Syntonic–diatonic equivalence continuum
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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 out) 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 out in the continuum.
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 | (22 digits) | [24 -21 4⟩ |
6 | 5 | Quintile | (24 digits) | [-28 25 -5⟩ |
7 | 6 | Hemiseven | (28 digits) | [-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.
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 | 33554432000/31381059609 | [28 -22 3⟩ |
… | … | … | … |
∞ | Superpyth | 20480/19683 | [12 -9 1⟩ |
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⟩ |
1/3 = 0.3 | −1/2 = −0.5 | Dirt | [28 -19 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⟩ |
Superpyth (5-limit)
- For extensions, see Archytas clan #Superpyth and Jubilismic clan #Bipyth.
In the 5-limit, superpyth tempers out 20480/19683. It has a fifth generator of ~3/2 = ~710 ¢ and ~5/4 is found at +9 generator steps, as an augmented second (C–D#). It corresponds to n = 1, meaning that the syntonic comma is equated with the diatonic semitone.
Subgroup: 2.3.5
Comma list: 20480/19683
Mapping: [⟨1 0 -12], ⟨0 1 9]]
- mapping generators: ~2, ~3
- WE: ~2 = 1197.6520 ¢, ~3/2 = 708.6882 ¢
- error map: ⟨-2.348 +4.385 -1.076]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.8213 ¢
- error map: ⟨0.000 +7.866 +2.078]
Optimal ET sequence: 5, 17, 22, 49, 120b, 169bbc
Badness (Sintel): 3.17
Uncle (5-limit)
- For extensions, see Trienstonic clan #Uncle.
The 5-limit version of uncle tempers out 4096/3645. It is generated by a fifth that is supposedly sharper than 3\5, so it leads to an oneirotonic scale, or otherwise a diatonic scale with negative small steps. The interval class of 5 is found at -6 fifths, as a major 2-step in oneirotonic, or a diminished fifth (C–Gb) in diatonic. It corresponds to n = -1/2 or m = 1/3.
Subgroup: 2.3.5
Comma list: 4096/3645
Mapping: [⟨1 0 12], ⟨0 1 -6]]
- mapping generators: ~2, ~3
- WE: ~2 = 1189.7544 ¢, ~3/2 = 724.6670 ¢
- error map: ⟨-10.246 +12.466 +4.210]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 731.7318 ¢
- error map: ⟨0.000 +29.777 +23.296]
Optimal ET sequence: 5, 13, 18, 23bc
Badness (Sintel): 6.33
Ultrapyth (5-limit)
- For extensions, see Archytas clan #Ultrapyth.
The 5-limit version of ultrapyth tempers out the ultrapyth comma. It is generated by a perfect fifth. The interval class of 5 is found at +14 fifths as a double-augmented unison (C–Cx). It corresponds to m = -1 and n = 1/2.
Subgroup: 2.3.5
Comma list: 5242880/4782969
Mapping: [⟨1 0 -20], ⟨0 1 14]]
- mapping generators: ~2, ~3
- WE: ~2 = 1196.4357 ¢, ~3/2 = 711.7085 ¢
- error map: ⟨-3.564 +6.189 -1.009]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.5968 ¢
- error map: ⟨0.000 +11.642 +4.041]
Optimal ET sequence: 5, 27c, 32, 37, 79bc, 116bbc
Badness (Sintel): 18.7
Dirt
Dirt tempers out the dirt comma, 1342177280/1162261467. It is generated by a perfect fifth. The interval class of 5 is found at +19 fifths, as a double-augmented seventh (C–Bx). It corresponds to n = 1/3 and m = -1/2.
Subgroup: 2.3.5
Comma list: [28 -19 1⟩
Mapping: [⟨1 0 -28], ⟨0 1 19]]
- mapping generators: ~2, ~3
- WE: ~2 = 1195.8566 ¢, ~3/2 = 713.0611 ¢
- error map: ⟨-4.143 +6.963 -0.863]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 715.3406 ¢
- error map: ⟨0.000 +13.386 +5.157]
Optimal ET sequence: 5, 42c, 47b, 52b, 109bbc
Badness (Sintel): 55.3
Rodan (5-limit)
- For extensions, see Gamelismic clan #Rodan.
