Brass Spectra

Somebody, in the course of the discussion on the possible
not-completely-harmonic nature of orchestral instruments, suggested that
brass instruments have more-detuned partials than average.

I don't personally have much information on that question, although I do
generally remember them to be more consistent than woodwinds from one cycle
to the next, which seems to suggest that that would not be the case.

Buuuuut, I do have very direct information on another interesting aspect
of brass instruments in general: Their amplitude spectra - almost always -
have a shape that might be described as a "truncated bell-curve" -
truncated at the fundamental. Something like this:
___
/ \
Amp / ` _
/ ` ________
\
-------------------------------
123456789...
harmonic#

That in contrast to woodwinds, which frequently have lumpy, jagged spectra.

And there are some general patterns to how this spectrum changes with
pitch and volume. What I drew above is more or less what a loud,
low-pitched tone would look like. A quiet tone at the same pitch has only
moderately lower energies in the lower harmonics, but that high-harmonic
shelf toward the right of the spectrum goes away.

Also, as you move toward higher notated pitches, that bell-curve narrows
(as if it represented a bell-curve with a lower standard deviation), and
also moves toward the left. So the spectrum of a higher note typically
looks somewhat like this:
__
/ \
Amp `_
`___
\
-------------------------------
123456789...
harmonic#

These are all only broad generalizations of course, and obviously the
details vary a lot between trumpets vs. F.horns, vs. trombones, etc., and
of course by different players and manufacturers of the instrument.

On Tue, 28 Apr 1998, Fred Kohler wrote:
> Because of the digital circuitry involved, synthesizer manufacturers will be
> inclined to select a number of steps per octave in which powers of 2 are a
> majority. Is there some mathematical reason that such numbers as 768, 864,
> 1024, 1152 and 1280 have poor consistency? [snip]

I don't have any deep or rigorous explanation for it, but whenever one
finds a ET number with high consistency for its size (like 12) its
multiples tend to drop off rapidly in consistency level.

> Is there a number that is mostly powers of 2 that would qualify as having
> high consistency that would make the synth manufacturers happy?

Here's how Fred's suggested numbers look:

(limit) 3 5 7 9
---------------------
768| . . .
864| . .
1024| 312 . .
1152| 4 3 2 .
1280| 2 .

(1152 looks the best of those, although 1024 is a power of 2 and has
great 3/2s--level 312 is a bit of overkill, though. 8-)> )

Of the multiple-of-12 numbers in my first list, their prime
factorizations are:

612 = 17 * 3^2 * 2^2
624 = 13 * 3 * 2^4
684 = 19 * 3^2 * 2^2

Compare these to the numbers in Fred's list:

768 = 3 * 2^8
864 = 3^3 * 2^5
1024 = 2^10
1152 = 3^2 * 2^7
1280 = 5 * 2^8

1200, BTW, is 5^2 * 3 * 2^4.

I don't have time just at the moment (maybe this weekend) to do a more
thorough search, but others are welcome to look through my consist.txt
table and suggest better compromises.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
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