Vocal tract resonances and the sound of the Australian didjeridu

(yidaki). III. Determinants of playing quality

John Smith,a兲 Guillaume Rey, and Paul Dickens
School of Physics, University of New South Wales, Sydney NSW 2052, Australia

Neville Fletcher
School of Physics, University of New South Wales, Sydney NSW 2052, Australia
and R.S.Phys.S.E., Australian National University, Canberra, ACT 0200, Australia

Lloyd Hollenberg
School of Physics, Melbourne University, Melbourne, Vic 3010, Australia

Joe Wolfe
School of Physics, University of New South Wales, Sydney NSW 2052, Australia
共Received 24 July 2006; revised 4 October 2006; accepted 9 October 2006兲
Traditional didjeridus have a broad range of bore geometries with many details not immediately
apparent to a player, and are therefore suitable for examining the relationship between perceived
quality and physical properties. Seven experienced players assessed the overall playing quality of 38
didjeridus that spanned a wide range of quality, pitch, and geometry, as well as 11 plastic cylindrical
pipes. The ranking of these instruments was correlated with detailed measurements of their acoustic
input impedance spectra. Most significantly, the ranked quality of a didjeridu was found to be
negatively correlated with the magnitude of its acoustic input impedance, particularly in the
frequency range from 1 to 2 kHz. This is in accord with the fact that maxima in the impedance of
the player’s vocal tract can inhibit acoustic flow, and consequently sound production, once the
magnitude of these impedance maxima becomes comparable with or greater than those of the
instrument. This produces the varying spectral peaks or formants in the sound envelope that
characterize this instrument. Thus an instrument with low impedance and relatively weak impedance
maxima in this frequency range would allow players greater control of the formants in the output
sound and thus lead to a higher perceived playing quality. © 2007 Acoustical Society of America.
关DOI: 10.1121/1.2384849兴
PACS number共s兲: 43.75.Fg 关DD兴 Pages: 547–558

I. INTRODUCTION interesting rhythm, the chief musical interest in performance

thus comes not from pitch variation, but from the contrasting
The didjeridu 共or didgeridoo兲 is an unusual and ancient and varying timbres that can be produced by either deliberate
musical instrument originally played in parts of Northern
movement of the player’s tongue and/or by movements as-
Australia, where the Yolngu people call it the yidaki or
sociated with “circular breathing.” There are also
yiraki. In didjeridu playing, the configuration of the player’s
vocalizations—sounds that are introduced when the player
vocal tract has a spectacular effect on the timbre, for reasons
deliberately sings simultaneously at a frequency different
that have been explained elsewhere 共Tarnopolsky et al.,
from that of the lips.
2005, 2006; Fletcher et al., 2006兲. In performance, the player
Acoustical aspects of the didjeridu previously studied
共usually a man兲 places his lips at the narrower end of the
include observations of the lip motion 共Wiggins, 1988; Tar-
instrument and uses the technique of “circular breathing”
that allows him to play without stopping. This involves fill- nopolsky et al., 2006兲, numerical modelling of the lip motion
ing his cheeks with air, then inhaling quickly through the 共Hollenberg, 2000兲, the linear acoustics of the instrument
nose 共and bypassing the mouth at the soft palate兲 to refill the 共Amir and Alon, 2001; Amir, 2003, 2004; Caussé et al.,
lungs with air. The contrast between the sounds produced 2004兲, the lip-air column interaction 共Fletcher, 1983; 1996兲
during inhalatory and exhalatory play is often used to estab- and the interactions among the vocal tract, the vocal folds,
lish a rhythm, which is believed to give the instrument its the lip motion, and the didjeridu air column 共Tarnopolsky et
onomatopoeic Western name. Unusual among wind instru- al., 2005, 2006; Fletcher et al., 2006兲.
ments, it usually only plays a single note at the frequency of Another unusual feature of the instrument is that its bore
its lowest resonance, although overblowing to produce notes is largely constructed by termites that eat the interiors of
at the second or very occasionally higher resonances is small eucalypt trees. This produces an irregular and some-
sometimes used to produce musical accents. Apart from the what flared bore. Suitable tree trunks are selected by listen-
ing to the sound made by tapping them, then cut to yield a
length of typically 1.2– 1.5 m 共different cultural groups have
a兲
Electronic mail: john.smith@unsw.edu.au different styles兲. The central bore is then cleared of debris

