TUNING digest 1438

from an exchange:

<> Here's my problem with that: the 28:25 is represented in 31TET by 193.=
5
<> cents, and that interval by itself doesn't really evoke a consonant 9:=
8.
<> If one really hears intervals at the 9-limit or higher, then both 9:8
<> and 10:9 must be considered consonant, and there should be a dissonant=

<> point about halfway between them.
<

Paul Erlich was describing the mean of two intervals while Graham Breed w=
as
describing the freshman sum.

This does, however, raise an interesting set of problems. Given two ratio=
s
(a) will the freshman sum of the ratios (notated in specific octaves)
always yield the peak consonance? and (b) how would one determine the pea=
k
dissonance? =


I suspect that (a) is true as long as the two intervals are within a
certain magnitude (thus the freshman sums of 1/1 and 2/1, 3/2, or 2/2 and=

3/2, which is 5/4, are clearly peaks, but that between 81/80 and 2/1,
83/81, is not) while (b) is rather more complicated. Depending on registe=
r,
timbre, amount of mistuning and musical context I might find a slightly
mistuned perfect fifth to be more dissonant than a 12tet augmented fourth=

or minor second. I don't have any elegant way of distinguishing these two=

kinds of dissonance. Any ideas?

Daniel Wolf
Frankfurt =