358, 60871, 63384, 65897, 68410, 70923, 73436, 75949, 78462

Badness: 0.134523

The temperament finder - 5-limit 2513 & 559

Quinla-tritrigu (5 & 118)

Subgroup: 2.3.5

Comma list: [-52 46 -9⟩

Mapping: [⟨1 -2 -16], ⟨0 9 46]]

Mapping generators: ~2, ~320/243

POTE generator: ~320/243 = 477.9609

Optimal ET sequence: 5, 108c, 113, 118, 1057, 1175, 1293, 1411, 1529, 1647, 1765, 1883, 2001b, 3884b

Badness: 0.617683

Tribilalegu (5 & 137)

Subgroup: 2.3.5

Comma list: [-60 54 -11⟩

Mapping: [⟨1 6 24], ⟨0 -11 -54]]

Mapping generators: ~2, ~320/243

POTE generator: ~320/243 = 481.7421

Optimal ET sequence: 5, 127c, 132, 137, 553, 690b, 827b, 964b

Badness: 3.620981

The temperament finder - 5-limit 5 & 137