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–dicot 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 dicot comma (or 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 number of specific reasons:
+* 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24.
+* 128/125 is fundamental because it uniquely defines the relatively-very-accurate (strongly form-fitting) representation of the 2.5 subgroup in 3edo.
+* 25/24 is fundamental because it gives the trivial way to relate ~5/4 = 1\3 to ~3/2 = 2\3 as 2 generators in 3edo. (By contrast, using 16/15 requires taking the octave-complement of one of the generators. There is also another stronger argument against using 16/15 detailed later in this list.)
+* Using 25/24 is also useful because we then know how many intervals between ~6/5 and ~5/4 are guaranteed in a nontrivial tuning; because 25/24 is divided into ''n'' equal parts, the answer is {{nowrap|''n'' - 1}}. Meanwhile, if {{nowrap|''n'' {{=}} ''a''/''b''}} is not an integer (meaning ''b'' > 1), then 25/24 is divided into ''a'' equal parts of {{nowrap|~(128/125)<sup>1/''b''</sup>}}, giving a clear meaning to the numerator and denominator (though more meanings are discussed later).
+* 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'' &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. In other words, for natural ''n'', the way to reach ~3/2 (up to octave equivalence) is ''always'' by flattening ((5/4)<sup>2</sup> =) 25/16 (by ''n'' dieses) into 3/2, where flattening by a diesis is equivalent to multiplying by (5/4)<sup>3</sup> (up to octave-equivalence).
+: * Therefore, if ''n'' = ''a''/''b'' is a rational with ''b'' > 1 and ''b'' not a multiple of 3 (so that 3''a''/''b'' + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather ''b'' equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in ''b''(3''a''/''b'' + 2) = 3''a'' + 2''b'' generators, and also means that ~128/125 is split into ''b'' equal parts.
+* 16/15 = 25/24 * 128/125, so that tempering out 16/15 (father) is found at ''n'' = -1. The reason it shouldn't be found at ''n'' = 0 instead is because ''n'' = -1 has an absurdly sharp tuning of ~5/4 because of being equated with ~4/3, which breaks the pattern from dicot at ''n'' = 0 having an absurdly flat tuning of ~5/4 because of being equated with ~6/5, and from ''n'' = 0 onwards, ~5/4 is tuned increasingly sharp. This observation is important enough for its own point:
+* For ''n'' a nonnegative integer, half-integer or third-integer('''*'''), increasing ''n'' corresponds to increasingly sharp tunings of ~5/4. In the limit, as ''n'' goes to infinity, these all approach ~5/4 = 1\3, corresponding to [[augmented temperament]].
+:: ('''*''' It is conjectured by [[User:Godtone]] that for a given choice of denominator ''b'' in ''n'' = ''a''/''b'', a larger value of ''a'' always corresponds to a sharper tuning of ~5/4, where the sharpness in a pure-octaves tuning is always strictly flat of 1\3, so that (more trivially) taking the limit as ''a'' goes to infinity, ~5/4 = 1\3. The intuition for why we might expect this to be true is that in a pure-2's pure-3's tuning, we are always constraining ~128/125's size to be equal to the appropriate relation to ~25/24, where as 2 and 3 are fixed, the ~5 is the only free variable and depending only on ''n'', with that ''n'' essentially indirectly specifying the degree of tempering.)
+* Also, if one is interested in what intervals are present between ~5/4 and ~4/3, it is simple to observe because (16/15)/(25/24) = 128/125, meaning for a nonnegative integer ''n'' there is exactly one more interval between ~5/4 and ~4/3 as between ~6/5 and ~5/4. As there is ''n'' - 1 intervals between ~6/5 and ~5/4 (because of splitting ~25/24 into ''n'' parts), that means that (for nonnegative integer ''n'') there is exactly ''n'' intervals between ~5/4 and ~4/3. More generally, for rational ''n'' = ''a''/''b'', we have ''a'' - 1 intervals between ~6/5 and ~5/4 and because there is another ~128/125 between ~5/4 and ~4/3 we have ''a''/''b'' + 1 = ''a''/''b'' + ''b''/''b'' for the translated coordinates so that we have ''a'' + ''b'' - 1 intervals between ~5/4 and ~4/3, corresponding to splitting ~16/15 into ''a'' + ''b'' equal parts.
