Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

Measurements of thehead-joint perturbation and

the embouchure-reactance offlutes

Fransson, F.

journal: STL-QPSRvolume: 9number: 4year: 1968pages: 015-022

http://www.speech.kth.se/qpsr

STL-QPSR 4/1968

111. MUSICAL ACOUSTICS

A. MEASUREMENTS OF THE HEAD-JOLNT PERTURBATION AND THE EMBOUCHURE-REACTANCE O F FLUTES

F. Fransson

Significant determinants for the tuning and the sca le of the modern

flute a r e the head-joint dimensions, the effective embouchure reactance,

and the effective length of the ~ / 4 pipe f r o m the embouchure up to the

cork. This i s well known by the instrument maker s and many improve-

ments a r e due to their grea t experience. However, a m o r e specific know-

ledge i s needed about (1) the head-joint dimensions and (2) the effective

embouchure reactance during appropriate blowing. The need of conveni-

ent measuring methods fo r such studies is apparent.

In a cylindrical flute the fundamental frequencies of the overblown

tones, i. e. the tones of the higher modes, a r e lower than the correspond-

ing mutliple frequencies of the f i r s t mode. Two useful compensation meth-

ods have been employed, The old flute has a cylindrical head-joint and

the body has a conical taper towards the end. The modern Boehm-flute

has a head-joint tapering towards the c o r k and a cylindrical body. This

l a t te r type will be investigated here.

Measurements were made on flutes with a conical head-joint f rom a

modern flute and a specially made cylindrical head-joint. The body con-

sisted of a cylindrical tube without finger holes. Four tubes of different

. lengths were used and in this way four flutes with cylindrical (NG I c to

NO 4c) and four with conical head-joints ( N O l p - NO 4p) could be tested.

The sine wave modulated Ionophone was used a s sound source and the

Ionophone electrodes were placed i n the center of the flute and a t the center

of the embouchure, The sound p res su re was measured with a 1/21' Briiel &

K j z r microphone about 4 c m f rom the open end, Measurements were made

with open a s well a s closed embouchure without moving the sound source,

In o r d e r to obtain a value of the effective embouchure reactance, the flutes

were blown mezzoforte by two flutists. The f i r s t four modes were blown

on each flute except on the two shortest (4e and 4p) where the fourth mode

was difficult to blow. F o u r tes t s by each flutist were recorded and meas-

ured. It i s known that a t moderate wind p res su re the fundamental f r e -

quency approximately equals a resonance frequency of the instrument. In

STL-GPSR 4/1968 16.

the present investigation the effective embouchure reactance is specified 7

a s the embouchure reactance calculated f r o m the above-mentioned blown

frequencies. In o r d e r to calculate a mean value of the sound velocity a

thermis tor was placed i n the center of each flute. The mean value of the 0

tempera ture was about 30°C inside the flute a t 25 ambient temperature.

The Ionophone measurements were correspondingly corrected with respect

to temperature.

Lis t of symbols (pertaining to the F igure on p. 17 and to Eqs. (1) - (8) ) --------- t - = The t r ans fe r function

'i

Pt = The sound p res su re a t the open end

Ut = The input volume velocity

r = The complex propagation constant = r +r + C = a L + j$L 1 2

where CY i s the attenuation constant and $ the phase constant

r ( h / 4 ) = The complex propagation constant f o r the flute with closed

embouchure ze = and the sound source a t the em-

bouchure = aL(h/4) + j@i(h/4)

r (A/2) = The complex propagation constant for the flute with open

embouchure and the sound source a t the embouchure =

= C Y A ( A / ~ ) + j$a(x/z)

w = Angular frequency = 2n.f

z = The character is t ic impedance = r + j x o z E 0 0

z = The embouchure impedance = r + jx e e e

p = Density of the a i r

c = Velocity of the sound

q = Cross-sectional a r e a

k(h/4) = Equivalent length of the flute with closed embouchure z = m e and the sound source a t the embouchure

R ( x / ~ ) = Equivalent length of the flute with open embouchure and the

sound source a t the embouchure

FLUTE AbJALOG

1: End

z t r = a rc tanh - t z 0

At resonance: s inh r (~ . /2 ) = 0

F o r closed embouchure , i . e . z = m: e

At. resonance: c o s h r ( ~ j 4 ) = 0

F r o m Eqs. ( 2 ) and (4 )