The 5-limit version of rodan tempers out the rodan comma, which is the difference between a stack of three retroptolemaic whole tones (729/640) and a perfect fifth (3/2). The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list. It corresponds to n = 3.
Subgroup: 2.3.5
Comma list: 131072000/129140163
Mapping: [⟨1 1 -1], ⟨0 3 17]]
- WE: ~2 = 1199.5618 ¢, ~729/640 = 234.4424 ¢
- error map: ⟨-0.438 +0.934 -0.355]
- CWE: ~2 = 1200.000 ¢, ~729/640 = 234.4999 ¢
- error map: ⟨0.000 +1.545 +0.185]
Optimal ET sequence: 5, …, 41, 46, 87, 220, 307
Badness (Sintel): 3.95
Laconic
- For extensions, see Gamelismic clan #Gorgo.
Laconic tempers out 2187/2000, which is the difference between a stack of three ptolemaic whole tones (10/9)'s and a perfect fifth (3/2). Although a higher-error temperament, it does pop up enough in the low-numbered edos to be useful, most notably in 16edo and 21edo. The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list. It corresponds to n = -3.
Subgroup: 2.3.5
Comma list: 2187/2000
Mapping: [⟨1 1 1], ⟨0 3 7]]
- WE: ~2 = 1203.1925 ¢, ~10/9 = 228.0305 ¢
- error map: ⟨+3.193 -14.671 +13.092]
- CWE: ~2 = 1200.000 ¢, ~10/9 = 228.0128 ¢
- error map: ⟨0.000 -17.917 +9.776]
Optimal ET sequence: 5, 11c, 16, 21, 37b
Badness (Sintel): 3.80
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]]
- WE: ~2 = 1186.1969 ¢, ~6/5 = 232.7334 ¢
- error map: ⟨-13.803 -17.558 +51.547]
- CWE: ~2 = 1200.000 ¢, ~6/5 = 231.4822 ¢
- error map: ⟨0.000 -7.509 +76.651]
Optimal ET sequence: 1b, …, 4bc, 5
Badness (Sintel): 2.39
Hemiseven (5-limit)
- For extensions, see Gamelismic clan #Hemiseven.
Subgroup: 2.3.5
Comma list: [32 -29 6⟩
Mapping: [⟨1 -2 -15], ⟨0 6 29]]
- mapping generators: ~2, ~243/160
- WE: ~2 = 1200.3725 ¢, ~243/160 = 716.9750 ¢
- error map: ⟨+0.373 -0.850 +0.376]
- CWE: ~2 = 1200.0000 ¢, ~243/160 = 716.7671 ¢
- error map: ⟨0.000 -1.352 -0.067]
Optimal ET sequence: 5, …, 72, 149, 221, 370, 591b
Badness (Sintel): 16.9
Counterpental
- For extensions, see Orwellismic temperaments #Pentorwell.
Subgroup: 2.3.5
Comma list: [36 -30 5⟩
Mapping: [⟨5 0 -36], ⟨0 1 6]]
- mapping generators: ~729/640, ~3
- WE: ~729/640 = 239.8575 ¢, ~3/2 = 704.1540 ¢
- error map: ⟨-0.712 +1.487 -0.535]
- CWE: ~729/640 = 240.0000 ¢, ~3/2 = 704.4446 ¢
- error map: ⟨0.000 +2.490 +0.354]
Optimal ET sequence: 5, …, 75, 80, 155, 390b, 545bbc
Badness (Sintel): 35.2
Septiquarter (5-limit)
- For extensions, see Hemifamity temperaments #Septiquarter.
Subgroup: 2.3.5
Comma list: [44 -38 7⟩
Mapping: [⟨1 -4 -28], ⟨0 7 38]]
- mapping generators: ~2, ~177147/102400
- WE: ~2 = 1199.7741 ¢, ~177147/102400 = 957.3630 ¢
- error map: ⟨-0.226 +0.490 -0.194]
- CWE: ~2 = 1200.0000 ¢, ~177147/102400 = 957.5367 ¢
- error map: ⟨0.000 +0.802 +0.082]
Optimal ET sequence: 5, …, 94, 99, 193, 292, 391, 1074b, 1465bb
Badness (Sintel): 22.8