RE: [tuning] A novice finds toothless wolves and so delves into t heory

William S. Annis wrote,

>Now this is no surprise that the addition of the 7/5 would
>improve the overall sensation of consonance, I thought, until I
>checked the intervocalic ratios: 56:45, 25:21. Now, 25:21 is only two
>cents off (301.85) a tempered minor third, but 56:45 is well away from
>anything at 378.6 cents (5/4 is 386.31). I've checked the tuning of
>the synth several times in desperation.

While you've accepted 301.85 cents as a decent representation of a minor
third, even though its 14 cents flat of 5:6, what you seem to miss is that
378.6 cents is less than 8 cents flat of 4:5. So you've divided the wolf
fifth into two essentially consonant intervals, "masking" the wolf
considerably.

I've made reference to this kind of effect before. For example, some years
ago I asked people on this list what would be the best major third to use
with a 720¢ fifth in a major triad. Though 720¢ is kind of hard to take as
a fifth on its own, those who did the experiment suggested major thirds in
the 395¢ range. As I see it, the major third is then about 9¢ sharp, and the
minor third is also about 9¢ sharp, both acceptable enough to make those
intervals recognizable, which goes a long way toward making the whole chord
a pretty undeniable major triad. The same thing is happening in your
example.

Once I have a workable model of triadic harmonic entropy, not just total
dyadic harmonic entropy, a portion of this sort of "masking" will be
quantifiable. The explanation of the portion I'm referring to is essentially
that although an out-of-tune 2:3 produces a not-too-strong sense of what the
fundamental at 1 is, addng a note that allows one to construe the 2 and 3 in
the 2:3 as the 4 in a pretty good 4:5 and the 5 in a pretty good 5:6, gives
you a pretty clear sense that you're listening to the 4th, 5th, and 6th
harmonics of a fundamental, and this "sureness" of fundamental sensation
contributes to increased consonance.

In order to get a sense of how far this explanation goes in solving your
dilemma, try the mirror-image version of this chord, with the "minor third"
on the bottom and the "major third" on top. How much is the dissonance of
the wolf fifth alleviated when this "minor third" is added? I would predict
not as much, because the additional intervals in the minor triad do not
affirm the same fundamental that the fifth is trying to affirm. However,
there could still be some alleviation, since some of the "masking" might not
be due to harmonic entropy but due to a more literal interpretation of
"masking". That would mean the suppression of sensitivity at frequencies
where the sensory roughness of the out-of-tune intervals takes place,
resulting from the additional frequencies.