so, i happen to be looking at psychoacoustic dissonance right now in a very different context. maybe i can take a stab at it.

first off, here’s William Holder writing in 1731:

From the Premises, it will be easie to comprehend the natural Reason, why the Ear is delighted with those forenamed Concords; and that is, because they all unite in the Motions often, and at the least at every Sixth Course of Vibration, which appears from the Rations by which they are constituted, which are all contained with that Number, and all Rations contained within that Space of Six, make Concords, because the Mixture of their Motions is answerable to the Ration of them, and are made at or before every Sixth Course."

this is pretty much the attitude from the renaissance through the early-mid 20th century, and for most of us it is still the baseline understanding of musical harmonic dissonance, as stated a couple times in this thread in different ways.

but in the 70s we get a more definitive correspondence between harmonics and psychoacoustics in the work of Ernst Terhardt (highly recommended reading btw.) here the discussion gets a little more complicated because actually we have to consider not just fundamental frequencies / pure tones, but harmonic series. assume that all tones in question are harmonic. take two simultaneous tones; each have a number of harmonic partials that are all combining with each other and producing interference tones. if the combining partials are close enough, then their interference frequency is low, and it registers as an amplitude modulation in one critical band on the cochlea’s basilar membrane. this produces a lot of neural activity and contributes to “roughness.”

it’s easy to see from this how pure tones at a perfect 5th apart are completely consonant, while even JI tones a minor 2nd apart (9/8) register as dissonant.

take a more complex example: JI perfect 4th (4/3) with a number of harmonics, played against a tone at the root with a number of harmonics, say 5. the P4th has harmonic numbers 4, 8, 12, 16, 20 (from the numnerator) and 3, 6, 9, 12, 15 (from the denominator.) so this system of interference includes 9/8 and 16/15, both of which are small enough to register as beating in the cochlea (9/8 is pretty fast so its marginal). that’s only 2 combinations out of… uh 25 right? so the JI 4th up to the 5th harmonic sounds “quite consonant.”

if you take a 6/5 minor sixth instead, you get 9/8, 16/15, 24/25, 18/20, and others. if your root is 400hz you get beating frequencies of 50hz and less; the minor 6th has a higher score for “roughness” and it makes sense that went back and forth being considered a “consonant” or “dissonant” interval in medeival / renaissance / baroque / classical theory.

this kind of analysis also makes it clear why just restraining pitch sets to low limits doesn’t really give you consonance as the ear understands it, because it forces you into these high orders of low primes that have complex beating relationships. (plus the music-theory limitations like being severly restrained in key / harmonic movement.)

think about bells, metallophones in general, even low strings on a piano (where the thickness of the string contributes partials.) these are *enharmonic* sounds - its generally not feasible to quantify their spectral content as a sum of integer ratios. in any pythagorean sense (or the Holder / Rameau sense) these are totally dissonant. its even sort of a mystery and source of debate over how people agree on the “fundamental pitch” for enharmonic sounds (aka *residue pitches*, aka *strike notes*.) (tuning bells and metallophones is an amazing art, of which gamelan builders demonstrate unparalleled mastery.)

so while i *love* working with tunings built on simple ratios, i’m skeptical of approaches that value number theory over sensory and *emotional* and other contexts. i think the shruti system has a value that goes far beyond how close it can be approximated by 5-limit tunings.

the 7th natural overtone (14 occurrences of this interval in the Semantic-53 scale) and the 17th and 19th overtones, to name a few, are naturally present in several ways, notably among indian shrutis, as well as the Semantic system.

i’ll have to think about what this could mean. makes no sense to me on the face of it - 17 and 19 just can’t be produced by multiplying and dividing the numbers 2, 3, and 5. (they can be arbitrarily approximated, as the greeks knew full well.) i guess the semantic system isn’t 5-limit?

anyway all of the above is just to support the idea that it doesn’t really matter if a ratio is prime, low-limit, or irrational, as far as its level of dissonance to the ear. in my teens and 20s i was a little obsessed with tuning in this “numerological” sense, with pythagoras and rameau. not anymore.