Xenharmonic Wiki:Cross-platform dialogue: Difference between revisions

Appreciate it. Thank you to Budjarn and Flora.

OK, now that that is settled, regarding the actual math involved for Godtone, Inthar, Flora and others:

Yes, it's true that one of the main reasons people liked POTE historically is because it approximates KE. If you want to declare KE some kind of best general-purpose tuning, or whatever, I would probably support that. However, we ought to compute POTE/KE tunings for a huge set of temperaments to see if for any they differ significantly (which would be very surprising to me).

The bigger picture: all of these tuning optimizations are imperfect because they only measure dyadic error. A 4:5:6 chord, flattened to the "isoharmonic"/"proportional"/"Mt. Meru"/whatever 4:4.98:5.96 chord, or 0-379-690, sounds less far off than something with a similar amount of error which is non-proportional, so we really care about which way the errors are oriented within the triad. If you want, you can look at the Hessian of simple chords in 3-HE, which models this correctly, to build a simple linear model for tuning error of some triad. Then, you can try to optimize the entire "Z-algebroid" of chords, rather than just the Z-module of monzos. In principle it can be done - I've played around with this kind of thing a bit, though don't have any firm results.

Or: just note empirically that TE seems to magically give pretty good results, even for triads, tetrads, etc. Why? Who knows. Maybe if you did the above thing out all the way you'd derive that TE, or something close enough to it, also happens to be the optimal tuning on the entire algebroid. Similarly, POTE gives good results for this but not CTE. Why? Again, who knows, but one idea is to note that stretching a chord isoharmonically/proportionally in Hz is very close to stretching it in cents, as a first order Taylor approximation. For instance, stretching 4:5:6 both ways so the outer dyad is 720 gives 0-397-720 (isoharmonic) vs 0-396-720 (stretching in cents). So scaling TE to POTE is approximately the same as isoharmonically stretching all chords in the entire tuning.

So that is the other reason POTE, or any tuning, is useful: empirically, we note it sounds good.

There are other reasons. Graham has expressed interest in it being the unique pure-octave tuning that minimizes the angle/dot product with the JIP. I don't remember what musical interpretation he gave to this angle; you'd have to ask him why. And there is always this "black magic" element to it where Gene knew a bunch of stuff about all of this, but has sadly passed on and we can't ask him about it. We have the same situation with, for instance, zeta integral and gap tunings - what theoretical justification do these things have, compared with something clear like zeta peak tunings? I don't know, but Gene did. Oh well.

: I have idiosyncratic reasons to favor CTE or a tuning that doesn't prioritize divisive ratios over multiplicative ratios based on where beating occurs in the frequency spectrum with any harmonic timbre. I documented the details in my essay (→ ''User:FloraC/Hard problems of harmony and psychoacoustically supported optimization''). The essay also contains a design around Hahn distance so you might be interested. Regardless, as I've related to you on FB, I think of CTE as the most straight-forward solution that is most likely to be understood by the widest audience, so the main appeal is its methodological transparency, which is somewhat shared by KE and is in contrast with the "black magic" of POTE.  

: My next plan is to add KE to later temperaments that never had POTE documented (and CTE is also to be added). [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:50, 17 March 2024 (UTC)