RE: Basic dimensionality, and related issues

Dan Wolf wrote,

>>(4) Using a TX81Z (with a resolution of 1.56 cent deviations from 12tet),=
>> I
>>perform
>>a melody based upon random walks over a lattice with two dimensions: 3s a=
>>nd
>>9s. The
>>best approximation of 9 does not coincide with the best approximation of
>>3^2. For all practical purposes, 3 and 9, in this temperament, are
>>relatively prime.

Graham replied,

>3*3=9 is a universal theorem. If you are using an interval that
>is not 2*log(3), that interval is not log(9).

Sorry, I agree with Dan here. What is true for rationals may not be true
for their best approximations in a set of irrationals. 768tET is a poor
example because you're likely to treat that many notes as a continuous
pitch spectrum. 18tET (Busoni's third-tone system) is a much better
example. By far the best and most consonant approximation of 9/4 is 21
steps. This can only be construed as 3/2*3/2 if two different values of
3/2 are used! In other words, the factorization of 9 into primes is at
best irrelevant and at worst counterproductive in some cases.

>Paul Erlich wrote:

>>Also, if 9/5 and 9/7 are to be used as distinctive harmonic entities (i.e.,
>>consonances), and not only arising from 3*3/5 and 3*3/7, then it is useful
to-

>>include a 9-axis into the triangular lattice.

>If the dissonance of a composite number is independent of its
>factors, why use lattices at all?

What an odd question! What could you possibly mean by it?

>If you do, though, this
>would introduce a concept of harmonic dimensionality different
>from scale dimensionality. What I previously called "n-D
>harmony" can be renamed "n-D JI" to avoid confusion here.

Yes, there are these different and equally valid concepts of
dimensionality. In my previous posts in which I talk about dimensions I
am more concerned with harmonic dimensionality.



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On Sat, 19 Jul 1997, Gary Morrison wrote:
> >The use of unvalved
> >brass in the lower harmonics will naturally restrict key choices
>
> 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.

Not exactly true. There were a variety of solutions to writing for horn
in the classic period (pre-valve). Mozart's G minor symphony calls for
on horn in G and the other in Bb. This allowed for tonic triads in both
G minor, and the relative major, Bb.

The french theorist, Gaveart, wrote a book on orchestration in the 1830s.
He says that when a piece is in minor, the best solution is to call for
horns in the sub-dominant key. (Ex: For g minor, use horns in C) He
reasons that a tonic triad will be playable. Of course. today we would be
horrified that the open triad would have the ratios of 6:7:9 because the
third of the triad would be so low. Gaveart proposes no solution, but it
is my guess that the minor third would be lip-tuned to a 5:6 ratio.

> 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".

The horn players did not choose the crooks. The composers always
specified which horn they were using, and it would be impossible to play
the part on any other horn. The technique of using the right hand
inserted into the bell to alter the pitch was invented in the 1770s, but
only used for solo horn playing (It is necessary in Mozart's four horn
concertos, but not in any of the symphonies or operas.) Fine tuning was
done with the lip.

BTW, horn players did not have unlimited crooks. I forget which were
considered standard (and some of them would be used in parallel to get new
possibilities) but four or five was normal. Incidentally, if you add too
much tubing to the basic horn, then the proportion of bell diameter to
tubing, and the degree of 'conicity' (ie: rate at which the bore
increases) get all messed up, and the harmonics go out of tune. (I once
played a REALLY bad trombone where the open notes were Bb, F, C! This was
because it was all banged up and poorly designed to begin with.)

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

Wrong. Berlioz used valved horns in the symphony fantastique.
Interestingly enough, he did not value them for their ability to play
chromatically, but rather for their ability to produce all three horn
timbres (open, muted and stopped) on any pitch.

Schumann had notorious troubles writing for horns (and other orchestral
instruments.) For a hilarious story, read in the 3rd edition of Groves
encylopaedia about the first rehearsal of the Eb major symphony. His
fanfare originally was on Eb horns the (written) notes C C C C C A B C.
Of course, the A is muted on the horn, and the B is completely stopped.
The fanfare was greeted by loud laughs (Schumann's included.) He, on the
spot, revised the fanfare up a diatonic third to: E E E E E C D E - all
open notes, and that is how it remains to this day! (Even though the
principal theme begins C C C C C A B C) You can imagine his joy when the
valved horn made it possible to play all notes with the same timbre. This
is the instrument he wanted to play the Konzertstuck for four horns.
(Although, he still had no idea of endurance! The piece is incredibly
difficult and HIGH!)


Joe Downing,
in Syracuse


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

Gary replied:
>
> Actually, that detail in particular isn't completely true, depending on
>what you mean by "restrict". [Gary goes on to describe crooks.]

Yes, obviously. I should have been clearer. I was speaking not only of the
keys in which the pieces were written, but also the ones to which they
modulated (and wanted to include brass). Even so, not all twelve horn
crooks were in common use in Mozart's time, and trumpets were generally
restricted to Bb, C, D, and Eb.

Other considerations affected key choices as well. Woodwinds of the time
did not sound the same in all different keys, even if they could play
chromatically. String players tended to play open strings when possible
(the opposite of today). Thus in the context of the previous discussion,
there were many considerations that determined key choices in ensemble
music in addition to temperament.

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

Just goes to show what a learned musicologist Haas is. According to my old
copy of the Carse History of Orchestration, parts for valved horns began to
appear in 1835, several years before Schumann began to write for orchestra.

Bill

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^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
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>Just goes to show what a learned musicologist Haas is. According to my old
>copy of the Carse History of Orchestration, parts for valved horns began to
>appear in 1835, several years before Schumann began to write for orchestra.

Thinking back, Haas may have been referring to usage of valved horns
specifically for use in solo work, because the work he referred to was
scored for solo horn with piano accompaniment. I don't recall the title or
any themes in it.



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On Sat, 19 Jul 1997, Gary Morrison wrote:

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

Being a horn player I must also add that the hand is a necessary element
in tuning with the natural horn. By slowly inserting the hand into the
bell the pitch can be lowered by as much as a half a step. However by
completely "stopping" the horn with the hand, the pitch is raised by about
half a step because you are effectively shortening the length of the tube.
Therefore the natural horn is quite flexible for use in chromatic playing,
though at times it is problematic with variations in the timbre, etc.

Bruce Kanzelmeyer



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