Flute measurements in a Physics of Music lab

Randy Worland

Citation: Proc. Mtgs. Acoust. 20, 035004 (2013); doi: 10.1121/1.4895818

View online: https://doi.org/10.1121/1.4895818
View Table of Contents: https://asa.scitation.org/toc/pma/20/1
Published by the Acoustical Society of America

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R. Worland

Proceedings of Meetings on Acoustics

Volume 20, 2013 http://acousticalsociety.org/

166th Meeting of the Acoustical Society of America

San Francisco, California
2 - 6 December 2013
Session 2aMU: Musical Acoustics

2aMU2. Flute measurements in a Physics of Music lab

Randy Worland*​

​ *Corresponding author's address: Physics, University of Puget S ound, 1500 N Warner S t., Tacoma, WA 98416, worland@pugetsound.edu
Physics of Music students often benefit from laboratory exercises that make use of real musical instruments, in addition to the more traditional labs that
are designed to illustrate physical principles as simply as possible. These "real instrument" labs help bridge the gap between idealized cases and the musical
instruments the students are familiar with. Modern woodwinds are particularly challenging in this regard due to the complex set of keys, levers, and
mechanical linkages that tend to obscure the underlying acoustics of these instruments. Among the woodwinds, the flute is relatively simple, both
mechanically and acoustically, and thus provides an excellent subject for a real world woodwind study. Laboratory exercises are described in which the
flute's tone hole locations and diameters are measured. The data are analyzed in terms of the acoustics of open cylindrical tubes, revealing the logical order
behind the spacing and use of the holes to play the lowest chromatic octave, as well as the higher registers of the flute. The measurement techniques and
analysis are presented along with the pedagogical role of the experiment.

Published by the Acoustical Society of America through the American Institute of Physics

© 2014 Acoustical Society of America [DOI: 10.1121/1.4895818]

Received 5 Jun 2014; published 5 Sep 2014
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1. Introduction and Pedagogy

A laboratory exercise for a “Physics of Music” course is presented in which students make
physical measurements on real flutes and interpret their data in terms of basic acoustic principles
discussed in class. We have found a high level of student engagement with labs that use real
instruments after the basic principles have been investigated. Previous such lab exercises have
used instruments from the string, brass,1 and percussion2 families, with measurements of fret
locations, bore profile, and Chladni sand mode patterns respectively (see Fig. 1). Engaging
students with hands-on lab exercises using familiar musical instruments is aligned with current
trends in physics education research,3 which show the benefits of an active-learning environment
that maximizes the students’ roles in the learning process.
The pedagogical plan in each case is to start with a discussion of ideal physical systems,
then proceed to measuring actual instruments, which are usually very different from simple
models of the ideal system. Finally, we return to the fundamental principles which may be
applied to a real instrument, at least to a good degree of approximation. In this way we attempt to
help non-science students make the transition from an idealized model to a real instrument, and
to better understand the role of models and measurements in the scientific process.

Figure 1. Physics of Music lab exercises for non-science majors, showing the use of string, brass, and
percussion instruments.

Another advantage of these labs is that they allow students to hold and manipulate
instruments, and to see aspects of construction and mechanisms that are not discussed as part of
the acoustic principles. Many students have never actually touched the particular instrument
being studied, and the need to make physical measurements focuses their attention closely on it.
For these hands-on instrument labs it is useful to have several instruments available that can be
handled without fear of being damaged. I have obtained many inexpensive instruments (brass,
woodwinds, etc.) from the Goodwill online auction site4 for this purpose. The price is typically
under $50 for instruments that are not always in playable condition, but are more than adequate
for our measurement purposes.
In looking to develop a hands-on lab for the woodwind family one is struck by the
mechanical complexity of these instruments, which can obscure their underlying physical
principles (see Fig. 2). Many woodwinds feature changing bore profiles, varying tone hole

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R. Worland

diameters, curved paths, complex mechanical linkages, and mouthpieces with either single or
double reeds.

