Father–3 equivalence continuum
The father–3 equivalence continuum is a continuum of 5-limit temperaments which equate a number of classical diatonic semitones (16/15) with the Pythagorean minor third (32/27).
Note that because 3et is a record equal temperament in the 2.5 subgroup, the continuum can be conceptualized as the augmented–dicot equivalence continuum, which Godtone argues is easier to understand, with characteristic 2.5-subgroup comma 128/125 as the interval with a single factor of 3 is 25/24.
All temperaments in the continuum satisfy (16/15)n ~ 32/27. Varying n results in different temperaments listed in the table below. It converges to father as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 3edo 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 2.63252…, and temperaments having n near this value tend to be the most accurate ones.
32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of 3 in the generator chain.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Alteraugment | 32/27 | [5 -3⟩ |
| 1 | Antonian | 10/9 | [1 -2 1⟩ |
| 2 | Dicot | 25/24 | [-3 -1 2⟩ |
| 3 | Augmented | 128/125 | [7 0 -3⟩ |
| 4 | Smate | 2048/1875 | [11 -1 -4⟩ |
| … | … | … | … |
| ∞ | Father | 16/15 | [4 -1 -1⟩ |
We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the antonian–3 equivalence continuum, which is essentially the same thing. The just value of m is 1.61255…
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Alteraugment | 32/27 | [5 -3⟩ |
| 1 | Father | 16/15 | [4 -1 -1⟩ |
| 2 | Dicot | 25/24 | [-3 -1 2⟩ |
| … | … | … | … |
| ∞ | Antonian | 10/9 | [1 -2 1⟩ |
| n | m | Temperament | Comma |
|---|---|---|---|
| 7/3 = 2.3 | 7/4 = 1.75 | Wesley | [-13 -2 7⟩ |
| 5/2 = 2.5 | 5/3 = 1.6 | Magic | [-10 -1 5⟩ |
| 21/8 = 2.625 | 21/13 = 1.615384 | Mutt | [-44 -3 21⟩ |
| 29/11 = 2.63 | 29/18 = 1.61 | Squarschmidt | [61 4 -29⟩ |
| 8/3 = 2.6 | 8/5 = 1.6 | Würschmidt | [17 1 -8⟩ |
| 19/7 = 2.714285 | 19/12 = 1.583 | Isnes | [41 2 -19⟩ |
| 11/4 = 2.75 | 11/7 = 1.571428 | Magus | [24 1 -11⟩ |
Mutt (5-limit)
- For extensions, see Horwell temperaments #Mutt.
Subgroup: 2.3.5
Comma list: [-44 -3 21⟩
Mapping: [⟨3 -2 6], ⟨0 7 1]]
- mapping generators: ~98304/78125, ~5/4
- WE: ~98304/78125 = 400.0227 ¢, ~5/4 = 386.0017 ¢ (~393216/390625 = 14.0210 ¢)
- error map: ⟨+0.068 +0.012 -0.176]
- CWE: ~98304/78125 = 400.0000 ¢, ~5/4 = 385.9858 ¢ (~393216/390625 = 14.0142 ¢)
- error map: ⟨0.000 -0.055 -0.328]
Optimal ET sequence: 84, 87, 171, 771, 942, 1113, 1284, 1455, 4194cc, 5649cc
Badness (Sintel): 3.81
Isnes
Isnes is so called because the generator is half of a 5/2 major tenth, in a similar way that sensi has a generator of half a 5/3 major sixth. This corresponds to n = 19/7 and m = 19/12.
Subgroup: 2.3.5
Comma list: [41 2 -19⟩
Mapping: [⟨1 -11 1], ⟨0 19 2]]
- mapping generators: ~2, ~3145728/1953125
- WE: ~2 = 1199.2782 ¢, ~3145728/1953125 = 794.4174 ¢
- error map: ⟨-0.722 -0.090 +1.799]
- CWE: ~2 = 1200.0000 ¢, ~3145728/1953125 = 794.8728 ¢
- error map: ⟨0.000 +0.628 +3.432]
Optimal ET sequence: 3, 71b, 74, 77, 157, 548ccc
Badness (Sintel): 30.4
Squarschmidt (5-limit)
- For extensions, see Hemimage temperaments #Squarschmidt.
A generator for the squarschmidt temperament is the fourth root of 5/2, (5/2)1/4, tuned around 396.6 cents.
Subgroup: 2.3.5
Comma list: [61 4 -29⟩
Mapping: [⟨1 -8 1], ⟨0 29 4]]
- mapping generators: ~2, ~98304/78125
- WE: ~2 = 1199.9653 ¢, ~98304/78125 = 396.6094 ¢
- error map: ⟨-0.099 +0.543 +0.029 -0.719]
- CWE: ~2 = 1200.0000 ¢, ~98304/78125 = 396.6201 ¢
- error map: ⟨0.000 +0.653 +0.253 -0.552]
Optimal ET sequence: 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc
Badness (Sintel): 5.12