Posted by John Swensen on June 23, 1999 at 17:28:40:
In Reply to: Partials ? posted by Paul H on June 23, 1999 at 15:07:13:
I pretty much agree with Bob, although I would like to add a few refinements to his definition.
Many authors, such as Benade,
use partials to denote the resonant frequencies of the horn with a particular valve combination,
and use harmonics to denote the (exact, integer) multiples of a particular
note's frequency.
For example, a CC tuba, tuned to A=440, with no valves down, might have partials at
32.7, 65.3, 105.2, 130.8, 162.5, 207.2, 228.1, 261.9, 294.3, 330.2, 350.2, 385.9, etc.,
but the harmonics of the pedal C would be (exactly)
32.7, 65.4, 98.1, 130.8, 163.5, 196.2, 228.9, 261.6, 294.3, 327.0, 359.7, 392.4, etc.
The partials are the frequencies at which the horn/mouthpiece resonates at,
when fed a pure, sinusoidal acoustical input.
For the example above, the horn has slightly sharp third and 6th partials.
A buzzing embouchure does not produce a pure sinusoidal input;
the nonlinearities of the buzzing process produce the harmonic frequencies.
If the harmonics of a note match the enough of the partials
of the horn/valve combination closely enough,
then that note resonates well.
If the pitch of a note is altered by lipping up or down,
the harmonics track the base (fundamental) frequency exactly.
If the note is lipped up or down too far, not enough of the new harmonics
will match the (unchanged) partials of the horn, and the tone will suffer,
as we have all heard ourselves.
If, instead, the pitch of the note is altered by pulling or pushing a slide,
the harmonics track the new base frequency as before,
but the partials of the horn change, also,
so the partials and harmonics match pretty much as before.
That, by the way, is why pulling a slide to fix a note's intonation
is better than lipping it very far.
Now, if a G below the staff is played at 98.1 Hz,
its harmonics will be (exactly) at
98.1, 196.2, 294.3, 392.4, 490.5, etc,
While the partials of the horn remain at
32.7, 65.3, 105.2, 130.8, 162.5, 207.2, 228.1, 261.9, 294.3, 330.2, 350.2, 385.9, etc.
To resonate well, enough of the harmonics of the played G must match partials of the horn;
98.1 vs 105.2, 196.2 vs 207.2, 294.3 vs 294.3, 392.4 vs 385.9, etc.
In this case, only harmonic 3, at 294.3 matches a partial exactly,
and most other harmonics are flatter than the horn's partials;
the note will resonate better if we let it go a bit sharper (about 7 cents, in this case).
In actual practice, the partials are not sharply defined frequencies,
but peaks in an impedance curve that vary in overall height and width.
A high, narrow peak has a sharp, well defined resonance,
whereas a lower peak with a more gentle slope on either side
has a broader, but less influential resonance.
On most horns, the partials are not nearly as close to the harmonics as shown here,
although the better the match, the better the horn plays.
Benade describes experiments with woodwind instruments,
where he deliberately altered the partial frequencies of woodwind instruments
(by carefully scraping or applying lacquer to the bore)
to make them match the harmonics of more notes;
the universal judgement of professionals playing the modified instruments
was that they were far better than before.
I suspect that Arnold Jacobs' York #1 has partials remarkably close to the harmonics
of each valve combinations' fundamental.
Conversely, a stuffy note or two could probably be traced to having a poor match
of the harmonics of the stuffy notes with the partials of the horn.
This message is already too long, but it is an oversimplification of a very complex subject. See Benade,
Fundamentals of Musical Acoustics,
for a bit more involved discussion (and without the math).