Figure 2. Woodwind instruments (bassoon, clarinet, saxophone,

and English horn) showing their high degree of mechanical complexity.

For our lab, the flute (see Fig. 3) offers several advantages over other common woodwinds, as it
is relatively simple, both mechanically and acoustically. It is very nearly an open, straight
cylinder (close to the simple tubes modeled in class). Furthermore, it has no reed and no bell,
both of which complicate the acoustics of an instrument. Finally, flutes are small, familiar, and
easy to obtain. With this in mind, the basic idea of our lab exercise is to measure the tone hole
positions and diameters along the length of the tube and apply fundamental acoustic principles in
an attempt to understand the notes played in the instrument’s lowest octave.

Figure 3. The modern flute (also called the Western concert flute or transverse flute).

2. Flute Background and Acoustics

The modern flute is based on the design of Theobold Boehm (1794-1881), a Munich flautist,
composer, and goldsmith, who designed and built flutes between 1832 and 1847 (see Fig. 4). His
1847 design is essentially the modern flute. Prior to Boehm, most tone holes on the flute were
designed to be covered directly by fingers, thus limiting their size, placement, and number (see
Fig. 5). Boehm’s revolutionary design was based on his acoustic experiments, which were

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devised to improve both the tone and intonation over the full range of the instrument. He chose a
cylindrical tube with large tone holes located for optimal intonation. The keys, mechanical
linkages, and fingering conventions were then designed to accommodate the acoustics.

Figure 4. Theobald Boehm (age 60).5

Figure 5. Top: A pre-Boehm flute by B. Pentenrieder (Munich, circa 1840).

Bottom: Boehm cylindrical flute Number 1 (1847).6

Boehm’s work is described in his book, “The Flute and Flute-Playing”5 in which he
describes his experimental process for developing the tone hole sizes, locations, and mechanisms
for fingering notes on the flute (see Fig. 6).

Figure 6. Images from Theobald Boehm’s book on flute design and performance.5

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2.1 Modern Concert Flute

The modern concert flute is cylindrical, with an inner diameter of 19 mm, except for a slight
taper in the head joint. The flute consists of three removable sections; the head joint, body joint,
and foot joint (see Fig. 7). The head joint contains the embouchure hole, across which the
musician blows, and an adjustable cork that defines the end of the air column. It also tapers
slightly from an inner diameter of 19 mm to about 17 mm. The body joint contains ten large
diameter tone holes as well as three smaller diameter holes toward the head joint end. The
standard “C-foot” joint contains three large tone holes, although higher end flute models often
have a “B-foot,” which contains four holes and extends the lower range by a semitone to B3. The
tone holes are covered by keys with pads in the flutes we use. In “open-hole” flutes five of the
holes have open rings that are sealed by the finger tips (see Fig. 7). The mechanism of keys,
levers, rods, and springs enables fingering the desired notes, using all fingers except the right
thumb. The standard playing range is from C4 (middle C, at 261.6 Hz) to C7, although this range
can be extended at the upper end.

(a)

(b) (c) (d)

Figure 7. (a) Flute with head, middle, and foot joints indicated (l-r), (b) C-foot (upper) and B-foot with
extra tone hole (lower), (c) open-hole keys (upper) vs. standard keys (lower), (d) range of the flute.

2.2 Flute Acoustics

Despite its mechanical complexity, the flute can be used to illustrate some basic acoustic
principles. It can be treated as an open cylinder, although the end correction at the embouchure
hole is not simple, and the effective acoustic length Leff is initially unknown. The flute produces
integer multiple harmonics, as predicted by simple tube theory,7 and therefore can be
“overblown” at an octave, twelfth, two octaves, etc. (see Fig. 8). As tone holes are successively
opened from the foot end, the fundamental frequencies should be inversely proportional to Leff,
where the effective length depends on both the location and diameter of the open holes. A small
diameter hole (relative to the tube diameter) shortens the effective acoustic length very little,
while a large diameter tone hole essentially shortens the tube to the location of the hole (see Fig.
9).8

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Figure 8. The first three standing wave patterns for pressure in an open cylinder.

