Father–3 equivalence continuum: Difference between revisions
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Because 3et is a record equal temperament in both 2.3 and [[2.5 subgroup]]s, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is [[128/125]], and the interval with a single factor of 3 is [[25/24]]. As such, Godtone has conceptualized this continuum as '' | Because 3et is a record equal temperament in both 2.3 and [[2.5 subgroup]]s, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is [[128/125]], and the interval with a single factor of 3 is [[25/24]]. As such, Godtone has conceptualized this continuum as ''augmented–dicot equivalence continuum''. See [[{{PAGENAME}}/Godtone's approach]]. | ||
Others prefer conceptualizing this continuum in terms of {{nowrap| ''k'' {{=}} {{sfrac|1|''n'' − 2}} }} such that temperaments satisfy {{nowrap|(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… | Others prefer conceptualizing this continuum in terms of {{nowrap| ''k'' {{=}} {{sfrac|1|''n'' − 2}} }} such that temperaments satisfy {{nowrap|(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… | ||
Revision as of 11:54, 28 April 2026
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 | 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⟩ |
| 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⟩ |
Because 3et is a record equal temperament in both 2.3 and 2.5 subgroups, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is 128/125, and the interval with a single factor of 3 is 25/24. As such, Godtone has conceptualized this continuum as augmented–dicot equivalence continuum. See Father–3 equivalence continuum/Godtone's approach.
Others 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 | Antonian | 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 (Sintel): 16.0
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 (Sintel): 30.4