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]]. | |||
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): | |||
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>. | |||
For integer {{nowrap|''n'' > 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. | |||
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 | |||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
| Line 21: | Line 20: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| | | −2 | ||
| [[Smate]] (14 & 17c) | | [[Smate]] ({{nowrap|14 & 17c}}) | ||
| [[2048/1875]] | | [[2048/1875]] | ||
| {{monzo| 11 -1 -4 }} | | {{monzo| 11 -1 -4 }} | ||
|- | |- | ||
| | | −1 | ||
| [[Father]] (5 & 8) | | [[Father]] ({{nowrap|5 & 8}}) | ||
| [[16/15]] | | [[16/15]] | ||
| {{monzo| 4 -1 -1 }} | | {{monzo| 4 -1 -1 }} | ||
|- | |- | ||
| 0 | | 0 | ||
| [[Dicot]] (7 & 10) | | [[Dicot]] ({{nowrap|7 & 10}}) | ||
| [[25/24]] | | [[25/24]] | ||
| {{ monzo| -3 -1 2 }} | | {{ monzo| -3 -1 2 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| [[Magic]] (19 & 22) | | [[Magic]] ({{nowrap|19 & 22}}) | ||
| [[3125/3072]] | | [[3125/3072]] | ||
| {{ monzo| -10 -1 5 }} | | {{ monzo| -10 -1 5 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| [[Würschmidt]] (31 & 34) | | [[Würschmidt]] ({{nowrap|31 & 34}}) | ||
| [[393216/390625]] | | [[393216/390625]] | ||
| {{ monzo| 17 1 -8 }} | | {{ monzo| 17 1 -8 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| [[Magus]] (43 & 46) | | [[Magus]] ({{nowrap|43 & 46}}) | ||
| [[50331648/48828125]] | | [[50331648/48828125]] | ||
| {{ monzo| 24 1 -11 }} | | {{ monzo| 24 1 -11 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| [[Supermagus]] (55 & 58) | | [[Supermagus]] ({{nowrap|55 & 58}}) | ||
| 6442450944/6103515625 | | 6442450944/6103515625 | ||
| {{ monzo| 31 1 -14 }} | | {{ monzo| 31 1 -14 }} | ||
|- | |- | ||
| 5 | | 5 | ||
| [[Ultramagus]] (67 & 70) | | [[Ultramagus]] ({{nowrap|67 & 70}}) | ||
| 824633720832/762939453125 | | 824633720832/762939453125 | ||
| {{ monzo| 38 1 -17 }} | | {{ monzo| 38 1 -17 }} | ||
| Line 67: | Line 66: | ||
|- | |- | ||
| ∞ | | ∞ | ||
| [[Augmented]] (12 & 15) | | [[Augmented]] ({{nowrap|12 & 15}}) | ||
| [[128/125]] | | [[128/125]] | ||
| {{ monzo| -7 0 3 }} | | {{ monzo| -7 0 3 }} | ||
| Line 83: | Line 82: | ||
|- | |- | ||
| -1/2 | | -1/2 | ||
| [[Very low accuracy temperaments#Yo (2c&3)|Yo]] | | [[Very low accuracy temperaments#Yo ({{nowrap|2c & 3}})|Yo]] | ||
| [[10/9]] | | [[10/9]] | ||
| {{monzo| 1 -2 1 }} | | {{monzo| 1 -2 1 }} | ||
|- | |- | ||
| 1/2 | | 1/2 | ||
| [[Wesley]] (26 & 29) | | [[Wesley]] ({{nowrap|26 & 29}}) | ||
| [[78125/73728]] | | [[78125/73728]] | ||
| {{ monzo| 13 2 -7 }} | | {{ monzo| 13 2 -7 }} | ||
|- | |- | ||
| 3/2 | | 3/2 | ||
| [[Ditonic]] (50 & 53) | | [[Ditonic]] ({{nowrap|50 & 53}}) | ||
| [[1220703125/1207959552]] | | [[1220703125/1207959552]] | ||
| {{ monzo| -27 -2 13 }} | | {{ monzo| -27 -2 13 }} | ||
|- | |- | ||
| 5/2 | | 5/2 | ||