Figure 9. Diagram from Benade8 showing the effective acoustic

length of a cylinder with a single tone hole of varying diameter.

The effective acoustic length Leff is the length of an ideal cylinder defined such that the
relation shown in Eq. (1) holds for the fundamental frequency,

, (1)

where c is the speed of sound in air, taken to be 345 m/s. For equal tempered chromatic
semitones, the frequencies go as fo * 2 n /12, where fo is the frequency with all holes covered and n
represents the number of open holes from the far end.9 Thus, we expect the (large) holes to be
located according to

Ln = Lo / 2 n / 12 , (2)

Where Lo is the effective tube length with all holes closed. In the lowest octave, tone holes
below the highest open hole have little effect and can often be either open or closed without

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changing the note perceptibly. The situation becomes more complex in the higher registers where
certain holes may be opened to serve as vent holes (there is no dedicated register key on the
flute) and “cross-fingerings” may be employed. These details are omitted from the current
exercises.

3. Student Laboratory Exercise

3.1 Pre-Lab Exercise
Having studied the behavior of ideal cylindrical tubes, students are asked to prepare for the flute
lab by designing a flute based on the simplest possible principles. They are to locate large
diameter holes such that one chromatic octave can be played, starting at C4. Taking the speed of
sound to be 345 m/s, this implies an ideal tube length of Lo = 660 mm, with hole positions
determined from Eq. (2) (see Table I). Students then prepare a 1:1 scale drawing of their design,
using the flute’s 19 mm inner diameter. The drawing makes it visually apparent that the holes get
closer together when ascending, as was observed for fret spacing in an earlier lab in which fret
positions on guitars were measured. Furthermore, as Boehm knew, they can see that the tone
holes are too numerous, and too far apart to be covered by fingers alone. At this point in the
investigation it remains to be seen how close this model will be to a real flute.

Table I. Theoretical tone hole locations in an open cylinder designed to produce a chromatic
octave ranging from C4 to C5. Tabulated distances are calculated using Eq. (2), with Lo = 660 mm.

n 0 1 2 3 4 5 6 7 8 9 10 11 12
L (mm) 660 623 588 555 524 494 467 440 416 392 370 350 330
Pitch C4 C #4 D4 D #4 E4 F4 F #4 G4 G#4 A4 A #4 B4 C5

3.2 Initial Flute Observations

The next step is to have students observe and handle the real flutes (we usually have one flute for
each lab group of four students). It is important that everyone gets their hands on the instruments,
as many students will have never actually touched a flute or looked closely at it. Going from the
ideal abstract classroom model to the real system is not easy for non-science students. This is
especially true due to the fact that the actual instrument contains many mechanical and
ergonomically inspired parts in addition to the acoustically required features.
During this qualitative observation stage students feel the weight of the instrument, the
tension in the springs, etc. They count the number of holes and notice that some are initially open
while others are closed. They also see that some finger keys operate more than one tone hole and
that the linkages are a bit mysterious at first. Some tone holes are operated directly with a finger
pushing on a key, while others are controlled at a distance. Careful observation reveals that the
tone holes are built up with flat-topped chimneys, so that flat pads can cover them. The height

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that the pads open above the tone holes can also be seen. All but the smallest pads contain
circular metal “resonators” (see Fig. 10). In the head joint, students observe the shapes of the
embouchure hole and lip plate. The cork can also be viewed (easiest if a flashlight as used)
behind the embouchure hole. Tiny screws for assembling and adjusting the mechanisms are also
visible. Several of the component pieces that students observe are shown in Figure 10. Given this
level of mechanical complexity, is it possible that the tone holes are simply opened one at a time
from the bottom to produce a chromatic scale? (Yes!)