Assuming low losses:

From, Eq. (4) at resonance:

@ j ( L / 4 ) .t{A/'4) - 2n-1 - 2n- 1 w c 4f o r k(h/4) = c . ~ T - - (8) n n

The equivalent length l(h/4) with the sound source a t the embouchure

is calculated using Eq. (8) and resonance frequencies measured with a

gliding s ine wave fo r closed embouchure z = a, Fig. 111-A- 1. A. The e

The resul ts a r e shown in Fig. 111-A-2. A. Extrapolation for intermediate

values was made by

The equivalent length of the quar te r wave pipe above the embouchure may

be calculated f rom the frequency of the f i r s t ze ro in Fig. 111-A-1. This

length was found to be 1 .82 c m and

The reactance f o r open embouchure and the effective reactance a t blow-

ing a r e calculated for respective resonance frequencies, sin@A(h/2)=0 by

Eq. (7). Values of $ R ( A / ~ ) and Pa for corresponding frequencies a r e cal- 1

culated f rom Fig. 111-A-2. A and Eq. (10). The resul ts a r e shown i n Fig.

111-A-3.A and Fig. 111-A-3. B. The reactance fo r the open embouchure,

Fig. 111-A-3. A, is simply x = k- f. The importance of this fact will be

evident la te r . The effective embouchure reactance, Fig. 111-A-3. B, in-

c r e a s e s monotonically but not l inearly with the frequency. The change i n

the slope of this curve is to a g rea t extent due to the well known fact that

the embouchure aper ture is m o r e covered by the f lut is t ' s lips f o r high

tones than fo r low tones. This means that a m o r e covered aper ture will

cause a n increased reactance. In a n ea r l i e r study, the frequencies of

blown tones were measured on three modern flutes. Each flute was played

by three professional flutists. No systematic difference in frequencies

was found. This means that no systematic difference i n the effective em-

bouchure reactance ex i s t s although the difference i n the open embouchure

reactance was considerable.

The effective embouchure reactance might be independent of the em-

bouchure dimensions, i. e. the player adjusts his covering to some op-

t imal value independent of the embouchure dimensions. However, this

should not be interpreted a s indicating that the embouchure design is of no

Fig. 111-A-1. T r a n s f e r function for a cyl indr ical flute.

F i g . 111-A-2. Equivalent lengths of exper imen ta l f lutes. A . Cylindrical head joint B. Conical head-joint , c losed emb. z = a (h/4 pipe) C. Conical head-joint , open cmb. z e = oo (h/2 pipe)

Fig. 111-A-3. Embouchure reactance. A. Open emb. reactance B. Eff. reactance at blowing

F i g . 111-A-4. C o n i c a l h e a d - joint . C o r r e c t i o n of l e n g t h d u e t o p e r t u r b a t i o n . A. A/4 p ipe m e a s u r e d v a l u e s B. A/2 I T I I I I

C . A/2 " c a l c u l a t e d I T

that the length correct ion A R i s , within measuring accuracy, the same fo r

the h/4 and the h/2 pipes but with reversed signs. A simple method,

called the "tuning-slide method" was tried. It i s a method for d i rec t

measurement of the length correct ion without computation. The cylind-

r ica l head-joint was provided with a tuning slide and was adjusted s o that

each corresponding resonance frequency was the same fo r both flutes.

Thin walled tubes with only 0.5 m m thickness were employed i n o rde r to

minimize the influence of the cavity i n the tuning slide. The length cor -

rection AQ could now be measured by a ru l e r a t the tuning slide. The

resul ts f r o m these measurements showed the same values a s shown in

Fig. III-&4. The calculated values f rom Eq. (1 1) a r e drawn by a broken

line in Fig. 111-A-4. C. The agreement between calculated and measured

values is reasonably good. The Rayleigh Eq, (11) i s only valid i f no d is -

continuities exist. In this case there is a slight discontinuity of the f i r s t

derivative a t the joint between the conical head-joint and the cylindrical

tube. Another important fac t , not accounted for i n the formula, i s that

the sound source i s placed a t the embouchure. The solution of the Ray-

leigh integral for a m o r e complicated head-joint shape i s facilitated by

means of a computer. A computer program for this purpose has been

written s o that the influence of different perturbations can conveniently

be studied i n advance of measurements .

