Klarinet Archive - Posting 001084.txt from 1999/05

From: James Pyne <jpyne@-----.edu>
Subj: [kl] Bradley's Question
Date: Sun, 23 May 1999 22:54:54 -0400

Reply to Mark Bradley's question,

>>Now, here are a couple of questions. Gibson mentions "modal frequency
>>ratios", can someone explain what this concept means?

I believe that Prof. Gibson refers to at least some of the following:

Let's start with a quote from "Vibration and Sound" Philip M. Morse,
American Institute of Physics.

"The air jet through the reed is modulated by the reed vibration, and the
reed vibration is maintained by the pressure fluctuation caused by the
modulation, so the complete analysis of the motion would be quite
intricate".

In Morse's statement "reed vibration" relates to the function and
dimensions of the reed (and mouthpiece to some degree) while "pressure
fluctuation" relates to the function and dimensions of the clarinet bore.
Both the (a) reed/mouthpiece and the (b) clarinet bore, have their
individual modal characteristics. When we actually produce tones these
characteristics are coupled. The (b) clarinet bore, dominates the (a)
reed/mouthpiece, in the basic pitch-selection process. The reed is "drawn"
to certain frequencies of periodic oscillation, as dictated by the
tube-length being established (fingered) by the player. Although the
variable-length bore dominates in establishing what tones are possible,
changes in reed/mouthpiece design can significantly effect the exact pitch
level (intonation) and timbre of produced tones.

Once this basic mode-coupling process is accomplished there are natural
laws that govern what tones (pitches) may be produced with any given tube
length. The clarinet, as a "closed pipe", produces only the odd numbered
partials (1,3,5,7...) in the form of overblown tones. However in the
SPECTRUM (timbre) of clarinet tones this is NOT true. Odd and even
(1,2,3,4,5...) partials are present, though the even partials are somewhat
subdued in the chalumeau. The ratios expressed below refer to the
fundamental and "overblown" modes of all clarinet tones.

To show how this would work we can consider a specific fingering (tube
length). The fundamental tone A3 (= 220 Hz) is produced by the tube length
fingered as the lowest C on the normal A clarinet (without basset
extension). That basic fingering produces written Chalumeau C, Clarion G
and Altissimo E and Altissimo+ A. For the tones above the fundamental
(Chalumeau C) the register vent is normally opened and for Altissimo E & A
the first finger of the left hand is lifted, or positioned for "half/hole",
to provide additional venting. These tones can be produced without any
venting but are musically less satisfactory both in pitch and quality.

Calculated as the harmonics appear within the SPECTRUM of that single tone (A3)

1:1 - 220 X 1 = 220 A3 (written C) facilitates playing Mode 1

1:3 - 220 x 3 = 660 E5 (written G) facilitates playing Mode 2

1:5 - 220 x 5 = 1100 C#6 (written E) facilitates playing Mode 3

1:7 - 220 x 7 @-----. ****
(see below)

The target pitches involved have the following frequencies in the equal
temperament system. 1 (A3 220) 2 (E5 659.26) 3 (C#6 1108.7) 4 (G6 1568)
or (F#6 1480)

**** In the last case above (1:7 - 220 x 7 = 1540 G6) the spectral
resonance peak is somewhat flat (G6 1540 Hz) to the equally tempered target
(G6 1568 Hz). And in its played form, at least on the French clarinet
designs most of us play, VERY flat to the required pitch for G6 (written
Bb). This mode is therefore used to produce F#6 (written A) rather than G6
(written Bb), the "acoustically correct" calculated spectral peak. In this
case we utilize a fingering that is technically 1/2 step higher than might
be expected to facilitate arriving at the equally tempered frequency
target. Thus an "acoustically incorrect" tube length yields production of
the wanted pitch.

Bore design parameters (inlet taper, perturbations) tone-holes (placement,
size, shape etc.) barrel bore (size, taper etc.) reed dimensions (shape,
vamp-contour, strength etc.) and many mouthpiece design characteristics can
affect how the ratios expressed in the Spectrum (1:1, 1:3, 1:5, 1:7) are
expanded or contracted when we actually produce the clarinet tones that
have a "window of opportunity" at, or near, the spectral "resonance peaks"
expressed by those ratios.

Design research involves "juggling" these interactive variables to
facilitate arriving at the equally tempered target frequencies for all
notes within the compass (all registers) of the clarinet. Because of the
complexity involved in a system with so many interactive variables,
progress has been slow in the exploration of better solutions.

---Jim Pyne

James Pyne, professor
Clarinet Studio/Research Group
School of Music
The Ohio State University
1866 College Road
Columbus, Ohio 43210
pyne.1@-----.edu
Tel: 614 292 8969
Fax: 614 292 1102
http://www.arts.ohio-state.edu/Music/Clarfest

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