RE: [tuning] Re: TD 750 -- Paul Erlich on tetrads

Margo wrote,

>One question might be how much the septimal schisma or "beta-2" (~3.80
>cents) affects the consonance rating of Xeno-Gothic tetrads approximating
>12:14:18:21 or 14:18:21:24. These are unstable sonorities with an effect
>at once "coloristic" and often cadentially oriented toward a stable trine
>or fifth.

The answer is, "not much" (hope that isn't too many decimal places for you!)
Since this model uses a 1% (about 17-18 cents) frequency resolution, changes
of 3.8 cents can only have minor effects on the results, and I would be
tempted to call the difference "negligible".

>Anyway, thanks for another fascinating survey. Incidentally regarding the
>not-so-consonant qualities of the minor sixth: it is often ranked by
>13th-century theorists along with the minor second, major seventh, and
>tritone or diminished fifth as a strong dissonance. I wonder if the
>tension between the third partial of the lower note and the second partial
>of the upper note could contribute to this result, not only with a more
>complex ratio like the Pythagorean 128:81, but also with a pure 8:5.

The magic of harmonic entropy is that it does nothing more than reflect the
mathematical fact that the more complex a ratio is (up to a point), the more
simple ratios are close to it and confuse its perception. So yes, 8:5 is not
one of the deepest minima in the harmonic entropy curve, because 3:2, 5:3,
and 13:8 are all close enough to confuse the sensation somewhat. For ratios
simpler than 8:5, such as 5:4 and 6:5, one will find less confusion from
nearby ratios, and for those more complex than 8:5 (but not so complex that
the ratio itself becomes irrelevant to the perception of the interval), one
will find more confusion from nearby ratios. I posted a proof of this
mathematical fact perhaps a month or two ago in response to a query by
Joseph Pehrson.