J. Acoust. Soc. Am. 121 共1兲, January 2007 0001-4966/2007/121共1兲/547/12/$23.00 © 2007 Acoustical Society of America 547
and a ring of beeswax placed on the smaller end of the in-
strument to allow a better and more comfortable seal around
the player’s mouth.
In the first two papers in this series 共Tarnopolsky et al.,
2006; Fletcher et al., 2006兲, it was shown how and why
changes in the geometry of a player’s vocal tract may have a
large effect on the timbre of the sound produced by the in-
strument. But what is the effect of the geometry of the bore?
More particularly, what makes a didjeridu a good or a bad
instrument in the judgment of performers?
One possibility is to examine the sound produced by a
didjeridu when played 共e.g. Amir 2003, 2004; Caussé et al.
2004兲: a good instrument should make a good sound. How-
ever, this approach has the disadvantage that the quality of
the didjeridu/player combination is being studied rather than
the intrinsic quality of the didjeridu. 共A superior player might
be capable of producing a superior sound from an inferior
instrument, whereas an inferior player might only produce an
inferior sound from a superior instrument.兲 Further, the very
large variation in timbre that an experienced player may pro-
FIG. 1. 共Color online兲 Photographs 共to scale兲 illustrating the difficulty of
duce on one instrument, using his vocal tract, could easily identifying the quality of a didjeridu from its external appearance and ge-
mask differences among instruments. ometry. The instruments labelled A–H were classified by the Didjshop as
The above problems can be avoided, in principle, by being high concert, medium-high concert, medium concert, medium-low
measuring the intrinsic acoustical properties of instruments, concert, low concert, first class, first-second class, and second class, respec-
tively. Their ranked qualities determined in this project were 5, 6, 7, 13, 21,
without players, and comparing these with subjective rank- 35, 37, and 26 respectively. As players know, it is difficult to judge a good
ings of quality provided by players. However, studies aimed instrument by inspection. This composite image was composed from images
at correlating the intrinsic acoustical properties of orchestral of the individual instruments that were adjusted to correct relative scale.
instruments with their quality as judged by players are
known to be difficult for a variety of reasons 共e.g., Pratt and
Bowsher, 1978, 1979兲. The difficulties may be less with the might affect a player’s assessment thus introduces
didjeridu for the following reasons: constraints upon performance, e.g., not being allowed
to hold the instrument normally, or playing in very
共i兲 Orchestral instruments have usually undergone a long dark rooms, or even being blindfolded 共e.g. Bertsch,
evolutionary period resulting in designs and manufac- 2003; Inta et al., 2005兲. The situation is quite different
turing processes that can produce instruments that are for the didjeridu; there is no simple, obvious correla-
close to optimal. Consequently the differences among tion between its appearance and its quality 共see Fig. 1兲
orchestral instruments of the same type are usually and consequently players can handle and play the in-
relatively subtle 共e.g., Widholm et al., 2001兲. Didjeri- strument normally. Furthermore, there are no manu-
dus are quite different. Although considerable skill is facturers’ insignia on a new instrument and the exter-
involved in selecting a trunk that might produce a nal shape of the instrument provides little information
good didjeridu, and in reaming out the bore appropri- on the critical bore profile and finish. Indeed, the de-
ately, the quality is virtually unknown until the did- tailed bore profiles are not known even to the experi-
jeridu is actually played. Because the shape of the menters. Perhaps the only information that can be
bore is largely determined by termites and the shape gleaned from a casual inspection is the size and de-
of the tree trunk, the variation in bore, and conse- gree of flaring in the bore at the end of the instrument.
quently in quality, is very great among didjeridus; far However one possible source of information might be
more than would normally be encountered with or- the tribal markings on some authentic instruments be-
chestral instruments. These extreme variations could cause some cultural groups and/or makers are known
help in deciding which acoustical properties are im- to prefer different bore profiles 共Knopoff, 1997兲.
portant to players 共e.g., Amir, 2004; Caussé et al., 共iii兲 Orchestral wind instruments usually play many differ-
2004兲. ent notes at frequencies that are determined by the
共ii兲 Much can be learned about an orchestral instrument extrema in the acoustic input impedance of the instru-
from its appearance. For example, a simple visual in- ment. 共The acoustic impedance is the complex,
spection of a brass instrument can provide informa- frequency-dependent ratio of acoustic pressure to vol-
tion on crucial parameters such as the bore dimen- ume flow.兲 The multiple pitches introduce a wide
sions and profile. Furthermore, the name of the range of desirable features that should be assessed for
instrument maker, the materials used, the degree of each note and combinations of notes including the
finish, and evidence of previous use can also influence intonation, stability, timbre, dynamic range, ease of
a player’s assessment. The avoidance of factors that transition, etc. 共e.g., Plitnik and Lawson, 1999兲. In

548 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality
 contrast, the didjeridu is usually played at only one sold is then assigned to one of five major categories accord-
pitch and at a limited range of volumes. Although the ing to its overall sound and performance quality 共see below兲.
didjeridu introduces a new set of qualities to be as- The 38 traditional instruments whose acoustical proper-
sessed that are associated with the influence of the ties were measured in the current study include 33 instru-
vocal tract and additional use of the vocal folds, the ments from a collection that is not for sale, and therefore in
qualities that need to be assessed are still rather fewer principle could be used in future studies. They were selected
than for orchestral instruments. This greatly reduces to cover a wide range of assessed qualities and nominal
the testing required for each instrument. pitches ranging from A1 共55 Hz兲 to G2 共98 Hz兲. There were
共iv兲 The effectiveness and ease of operation of the keys or five instruments at each of F2, E2, and C2; 4 instruments at
valves on a traditional wind instrument, and even their D#2 and D2; 3 instruments at G2, C#2, B1, and A#1; 2
“feel” under the player’s fingers, may influence judg- instruments at A1.
ments about quality. Didjeridus have no keys or me- The internal diameter at the mouth end of the instru-
chanical parts. ment, din, had been measured previously for each instrument
prior to the application of the beeswax. The maximum and
Any decision on the quality of a musical instrument minimum internal diameters of the often asymmetric bore at
should be made by experienced players. In this paper the the terminal end of the instrument were measured using cali-
quality of 38 naturally made didjeridus, spanning a wide pers, and the effective output diameter, dout, taken as their
range of quality and geometry, and a typical range of pitch, geometric mean.
was assessed by two very experienced players involved in The perceived quality of a didjeridu might depend upon
selling didjeridus 共from a wholesale and mail order enter- its playing pitch. This would be difficult to investigate using
prise called “the Didjshop”兲 and also a panel of five experi- traditional instruments because of the wide variations in bore
enced players with a range of styles from traditional to con- profile between instruments. Consequently a study was also
temporary. To determine the possible influence of pitch on made on 9 PVC tubes with the same diameter 共38 mm兲, but
assessed quality, a control study was also made on nine did- different lengths, that covered the pitch range from A1 to F2.
jeridus made from plastic pipe, which differed only in their A study was also made on a set of 3 PVC tubes with different
length. The subjective assessments of players were then cor- diameters 共25, 38, and 50 mm兲, but the same pitch—C2
related with detailed measurements of the acoustic input im- 共nominally 65.4 Hz兲.
pedance spectra and geometrical parameters of the 38 did-
jeridus to determine which acoustical properties are B. The players
important to players.
In addition to the two experienced members at the Didj-
shop, five other experienced players volunteered for this
II. MATERIALS AND METHODS study to assess the sound quality of each didjeridu on each of
A. The didjeridus seven criteria. They then also ranked each didjeridu accord-
ing to its overall quality. These five players had an average of
The 38 traditional didjeridus studied were all from the 11 years playing experience, in styles ranging from tradi-
collection of the “Didjshop” 共www.didjshop.com兲. This is a tional to contemporary.
commercial enterprise located near Kuranda in North Queen-
sland, approximately 2000 km to the north of Brisbane, the
C. The assessed quality of each didjeridu
state capital. It handles thousands of didjeridus each year;
each one has been harvested and crafted by a local indig- The classification procedure used by the Didjshop in-
enous craftsperson from a termite-eaten Eucalypt. This re- volves the staff assessing the sound quality of each instru-
sults in instruments with a very wide range of geometries ment by awarding a score out of five in each of the following
and consequently there is a large variation in quality among seven categories; backpressure, clarity, resonance, loudness,
these instruments. During the last 13 years the Didjshop has overtones, vocals, and speed. Each instrument is then as-
developed a standardised procedure to classify these instru- sessed for its overall quality by awarding a score out of ten.
ments and maintains a large database including these classi- This classification system has been used by the staff for
fications and other information. 13 years to describe thousands of instruments. For each in-
Each didjeridu used in the current study has undergone strument, these and other data, a brief sound file and a pho-
the standard preparation of instruments handled by the enter- tograph are recorded. Because this database was available,
prise. Each is subjected to a proprietary treatment to stabilize and because the classifications were familiar to the standard
the wood against cracking, and then varnished or painted if panel players, we retain this classification system for the cur-
required. 共For painting, the instruments are returned to local rent study. However, in this paper we concentrate on the
indigenous artists.兲 Each is then played and assigned a nomi- overall quality awarded.
nal playing pitch 共henceforth denoted as the pitch兲. Two ex- The five major categories used by the Didjshop are
perienced musicians then play the instrument and assess its given names chosen for commercial reasons. They are 共with
sound quality in one of eight categories. They also compare the associated scores兲:
it with a permanent set of reference instruments consisting, High concert class. These didjeridus represent the best
for each note over the normal pitch range, of several instru- 1%–2% of the instruments sold/handled by the Didjshop.
ments with different degrees of quality. The didjeridu to be 共They are judged to score 10 out of 10 for overall quality兲.