Figure 10. Photos showing several features of the flutes observed by students.

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3.3 Measurements
Simple dimensional measurements are made next, including tube length and inner diameter, as
well as tone hole locations and diameters. The first issue is what to measure as the total tube
length. The tube is open at the foot end and the embouchure hole. However, the effective
acoustic length depends on the details of the head joint (embouchure hole, player’s mouth
position, cork position, taper) as well as the tone holes (even when closed), and the end
correction at the foot end. Most of these effects go beyond the level of our course, but fortunately
the full physical length of the flute turns out to be very nearly equal to the 660 mm predicted for
the effective length. This fortuitous result means that students can measure all distances from the
closed end (far left in Fig. 11) of the instrument, rather than from the embouchure hole, or the
internal cork. Students are told this, and then encouraged to simply measure from the closed end
of the flute to the center of each tone hole.
Measuring the tone hole locations and diameters can be challenging due to the
mechanisms that surround most of the holes. One fairly simple technique we have used is to
trace the outline of the flute on a piece of graph paper (we use old chart recorder rolls that are no
longer needed) as shown in Figure 11. Students then make marks on the paper corresponding to
the center of each hole. With the flute removed these marks are extended into lines that can
easily be measured as shown in Figure 11. This method yields measurements good to within
about one millimeter, which is sufficient for our purposes. Tone hole diameters are measured by
placing paper under each pad and pressing down firmly. The rim of the chimney leaves a circular
mark which is then easily measured with a ruler (see Fig. 12).

Figure 11. The flute is placed on chart recorder paper for marking and measuring the locations of the tone
holes.

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Figure 12. Keys are pressed down firmly on paper, leaving a circular
imprint that can then be used to measure the diameter of the tone hole.

The data are collected in a table displaying tone hole number, distance, and diameter (see
Table II). There are 15 tone holes listed in the table, although there are actually 16 holes in the
flute (in addition to the embouchure hole). Although not initially obvious due to the placement of
tone holes at different angles around the tube, there are two holes at the position of hole #8, both
of the same diameter. This is a product of the fingering mechanism, with one hole initially open
and one closed. Both holes have the same acoustic effect, however, and are only entered once.

Table II. Measured values of tone hole distances and diameters.

Tone Hole n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Distance (mm) 660 616 582 552 517 487 461 436 414 391 367 346 329 296 278 263
Hole dia. (mm) 16 16 16 16 15 15 15 14 14 14 14 13 7 7 7

3.4 Data Analysis

Students recognize from their data that the first 12 holes are fairly large (13-16 mm diameter)
compared to the 19 mm tube diameter. They also observe that the 12th hole is very nearly
halfway up the tube, as predicted for a note one octave above the lowest note. In fact, the first
twelve measured hole locations compare very favorably with the initial model they developed in
the pre-lab exercise (compare Tables I and II).
Students then plot the data using a spreadsheet (e.g. Excel) to visually see the comparison
between measurement and the simple model of hole placement. The first plot (see Fig. 13) shows
length vs. hole number, for the first twelve holes. Visually, the agreement is excellent. This plot
also looks very much like the graph of fret position vs. fret number that they prepared in their
earlier lab on guitar measurements.
Depending on the mathematical level of the class, the data can also be plotted on a semi-
log plot displaying Log (Ln / Lo) vs. hole number n, as seen in Figure 14. From Eq. (2) we can
derive,

Log (Ln / Lo) = – n Log (2) / 12 . (3)

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Figure 13. A plot showing distance to tone hole center vs. hole number. Measured data
(blue diamonds) for the first twelve holes are compared to the theoretical prediction (solid red line).

This model predicts a linear relationship with a slope of – Log (2) / 12 – 0.0251, in excellent
agreement with the trend line shown in Figure 14. (Note that if holes 13-15 are included, they
will fall below the trend line seen in Figure 14 due to their significantly smaller diameters.)

