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 | 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 ''n'' - 1. Meanwhile, if ''n'' = ''a''/''b'' is not an integer (meaning ''b'' > 1), then 25/24 is divided into ''a'' equal parts of ~(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'' > 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, and 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. | |||
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. | |||
The just value of ''n'' is {{nowrap|log(25/24) / log(128/125) {{=}} 1.72125…}} where {{nowrap|''n'' {{=}} 2}} corresponds to the [[Würschmidt comma]].{| class="wikitable center-1" | |||
The just value of ''n'' is {{nowrap|log(25/24) / log(128/125) {{=}} 1.72125…}} where {{nowrap|''n'' {{=}} 2}} corresponds to the [[Würschmidt comma]]. | |||
{| class="wikitable center-1" | |||
|+ style="font-size: 105%;" | Temperaments with integer ''n'' | |+ style="font-size: 105%;" | Temperaments with integer ''n'' | ||
|- | |- | ||