Re: [tuning] Reply to Keenan Pepper

>I agree, and I beat you to it, Keenan -- these are the definitions of
>"prime-limit" and "odd-limit" I contributed to the Tuning Dictionary
> http://www.ixpres.com/interval/dict/index.htm
> http://www.ixpres.com/interval/dict/limit.htm

When the same conclusion is arrived at independantly by several different
sources, it's a sign of its inherent truth.

>But Keenan, wouldn't the out-of-tuneness of intervals at a certain
odd-limit
>(say 7) help to project the logic piece if the piece is in that odd-limit
>(say 5)? Wouldn't you think the piece could more easily slip into a 7-limit
>of consonance if the 7-limit intervals were all very much in tune?

For one thing, "slipping into" 7-limit consonance is far from an involentary
reaction for me, and many say I have a good ear. If I'm not concentrating, a
piece ending with a 7-limit sonority will leave me saying, "Where's the last
chord?" (good example - http://www.xs4all.nl/~huygensf/midi/prelud.mid). If
a composer can't get 5-consonance across with the music, they shouldn't
depend on tuning to do it for them.
Secondly, though proper intonation has a different importance for simple and
complex intervals, niether is more important than the other. Out-of-tuneness
is definately more noticable for simple consonances, but in Partch's own
words, "The higher the numbers of a ratio the more subtle its effect, and
the more scrupulously should we try to foretall its dissipation in the
stronger ratios surrounding it."

>In my opinion, since resolutions operate melodically as well as
>harmonically, one should allow melodic considerations to determine the best
>tuning for the dissonant intervals, rather than tuning them to a just
>interval standard beyond the consonance limit of the composition.

If harmonic intervals are wanted to be in JI (not saying they always are),
why shouldn't melodic ones? I always make sure my compositions have not only
vertical but horizontal JI: they sound better to me that way. In a
1/1-5O>4/3-5O>3/2-7O>1/1-5O (I-IV-V7-I) cadence, I think the 64/63
difference between the 4/3 and the 21/16 shouldn't be hidden, but celebrated
and exploited as a subtle nuance. Don't call it "microtonal" if you're
afraid to use microtones!

Ooh, I forgot to add that Muggles (If you aren't familiar with this term,
read more Harry Potter; it makes perfect sense when you consider JI as
somewhat "magical" :-) ) always judge in-tune-ness by comparison to 12-eq,
so JI sounds out of tune. Thus something far from equal temperament (like
the 21/16 in that cadence) would sound out of tune to them and be more
"dissonant" while still being really in tune.

>For one thing, "slipping into" 7-limit consonance is far from an
involentary
>reaction for me, and many say I have a good ear. If I'm not concentrating,
a
>piece ending with a 7-limit sonority will leave me saying, "Where's the
last
>chord?" (good example - http://www.xs4all.nl/~huygensf/midi/prelud.mid). If
>a composer can't get 5-consonance across with the music, they shouldn't
>depend on tuning to do it for them.

I have to admit, I didn't put that point very well. Though a historical
example might be appropriate (read Margo Schulter's page for background). In
the early and middle medieval period, Pythagorean tuning was the norm,
because there was a consonance limit of 3, and therefore no need for prime
factors above 3 (see below). Harmony was dyadic (or trinic, to use Margo's
term, referring to the dyad plus the octave doubling). Thirds and sixths,
which were not simple ratios, were used as dissonances, avoided in final
chords. The tuning they had was felt to be just right for the feeling of the
music -- in fact, any tuning inflections that were suggested for cadential
purposes tended to tune them _away_ from the nearest 5-limit consonances.
Then, as a new, 5-limit, triadic, style began to spread from the Celts, the
Pythagorean tuning was first adapted with flats in place of sharp to exploit
schismatic relationships, and then replaced outright with a tuning in which
the thirds and sixths were far closer to the simple 5-limit ratios:
meantone.

>Secondly, though proper intonation has a different importance for simple
and
>complex intervals, niether is more important than the other.
Out-of-tuneness
>is definately more noticable for simple consonances, but in Partch's own
>words, "The higher the numbers of a ratio the more subtle its effect, and
>the more scrupulously should we try to foretall its dissipation in the
>stronger ratios surrounding it."

But I see no need to _use_ any ratios with prime numbers above the
consonance limit, since the only audible meaning that ratio-complexity has
is: if an interval is close to a simple ratio, it's consonant. And though
Partch's statement is absolutely true, it doesn't contradict the following
statement: The more complex a ratio, the less mistuning affects its
consonance. See the harmonic entropy graphs and/or Sethares' dissonance
graphs to understand how both statements can be true at once. Finally, with
13-(odd)-limit dyads, in most ways mistuning makes no difference, because
(a) you can't perceive the ratio in terms of an implied fundamental without
help from combination tones, and (b), though the just ratio minimizes one
beat rate, other beat rates (of stronger partials) will be very disturbing
at the same time and the perception of consonance associated with
beat-elimination fails to "make it through"; e.g. for a 13:11 or near-13:11
the beatings of 6:5 and of 7:6 will tend to be the prominent elements.

>If harmonic intervals are wanted to be in JI (not saying they always are),
>why shouldn't melodic ones?

Because for melodic ratios more complex than 4:3, it makes no acoustical or
psychoacoustical difference. Let's go back to the historical example --
we're now in the Renaissance and Baroque periods. The major seconds in
meantone are halfway between 10:9 and 9:8: one could use the usual,
irrational, geometric mean; or one could use the JI interval 19:17; it
really makes no difference in how the music sounds. As long as all the
melodic major seconds are perceptually the same, the motivic structure of
the music will remain intact, one will never encounter comma drifts or
shifts, and one is even free to use melodically sub-noticeable 6¢ shifts to
achieve perfect JI harmonies as in Vicentino's second tuning of 1555.

>I always make sure my compositions have not only
>vertical but horizontal JI: they sound better to me that way. In a
>1/1-5O>4/3-5O>3/2-7O>1/1-5O (I-IV-V7-I) cadence, I think the 64/63
>difference between the 4/3 and the 21/16 shouldn't be hidden, but
celebrated
>and exploited as a subtle nuance. Don't call it "microtonal" if you're
>afraid to use microtones!

Huh? I'm certainly not afraid to use microtones, especially those in 22-tET.
See my paper (http://www-math.cudenver.edu/~jstarret/22ALL.pdf, ignore page
20) for more on my philosophy on the above. But the I-IV-V7-I cadence is, in
many ways, the antithesis of what the goal of "microtonal" music is to many
of us -- to provide us with a new language that allows for a new form of
tonality, as opposed to the abandonment of tonality that occured in the last
century.