J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality 549
 Medium concert class. These didjeridus will have rated used in conjunction with a deltatron preamplifier 共Bruel &
4 out of 5 in most sound attributes and should not score less Kjaer 2693 OS2兲. The stimulus waveform contained 2194
than 4 in either clarity or resonance 共overall quality score 8 harmonics with a fundamental frequency of 1.346 Hz span-
of 10兲. ning the frequency range from 50 to 3000 Hz.
Low concert class. These didjeridus are guaranteed to
have a better sound quality than that of a PVC pipe 共overall III. RESULTS AND DISCUSSION
quality score 6 out of 10兲. A. Consistency of the subjective assessments
First class. These are of sound quality similar to or bet- of overall quality
ter than that of a PVC pipe 共overall quality score 4 out of
10兲. Musicians have different musical backgrounds, prefer-
Second class. These instruments display a major short- ences and individual styles. This observation is at least as
coming in one or more categories, usually clarity. It is sug- true of the didjeridu players as of most other instrumental-
gested that these are more suited to being used as artifacts for ists; indeed the variation may be greater for the didjeridu
display rather than as musical instruments 共overall quality because of the absence of a formal academic tuition system
score 2 out of 10兲. with examinations. Consequently, a player’s assessment of
Instruments are also often assigned to intermediate the quality of an instrument could be influenced by its suit-
classes resulting in nine possible classes. Examples of the ability to his particular style of playing. It is thus important
sound produced by a representative instrument belonging to to examine how consistently the instruments are judged by
each of the above five main classes when played by the same the players. Because it is difficult to maintain an absolute
musician are available on the web: see http://www.didjshop. scale for these subjective parameters, the relative rankings
com/shop1/soundscapescart.html; The instruments studied will be used for analysis rather than the individual scores.
spanned the whole range from “second” to “high concert.”
1. Ranking of traditional instruments
D. Measurements of the input impedance spectra by the players
of each didjeridu The Kendall coefficient of concordance, W, calculated
An impedance spectrometer described previously 共Smith from the individual ranking of overall quality by the 5 play-
et al., 1997; Epps et al., 1997兲 was adapted for this study, ers for the 38 traditional didjeridus was calculated to be W
using some of the techniques of Gibiat and Laloe 共1990兲 and = 0.335. 关W is a parameter used to rank the concordance of
of Jang and Ih 共1998兲, and used to measure the input imped- judgements; W = 1 if all players ranked the instruments in
ance spectrum of each instrument, i.e., the impedance at the exactly the same order, W = 0 if the rankings were essentially
end that is blown by the player. Briefly, a waveform is syn- at random, see Kendall and Babington Smith 共1939兲.兴 This
thesized from harmonic components, amplified, and input via value corresponds to a ␹2 = 62 and a probability p = 0.006 that
a loudspeaker and impedance matching horn to a waveguide such a result might arise by chance. 共The “probability” p is
of known geometry with three microphones. The spectrom- used hereafter in this sense, subject to the usual assump-
eter is calibrated by connecting this waveguide to two refer- tions.兲
ence impedances. One reference is an acoustic open
circuit—a rigid seal located at the measurement plane. The 2. Ranking of PVC pipes by the players
other reference is an acoustically quasi-infinite cylindrical Nine of the 11 PVC pipes studied had the same internal
pipe whose impedance is assumed to be real and equal to its diameter and were identical except for their length; this
calculated characteristic impedance. The spectrometer is then should only substantially affect the sounding pitch of each
connected to the impedance to be measured, and the un- instrument. Providing the assessed overall quality is indepen-
known impedance spectrum calculated from the pressure dent of pitch, the scores assigned by the five players to these
components measured in the measurement and two calibra- nine instruments would be expected to be very similar, and
tion stages. ranked in no particular order. Indeed the Kendall coefficient
The bore diameter of the didjeridu is larger than that of of concordance, W, showed no significant correlation in the
the instruments we have studied previously 共Smith et al., ranking of overall quality for these instruments. There was
1997; Wolfe et al., 2001兲 and consequently a lower imped- also no significant correlation between overall quality and
ance reference was required. The acoustically semi-infinite pitch. The average score for overall quality for the PVC
cylindrical pipe used for calibration in this study had an in- pipes was 共3.0/ 10兲 which would have placed them in a tra-
ternal diameter of 26.2 mm and a length of 194 m. Because ditional didjeridu category midway between first class
the first curve in the pipe occurs at 40 m from the impedance 共4 / 10兲 and second class 共2 / 10兲. This is consistent with the
spectrometer and because any curves have a radius of 5 m or definitions given in Sec. II C.
greater, the effects of reflections from these curves are ex- Three of the PVC pipes were of the same length, but had
pected to be negligible and this reference impedance should different internal diameters 共25, 38, and 50 mm兲. There was
be purely resistive. a significant difference in their scores and in their rankings.
Measurements were made at a sample rate of 44.1 kHz The value of W = 0.88 indicated a significant correlation be-
using a MOTU 828 audio converter 共nominal 24-bit兲 con- tween the rankings for overall quality, with a probability p
nected via a Firewire serial connection to a MacIntosh G4 = 0.012 that this correlation might arise by chance. Players
iBook. Small electret microphones 共Tandy #33-1052兲 were showed a distinct preference for a large internal diameter