-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>.
+Therefore, if ''n'' = ''a''/''b'' is a rational with ''b'' > 1 and ''b'' not a multiple of 3 (so that 3''a''/''b'' + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather ''b'' equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in ''b''(3''a''/''b'' + 2) = 3''a'' + 2''b'' generators, and also means that ~128/125 is split into ''b'' equal parts.
-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.
 The just value of ''n'' is {{nowrap|log(25/24) / log(128/125) {{=}} 1.72125…}} where {{nowrap|''n'' {{=}} 2}} corresponds to the [[Würschmidt comma]].
@@ Line 70: / Line 80: @@
 | {{ monzo| -7 0 3 }}
 |}
+Notice that (as mentioned), as ''n'' increases, we temper ~5/4 sharper and ~128/125 flatter (closer to unison), so that as ''n'' goes to infinity, ~5/4 goes to 1\3.
 {| class="wikitable center-1"
@@ Line 125: / Line 136: @@
 | [[mutt comma]]
 | {{ monzo| -44 -3 21 }}
+|-
+| 12/7
+| {{nowrap|202 &amp; 205 {{=}} 3 &amp; 612}}
+| [[88817841970012523233890533447265625/88715259606372406434345277125033984|(70 digits)]]
+| {{ monzo| -105 -7 50 }}
 |-
 | 7/4
-| {{nowrap|3 &amp; 118}}
+| [[Squarschmidt]] ({{nowrap|3 &amp; 118}})
-| (42 digits)
+| [[186773283746309210112/186264514923095703125|(42 digits)]]
 | {{ monzo| 61 4 -29 }}
 |}
-The simplest of these is [[mutt]] which has interesting properties discussed there. In regards to mutt, the fact that the denominator of ''n'' is a multiple of 3 tells us that it has a 1\3 period because it's contained in 3edo. The fact that the numerator is 5 tells us that 25/24 is split into 5 parts. From {{nowrap|(128/125)<sup>n</sup> {{=}} 25/24}} we can thus deduce that each part is thus equal to ~cbrt(128/125) = (128/125)<sup>1/3</sup>, so that ~5/4 is found at 1\3 minus a third of a diesis, so that ~125/64 is found at thrice that.
+The simplest of these is [[mutt]] which has interesting properties discussed there. In regards to mutt, the fact that the denominator of ''n'' is a multiple of 3 tells us that it has a 1\3 period because it's contained in 3edo. The fact that the numerator is 5 tells us that 25/24 is split into 5 parts. From {{nowrap|(128/125)<sup>n</sup> {{=}} 25/24}} we can thus deduce that each part is thus equal to ~cbrt(128/125) = (128/125)<sup>1/3</sup>, so that ~5/4 is found at 1\3 minus a third of a diesis, so that ~125/64 is found at thrice that. This observation is more general, leading to consideration of temperaments of third-integer ''n''.
+The 3 & 118 microtemperament [[squarschmidt]] is at ''n'' = 7/4. Its generator is approximately 397{{cent}} so that four generators reaches 5/2, corresponding to the denominator of 4. The number of generators of ~(5/2)<sup>1/4</sup> needed to find prime 3 is thus four times the result of plugging ''n'' = 7/4 into 3''n'' + 2 , which is 3(7/4) + 2 = 21/4 + 8/4 = 29/4, that is, 29 generators.
+Finally, the 3 & 612 microtemperament at ''n'' = 12/7 is extremely complex, because to find prime 5, you need 7 times 3(12/7) + 2 = 36/7 + 14/7 = 50/7, that is, 50 generators, and is noted only because of being extremely close to the JIP and being supported by the 5-limit microtemperament [[612edo]]. The denominator (7) indicates that 128/125 is split into 7 equal parts, while the numerator indicates that each (128/125)<sup>1/7</sup> part represents (25/24)<sup>1/12</sup>, that is, a twelfth of 25/24.
