Musical Mathematics
on the art and science of acoustic instruments
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CHAPTER 8: SIMPLE FLUTES
Part I: Equations for the Placement of Tone Holes on Concert Flutes and Simple Flutes
Flutes, harps, and drums are the oldest musical instruments created by man. Wind instruments are unique, however, because they alone embody the physical dimensions of scales and tunings. In a work entitled The Greek Aulos, Kathleen Schlesinger (1862–1953) attempted to reconstruct Greek music theory by analyzing the remains of ancient reed flutes.[1] It is possible to approach the subject of flute tunings from two different perspectives. We may predict the tuning of an existing instrument by first measuring various flute bore, embouchure hole, and tone hole dimensions, and then substituting these data into a sequence of equations. This method provides convenient solutions when a flute is extremely fragile and cannot be played, or is simply not available for playing. On the other hand, we may realize a given tuning by making a flute according to another sequence of equations. From a mathematical perspective, these two approaches are distinctly different and require separate discussions.
As all experienced flute players know, the intonation of a transverse flute — with either a very simple or a very complex embouchure hole — depends not only on the precision of instrument construction, but also on the performer. The mathematics of flute tubes, embouchure holes, and tone holes does not necessarily produce an accurate sounding instrument. The intonation of a flute is also governed by the strength of the airstream, and by the amount the lips cover the embouchure hole. Because these two variables exist beyond the realm of mathematical predictability, they depend exclusively on the skill of the performer.
Due to the overall complexity of flutes, this chapter is divided into three parts. Part I investigates equations for the placement of tone holes, and Part II, mathematical procedures required to analyze existing flutes. Since Part II is unintelligible without a thorough understanding of Part I, the reader should study this chapter from beginning to end. Finally, Part III gives some suggestions on how to make very inexpensive yet highly accurate simple flutes.
Section 8.1
In writing this chapter, I am indebted to Cornelis J. Nederveen. In his book
entitled Acoustical Aspects of Woodwind Instruments,[2] Nederveen
carefully defines all the mathematical variables needed for a thorough
investigation into woodwind acoustics. Numerous tables of woodwind instrument
dimensions are included at the end of the book. Because a full description of
flute acoustics requires many different variables, this discussion begins with
a list of symbols originally defined by Nederveen. Since most of these symbols
appear only in this chapter, they are not included in the List of Symbols at
the beginning of this book. Furthermore, the List of Flute Symbols below gives
seven symbols that do not appear in Nederveen’s book: effective length
Recall that in Chapter 7, Section 12, we discussed the great convenience of centimeter measurements for cutting resonator lengths. A similar advantage also applies to flute construction. However, because many flute dimensions are much smaller than resonator dimensions, we will use millimeter measurements throughout this chapter. To convert inches to millimeters, multiply inches by 25.40; and to convert millimeters to inches, multiply millimeters by 0.039370.[3]
Finally, we must also define a dimensionally consistent speed of sound ( c ) inside a flute. Heat, water vapor, and carbon dioxide from the player’s breath, as well as mechanical boundary layer effects that are frequency sensitive, all affect the value of c.[4] For simplicity, assume an average temperature of 25°C (or 77°F) inside the flute, and not accounting for changes due to frequency,[5] we will therefore use c = 346,000 mm/s in all future calculations.[6]
Section 8.2
We begin this discussion on the mathematics of flute construction by carefully
distinguishing between a tube’s invisible acoustic length and its
visible geometric length. Figure 8.1 shows that for the fundamental mode
of vibration, the acoustic length of a tube open at both ends spans the
distance of a half-wavelength. Similarly, Chapter 7, Figures 12(a) and 12(b),
show that the acoustic length of a resonator closed at one end spans the
distance of a quarter-wavelength. The latter three figures illustrate that a
tube’s acoustic or effective length (
Until now, we have not considered tubes closed at both ends because they serve
no practical musical purpose. However, such tubes are fundamentally important
because they provide a theoretical basis for flute length calculations. Figure
8.2
shows a tube closed at both ends, also called a closed-closed tube. For a
closed-closed tube, the distinction between an acoustic/effective length and a
geometric/measured length does not exist. Since there are no
openings, pressure waves or displacement waves inside the tube do not reflect
a small distance beyond the tube’s physical length. Instead, all waves reflect
precisely at the tube’s closed ends. Consequently, end corrections are not a
factor,[7] which
means that for a tube closed at both ends,
where L is the tube’s inner measured length.