Figure 14. Semi-log plot of the measured tone hole distances (blue diamonds) used in Figure 13 vs.
hole number. Solid is a best-fit line, with a slope matching the theoretical prediction shown in Eq. (3).

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Thus it would appear that a chromatic octave can be played by simply opening the tone
holes in sequence. But is that the case in practice? Most fingering charts show which fingers to
depress for each note, but do not show which holes are open; even flute players may not know
which holes are open when playing a particular note. Due to the mechanism, it is far from
obvious how the fingerings relate to the tone holes.
A valuable resource at this point is the University of New South Wales (UNSW) Music
Acoustics site.10 In addition to providing an excellent discussion of flute acoustics, the site
contains fingering diagrams for every note on several types of flutes. These diagrams show
which holes are open and closed, in addition to which keys are depressed by the fingers (see Fig.
15). Students use these diagrams to verify that the lowest chromatic octave does indeed
correspond to opening the tone holes consecutively from the foot end of the flute. Diagrams for
the first few notes are shown in Figure 16.

Figure 15. Sample page from the UNSW web site10 showing flute fingering and tone hole diagram.

Figure 16. Diagrams taken from the UNSW Music Acoustics web site10 showing the sequence of open
tone holes for chromatically ascending notes from C4 to F4. Red arrows indicate highest open tone hole.

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3.5 Additional Resource

After the lab exercise is completed, it is useful to play a short video showing how flutes are
manufactured11 (see Fig. 17). Having worked closely with the instruments during the lab period,
students can now relate to many of the construction details that are not directly based on the
acoustic principles. Ultimately it is hoped that students recognize that the real instrument is
based on a combination of acoustic, ergonomic, and mechanical requirements and compromises.

Figure 17. Screen shot from the YouTube “How It’s Made” video11 on flutes.

4. Summary
The lab exercise presented here encourages students to look closely at the various parts that
comprise the flute, and to make simple dimensional measurements of the acoustically relevant
locations and diameters of the tone holes. The analysis helps make sense of the complicated
mechanisms and fingerings with the use of a small number of acoustic principles. In this way we
hope that students can bridge the gap between the physicist’s ideal cylinder model and a real
woodwind instrument. The exercise engages the Physics of Music students – and even flautists
will notice new elements of their instrument.

References
[1] R. Worland, “Measuring brass instruments: A ‘Physics of Music’ lab exercise,”
J. Acoust. Soc. Am. 132, 1958 (2012).

[2] R. Worland, “Chladni patterns on drumheads: A ‘Physics of Music’ experiment,”

Phys. Teach. 49, 24-27 (2011).

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R. Worland

[3] T.B. Neilsen, W.J. Strong, B.E. Anderson, K.L. Gee, S.D. Sommerfeldt, and
T.W. Leishman, “Creating an active-learning environment in an introductory
acoustics course,” J. Acoust. Soc. Am. 131, 2500-2509 (2012).

[4] Goodwill online auction site: http://www.shopgoodwill.com/

[5] Theobald Boehm, The Flute and Flute Playing (1871), translated by Dayton C. Miller
(1922), (Dover, 2011).

[6] Images from Rick Wilson's Historical Flutes Page: http://www.oldflutes.com/

[7] T.D. Rossing, F.R. Moore, and P.A. Wheeler, The Science of Sound, 3rd ed. (Addison
Wesley, San Francisco, 2002), Section 9.5.

[8] A.H. Benade, “Physics of wood winds,” Sci. Am. 203(4):144-153 (October 1960).

[9] T.D. Rossing, F.R. Moore, and P.A. Wheeler, The Science of Sound, 3rd ed. (Addison
Wesley, San Francisco, 2002), Section 4.5.

[10] University of New South Wales music acoustics site:

http://www.phys.unsw.edu.au/music/flute/

[11] “How It’s Made” video on flute manufacturing (YouTube):

http://www.youtube.com/watch?v=DHSu0trGkRg

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