+{| class="wikitable center-1"
+|+ style="font-size: 105%;" | Temperaments with third-integer ''n''<br />aka temperaments with 1\3 period
+|-
+! rowspan="2" | ''n''
+! rowspan="2" | Temperament
+! colspan="2" | Comma
+|-
+! Ratio
+! Monzo
+|-
+| -2/3
+| [[Very low accuracy temperaments#Alteraugment|32/27]] ({{nowrap|3 & 3c}}) {{nowrap|(generator {{=}} [[16/15|father comma]])}}
+| 32/27
+| {{ monzo| 5 0 -3 }}
+|-
+| -1/3
+| {{nowrap|9c &amp; 12c}} {{nowrap|(generator {{=}} negative ~[[25/24|dicot comma]])}}
+| 125/108
+| {{ monzo| -2 -3 3 }}
+|-
+| 1/3
+| {{nowrap|33c &amp; 36c}} {{nowrap|(generator {{=}} ~[[25/24|dicot comma]])}}
+| 1953125/1769472
+| {{ monzo| -16 -3 9 }}
+|-
+| 2/3
+| {{nowrap|48 &amp; 51}} {{nowrap|(generator {{=}} negative ~[[magic comma]])}}
+| 244140625/226492416
+| {{ monzo| -23 -3 12 }}
+|-
+| 4/3
+| {{nowrap|72 &amp; 75}} {{nowrap|(generator {{=}} ~[[magic comma]])}}
+| 3814697265625/3710851743744
+| {{ monzo| 41 2 -19 }}
+|-
+| 5/3
+| [[Mutt]] ({{nowrap|84 &amp; 87}}) {{nowrap|(generator {{=}} ~[[Würschmidt's comma]])}}
+| [[476837158203125/474989023199232|mutt comma]]
+| {{ monzo| -44 -3 21 }}
+|-
+| 7/3
+| {{nowrap|108 &amp; 111}} {{nowrap|(generator {{=}} negative ~[[Würschmidt's comma]])}}
+| [[7782220156096217088/7450580596923828125|(38 digits)]]
+| {{ monzo| 58 3 -27 }}
+|-
+| 8/3
+| {{nowrap|120 &amp; 123}} {{nowrap|(generator {{=}} ~[[magus comma]])}}
+| [[996124179980315787264/931322574615478515625|(42 digits)]]
+| {{ monzo| 65 3 -30 }}
+|-
+| 10/3
+| {{nowrap|147c &amp; 150c {{=}} 291cc &amp; 297cc }} {{nowrap|(generator {{=}} negative ~[[magus comma]])}}
+| [[16320498564797493858533376/14551915228366851806640625|(52 digits)]]
+| {{ monzo| 79 3 -36 }}
+|-
+| 11/3
+| {{nowrap|156c &amp; 159c}} {{nowrap|(generator {{=}} ~[[6442450944/6103515625|supermagus comma]])}}
+| [[2089023816294079213892272128/1818989403545856475830078125|(56 digits)]]
+| {{ monzo| 86 3 -39 }}
+|}
+Notice the alternating pattern of comma offsets from 1\3, where those commas are themselves in the pattern present in the continuum of integer ''n''.
-The 3 & 118 microtemperament is at ''n'' = 7/4. Its generator is approximately 397{{cent}} so that four generators reaches 5/2, corresponding to the denominator of 4. The number of generators of ~(5/2)<sup>1/4</sup> needed to find prime 3 is thus four times the result of plugging ''n'' = 7/4 into 3''n'' + 2 , which is 3(7/4) + 2 = 21/4 + 8/4 = 29/4, that is, 29 generators.
+Also notice that we always find ~5/4 in terms of 1\3 minus the generator, which is a tempered version of the aforementioned comma offset, which is either positive or negative, and that as ''n'' grows, the generator becomes smaller so that ~5/4 becomes sharper.
-Also note that at {{nowrap|''n'' {{=}} −{{frac|2|3}}}}, we find the exotemperament tempering out [[32/27]].
 [[Category:3edo]]
 [[Category:Equivalence continua]]