| [[Novamajor]]* (77 & 80) | | [[Novamajor]]* ({{nowrap|77 & 80}}) | ||
| 19791209299968/19073486328125 | | 19791209299968/19073486328125 | ||
| {{ monzo| 41 2 -19 }} | | {{ monzo| 41 2 -19 }} | ||
|- | |- | ||
| 7/2 | | 7/2 | ||
| 3 & 101 | | {{nowrap|3 & 101}} | ||
| (36 digits) | | (36 digits) | ||
| {{ monzo| 55 2 -25 }} | | {{ monzo| 55 2 -25 }} | ||
| Line 121: | Line 120: | ||
|- | |- | ||
| 5/3 | | 5/3 | ||
| [[Mutt]] (84 & 87) | | [[Mutt]] ({{nowrap|84 & 87}}) | ||
| [[mutt comma]] | | [[mutt comma]] | ||
| {{ monzo| -44 -3 21 }} | | {{ monzo| -44 -3 21 }} | ||
|- | |- | ||
| 7/4 | | 7/4 | ||
| 3 & 118 | | {{nowrap|3 & 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'' = | 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]] | ||
Revision as of 13:51, 27 February 2025
The augmented–chromatic equivalence continuum is a continuum of 5-limit 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 25/24 = (25/16)/(3/2), this has the consequence of clearly relating the n in (128/125)n = 25/24 with how many 5/4's are used to reach 3/2 (when octave-reduced):
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)2.
For integer n > 0, we always reach 25/24 via (25/16)/(128/125)n because of (128/125)n ~ 25/24 by definition, meaning that we reach 3/2 at 3n + 2 generators of ~5/4, octave-reduced.
The just value of n is log(25/24) / log(128/125) = 1.72125… where n = 2 corresponds to the Würschmidt comma.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| −2 | Smate (14 & 17c) | 2048/1875 | [11 -1 -4⟩ |
| −1 | Father (5 & 8) | 16/15 | [4 -1 -1⟩ |
| 0 | Dicot (7 & 10) | 25/24 | [-3 -1 2⟩ |
| 1 | Magic (19 & 22) | 3125/3072 | [-10 -1 5⟩ |
| 2 | Würschmidt (31 & 34) | 393216/390625 | [17 1 -8⟩ |
| 3 | Magus (43 & 46) | 50331648/48828125 | [24 1 -11⟩ |
| 4 | Supermagus (55 & 58) | 6442450944/6103515625 | [31 1 -14⟩ |
| 5 | Ultramagus (67 & 70) | 824633720832/762939453125 | [38 1 -17⟩ |
| … | … | … | … |
| ∞ | Augmented (12 & 15) | 128/125 | [-7 0 3⟩ |
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1/2 | [[Very low accuracy temperaments#Yo (2c & 3)|Yo]] | 10/9 | [1 -2 1⟩ |
| 1/2 | Wesley (26 & 29) | 78125/73728 | [13 2 -7⟩ |
| 3/2 | Ditonic (50 & 53) | 1220703125/1207959552 | [-27 -2 13⟩ |
| 5/2 | Novamajor* (77 & 80) | 19791209299968/19073486328125 | [41 2 -19⟩ |
| 7/2 | 3 & 101 | (36 digits) | [55 2 -25⟩ |
* Note that "novamajor" (User:Godtone's name) is also called "isnes"; both names are based on the size of the generator being around 405 cents, but "isnes" was discovered as a point in the continuum while "novamajor" was discovered as one temperament in the fifth-chroma temperaments.
If we approximate the JIP with increasing accuracy, (that is, using n a rational that is an increasingly good approximation of 1.72125...) we find these high-accuracy temperaments:
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 5/3 | Mutt (84 & 87) | mutt comma | [-44 -3 21⟩ |
| 7/4 | 3 & 118 | (42 digits) | [61 4 -29⟩ |
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.