Historical temperaments

>The use of unvalved
>brass in the lower harmonics will naturally restrict key choices

Actually, that detail in particular isn't completely true, depending on
what you mean by "restrict". With natural horns they inserted a
carefully-selected length of tubing between the mouthpiece and the rest of
the horn, based upon the key of the music. They literally had such a
"crook" to set the fundamental of the horn to a wide variety of keys.

This is why, by the way, natural horn parts notated as though they were
in C-Major (even if they were in a minor key). In the example Mozart's
horn quintet, which is in Eb, they'd put on the Eb crook, and pretend like
the fundamental were C. A 3rd or 6th harmonic, notated as a G, would sound
as a Bb.

Now I don't know for sure, but I doubt if natural hornists had literally
twelve (or more) crooks, in which case I suppose that they might use, for
example, an A crook for something in E if they couldn't afford both. So in
that sense their ranges of keys could have been "restricted".

By the way, Karl Haas once said that (Robert) Schumann was the first
composer to specifically call for a valved horn.



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Let me start by reiterating that I find it perfectly acceptable to consider 9
or other odd numbers as a separate harmonic dimension, thereby validating the
usefulness of odd-number limits. What I'm maintaining is that there is also
a valid prime dimensionality in which primes^x and composite numbers are
included in the dimensions of the prime numbers of which they are composed,
and that prime limits can be used instead of, or in conjunction with, odd
limits.

Daniel Wolf writes:

>Two numbers sharing no prime factors other than 1 are relatively prime.

in response to my response to:

>... then 9 is being articulated
>independently of 3. Since
>9 is here relatively prime, how could 9 be here distinguished a 'real'
>prime?

3 and 9 certainly do share a prime factor, unless, as in your 4th example,
"3" and "9" are really only rough approximations of 3 and 9, in which case
one or both should be given a more specific designation to clarify their
geometrical relationships.

>The V chord is in a V of V relationship to the IV chord.

OK. If the IV were I, then the I would be V, and the V would be the V of V.
So we pretend that the IV is a I and analyze the V of the I in terms of the
IV instead of in terms of the I. Why? Forgive my denseness - How does this
mean that 9 should not be considered part of the prime dimension on 3?

>You will encounter wave fronts for 9 alone in thirds of the distance
>between fronts shared by 3 and 9.

I think I'm getting your point of view here. It's like the previous example.
If the sine wave tuned to the 3rd overtone were temporarily considered the
fundamental of its own tone, then the wave that is the 9 of the fundamental
of which the 3 is the 3 would be the 3 of it. That's just a really hard way
of saying that 3*3=9. The fact is, if several other primes are present, they
cannot all cancel at once (unless they are very carefully manipulated to do
so via complete and separate control over the phase of each sine) and when
some of them combine with the 9, it will serve to reinforce the implied 1,
whether or not the 3 is cancelled.

>a real example where the particular tuning is at a
>relatively low level in the identity of a piece of music and that certain
>relationships - which may be paralleled by mathematical ideals - are not
>necessarily projected materially, but are understood by listeners to be
>present through cultural or compositional means.

I'm sure that is often the case. Virtually nobody in the country music
business here in Nashville understands why I, IV, and V are the primary
chords - they just use 'em 'cause it's the traditional country thang to do.
But anyone who has studied JI and American culture knows why country writers
wear out those chords - there are clear mathematical and cultural reasons.
Yer average country listener would not likely appreciate the projection of
prime dimensionality even if it were made through a country-style recording.
However that may be, it doesn't mean that purely prime dimensionality cannot
be projected materially through any composition.

David



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David wrote:

< However that may be, it doesn't mean that purely prime dimensionality
cannot

Which is my point: the composition articulates the material, not vice
versa.

and:

<3 and 9 certainly do share a prime factor, unless, as in your 4th exampl=
e,
<"3" and "9" are really only rough approximations of 3 and 9, in which ca=
se


In my first example, I gave chords and progressions without fifths, thus
the factor three was not represented and 9 was relatively prime to the
remaining tones.

and:

f
V.
< So we pretend that the IV is a I and analyze the V of the I in terms of=

the
this


This example was the single one where the composite reading of 9 was more=

likely to be understood than and independent axis reading. Incidentally,=

one compositional alternative is to add a suspended ninth to the IV chord=

which is held through the V chord and then to voice the I without a fifth=
,
possibly suspending a ninth in the I as well. This is the best I can come=

up with as a way to articulate 'nineness' in such a fifth-oriented
environment. Perhaps someone else has a better idea... =



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