Fifth-chroma temperaments: Difference between revisions

The spacing of these intervals also implies a certain "resolution", that is, a certain precision of distinguishing different intervals from each-other, which for a given edo can be made precise by analysing its [[S-expression]]-based comma list. Therefore, common to all of these is that [[144/143|S12]] = ([[12/11]])/([[13/12]]) = ([[16/13]])/([[11/9]]) is observed, which can therefore be considered the generalized "comma", so that we temper out S5 / S12⁵ = 1494927723575/1486016741376.

These two commas actually define the 13-limit of the rank 4 temperament that relates all temperaments discussed here. It can be described with any of the following four edo joins:

* 77 & 80 & 84e & 87 (omitting 94)

:: <nowiki>**</nowiki> [[84edo]] is peculiar because though using the flat 11 makes some sense in lower limits, in higher limits using the patent val tends to be more performant, so that though one is likely mapping 11/9 as the subneutral third, this mapping may sometimes be inconsistent with the mapping of 11 used when building chords, corresponding to multiple possible tunings of ~8:9:11; the sharp tuning uses 28edo's sharp ~9/8 and ~11/8 while the flat tuning uses the flat ~11/8 and 12edo's major second; interestingly, both of these map 11/9 to the subneutral third, with the trick being that the 28edo rendition uses a sharp ~9/8 that can't be achieved from the 12edo circle of fifths.

This temperament, which could uncreatively be called "fifthchroma", admits multiple similar mappings of the [[29-limit]] and higher. The three smallest edos, 77, 80 and 84, are of note as appearing in [[User:Godtone]]'s <code>optimal_edo_sequence</code>s for all odd-limits 23 through 51 using the patent mappings for 77edo and 84edo and using the Ringer 80 val (80koprsuvBC using extended warts where o=47, p=53, q=59, .., y=97, z=101, A=103, B=107, C=109, ...) for 80edo, though fittingly with the analysis, 87edo and 94edo also make their appearance; the reason for them being less optimal then is because the supraminor and submajor thirds start to become slightly too close to the neutral third for the bundle of high-limit compromises represented by the fifth-chroma resolution, but their slightly higher resolution causes them to appear in some larger odd-limits; in fact, 87edo and 94edo appear in the <code>optimal_edo_sequence</code> of all odd limits 77 through 123, while 77edo and 80edo do even better by appearing in all odd limits 69 through 123, as well as in the 51-, 61- and 65-odd-limit. 77edo, perhaps surprisingly, does the best, appearing by patent val in the <code>optimal_edo_sequence</code> of ''all'' odd limits 23 through 123, except for 25.

The default badness (out-of-tune-ness) measure is a type of Mean-Square-Error applied to the set of intervals given, where the "error" in question is cent error weighted proportional to the square-root of the odd-limit of the interval, so that it can also be thought of as minimizing the square of the cent error with each squared error contribution of a given interval weighted proportional to its odd-limit. The mean-square-error is standard, but the odd-limit weighting needs explanation: a more in-depth discussion of its principles can be found in [[Talk:Marvel#Challenge on optimality of 53edo for FloraC|the discussion page of marvel]] as the discussion also covers various tuning principles User:Godtone believes in, but the basic motivation is that the more complex an interval is, the more precise the tuning needs to be to approximate it; initially this seems extremely harsh, because one would expect that the precision required goes up with the square of the odd-limit, and though this is true for dyads, it needn't be true in practice when constructing harmonies, because the other notes that a tempered interval is placed with quickly eliminate possibilities for what the tempered interval might be approximating in favor of a comparatively small number of possible harmonic series chords, so that even weighting the unsquared error proportional to the odd-limit can give strangely biased results, but clearly more complex intervals need more tuning fidelity* even if we have a relatively forcing harmonic context, so the next best thing was the square-root which also can be interpreted as weighting the squared error proportional the odd-limit, which is how it was originally conceived.