Yahoo Tuning Groups Ultimate Backup mills-tuning-list Odd vs. Prime

The debate is being pushed to the limit. What do I think? It depends on
how you're using "limit"...

a) For describing music tuned in Just Intonation
b) For describing music in root-of-two equal-step tunings

For "a" I prefer prime, and for "b" I prefer odd. Now I say I use "limit"
for "describing music", NOT for creating an acoustical theory of interval
perception. I admit the two are intimately tied up, and I will address the
latter only as the former requires.

~~~~~
(a)
~~~~~
In Just Intonation, no interval with a given prime factor in it will ever
hit an interval without that prime factor, no matter how many times you
stack the intervals. As Gary Morrison points out...

>>Admittedly, 2 is a very special prime number,
>>in that it is the only one which is even (not odd), so
>>most likely, this fact can go a long way to explain
>>why we hear these similarities.
>
>You think so? 2 is the only prime that's even, but is that really all
that >important? 3 is the only prime that's a multiple of 3, and 5 is the
only >prime that's a multiple of 5.

..this even applies to factors of 2. That we consider intervals related
by factors of 2 to be more or less equivalent most of the time supports the
idea that prime-limit interval groups (not odd-limit, since 2 is even)
share subjective qualities. Against this, Paul E. argues...

>"An arbitrary power of two produces an effect of equivalence." This must
>of course include the case where the power is zero, and the equivalence
>is greatest. The equivalence evidently falls off as the power increases.
>
>But what about steely coldness? Is it there when the power is zero? Does
>a unison contain all potential interval qualities, to a greater degree
>than the intervals themselves? It would seem hard to argue that way.

..And I wonder if he is talking powers or factors. Factors are involved
in the ratio of frequencies in a given interval, and powers are involved in
the stacking of any interval. The idea is that adding a factor of 2 to an
interval produces a new interval with a certain "sameness" to the first,
because 2 is the "sameness factor". And stacking any interval will produce
a new interval with a certain "sameness" to the first, simply because
stacking doesn't add any new factors of any kind.

If he's means what he says ("powers"), then his first paragraph is correct,
but his second is not, as "steely coldness" is an attribute of the factor
3, not the power 3.

If he means factors instead of "powers", then I suppose his whole thing is
correct. Although I do not see why he insists on making the unison into
some kind of "white light", I am willing to oblige. As far as factor
determines which partials line up, and all of the partials in a unison line
up, the analogy holds...

2 - sameness
3 - steely coldness
5 - sweetness
7 - flourescent lightingness
[etc]

Now since we've brought stacking up, I've said I agree with his first
paragraph (taken as writen ) in that equivalence does fall off as you
stack. Paul offers...

>There seems to be a pervasive (mis)conception that, for
>example, an 81/64 pythagorean major third has some of the
>"strong/steely" characteristics of the perfect fifth and fourth, while
>the 5/4 just major third has a different, "sweet/emotional" character.
>Supposedly, this is due to the "character" of the prime numbers 3 and 5,
>respectively.

..except the particular example of 81/64 does not involve enough stackings
to take away the "3-ness" of the interval. I don't know how many stacking
ya need, and it's probably different for every listener, for every timbre,
for every limit and for every musical context. But if I'm breathing, I'm
not at a loss to hear the "3-ness" in the 81/64 or the "5-ness" in the 5/4.

Although Paul E. adds...

>To me, this sounds preposterous, as there seems to be no
>mechanism or reason for the auditory system to be performing prime
>factorizations.

..I doubt the current understanding of the auditory system (or, more
likely to be needed, the cognitive system that interprets the data) is
sufficient to declare the existence or non-existence of such a mechanism.
Talk of chaos theory notconvincing.

So, with the "ness" of the prime, non-odd 2 firmly in hand, we can get back
to the original point, shunning interval acoustics in favor of describing
music. We have the fact that no stacking (powers) of an interval will
change the prime factors of the interval. It will change the odd factors,
however. That's the point. Why is it musically significant? Because in
music, we stack intervals. And here's how it works.

Mideval music choral music has ratios of 3 and 9, but not of 5 or 7. So is
it 9-limit? Now you might say that we can just as well have music with 7's
and no 5's, creating an ill-defined prime limit. And true, we can have
such a music, but we don't. Except for isolated experiments with
fixed-pitch instruments (I believe Fokker did work with such tunings),
music has evolved by prime limit. Pull out a CD of English tudor music, as
sung by the King's Singers, for example, and try to add the 7's. You
won't. They didn't, not for hundreds of years. They've got ratios of 25
and everything else needed to modulate around the 5-limit, throwing commas
around like frisbees, but no 7's. Barbershop's got 7's and 28's, and 63's
a-plenty. And 9's, and 18's, and 27's. But no 11's. Never will you hear
an 11 used harmonically in Barbershop music. And the first time you do,
you'll be hearing them again soon and often :~)

And that's the case for prime limit. It's good because it describes music
made in JI.

~~~~~
(b)
~~~~~

In root of two equal temperaments, stacking CHANGES THE INTERVALS. Except
when dealing with 2's, becuase they're just. This means odds are all fair
play. Witness Paul Hahn's measurement of level 2 and greater consistency:
far from meaning he's on the "other side", it only makes sense at an odd
limit.

Because ET's make every interval in all the modes available to the 1/1
(like a tonality diamond does in Just Intonation), you can measure lots of
intervals just by looking at the primary intervals in each limit. Say
you've got good 3/2's and good 7/4's, you can bet you'll have good 7/6's
too. But that doesn't make the 6-7-9 triads consistant; there could be a
better 9/7 that isn't 3/2 minus 7/6. So you need seperate consistency
measurements at both the 3 and 9 limits.

The reason, I suspect, that Paul Erlich fights for odd limits: He is a man
of equal temperaments :~) The idea that 9's do not become harmonically
significan't until we have 7's does not hold in my experience. Try playing
just 4-5-6-8-9 chords and see if the 9 doesn't serve the same purpose as it
does in a 4-5-6-7-8-9 chord. I use lots of 9 in my 12-tone compositions
because it's so well represented in 12. Listen to my compositions on my
web page and see if the 9's don't work.

Carl

Odd vs. Prime

🔗Carl Lumma <clumma@...>

🔗Paul Hahn <Paul-Hahn@...>