550 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality
with the widest tube ranked as the best and the narrowest
tube ranked as the worst of the 11 PVC pipes.

3. Comparison with the Didjshop rankings

The assessments made by the Didjshop involve musi-
cians with experience of thousands of instruments. It is thus
quite likely that their ranking might differ from those made
by the five volunteer players. However the Spearman rank-
order correlation coefficient, rS 共see Kendall and Babington
Smith, 1939兲 calculated between the average ranking
awarded by the Didjshop and the average ranking of the five
players for the traditional instruments was rS = 0.51 with a
probability p = 5 ⫻ 10−4 that such a non-null result might
arise by chance. There was thus good agreement between the
rankings of the Didjshop and the players.
The “ranked quality” of each instrument used for the
remainder of this paper was derived by averaging the Didj-
shop ranking for overall quality and the mean ranking of the
panel of five players for overall quality. There were only two
ties in the ranked quality of the traditional instruments 共one
pair with rank 17 and one pair with rank 30兲.

B. Does assessed quality depend upon pitch?

A most obvious difference among the 38 traditional did-
jeridus studied is that they play at different pitches. It is thus
important to check first whether the pitch influences the
ranked quality. For example, it is possible that players might
prefer didjeridus in a certain pitch range, perhaps the range
they have most commonly used or heard. However the fol-
lowing results indicate that, over the range studied 共A1–G2兲,
pitch does not influence the ranked quality: FIG. 2. Semilogarithmic plots of the measured acoustic impedance as a
function of frequency for three didjeridus with the highest ranked quality
共i兲 the absence of any significant correlation between 共top兲, an intermediate ranked quality 共middle兲, and worst ranked quality
共bottom兲 from the 38 traditional instruments. The Didjshop classifications of
pitch and ranked quality for the 38 traditional didjeri-
these instruments were high concert, medium-low concert, and first-second
dus; class, respectively. The arrow on the right of each figure indicates the esti-
共ii兲 the absence of any significant correlation between mated value of the characteristic impedance, Z0, for that instrument.
pitch and ranked quality for the 9 PVC pipes;
共iii兲 the absence of any significant concordance in the or- emphasized in performance. 共Throughout this paper, we use
der in which players ranked the quality of the 9 PVC “resonance” to refer to a passive acoustic property of a sys-
pipes of the same diameter with different pitch 共W tem and “formant” to refer to a consequent broad spectral
= 0.11, p = 0.84兲. peak in the envelope of the sound spectrum. Although some
writers use “formant” in both senses, in this paper it is nec-
Consequently, we did not divide the didjeridus into essary to distinguish carefully the two different phenomena.兲
groups with limited pitch range, and instead used all of the In general, the magnitudes of the first few peaks in the
38 traditional didjeridus studied for the correlation studies spectrum decrease monotonically with increasing frequency.
For brevity, we write Zn to denote the magnitude of the nth
C. The acoustic input impedance maximum in Z共f兲 with frequency f n. In only one of the 38
instruments studied was Z2 significantly higher than Z1 共in-
Figure 2 shows the input impedance spectrum Z共f兲 mea-
strument #5兲. The 38 traditional didjeridus studied here are
sured on three of the instruments. Of the 38 studied, they
thus quite different from the instrument E1 studied by Amir
were the instruments with the best, an intermediate, and the
共2004兲 for which Z3 was noticeably higher than Z1 and Z2.
worst overall ranking for quality. All the impedance spectra
showed the expected strong resonance at the fundamental
D. The relationship between quality and the acoustic
frequency that is required for maintenance of the vibration
impedance of the fundamental resonance
regime. However the amplitude of the higher resonance
peaks decreased rapidly with increasing frequency in the The low frequency impedance of the instrument is
Z共f兲 spectra of didjeridus of higher ranked quality, leaving mainly responsible for controlling the vibration regime. Al-
spectra with little structure and only maxima of low magni- though a strong low frequency resonance is obviously essen-
tude in the region of 1 – 2 kHz, which is where formants are tial, the strength of the resonance 共however defined兲 might