The effective length of the h/4 flutes with conical head-joint and closed

embouchure ( z = co) cos@k(h/-l) = 0, a r e obtained f rom Fig. 111-A-4.A e o r can be measured in the same way a s by the cylindric head-joint. These

effective lengths a r e shown in Fig. 111-f~2. B. The effective lengths for

the h/2 pipe and ze = m a r e obtained f rom Fig. 111-A-4. B o r f rom the

difference A - B in Fig. 111-A-2.

The embouchure reactance for a flute with a perturbed head-joint can

be obtained f rom Eq. (7) by using the curve C i n Fig. 111-A-2 for the ef-

fective length ,E(h/2)ze = m, Eq. (8) will then be modified and

STL-QPSR 4/1968 21.

The necessity of using an auxiliary cylindrical head-joint fo r measure-

ments on flutes i s a disadvantage. It can, however, be seen f rom Fig.

111-A-4 that the length correct ions fo r h/4 and h/2 pipes (curves A and B)

a r e very nearly the same and S 0 a t high frequencies f 2 1700 Hz. F o r

these measurements a cylindrical tube i s attached to the head-joint.

Measurements a r e made with closed and open embouchure.

At resonance frequencies between 1600 and 2000 Hz the values of

$ ~ ( h / 2 ) and $R(h/4) a r e almost equal. x is obtained f rom Eq. (12). The e

ea r l i e r mentioned important fact that x = kg f now makes it possible to e

calculate a curve corresponding to C in Fig. 111-A-2, z = oo. The length C-B e

correct ion f rom the perturbation i s then AQ = - 2

The cross-sect ional a r e a q i s not constant along the whole length of the

flute and the mean value of q depends on the length of the flute, i. e. on

how many holes o r keys a r e open. The character is t ic impedance zo will

increase when the flute length decreases but only slightly. The correct ion

is in a l l ca ses l e s s than 5 70.

F r o m Eq. (12):

'i

X e eff - -- -tg B = tg(nrr-B)

z 0

The resonance frequencies f rom Ionophone measurements on an actual

flute with open embouchure a r e f and a t blowing fbln In

fln ' fbln then:

" '~n* ' w * R A=- bln

C a n d B =

The frequencies a t blowing a r e then:

X e eff

nn-arctg - z -

STL-QPSR 4/1968 22.

F r o m Ionophone measurements with open embouchure for each s top o r

s tops of in te res t , the actual resonance frequencies may b e obtained f r o m

Eq. (14) i f the reactance function fo r open embouchure and the correspond-

ing effective reactance function a r e known. It i s l ikely that the curve B

i n Fig. 111-A-3 i s representat ive fo r x but fur ther t e s t s will be made e eff to verify this proposition.

Acknowledgments

This work was made possible by support f r o m the Dept. of Speech

Communication, KTH and the Tri-Centennial Fund of the Bank of Sweden,

The author wishes to thank Prof. G. Fant , J. Sundberg, E. Jansson and

T. Murray for valuable suggestions and ass is tance.

References:

(1) Backus, J. : "Vibrations of the Reed and Ai r Column in the Clarinet", J. Acoust. Soc. Am. - 33(196 l ) , pp. 806-809.

(2 ) Backus, J. : "Small-Vibration Theory of the Clarinet", J. Acoust.Soc. Am. - 35 (1963), pp. 305-313.

(3) Benade, A.H.: "On Woodwind Instrument Bores", J.Acoust. Soc.Arn. 31 (1959), pp. 137-146. -

(4) Benade, A. H. : "On the Mathematical Theory of Woodwind F inge r Holes", J. Acoust. Soc. Am. 32 ( 1 9 6 0 ) ~ pp. - 1591-1608.

(5) Benade, A. H. : Horns, Strings and Harmony (Garden City, N. Y. 1960).

(6) Benade, A. H. and French , J. W.: "Analysis of the Flute Head-Joint", J. Acoust. Soc. Am. 37 (1 965), pp. 679-691. -