Father–3 equivalence continuum: Difference between revisions
m i will at least make it more clearly linked to |
Remove the k-continuum since no one is actively arguing for it. Also remove the 3 & 33c temp, which is unenlighted result of looking at the continuum that way |
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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 [[ | 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 {{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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! ''n'' !! ''m''!! Temperament || Comma | ! ''n'' !! ''m''!! Temperament || Comma | ||
|- | |- | ||
| 7/3 = 2.{{overline|3}} || 7/4 = 1.75 || [[Wesley]] || {{monzo| 13 2 | | 7/3 = 2.{{overline|3}} || 7/4 = 1.75 || [[Wesley]] || {{monzo| -13 -2 7 }} | ||
|- | |- | ||
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| 10 1 - | | 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| -10 -1 5 }} | ||
|- | |||
| 21/8 = 2.625 || 21/13 = 1.{{overline|615384}} || [[Mutt]] || {{monzo| -44 -3 21 }} | |||
|- | |- | ||
| 29/11 = 2.{{overline|63}} || 29/18 = 1.6{{overline|1}} || [[Squarschmidt]] || {{monzo| 61 4 -29 }} | | 29/11 = 2.{{overline|63}} || 29/18 = 1.6{{overline|1}} || [[Squarschmidt]] || {{monzo| 61 4 -29 }} | ||
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== Mutt (5-limit) == | |||
{{Main| Mutt }} | |||
: ''For extensions, see [[Horwell temperaments #Mutt]].'' | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: | [[Comma list]]: {{monzo| -44 -3 21 }} | ||
{{Mapping|legend=1| 3 2 6 | 0 | {{Mapping|legend=1| 3 -2 6 | 0 7 1 }} | ||
: mapping generators: ~ | : mapping generators: ~98304/78125, ~5/4 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~ | * [[WE]]: ~98304/78125 = 400.0227{{c}}, ~5/4 = 386.0017{{c}} (~393216/390625 = 14.0210{{c}}) | ||
: [[error map]]: {{val| + | : [[error map]]: {{val| +0.068 +0.012 -0.176 }} | ||
* [[CWE]]: ~ | * [[CWE]]: ~98304/78125 = 400.0000{{c}}, ~5/4 = 385.9858{{c}} (~393216/390625 = 14.0142{{c}}) | ||
: error map: {{val| 0.000 - | : error map: {{val| 0.000 -0.055 -0.328 }} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455, 4194cc, 5649cc }} | ||
[[Badness]] (Sintel): | [[Badness]] (Sintel): 3.81 | ||
== Isnes == | == Isnes == | ||
Latest revision as of 14:43, 22 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 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