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 has the advantage of being the characteristic 3-limit comma tempered out in 5edo. For 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 MOS scale. 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, which might be a preferred way of conceptualising it because:

• 16/15 is the 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 in the continuum

k = n + 1	n = k − 1	Temperament	Comma
			Ratio	Monzo
-3	-4	Laquadgu	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⟩

Examples of temperaments with fractional values of n:

• University (n = -1.5)
• Uncle (n = -0.5)
• 5 & 32 (n = 0.5)
• 5 & 56 (n = 1.5)
• Counterpental (n = 2.5)
• Septiquarter (n = 3.5)
• 2513 & 559 (n = 4.2)
• 5 & 118 (n = 4.5)
• 5 & 137 (n = 5.5)

Hemiseven (5-limit)

See also: Gamelismic clan § Hemiseven

Comma: [32 -29 6⟩

Mapping: [⟨1 4 14], ⟨0 -6 -29]]

POTE generator: ~320/243 = 483.2474 cents

Vals: Template:Val list

Badness: 0.720465

The temperament finder - 5-limit 5 & 72

Sasayo (5 & 32)

Comma: [20 -14 1⟩ = 5242880/4782969

Mapping: [⟨1 2 8], ⟨0 -1 -14]]

POTE generator: ~4/3 = 486.1713 cents

Vals: Template:Val list

Badness: 0.795243

The temperament finder - 5-limit 5 & 32p

Trisatriyo (5 & 56)

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

Mapping: [⟨1 1 -2], ⟨0 3 22]]

POTE generator: ~2560/2187 = 235.8673 cents

Vals: Template:Val list

Badness: 1.323443

The temperament finder - 5-limit 5 & 56

Counterpental

Comma: [36 -30 5⟩

Mapping: [⟨5 8 12], ⟨0 -1 -6]]

POTE generator: 15.4278 cents

Vals: Template:Val list

Badness: 1.500224

The temperament finder - 5-limit 5 & 75

Septiquarter (5-limit)

See also: Hemifamity temperaments § Septiquarter

Comma: [44 -38 7⟩

Mapping: [⟨1 3 10], ⟨0 -7 -38]]

POTE generator: ~204800/177147 = 242.4567 cents

Vals: Template:Val list

Badness: 0.971284

The temperament finder - 5-limit 99 & 94

559 & 2513

Comma: [-124 109 -21⟩

Mapping: [⟨1 10 46], ⟨0 -21 -109]]

POTE generator: ~3355443200000/2541865828329 = 480.8595 cents

Vals: Template:Val list

Badness: 0.134523

The temperament finder - 5-limit 2513 & 559

Quinla-tritrigu (5 & 118)

Comma: [-52 46 -9⟩

Mapping: [⟨1 -2 -16], ⟨0 9 46]]

POTE generator: ~320/243 = 477.9609 cents

Vals: Template:Val list

Badness: 0.617683

Tribilalegu (5 & 137)

Comma: [-60 54 -11⟩

Mapping: [⟨1 6 24], ⟨0 -11 -54]]

POTE generator: ~320/243 = 481.7421 cents

Vals: Template:Val list

Badness: 3.620981

The temperament finder - 5-limit 5 & 137