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Father–3 equivalence continuum: Difference between revisions

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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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{{ See also | Augmented-chromatic equivalence continuum }}
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''' (which is equivalent to the [[augmented-chromatic equivalence continuum]] but focused on lower-accuracy temperaments) 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 [[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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|-
|-
| 0
| 0
| [[Very low accuracy temperaments#Alteraugment|Alteraugment]]
| [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
| [[32/27]]
| [[32/27]]
| {{monzo| 5 -3 }}
| {{Monzo| 5 -3 }}
|-
|-
| 1
| 1
| [[Very low accuracy temperaments#Yo (2c&3)|Yo]]
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[10/9]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
| {{Monzo| 1 -2 1 }}
|-
|-
| 2
| 2
| [[Dicot]]
| [[Dicot]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
| {{Monzo| -3 -1 2 }}
|-
|-
| 3
| 3
| [[Augmented]]
| [[Augmented (temperament)|Augmented]]
| [[128/125]]
| [[128/125]]
| {{monzo| 7 0 -3 }}
| {{Monzo| 7 0 -3 }}
|-
|-
| 4
| 4
| [[Smate]]
| [[Smate]]
| [[2048/1875]]
| [[2048/1875]]
| {{monzo| 11 -1 -4 }}
| {{Monzo| 11 -1 -4 }}
|-
|-
| …
| …
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| [[Father]]
| [[Father]]
| [[16/15]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
| {{Monzo| 4 -1 -1 }}
|}
|}


We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''yo-3 equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.61255…  
We may invert the continuum by setting ''m'' such that {{nowrap| 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…  


{| class="wikitable center-1"
{| class="wikitable center-1"
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|-
|-
| 0
| 0
| [[Alteraugment]]
| [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
| [[32/27]]
| [[32/27]]
| {{monzo| 5 -3 }}
| {{Monzo| 5 -3 }}
|-
|-
| 1
| 1
| [[Father]]
| [[Father]]
| [[16/15]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
| {{Monzo| 4 -1 -1 }}
|-
|-
| 2
| 2
| [[Dicot]]
| [[Dicot]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
| {{Monzo| -3 -1 2 }}
|-
|-
| …
| …
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|-
|-
| ∞
| ∞
| [[Very low accuracy temperaments#Yo (2c&3)|Yo]]
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[10/9]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
| {{Monzo| 1 -2 1 }}
|}
|}


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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 }}
| 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 }}
|-
| 21/8 = 2.625 || 21/13 = 1.{{overline|615384}} || [[Mutt]] || {{monzo| -44 -3 21 }}
|-
|-
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| 10 1 -5 }}
| 29/11 = 2.{{overline|63}} || 29/18 = 1.6{{overline|1}} || [[Squarschmidt]] || {{monzo| 61 4 -29 }}
|-
|-
| 8/3 = 2.{{overline|6}} || 8/5 = 1.6 || [[Würschmidt]] || {{monzo| 17 1 -8 }}
| 8/3 = 2.{{overline|6}} || 8/5 = 1.6 || [[Würschmidt]] || {{monzo| 17 1 -8 }}
|-
|-
| 19/7 = 2.{{overline|714285}} || 19/12 = 1.58{{overline|3}} || [[Isnes]] || {{monzo| 41 2 -19 }}
| 19/7 = 2.{{overline|714285}} || 19/12 = 1.58{{overline|3}} || [[#Isnes|Isnes]] || {{monzo| 41 2 -19 }}
|-
|-
| 11/4 = 2.75 || 11/7 = 1.{{overline|571428}} || [[Magus]] || {{monzo| 24 1 -11 }}
| 11/4 = 2.75 || 11/7 = 1.{{overline|571428}} || [[Magus]] || {{monzo| 24 1 -11 }}
|}
|}


Some prefer conceptualizing this continuum in terms of {{nowrap|''k'' {{=}} {{sfrac|1|''n'' &minus; 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…
== Mutt (5-limit) ==
 
