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Father–3 equivalence continuum: Difference between revisions

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The '''chromatic-diatonic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[25/24|chromatic semitones (25/24)]] with [[16/15|diatonic semitones (16/15)]].
The '''chromatic-diatonic equivalence continuum''', despite its name, 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)]].


All temperaments in the continuum satisfy (25/24)<sup>''n''</sup> ~ 16/15. Varying ''n'' results in different temperaments listed in the table below. It converges to [[dicot]] 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 1.58097..., and temperaments having ''n'' near this value tend to be the most accurate ones.  
All temperaments in the continuum satisfy (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.  


{| class="wikitable center-1 center-2"
32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, we notice that ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|3]] in the generator chain. Such an equivalence continuum is more properly called the ''father-3 equivalence continuum''.
|+ Temperaments in the continuum
 
{| class="wikitable center-1"
|+ Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Alteraugment]]
| [[32/27]]
| {{monzo| 5 -3 }}
|-
| 1
| [[Very low accuracy temperaments #Yellow (2c&3)|Yellow]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
|-
| 2
| [[Dicot]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
|-
| 3
| [[Augmented]]
| [[128/125]]
| {{monzo| 7 0 -3 }}
|-
| 4
| [[Smate]]
| [[2048/1875]]
| {{monzo| 11 -1 -4 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Father]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
|}
We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''yellow-3 equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.61255…
{| class="wikitable center-1"
|+ Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Alteraugment]]
| [[32/27]]
| {{monzo| 5 -3 }}
|-
| 1
| [[Father]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
|-
| 2
| [[Dicot]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Very low accuracy temperaments #Yellow (2c&3)|Yellow]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
|}
{| class="wikitable"
|+ Temperaments with fractional ''n'' and ''m''
|-
! Temperament !! ''n'' !! ''m''
|-
| [[Wesley]] || 7/3 = 2.{{overline|3}} || 7/4 = 1.75
|-
| [[Magic]] || 5/2 = 2.5 || 5/3 = 1.{{overline|6}}
|-
| [[Würschmidt]] || 8/3 = 2.{{overline|6}} || 8/5 = 1.6
|-
| [[Isnes]] || 19/7 = 2.{{overline|714285}} || 19/12 = 1.58{{overline|3}}
|-
| [[Magus]] || 11/4 = 2.75 || 11/7 = 1.{{overline|571428}}
|}
Some prefer conceptualizing this continuum in terms of ''k'' = 1/(''n'' - 2) such that temperaments satisfy (25/24)<sup>''k''</sup> = 16/15. This is the source of 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…
{| class="wikitable center-1"
|+ Temperaments with integer ''k''
|-
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
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| [[Very low accuracy temperaments#Yellow (2c&3)|Yellow]]
| [[Very low accuracy temperaments#Yellow (2c&3)|Yellow]]
| [[10/9]]
| [[10/9]]
| {{monzo|1 -2 1}}
| {{monzo| 1 -2 1 }}
|-
|-
| 0
| 0
| [[Father]]
| [[Father]]
| [[16/15]]
| [[16/15]]
| {{monzo|4 -1 -1}}
| {{monzo| 4 -1 -1 }}
|-
|-
| 1
| 1
| [[Augmented]]
| [[Augmented]]
| [[128/125]]
| [[128/125]]
| {{monzo|7 0 -3}}
| {{monzo| 7 0 -3 }}
|-
|-
| 2
| 2
| [[Magic]]
| [[Magic]]
| [[3125/3072]]
| [[3125/3072]]
| {{monzo|10 1 -5}}
| {{monzo| 10 1 -5 }}
|-
|-
| 3
| 3
| [[Wesley family|Wesley]]
| [[Wesley family|Wesley]]
| 78125/73728
| 78125/73728
| {{monzo| 13 2 -7}}
| {{monzo| 13 2 -7 }}
|-
|-
| 4
| 4
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| [[Dicot]]
| [[Dicot]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2}}
| {{monzo| -3 -1 2 }}
|}
|}
Examples of temperaments with fractional values of ''n'':
* [[Very low accuracy temperaments#Alteraugment|Alteraugment]] (''n'' = -0.5)
* [[Mint temperaments#Smate|Smate]] (''n'' = 0.5)
* [[Würschmidt]] (''n'' = 1.5)
* Isnes (''n'' = 1.6)
* [[Sensamagic clan#Magus|Magus]] (''n'' = 5/3 = 1.{{Overline|6}})


== 3 & 33c ==
== 3 & 33c ==
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