J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality 551
 and Z0, higher quality instruments being associated with a
lower Z0. It could be argued that the effective value of Z0
defined above is a slight overestimate because the magnitude
of successive extrema in Z共f兲 will decrease with increasing
frequency. This can be a consequence of increased losses due
to wall damping and/or radiation losses from the open end of
the instrument. The decrease in extrema with increasing fre-
quency could be described in the form
Zi = Z1 exp关− ␭共f i − f 1兲兴, 共1兲
where ␭ was estimated by linear regression to the first 3–6
maxima of log Z共f兲. The regression was always significant
to 5% or better. The effect of a decrease in extrema with
increasing frequency upon Z0 can then be allowed for us-
ing the relationship,
log10共Z0兲 = 关log10共Z1兲 + log10共Zmin兲 − ␭f diff/loge共10兲兴/2,
共2兲
where f diff is the separation in frequency between Z1 and
Zmin. Correction of Z0 for the decrease in extrema with
increasing frequency was found to reduce Z0 by an aver-
age value of 5% and to make no significant difference to
any of the correlations studied.

2. Z1: The maximum impedance of the first resonance

Z1 was estimated from the largest magnitude measured
around the first resonance peak and was found to range from
approximately 5 to 40 MPa s m−3. Figure 3共b兲 shows there is
a negative correlation between ranked quality and Z1, higher
quality instruments being associated with a lower value of
Z1. The impedance spectrum was measured with a frequency
resolution of 1.346 Hz and consequently a value of Z1 esti-
mated from the largest measured value can be an underesti-
mate because the peak value of Z1 is unlikely to be mea-
FIG. 3. The relationships between measured features in the low frequency sured. Consequently Z1 was also estimated by fitting a
acoustic impedance and the ranked quality of the 38 traditional instruments. polynomial function to the Z共f兲 data on each side of the
共a兲 displays Z0, the characteristic impedance, on a logarithmic scale. 共b兲 measured maximum, and then finding their intersection. Z1
displays Z1, the value of the first maximum, on a logarithmic scale. 共c兲
displays the ratio Z1 / Z0. The datum with an anomalously low value in 共a兲
estimated in this fashion was only an average of 12% greater
and 共b兲 was measured on didjeridu #5. This instrument was ranked as #23 than the maximum measured value. This made no significant
and was the only didjeridu where Z1 was significantly less than Z2. The difference to any of the correlations studied.
dashed lines indicate the correlations. The probabilities were p = 5 ⫻ 10−6,
p = 1 ⫻ 10−4, and p = 0.04, respectively, that the correlation might occur by
chance. 3. Z1 / Z0: The ratio of the maximum value of the first
resonance to the characteristic impedance

be less important for the didjeridu than for other instruments Both Z0 and Z1 are negatively correlated with ranked
where precise and rapid changes in pitch are important. quality. The ratio Z1 / Z0 was investigated as a measure of the
“strength” of the resonance. It varied from approximately 20
to 70, and was positively correlated with ranked quality, but
1. Z0: The characteristic impedance to a smaller degree than the negative correlations for Z0 and
An effective value for Z0, deliberately biased by fre- Z1 关see Fig. 3共c兲兴. The correction of Z0 for a decrease in
quencies near that of the playing pitch, was defined as the extrema with increasing frequency, and Z1 for finite fre-
geometric mean of the magnitudes of the first maximum 共Z1兲 quency resolution, only slightly reduced the correlation of
and first minimum 共Zmin兲 in the impedance spectrum. 关On the the ratio Z1 / Z0 with ranked quality.
semilogarithmic plots of Z共f兲 shown in Fig. 2 it falls halfway
between the first maximum and minimum.兴 Z0 was found to 4. Q1: The quality factor of the first resonance
vary from approximately 0.2 to 1 MPa s m−3. This corre- Q1 was examined as a measure of the “sharpness” of the
sponds to the expected values for cylindrical pipes with di- fundamental resonance. A precise estimation required apply-
ameters in the approximate range 25– 50 mm. Figure 3共a兲 ing a polynomial interpolating function to the Z共f兲 data on
shows there is a negative correlation between ranked quality each side of the first measured maximum, and then finding

