RE: [tuning] Hello and a Question

I wrote,

>>The main reason your metric doesn't "work" for me is that it assigns the
>>same distance for 15/8 as for 5/3. 5/3 is much more consonant than 15/8
and
>>thus should receive a shorter distance.

Keenan wrote,

>I disagree. This is just my opinion, but 15/8 and 5/3 are both a third from
>a fifth, and are the same distance from 1/1 (along with 16/15 and 6/5).

That's true on the rectangular lattice, but on the triangular lattice, the
minor third or major sixth gets a line of its own. It occurs very early in
the harmonic series, and so is quite consonant, regardless of whether one of
the notes is octave-equivalent to the fundamental or not.

Try to listen to the two intervals objectively and tell me you don't think
one blends a lot more smoothly. Also include 16/15 and 6/5 in your
comparisons.

>Consider the chords - maj7 and 6. They certaintly have different flavors,
>but one is not more dissonant than the other.

That's a different issue. I actually do tend to agree with this assessment
of these tetrads -- look at these chords on the triangular lattice:

Major 7:

5/4------15/8
/ \ /
/ \ /
/ \ /
/ \ /
1/1-------3/2

6:

5/3-------5/4
\ / \
\ / \
\ / \
\ / \
1/1-------3/2

They are about equally compact on the lattice, which reflects their similar
degree of dissonance. However, observe that 5/3 to 1/1 is shorter than 15/8
to 1/1. That reflects the relative dissonance of these dyads, while the
rectangular lattice would lead you to believe that these dyads are equally
consonant.

>Besides, I designed my system
>not to be so much a measure of consonance as a measure of how much
>modulation one would need to do to "get there".

So you believe that modulating by a major third is a "shorter distance" than
modulating by a minor third? I certainly don't find that at all, especially
when the music has a diatonic basis, in which case minor third modulations
are particularly easy to hear.