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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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! ''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 }}
| 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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|}
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Because 3et is a record equal temperament in 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]].
== Mutt (5-limit) ==
 
{{Main| Mutt }}
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…
: ''For extensions, see [[Horwell temperaments #Mutt]].''  
 
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with integer ''k''
|-
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| -1
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[10/9]]
| {{Monzo| 1 -2 1 }}
|-
| 0
| [[Father]]
| [[16/15]]
| {{Monzo| 4 -1 -1 }}
|-
| 1
| [[Augmented (temperament)|Augmented]]
| [[128/125]]
| {{Monzo| 7 0 -3 }}
|-
| 2
| [[Magic]]
| [[3125/3072]]
| {{Monzo| 10 1 -5 }}
|-
| 3
| [[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|legend=1| 3 -2 6 | 0 7 1 }}
: mapping generators: ~125/96, ~5/4
: mapping generators: ~98304/78125, ~5/4


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~125/96 = 401.2633{{c}}, ~5/4 = 367.0585{{c}} (~25/24 = 34.2047{{c}})
* [[WE]]: ~98304/78125 = 400.0227{{c}}, ~5/4 = 386.0017{{c}} (~393216/390625 = 14.0210{{c}})
: [[error map]]: {{val| +3.790 +1.747 -11.676 }}
: [[error map]]: {{val| +0.068 +0.012 -0.176 }}
* [[CWE]]: ~125/96 = 400.0000{{c}}, ~5/4 = 366.8103{{c}} (~25/24 = 33.1897{{c}})
* [[CWE]]: ~98304/78125 = 400.0000{{c}}, ~5/4 = 385.9858{{c}} (~393216/390625 = 14.0142{{c}})
: error map: {{val| 0.000 -1.524 -19.503 }}
: 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]] (Sintel): 16.0
[[Badness]] (Sintel): 3.81


== Isnes ==
== Isnes ==
Retrieved from "https://en.xen.wiki/w/Father–3_equivalence_continuum"