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Father–3 equivalence continuum/Godtone's approach: Difference between revisions

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The '''augmented–chromatic equivalence continuum'''  is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equates a number of [[128/125]]'s (augmented commas) with the chroma, [[25/24]]. As such, it represents the continuum of all 5-limit temperaments supported by [[3edo]].


The '''augmented-chromatic equivalence continuum'''  is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equates a number of [[128/125]]'s (augmented commas) with the chroma, [[25/24]]. As such, it represents the continuum of all 5-limit temperaments supported by [[3edo]].
This formulation has a specific reason: 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24, but because {{nowrap|25/24 {{=}} ([[25/16]])/([[3/2]])}}, this has the consequence of clearly relating the ''n'' in {{nowrap|(128/125)<sup>n</sup> {{=}} 25/24}} with how many 5/4's are used to reach 3/2 (when octave-reduced):


This formulation has a specific reason: 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24, but because 25/24 = ([[25/16]])/([[3/2]]), this has the consequence of clearly relating the ''n'' in (128/125)<sup>n</sup> = 25/24 with how many 5/4's are used to reach 3/2 (when octave-reduced):
If {{nowrap|''n'' {{=}} 0}}, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)<sup>2</sup>.


If ''n'' = 0, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)<sup>2</sup>.
For integer {{nowrap|''n'' &gt; 0}}, we always reach 25/24 via (25/16)/(128/125)<sup>''n''</sup> because of {{nowrap|(128/125)<sup>''n''</sup> ~ 25/24}} by definition, meaning that we reach 3/2 at {{nowrap|3''n'' + 2}} generators of ~5/4, octave-reduced.


For integer ''n'' > 0, we always reach 25/24 via (25/16)/(128/125)<sup>''n''</sup> because of (128/125)<sup>''n''</sup> ~ 25/24 by definition, meaning that we reach 3/2 at 3''n'' + 2 generators of ~5/4, octave-reduced.
The just value of ''n'' is {{nowrap|log(25/24) / log(128/125) {{=}} 1.72125…}} where {{nowrap|''n'' {{=}} 2}} corresponds to the [[Würschmidt comma]].
 
The just value of ''n'' is log<sub>2</sub>(25/24) / log<sub>2</sub>(128/125) = 1.72125... where ''n'' = 2 corresponds to [[Würschmidt's comma]].


{| class="wikitable center-1"
{| class="wikitable center-1"
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! Monzo
! Monzo
|-
|-
| -2
| −2
| [[Smate]] (14 & 17c)
| [[Smate]] ({{nowrap|14 &amp; 17c}})
| [[2048/1875]]
| [[2048/1875]]
| {{monzo| 11 -1 -4 }}
| {{monzo| 11 -1 -4 }}
|-
|-
| -1
| −1
| [[Father]] (5 & 8)
| [[Father]] ({{nowrap|5 &amp; 8}})
| [[16/15]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
| {{monzo| 4 -1 -1 }}
|-
|-
| 0
| 0
| [[Dicot]] (7 & 10)
| [[Dicot]] ({{nowrap|7 &amp; 10}})
| [[25/24]]
| [[25/24]]
| {{ monzo| -3 -1 2 }}
| {{ monzo| -3 -1 2 }}
|-
|-
| 1
| 1
| [[Magic]] (19 & 22)
| [[Magic]] ({{nowrap|19 &amp; 22}})
| [[3125/3072]]
| [[3125/3072]]
| {{ monzo| -10 -1 5 }}
| {{ monzo| -10 -1 5 }}
|-
|-
| 2
| 2
| [[Würschmidt]] (31 & 34)
| [[Würschmidt]] ({{nowrap|31 &amp; 34}})
| [[393216/390625]]
| [[393216/390625]]
| {{ monzo| 17 1 -8 }}
| {{ monzo| 17 1 -8 }}
|-
|-
| 3
| 3
| [[Magus]] (43 & 46)
| [[Magus]] ({{nowrap|43 &amp; 46}})
| [[50331648/48828125]]
| [[50331648/48828125]]
| {{ monzo| 24 1 -11 }}
| {{ monzo| 24 1 -11 }}
|-
|-
| 4
| 4
| [[Supermagus]] (55 & 58)
| [[Supermagus]] ({{nowrap|55 &amp; 58}})
| 6442450944/6103515625
| 6442450944/6103515625
| {{ monzo| 31 1 -14 }}
| {{ monzo| 31 1 -14 }}
|-
|-
| 5
| 5
| [[Ultramagus]] (67 & 70)
| [[Ultramagus]] ({{nowrap|67 &amp; 70}})
| 824633720832/762939453125
| 824633720832/762939453125
| {{ monzo| 38 1 -17 }}
| {{ monzo| 38 1 -17 }}
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|-
|-
| ∞
| ∞
| [[Augmented]] (12 & 15)
| [[Augmented]] ({{nowrap|12 &amp; 15}})
| [[128/125]]
| [[128/125]]
| {{ monzo| -7 0 3 }}
| {{ monzo| -7 0 3 }}
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|-
|-
| -1/2
| -1/2
| [[Very low accuracy temperaments#Yo (2c&3)|Yo]]
| [[Very low accuracy temperaments#Yo ({{nowrap|2c &amp; 3}})|Yo]]
| [[10/9]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
| {{monzo| 1 -2 1 }}
|-
|-
| 1/2
| 1/2
| [[Wesley]] (26 & 29)
| [[Wesley]] ({{nowrap|26 &amp; 29}})
| [[78125/73728]]
| [[78125/73728]]
| {{ monzo| 13 2 -7 }}
| {{ monzo| 13 2 -7 }}
|-
|-
| 3/2
| 3/2
| [[Ditonic]] (50 & 53)
| [[Ditonic]] ({{nowrap|50 &amp; 53}})
| [[1220703125/1207959552]]
| [[1220703125/1207959552]]
| {{ monzo| -27 -2 13 }}
| {{ monzo| -27 -2 13 }}
|-
|-
| 5/2
| 5/2
| [[Novamajor]]* (77 & 80)
| [[Novamajor]]* ({{nowrap|77 &amp; 80}})
| 19791209299968/19073486328125
| 19791209299968/19073486328125
| {{ monzo| 41 2 -19 }}
| {{ monzo| 41 2 -19 }}
|-
|-
| 7/2
| 7/2
| 3 & 101
| {{nowrap|3 &amp; 101}}
| (36 digits)
| (36 digits)
| {{ monzo| 55 2 -25 }}
| {{ monzo| 55 2 -25 }}
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|-
|-
| 5/3
| 5/3
| [[Mutt]] (84 & 87)
| [[Mutt]] ({{nowrap|84 &amp; 87}})
| [[mutt comma]]
| [[mutt comma]]
| {{ monzo| -44 -3 21 }}
| {{ monzo| -44 -3 21 }}
|-
|-
| 7/4
| 7/4
| 3 & 118
| {{nowrap|3 &amp; 118}}
| (42 digits)
| (42 digits)
| {{ monzo| 61 4 -29 }}
| {{ monzo| 61 4 -29 }}
Line 132: Line 131:
The simplest of these is [[mutt]] which has interesting properties discussed there.
The simplest of these is [[mutt]] which has interesting properties discussed there.


Also note that at ''n'' = -2/3, we find the exotemperament tempering out [[32/27]].
Also note that at {{nowrap|''n'' {{=}} −{{frac|2|3}}}}, we find the exotemperament tempering out [[32/27]].


[[Category:3edo]]
[[Category:3edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
Retrieved from "https://en.xen.wiki/w/Father–3_equivalence_continuum/Godtone%27s_approach"