{{Main| Mutt }}
{| class="wikitable center-1"
: ''For extensions, see [[Horwell temperaments #Mutt]].''  
|+ style="font-size: 105%;" | Temperaments with integer ''k''
|-
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| -1
| [[Very low accuracy temperaments#Yo (2c&3)|Yo]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
|-
| 0
| [[Father]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
|-
| 1
| [[Augmented]]
| [[128/125]]
| {{monzo| 7 0 -3 }}
|-
| 2
| [[Magic]]
| [[3125/3072]]
| {{monzo| 10 1 -5 }}
|-
| 3
| [[Wesley family|Wesley]]
| 78125/73728
| {{monzo| 13 2 -7 }}
|-
| 4
| 3 & 33c
| 1953125/1769472
| {{monzo| 16 3 -9 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Dicot]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
|}
 
== 3 & 33c ==
This low-accuracy high-complexity temperament corresponds to {{nowrap|''n'' {{=}} 9/4}} and {{nowrap|''m'' {{=}} 9/5}}.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 1953125/1769472
[[Comma list]]: {{monzo| -44 -3 21 }}
 
{{Mapping|legend=1| 3 2 6 | 0 3 1 }}


: mapping generators: ~125/96, ~5/4
{{Mapping|legend=1| 3 -2 6 | 0 7 1 }}
: mapping generators: ~98304/78125, ~5/4


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~125/96 = 1\3, ~5/4 = 368.2534 (~25/24 = 31.7466)
* [[WE]]: ~98304/78125 = 400.0227{{c}}, ~5/4 = 386.0017{{c}} (~393216/390625 = 14.0210{{c}})
* [[CWE]]: ~125/96 = 1\3, ~5/4 = 366.8103 (~25/24 = 33.1897)
: [[error map]]: {{val| +0.068 +0.012 -0.176 }}
* [[CWE]]: ~98304/78125 = 400.0000{{c}}, ~5/4 = 385.9858{{c}} (~393216/390625 = 14.0142{{c}})
: error map: {{val| 0.000 -0.055 -0.328 }}


{{Optimal ET sequence|legend=1| 3, …, 33c, 36c, 69cc }}
{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455, 4194cc, 5649cc }}


[[Badness]]: 0.682
[[Badness]] (Sintel): 3.81


== Isnes ==
== 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 {{nowrap|''n'' {{=}} 19/7}} and ''m'' {{=}} 19/12}}.  
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 {{nowrap|''n'' {{=}} 19/7 }} and {{nowrap| ''m'' {{=}} 19/12 }}.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Comma list]]: {{monzo| 41 2 -19 }}
[[Comma list]]: {{monzo| 41 2 -19 }}


{{Mapping|legend=1| 1 8 3 | 0 -19 -2 }}
{{Mapping|legend=1| 1 -11 1 | 0 19 2 }}
: mapping generators: ~2, ~3145728/1953125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2782{{c}}, ~3145728/1953125 = 794.4174{{c}}
: [[error map]]: {{val| -0.722 -0.090 +1.799 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3145728/1953125 = 794.8728{{c}}
: error map: {{val| 0.000 +0.628 +3.432 }}
 
{{Optimal ET sequence|legend=1| 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)<sup>1/4</sup>, tuned around 396.6 cents.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 61 4 -29 }}


: mapping generators: ~2, ~1953125/1572864
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
: mapping generators: ~2, ~98304/78125


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~1953125/1572864 = 405.1689
* [[WE]]: ~2 = 1199.9653{{c}}, ~98304/78125 = 396.6094{{c}}
* [[CWE]]: ~2 = 1\1, ~1953125/1572864 = 405.1272
: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98304/78125 = 396.6201{{c}}
: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}


{{Optimal ET sequence|legend=1| 3, 71b, 74, 77, 157, 548ccc }}
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc }}


[[Badness]]: 1.30
[[Badness]] (Sintel): 5.12


[[Category:3edo]]
[[Category:3edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
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