552 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality
 bore operates as an impedance transformer, efficiently
matching the relatively high acoustic impedance at the lips or
reed to the low impedance of the radiation field.
A simple cylindrical pipe has strong impedance maxima
with frequencies that fall close to the ratio 1:3:5. Thus the
third and fifth harmonics of a note played near the frequency
of the first maximum fall near resonances of the bore and so
are radiated efficiently. When such a pipe is used as did-
jeridu, the spectrum of the sound inside the player’s mouth
has many harmonics, in the ratio 1:2:3 etc., whose ampli-
tudes fall almost monotonically as frequency increases. The
spectrum of radiated sound, on the other hand, has relatively
weak second, fourth, and sixth harmonics 共Tarnopolsky et
FIG. 4. The relationship between the Q1, the measured Q factor of the first al., 2006兲.
impedance maximum, and the ranked quality of the 38 traditional instru- A simple cylindrical pipe is classed as a relatively poor
ments. The dashed line indicates the correlation. The probability was p didjeridu. Could the quality of a noncylindrical traditional
= 0.02 that this correlation might occur by chance.
didjeridu be related to how well the first several harmonics
are efficiently radiated as sound? Efficient radiation of these
their intersection. Values ranged from 12 to 86. There was a low harmonics can be reduced in two distinct ways, or by a
modest positive correlation between Q1 and ranked quality combination of them:
共see Fig. 4兲 with players preferring instruments with a
共i兲 a bore whose impedance maxima were harmonically
sharper peak in Z共f兲.
related, but which decreased rapidly with increasing
frequency, would be less efficient as an impedance
5. f1: The frequency of the first resonance transformer, or
f 1 ranged from approximately 50 to 105 Hz. No signifi- 共ii兲 in a bore whose impedance maxima were not close to
cant correlation was found between ranked quality and f 1 harmonic ratios, the harmonics of a note played near
共data not shown兲. This is consistent with the results found the frequency of the lowest resonance would not fall
with the 9 PVC pipes of different length, but the same diam- at impedance maxima, and so there would be ineffi-
eter. cient radiation, even if impedance maxima did not
The average difference between the frequency of the decrease significantly with increasing frequency.
nominal playing pitch and f 1 was 2%, with the pitch fre-
quency usually slightly above f 1. In only one case was the We investigate the second effect first: examining the cor-
pitch frequency significantly below f 1, this was instrument relation between ranked quality and the resonance frequen-
#5 which was noticeably anomalous in that it was the only cies of the instrument bore.
instrument for which Z1 was significantly lower than Z2. In
this case the pitch frequency 共98 Hz兲 fell between f 1 1. f2 / f1: The frequency ratio of the second to the first
共102 Hz兲 and f 2 / 3 共92 Hz兲. resonance
The relationships between ranked quality and the char-
acteristics of the first resonance show that players strongly The ratio f 2 / f 1 ranged from approximately 2.6 to 3.0.
prefer instruments with an overall low impedance and, to a The third harmonic would be enhanced in the radiated sound
lesser degree, a relatively narrow and strong fundamental of an instrument with a value close to 3 compared to an
resonance. instrument with a value significantly less than 3. No signifi-
cant correlation was found between ranked quality and f 2 / f 1
关see Fig. 5共a兲兴.
E. The relationship between quality and the low
frequency impedance maxima
2. Harmonicity of impedance maxima
It has been suggested previously that the characteristic
sound of a didjeridu is partly explained by the presence of a The degree of inharmonicity 共Fletcher, 2002兲 was exam-
characteristic “hole” in the sound spectrum: in other words, ined by comparing the didjeridu resonance frequencies to
the first few overtones above the fundamental are weak those expected for a cylinder. A parameter denoted as the
共Amir, 2003, 2004兲. Could this feature of the spectrum be harmonicity was estimated as the slope obtained by fitting a
related to the perceived quality? In most musical reed and linear relationship between f n and 共2n − 1兲f 1. There was al-
lip-valve instruments, at least in the low region of their pitch ways a good fit with p ⬍ 10−5. Values for the harmonicity
range, several of the first few maxima in the impedance spec- ranged from 0.72 to 1.0 共a value of 1 would be expected for
trum are in nearly harmonic ratios: either 1:2:3, etc. 共oboe, a perfect cylinder without end effects兲. There was no signifi-
saxophone etc.兲, 2:3:4 etc. 共brass instruments兲 or 1:3:5 etc. cant correlation with ranked quality 关see Fig. 5共b兲兴. This is
共clarinet兲. This means that, at least for low notes, harmonics interesting because it has been previously suggested that in-
of the vibration of the reed or lips fall very close to the harmonicity can play an important positive role in determin-
frequencies of impedance maxima. At these frequencies, the ing quality 共Amir, 2004兲.

J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality 553
FIG. 5. The relationships between the frequencies of the maxima in Z共f兲 and FIG. 6. The relationships between the magnitudes of the maxima in Z共f兲
the ranked quality of the 38 traditional instruments. 共a兲 and 共b兲 indicate no and the ranked quality of the 38 traditional instruments. No significant cor-
significant correlation with f 2 / f 1 共the frequency ratio of the second to the relation was found for Z2 / Z1 共the ratio of the magnitudes of the second to
first resonance兲 or the harmonicity 共see text兲, respectively. the first resonance兲 共a兲. No significant correlation was also found for ␭ 共b兲.
关␭ describes how rapidly the magnitude of the low frequency extrema de-
crease with increasing frequency, see Eq. 共1兲兴. The datum with an anoma-
The ratio f 2 / f 1 and the harmonicity can be significantly lously high value for Z2 / Z1 was measured on didjeridu #5, the only didjeridu
where Z1 was significantly less than Z2.
affected by the bore angle. An instrument with a larger bore
angle 共e.g., a truncated cone rather than a cylinder兲 might be
expected to radiate more strongly, which might be a desirable
ments show the entire spectrum of Z共f兲, the only frequencies
characteristic. Although it is difficult to define a useful bore
present when a note is played at f o are integral multiples of
angle for such an irregular shape, f 2 / f 1 is a measure of the
that frequency 共nf o兲.
acoustic property usually associated with bore angle.
Although the resonance frequencies appear to be uncor-
related with ranked quality, it is possible that the magnitudes
of the maxima in Z共f兲 might be important.
5. Dependence on Z„nfo…

3. Z2 / Z1: The ratio of the magnitudes of the second The important parameters in determining the spectrum
to the first resonances of the sound of the instrument playing at f o will be the im-
The ratio Z2 / Z1 ranged from approximately 0.2 to 1.4. pedance at harmonics of the fundamental frequency, i.e.,
No significant correlation was found between ranked quality Z共2f o兲, Z共3f o兲 , . . ., Z共nf o兲, etc. The value of f o will probably
and Z2 / Z1 关see Fig. 6共a兲兴. vary from player to player, and may indeed be deliberately
varied by a player to produce different interactions with Z共f兲.
To allow for this possibility the values of Z共nf o兲 quoted are
4. The decrease in impedance extrema with the maximum values calculated when f o was varied over a
increasing frequency
range of plus or minus one semitone in half cent steps from
Could the ranked quality depend upon how rapidly the the measured value. There was no significant correlation be-
impedance extrema decrease with increasing frequency? Val- tween ranked quality and the individual magnitudes of either
ues for ␭ 关see Eq. 共1兲兴 ranged from 1 to 6 kHz−1. There was Z共nf o兲 or the normalized ratio Z共nf o兲 / Z共f o兲 for n = 2 – 11
no significant correlation between ␭ and ranked quality 关see 共data not shown兲. However there was a good negative corre-
Fig. 6共b兲兴. lation with the sum ⌺Z共nf o兲 for n = 2 – 11; high quality instru-
It could be argued that there is no need to look at the ments being associated with a low value for ⌺Z共nf o兲 共see
overall structure of the impedance spectrum. The shape of Fig. 7兲. The absolute value was important rather than the
the first resonance peak 共and perhaps one or two more兲 may relative value: there was no correlation of ranked quality
influence the vibration regime but, although our measure- with the normalized sum ⌺Z共nf o兲 / Z共f o兲 共data not shown兲.

