Father–3 equivalence continuum: Difference between revisions
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The ''' | The '''father-3 equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[16/15|classical diatonic semitones (16/15)]] with the [[32/27|Pythagorean minor third (32/27)]]. | ||
All temperaments in the continuum satisfy (16/15)<sup>''n''</sup> ~ 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. | All temperaments in the continuum satisfy (16/15)<sup>''n''</sup> ~ 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, | 32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|3]] in the generator chain. | ||
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Some prefer conceptualizing this continuum in terms of ''k'' = 1/(''n'' - 2) such that temperaments satisfy (25/24)<sup>''k''</sup> = 16/15. This | Some prefer conceptualizing this continuum in terms of ''k'' = 1/(''n'' - 2) such that temperaments satisfy (25/24)<sup>''k''</sup> = 16/15. This gives rise to the name ''chromatic-diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097… | ||
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Revision as of 07:51, 23 July 2024
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).
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 | Yellow | 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 yellow-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⟩ |
| … | … | … | … |
| ∞ | Yellow | 10/9 | [1 -2 1⟩ |
| Temperament | n | m |
|---|---|---|
| Wesley | 7/3 = 2.3 | 7/4 = 1.75 |
| Magic | 5/2 = 2.5 | 5/3 = 1.6 |
| Würschmidt | 8/3 = 2.6 | 8/5 = 1.6 |
| Isnes | 19/7 = 2.714285 | 19/12 = 1.583 |
| Magus | 11/4 = 2.75 | 11/7 = 1.571428 |
Some prefer conceptualizing this continuum in terms of k = 1/(n - 2) such that temperaments satisfy (25/24)k = 16/15. This gives rise to the name chromatic-diatonic equivalence continuum, where both chromatic and diatonic refer to the classical versions of semitones. The just value of k is approximately 1.58097…
| k | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1 | Yellow | 10/9 | [1 -2 1⟩ |
| 0 | Father | 16/15 | [4 -1 -1⟩ |
| 1 | Augmented | 128/125 | [7 0 -3⟩ |
| 2 | Magic | 3125/3072 | [10 1 -5⟩ |
| 3 | Wesley | 78125/73728 | [13 2 -7⟩ |
| 4 | 3 & 33c | 1953125/1769472 | [16 3 -9⟩ |
| … | … | … | … |
| ∞ | Dicot | 25/24 | [-3 -1 2⟩ |
3 & 33c
This low-accuracy high-complexity temperament corresponds to n = 9/4 and m = 9/5.
Subgroup: 2.3.5
Comma list: 1953125/1769472
Mapping: [⟨3 2 6], ⟨0 3 1]]
- mapping generators: ~125/96, ~5/4
- CTE: ~125/96 = 1\3, ~5/4 = 368.2534 (~25/24 = 31.7466)
- CWE: ~125/96 = 1\3, ~5/4 = 366.8103 (~25/24 = 33.1897)
Optimal ET sequence: 3, …, 33c, 36c, 69cc
Badness: 0.682
Isnes
Isnes is so called because the generator is half of a 8/5 minor sixth, in a similar way that sensi has a generator of half a 5/3. This corresponds to n = 19/7 and m = 19/12.
Subgroup: 2.3.5
Comma list: [41 2 -19⟩
Mapping: [⟨1 8 3], ⟨0 -19 -2]]
- mapping generators: ~2, ~1953125/1572864
Optimal ET sequence: 3, 71b, 74, 77, 157, 548ccc
Badness: 1.30