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 Pythagorean limma (256/243). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 5edo.

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 out both commas and thus tempers out 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, 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 generators to obtain a harmonic 3 in the generator chain. For example:

Superpyth (n = 1) is generated by a fifth;
Immunity (n = 2) splits its twelfth in two;
Rodan (n = 3) splits its fifth in three;
Etc.

At n = 5, the corresponding temperament splits the octave into five instead, as after a stack of five syntonic commas, both the orders of 3 and 5 are multiples of 5 again.

If we let k = n + 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 = ∞). (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
k	n	Temperament	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	847288609443/838860800000	[-28 25 -5⟩
7	6	Hemiseven	68630377364883/67108864000000	[-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… The superpyth comma is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.

Temperaments with integer m
m	Temperament	Comma
m	Temperament	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	33554432000/31381059609	[28 -22 3⟩
…	…	…	…
∞	Superpyth	20480/19683	[12 -9 1⟩

Temperaments with fractional n and m
n	m	Temperament	Comma
−3/2 = −1.5	3/5 = 0.6	University	[4 2 -3⟩
−1/2 = −0.5	1/3 = 0.3	Uncle	[12 -6 -1⟩
5/2 = 2.5	5/3 = 1.6	Counterpental	[36 -30 5⟩
7/2 = 3.5	7/5 = 1.4	Septiquarter	[44 -38 7⟩
21/5 = 4.2	21/16 = 1.3125	559 & 2513	[-124 109 -21⟩
9/2 = 4.5	9/7 = 1.285714	5 & 118	[-52 46 -9⟩
11/2 = 5.5	11/9 = 1.2	5 & 137	[-60 54 -11⟩

Uncle (5-limit)

For extensions, see Trienstonic clan #Uncle.

Subgroup: 2.3.5

Comma list: 4096/3645

Mapping: [⟨1 0 12], ⟨0 1 -6]]

mapping generators: ~2, ~3

Optimal tunings:

CTE: ~2 = 1200.000, ~3/2 = 733.721

error map: ⟨0.000 +31.766 +11.362]

CWE: ~2 = 1200.000, ~3/2 = 731.732

error map: ⟨0.000 +29.777 +23.296]

Optimal ET sequence: 5, 13, 18, 23bc

Badness:

Smith: 0.270
Dirichlet: 6.33

Ultrapyth (5-limit)

For extensions, see Archytas clan #Ultrapyth.

Subgroup: 2.3.5

Comma list: 5242880/4782969

Mapping: [⟨1 0 -20], ⟨0 1 14]]

mapping generators: ~2, ~3

Optimal tunings:

POTE: ~2 = 1200.000, ~3/2 = 713.829

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

Badness (Smith): 0.795243

Rodan (5-limit)

For extensions, see Gamelismic clan #Rodan.

Subgroup: 2.3.5

Comma list: 131072000/129140163

Mapping: [⟨1 1 -1], ⟨0 3 17]]

Optimal tunings:

POTE: ~2 = 1200.000, ~729/640 = 234.528

Optimal ET sequence: 5, 31c, 36c, 41, 46, 87, 220, 307

Badness: 0.168264

University

Main article: University

For extensions, see Gamelismic clan #Gidorah and Mint temperaments #Penta.

Named by John Moriarty, university is the 5 & 6b temperament, and tempers out 144/125, the triptolemaic diminished third. It corresponds to n = −3/2 and m = 3/5. In this temperament, two instances of 6/5 make a 5/4, and three make a 3/2. Equating 6/5 with 8/7 (which makes sense since it is already very flat in the most accurate tunings of this temperament) leads to gidorah, and 6/5 with 7/6 leads to penta.

Subgroup: 2.3.5

Comma list: 144/125

Mapping: [⟨1 1 2], ⟨0 3 2]]

Optimal tunings: