Father–3 equivalence continuum: Difference between revisions
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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)]]. | 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)]]. | ||
Note that because 3et is a record equal temperament in the [[2.5 subgroup]], the continuum can be conceptualized as the [[{{PAGENAME}}/Godtone's approach]|''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 {{nowrap|(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 {{nowrap|(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. | ||
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Because 3et is a record equal temperament in | Because 3et is a record equal temperament in the 2.3 subgroup and (especially) the [[2.5 subgroup]], 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 19:30, 10 June 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).
Note that because 3et is a record equal temperament in the 2.5 subgroup, the continuum can be conceptualized as the [[Father–3 equivalence continuum/Godtone's approach]|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⟩ |
| 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⟩ |
Because 3et is a record equal temperament in the 2.3 subgroup and (especially) the 2.5 subgroup, 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
- WE: ~125/96 = 401.2633 ¢, ~5/4 = 367.0585 ¢ (~25/24 = 34.2047 ¢)
- error map: ⟨+3.790 +1.747 -11.676]
- CWE: ~125/96 = 400.0000 ¢, ~5/4 = 366.8103 ¢ (~25/24 = 33.1897 ¢)
- error map: ⟨0.000 -1.524 -19.503]
Optimal ET sequence: 3, …, 33c, 36c, 69cc
Badness (Sintel): 16.0
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