Fifth-chroma temperaments

This page is 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, and for which their significance is unclear without analysis. These temperaments have been named "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 organization 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

* of the edos discussed, only 94edo does not temper out (50/41)/(39/32) = 1600/1599 = S40, corresponding to its inconsistently-flat mapping of 25/16; this is noteworthy only because the subneutral and superneutral categories generally are supposed to be unambiguous in their interpretation; an asterisk with 84edo's mapping of prime 11 is discussed later

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.

Fifthchroma

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 at least* 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.

* There is a more idiosyncratic edo that fits into this scheme as well: 70edo also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of 140edo, so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section #Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness.

** 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.

If we also consider the aforediscussed idiosyncratic tuning 70edo, then many more edo joins are possible for describing this rank 4 (= 4-dimensional) 13-limit temperament:

• 70 & 77 & 80 & 84e
• 70 & 77 & 80 & 94
• 70 & 77 & 84e & 87
• 70 & 77 & 84e & 94
• 70 & 77 & 87 & 94
• 70 & 80 & 84e & 87
• 70 & 80 & 87 & 94
• 70 & 84e & 87 & 94

Note that 70 & 77 & 80 & 87 simplifies to 77 & 80 & 87, which is notable as being the temperament that imposes another restriction on the representation of the 13-limit, by assuming finding exactly one interval between ~5/4 and ~14/11, distinct from both, by tempering (56/55)/(144/143)² = 13013/12960, where 56/55 = (14/11)/(5/4).

Another omission is 70 & 80 & 84e & 94 which reduces by tempering out S12/S14; though the significance of this is less clear, it can be observed that it is similar to the other omission by having the latter two edos being an offset of 10 from the former two edos.

As a reminder then, the remarkable feature is that all of these edo joins are equivalent* and refer to the same 13-limit tempered structure: the one which splits 25/24 into five equal parts of ~144/143 so that 11/9 and 16/13 are made fifth-complements. This structure maps various high-limit things onto the 2.3.7.11 subgroup, and is home to a family of lower-rank temperaments that introduce various extra equivalences, discussed in #Fifth-chroma temperaments.

* (other than the ones indicated as specifying something different/more specific)

Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness

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 optimal_edo_sequences 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 optimal_edo_sequence 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 optimal_edo_sequence of all odd limits 23 through 123, except for 25.

On the measure

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 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.

* This is not to touch on the debate of what intervals it is aesthetically more important to get right, where one might consider 3/2 as requiring more precise tuning simply because it's more obvious when mistuned due to its simplicity; the perspective of the default used by optimal_edo_sequence is that different people often prefer different temperings of simple intervals - and not necessarily the most pure ones - so that its goal is instead trying to maximize psychoacoustic plausibility of the approximations rather than making a judgement on aesthetic, which is something that only the musician (or tuning theorist) can make a judgement on.

Fifth-chroma temperaments

Overview

Primary

• 77 & 80: Novamajor, a temperament with a generator of 5/4 * S12, named because the interval colour of ~405 ¢ is named a "novamajor third" by User:Godtone

* The "5-limit seed" of novamajor is by splitting 25/24 into five parts of ~sqrt(128/125) = (128/125)^1/2.

• 77 & 84: Absurdity, the 1\7-period (at least) 29-limit temperament; note that 70edo also supports it in a reduced/slightly modified form

* The "5-limit seed" of absurdity is by tempering out (81/80)⁵ / (25/24) (the absurdity comma) so that 25/24 is split into five flattened (underexaggerated) ~81/80's, which is equivalent to interpreting 1\7 as (10/9)/(81/80) because 2 / ((10/9)/(81/80) = 800/729)⁷ is equal to the square of the absurdity comma.

• 80 & 84: unnamed? proposed name: "quarterchromatic" from quartering the octave and having a comma-sized generator, four of which functions melodically as a type of chroma (though not as 25/24)

Secondary

• 77 & 87: Restles

* The "2.3.13 seed" of restles is lesser tendoneutralic, which observes (16/13)/(39/32) = 512/507 so that this interval is what divides 25/24 into five in 2.3.5.13 restles.

• 84 & 87: Mutt

* The "5-limit seed" of mutt is by splitting 25/24 into 5 parts of ~cbrt(128/125) = (128/125)^1/3.

• 77 & 94: Tsaharuk
• 80 & 87: "alt artoneutral": a version of artoneutral that uses a different mapping for prime 7

* This has the same "5-limit seed" as artoneutral, but is actually more optimal for the 5-limit.

• 80 & 94: unnamed? proposed name: "pseudohemipyth"
• 87 & 94: Artoneutral

* The "5-limit seed" of artoneutral is by splitting 25/24 into 5 parts of ~sqrt(81/80) = (81/80)^1/2.

New temperament definitions

Absurdity

See also: Absurdity

Absurdity was extended to the 29-limit; the mapping for prime 29 is obvious via 1\7 = 32/29 being accurate. The mapping for prime 23 is via 23/20 being equated with 8/7, which isn't ideal, but both edo tunings (77edo and 84edo) have resources to make various elements of 29-limit harmony possible.

Novamajor

See also: Augmented–chromatic equivalence continuum

This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering (128/125)⁵ / (25/24)²), is a full no-31's 37-limit temperament (and possibly/likely higher) that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):

• inframinor third: ~15/13
• subminor third: ~7/6
• neominor third: ~20/17 (note the absence of ~13/11, which here is ambiguous in mapping between novaminor third (77edo) and neominor third (80edo))
• novaminor third: ~19/16 (note the absence of ~25/21, which here is ambiguous in mapping between minor third (77edo) and novaminor third (80edo))
• minor third: ~6/5
• supraminor third: ~35/29~29/24~23/19 (note the absence of ~17/14~40/33, which here is ambiguous in mapping between supraminor third (77edo) and subneutral third (80edo))
• subneutral third: ~39/32~50/41~11/9
• superneutral third: ~27/22~16/13
• submajor third: ~36/29~46/37~56/45 (note the absence of ~26/21, which here is ambiguous in mapping between submajor third (77edo) and superneutral third (80edo))
• major third: ~5/4
• novamajor third: ~24/19~43/34~19/15
• neomajor third: ~14/11~37/29~23/18
• supermajor third: ~9/7
• ultramajor third: ~13/10

Therefore, the reason for the name "novamajor" is that the structure that the aforementioned "guarantee of a certain kind of metaphorical colour palette with certain properties" via making those intervals equidistant relates it directly to the interval colour names devised by User:Godtone, with the most idiosyncratic term (that has seen least adoption by others) being "novamajor" and "novaminor", and as the novamajor third is the generator of this temperament, it seemed a fitting name. (The associated 5-limit temperament was discovered independently by User:Godtone with this high-limit application in mind, but isnes wasn't found at the time for various reasons including the ratio not being shown.)