554 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality
 maximum impedance is apparent; the highest ranking instru-
ments have Z 艋 1 MPa s m−3, whereas the instruments with
lowest ranked quality have Z 艌 3 MPa s m−3. This is consis-
tent with our measurements that found the vocal tract imped-
ance measured just inside the lips needed to be several
MPa s m−3 before strong formants appeared in the output
sound 共Tarnopolsky et al., 2005, 2006兲.
This observation suggests two more specific questions:
共i兲 Does a high quality instrument need to display a low
value of Z共f兲 in a particular narrow frequency range
f r?
共ii兲 Is it the maximum value ZMAX, or the average value
FIG. 7. A semilogarithmic plot showing the relationship between the sum ZAVG, or the rms variation ZRMS, in Z共f兲 over a par-
兺Z共nf o兲 for n = 2 – 11 and the ranked quality. The dashed line indicates the
correlation. There is a probability p = 0.001 that the observed correlation ticular frequency range that is important?
might occur by chance.
These were investigated by calculating the three above
parameters for a number of narrower frequency ranges. A
F. Relationship between quality and mid range
impedance frequency range f r = 200 Hz was selected to ensure that at
least one maximum in Z共f兲 was always present. In each fre-
The vocal tract influences significantly the sound of the quency range and for each instrument, the slope and inter-
didjeridu when its impedance 共measured just inside the play- cept 共both with errors兲 of the correlation of the logarithm of
er’s lips, during performance兲 is comparable to or greater the calculated parameter with ranked quality were calculated
than that of the instrument 共Tarnopolsky et al., 2005, 2006; for each different frequency range using a plot similar to Fig.
Fletcher et al., 2006兲. The main influence of the tract on the 8. The slope and intercept were then used to calculate two
sound appears to be the production of formants in the radi- values of Z predicted by this relationship and their associated
ated sound in the frequency range from about 1 to 2 kHz. errors; one that would be associated with the lowest ranked
Sufficiently high peaks of acoustic impedance in the player’s instrument and the other with the highest ranked instrument.
vocal tract inhibit acoustic airflow entering the instrument at These data are plotted in Fig. 9 on a logarithmic scale as a
those frequencies, by typically 20 dB 共Tarnopolsky et al., function of frequency. For each frequency range and param-
2006, Figs. 4 and 6兲. These peaks in the tract impedance thus eter it is apparent that a decrease in ZMAX, ZAVG, or ZRMS is
produce minima in the sound radiated from the instrument. always associated with an improvement in ranked quality.
The resultant formants in the instrument’s sound 共the fre- Figure 9共a兲 indicates that the improvement is most sensitive
quency bands near tract impedance minima and not thus in- to fractional changes in Zmax around 1200 Hz. An improve-
hibited兲 are important in idiomatic performance. Conse- ment in ranked quality produced by a decrease in ZAVG is
quently a high quality instrument might be expected to have significant over a wider frequency range 共200– 1600 Hz兲
a low acoustic input impedance in this frequency range. 关see Fig. 9共b兲兴. Figure 9共c兲 indicates that the ranked quality
Comparison of the three examples shown in Fig. 2 suggests was most sensitive to fractional changes in ZRMS around
this might be the case. To quantify this hypothesis, Fig. 8 1600 Hz.
shows the relationship between the ranked quality and the The results shown in Fig. 9 indicate that, although a
maximum impedance in the range 1 – 2 kHz. A clear negative decrease in Z is always associated with an improvement in
correlation between the ranked quality and the logarithm of quality, low values in the frequency range 1 – 2 kHz are the
best indicators of quality.

G. Effects of harmonic coincidence

We showed previously 共Tarnopolsky et al., 2006兲 that
“harmonic coincidence” in the 1 – 2 kHz region can contrib-
ute in a modest way to the strength of formants in the output
sound. A set of harmonically-related maxima in the imped-
ance of the instrument in this frequency range might, in prin-
ciple, be excited by the appropriate playing pitch, or
“switched off” by a relatively small change in the playing
pitch. In some of the sound samples studied, this effect con-
tributed to the magnitude of formants, although the effect
was rather smaller than that produced by vocal tract reso-
nances. Because the formants in question typically involve
FIG. 8. A semilogarithmic plot showing the relationship between the maxi-
the 20 to 30th harmonics, changes in the playing frequency
mum value of Z共f兲 in the range 1 – 2 kHz and the ranked quality. The dashed
line indicates the correlation. There is a probability p = 5 ⫻ 10−7 that the of only a couple of percent 共a fraction of a semitone兲 can
observed correlation might occur by chance. produce or remove the harmonic coincidence effect.

