Fifth-chroma temperaments

This page is primarily the work of User:Godtone which they believe to be valuable due to surprisingly strong results in tuning theory (which will be discussed afterwards), but is placed in the public namespace due to it concluding in relating a number of medium-accuracy high-limit temperaments that are at first glance not obviously related. These temperaments have been called "fifth-chroma temperaments", because they all share the following foundational feature: in all nontrivial (reasonable) edo tunings < 100, they find exactly four intervals between ~5/4 and ~6/5, so that the chroma, 25/24, is split into 5 equal parts, and they are all based on certain observations and constraints about the organisation of various thirds and their fifth-complements. So the analysis begins by addressing why.

The base observation is that it is in some sense optimal to find exactly four intervals between 6/5 and 5/4, which can be called:

• "supraminor", a general bucket/category for aggregating ratios like 35/29, 64/53, 29/24, 52/43, 23/19, 40/33, 17/14
• "subneutral", which principally represents ~39/32~50/41~11/9
• "superneutral", which principally represents ~27/22~16/13
• "submajor", a general bucket/category for aggregating ratios like 21/17, 26/21, 31/25, 36/29, 41/33, 46/37

The lists of ratios for supraminor and submajor are not complete and a given edo tuning will not necessarily find contexts where all of these ratios make sense as interpretations, but the principle is that between 6/5 and 11/9 are many ratios that are various mediants of 6/5 and 11/9, and between 16/13 and 5/4 are many ratios that are various mediants of 16/13 and 5/4, so that both general areas represent places where the source of concordance (if any) is not necessarily obvious, so that any mediant therein can potentially be suggested with sufficiently forcing amounts of harmonic context (notes in a chord with approximate frequency ratios suggesting a certain otonal/harmonic series interpretation). Therefore, it represents a flexible melodic category able to represent a wide variety of tempered harmonies contextually.

The structure therefore assumes tempering out 352/351 = (11/9)/(39/32) = (32/27)/(13/11).

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

That is, it represents the shared harmonic and spacing logic between 77edo, 80edo, 84edo*, 87edo and 94edo, so that its join can be formed by any four of these five edos except for the one formed by omitting 80edo, because 77 & 84e & 87 & 94 reduces to something simpler (77 & 87 & 94), corresponding to 80edo being the only one that doesn't temper out 385/384.

* 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 which are mostly based on various ways of interpreting the possible harmonies of the supraminor and submajor thirds, as different interpretations will cause different edos to be the "odd one out" by failing to map a specific JI supraminor or submajor third as such, or occasionally, as in the case of 84edo.