Revised: Super Particular Stepsize

In: /tuning/topicId_31138.html#31404

Paul wrote:

<< Again, from the ancient Greeks to today, superparticularity was
considered a desirable feature to have *in the ratios describing the
step sizes in the scale*, NOT *in the ratios describing the pitches
in the scale*. >>

Paul,

Thank you for the information! It seems that my mind now asks similar questions as before (where the pitch of scale intervals relative to 1/1 was the item of interest), but now (considering your and Kraig's comments) directed towards the ratiometric "step sizes" between the individual pitches of a scale.

Indeed, once one begins to "rotate" the scale (thus referencing from other individual pitches within a scale), such stepsizes *become* the "interval of interest" (relative to each pitches adjacent neighbors) ...

And stepsizes of stepsizes, etc. Is there any "functional" meaning to the "stepsizes of the stepsizes of the stepsizes" (and onward), until one becomes "comma-tose"? :)

What interests me, in all this ratio stuff, are the spectral implications of the choices of utilizing this ratio or that [relevant to whatever ratiometric level is associated with the calculation and analysis of the resultant (harmonic) spectrums resulting from the potential existence of such overtones], that is - energy appearing at integer multiples of the fundamental frequency of the sin/cos functions (when the frequency of such sin/cos fundamentals is described by rational numbers taken to integer-valued powers in the exponent).

While I can no more mathematically model the composite spectra of my guitar's ringing, sustaining strings sweet tones in time/frequency than any other mortal, recognizing that the entirety of the synthesis, acoustic, and perceptual processes at play evade such "in vitro" atomizations, I would hope, anyway, that this "superparticularity" bit, which many seem to prefer, have at least some sort of grounding possible in theoretical numerical modelling which *might* further enlighten us as to its origins (and, possibly, its applications). This is, admittedly, a fully "left-hemisphere" endeavor for me. With or without the existence of such structures, improvisor that I am, such platitudes will not demarcate what pitches my ear chooses when I play my instrument. It may not yield interesting musical composition on my part. It may just be a pretty diagram, an object of "in vitro" mathematical "worship"... Yet mathematical curiosity drives me - to find out what, if anything, is there.

"Because it's (possibly) there"... If not, I play on, no worse for the wear...

What can we definitively state about the resulting harmonic spectrums resulting from such a (perhaps feeble, perhaps instructive) model of musical spectrums. I find William Sethares' work fascinating, and have much to read and think about in regards to his ideas. Meanwhile, I have been exploring some numerical relationships surrounding the above stated specific interests, and wondered if you (or others) have, in the many hours some have spent pondering matters of (JI) scale intervals (and step sizes, commas, etc.), endeavored to find (comprehensive, rather than anecdotal or isolated) pattern and periodicity - specifically, in the actual *harmonic* spectrums (potentially) resulting from the various combinations of the pitches of a given musical scale(s)?

Maybe I'm just not reading the right folks, but I (in my short career thinking about such mathematical analysis of musical structures, as opposed to the analysis of already existing musical waveforms and signals) do not see much out there which offers much more than a fleeting and "partial" (bad pun) glimpse of the coincident harmonic structures which evolve out of the utilization of rational scale intervals, in general. So I endeavor to (re -) chart the pathways, and find the "golden fleece" of harmonicity (if there is any) :)

There seems to be as much (if not more) stuff on "new-age" and "astrology" websites utilizing the "phraseology" of "harmonics" and "overtones" (most often absurdly, and ignorantly "whoring" these words to any manner of flaky ends) than stuff to be found relating to musical applications which is not largely "Byzantine" in its convoluted, spotty, anecdotal, and narrow focus.

It (almost) always strikes me as "tunneled" and "isolated" information, leaving me feeling like I never know whether begin to try to piece together a number of such "partial pieces" of the pie, or just figure it out for myself as I choose.

Why the (apparent) scarcity of (unified and comprehensive) theories relating to such multi-complex-tone resultant harmonic spectra? Is there no pattern?

David Canright's stuff has been some of the most promising and fruitful found (in regards to clarity and the citation of useful information). Perhaps you or others could make me aware of the existence of the work of others who have attempted to follow such complicated harmonic pathways before, and lived to tell the tale of pattern, structure, symmetry, beauty ... or random chaos devoid of coherent or repeatable structures ...

This is an octave specific world (in addition to being, in other levels of interpretation, octave- invariant). It is a world where the harmonic spectrum of an interval inversion around a given pitch is non-equivalent musically (harmonically, that is). It is not a world which I have seen anyone visiting (but, then again, I have not seen much of the world). Is anybody out there?

Sincerely, J Gill