J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality 555
 FIG. 10. The relationship between the magnitude of “harmonic coinci-
dence” in the range 1 – 2 kHz and the ranked quality of the 38 traditional
instruments. 共a兲 shows ZHC, the maximum value of the sum of Z共nf o兲 over
FIG. 9. Semilogarithmic plots showing the relationships between param- the frequency range 1 – 2 kHz when f o is varied over a range of plus or
eters associated with the impedance and the ranked quality as a function of minus one semitone in half cent steps. 共b兲 shows Z⌬HC, the difference be-
frequency. Each figure shows the idealized behavior obtained from the cor- tween the maximum and minimum values of the sum of Z共nf o兲 under the
relations between that parameter and ranked quality over a frequency range same conditions. The dashed lines indicate the correlations. The probabili-
of 200 Hz. Values towards the top of each figure indicate the high imped- ties are p = 2 ⫻ 10−4 and p = 0.01, respectively that the correlations might
ances associated with the lowest quality instruments. Values on the lower occur by chance.
part of each figure indicate the low impedances associated with the highest
quality instruments. ZMAX indicates the maximum value, ZAVG indicates the
H. Relationships among geometry, quality, and
average value, and ZRMS indicates the rms variation in Z共f兲 over that fre-
quency range. Error bars indicate the s.e. acoustic impedance
The bore profiles of traditional, termite-eaten didjeridus
are much more complicated than those of any other wind
Could the availability of resonances in the 1 – 2 kHz re- instrument. So it is difficult to estimate acoustic parameters
gion contribute to perceived quality? This was tested by cal- precisely from geometry. However we can look at some fac-
culating ZH defined as the sum of all the values of Z共f兲 tors.
within a certain frequency range that corresponded to har- A player might possibly be influenced by the size and
monics of the fundamental frequency f 0. 关Interpolation be- heft of an instrument. However there was no statistically
tween discrete values of Z共f兲 was used where necessary兴. significant correlation between length and ranked quality.
ZHCmax and ZHCmin were defined as the maximum and mini- Z0 of a cylindrical pipe is inversely proportional to the
mum values of ZH, respectively, when f 0 was varied over a cross-sectional area. The didjeridu bore-profile is much more
frequency range of plus or minus a semitone in half cent complicated, but one might expect that Z0 is proportional to
steps. In all cases there was a clear negative correlation be- d−2 −2
in . We found that Z0 varies with din with a slope
tween ZHC and ranked quality 关see Fig. 10共a兲兴. Of course the 190± 30 MPa s m 共p = 5 ⫻ 10 兲. For a cylindrical tube the
−5 −7

difference in ZH as f o is varied might be the important pa- expected slope is around 500 MPa s m−5.
rameter. Figure 10共b兲 shows Z⌬HC defined as the difference Consequently a correlation between ranked quality and
ZHCmax − ZHCmin. Again there is a clear negative correlation din, the bore diameter just inside the beeswax ring, might be
between Z⌬HC and ranked quality. It thus appears that any expected. Figure 11共a兲 shows in this case, that higher quality
timbral variations produced by harmonic coincidence are instruments tend to have a larger diameter at the mouth end.
more than countermanded by the necessary associated in- A correlation between ranked quality and dout might also
crease in Z共f兲, which in turn reduces the effect of changes in be expected. This is because waves in the bore are more
the vocal tract impedance. readily transmitted to the radiation field if the instrument has

556 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality
 FIG. 12. 共Color online兲 Photographs 共to scale兲 illustrating the seven didjeri-
dus with the highest ranked quality of the 38 studied. Instruments labelled
A–G were ranked from 1 to 7, respectively. All instruments were classified
by the Didjshop as being high concert, medium-high concert, or medium
concert. This composite image was composed from images of the individual
instruments that were adjusted for correct relative scale.

variations in external geometry illustrate just how difficult it

is to judge a good didjeridu by inspection. The players’ judg-
ments of the overall quality of the instruments were signifi-
cantly concordant and allow the following conclusions.
The dependence of ranked quality on a number of pa-
rameters is summarized in Table I. Most of the effects listed,
however, are relatively small. Perhaps the most important
conclusions are these:
共1兲 Ranked quality is not significantly related to pitch, nor to
the frequencies of the resonances.
共2兲 Providing the characteristic input impedance at low fre-
quencies is sufficiently low, players prefer a relatively
FIG. 11. The relationships between instrument geometry and ranked quality narrow and strong 共i.e., high Q兲 fundamental resonance.
for the 38 traditional instruments. 共a兲 shows din, the bore diameter just inside 共3兲 Quality is inversely related to the magnitude of the
the beeswax ring. 共b兲 shows the effective output diameter, dout. 共c兲 shows the
acoustic impedance in all frequency ranges. These re-
overall flare ratio dout / din. The dashed lines indicate the correlations. The
probabilities were p = 2 ⫻ 10−4, 7 ⫻ 10−6, and 0.044, respectively, that the sults show that players strongly prefer instruments with
correlations might occur by chance. 共Values of din were not available for 3 an overall low impedance and the most important deter-
instruments, and dout was not available for one instrument; hence the miss- minant of ranked quality is the impedance in the fre-
ing points.兲
quency range about 1 – 2 kHz, where strong resonances
and high values of impedance are strongly negatively
a large cross-sectional area at the output end. This has two correlated with ranked quality.
effects: it reduces the magnitude of the standing wave and
therefore the acoustic impedance associated with resonances. This last point can be explained in terms of the tech-
As explained above, this is a desirable consequence. Further- niques used for sound production and variation in the instru-
more, the increased radiation increases the loudness, a prop- ment 共Tarnopolsky et al., 2005, 2006; Fletcher et al., 2006兲.
erty that is usually regarded by musicians as desirable in an At the frequencies around the maxima in the impedance
instrument. Figure 11共b兲 shows that this is the case: higher spectra of the player’s vocal tract, acoustic airflow into the
quality instruments tend to have a larger diameter at the far instrument is inhibited and, consequently, output power from
end. There is also a positive correlation between the overall the instrument is weak in these frequency bands. The for-
flare ratio dout / din and ranked quality 关see Fig. 11共c兲兴. mants in the output sound are the remaining bands of
frequencies—those not inhibited by high impedance in the
IV. CONCLUSIONS
tract. This inhibition effect requires that the maxima in the
Figure 12 presents photographs of the seven didjeridus impedance of the vocal tract have magnitudes comparable
with the highest ranked quality of the 38 studied. The large with or greater than the input impedance of the didjeridu.

J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Smith et al.: Acoustic determinants of didjeridu quality 557
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