Nederveen refers to a closed-closed tube as a substitution tube, and
notates its inner geometric length L, or its exact acoustic
half-wavelength, as
A transverse flute consists of a tube open at both ends. The embouchure hole
provides the first opening, and the far end of the flute, or an open tone
hole, provides the second opening. Turn to Figures
8.3(a),
8.3(b),
and 8.3(c),
and observe that each illustration consists of two sections. The top portion
shows a substitution tube closed at both ends, and the bottom portion shows a
flute tube open at both ends. Nederveen’s substitution tube exemplifies a tube
without any openings that generates exactly the same frequency as a
flute tube with two openings.[10]
Stated another way,
This expression states that the acoustic/geometric length of a substitution tube, or of a tube closed at both ends, is by definition identical to the acoustic/effective length of a tube open at both ends.
Due to the reflection of waves a small distance beyond the flute openings, the
geometric/measured length from the embouchure hole center to the open tube end
(
Since the
and (4) determine
In Section
8.3,
a solution to Equation 8.3 for a concert flute with keys predicts
Before we continue, it is important to first clear up a semantic problem. In
Section
8.1,
the List of Symbols defines four corrections:
where
Section 8.3
We turn our attention now to solutions for
Unfortunately, an exact equation for the embouchure correction does not exist.
where
To understand the origins of Equation 8.6, we must first consider the acoustic
admittance (
As the impedance of a duct increases, the admittance decreases; vice versa. A non-complex definition[15] of acoustic impedance is
Therefore, a non-complex definition of acoustic admittance is
where S is the surface area of the duct, in square millimeters; ρ is the density of air, in kilograms per cubic millimeters; c is the speed of sound, in millimeters per second; k is the angular wave number,[16] in radians per millimeter; and κ is the conductivity of the hole, in millimeters. Conductivity is the reciprocal (or opposite) of resistance. We define conductivity as the ratio of a duct’s surface area divided by its effective length[17]
where S is the surface area of a
bore or hole, in square millimeters;
Consider now the locations on a flute where two admittances (of conductivities
κ
) are simultaneously present. Such a place exists at the top end of the
flute. Here the conductivity of the embouchure hole interacts with the
conductivity of the flute bore at the location of the embouchure hole. The
flute provides us with the dimensions of the diameter and the length of the
embouchure hole, and the diameter of the bore. However, because the effective
length of the bore at the location of the embouchure hole remains unknown, we
may solve for
To solve for
Equation 8.10 defines the conductivity of one duct in terms of another duct.
The difference between these two ducts is that the left side of Equation 8.10
represents the actual duct of the embouchure hole, whereas the right side of
Equation 8.10 represents a fictitious duct associated with the bore. This
fictitious duct has the same diameter as the flute tube, and by definition,
the same conductivity as the embouchure hole. Figures
8.3(a),
8.3(b),
and 8.3(c)
show that
According to Arthur H. Benade,[20] a
good estimate for the coefficient of both the inside and outside corrections[21] of
an embouchure hole or tone hole is
A substitution of these values into
Equation 8.6 gives
Four years after the publication of his book, Nederveen concluded that
empirical observations over a wide frequency range indicate
and assign the value
Next, substitute a 19.0 mm diameter bore into Equation 8.5 to obtain the end correction:
Finally, substitute the values for
Section 8.4
On the Armstrong Concert Flute (Serial Number 104-25-29992),
During the early stages of making a simple flute, mark the location of the
embouchure hole 32 mm (1 1/4 in.) from the upper end of the tube; now cut the
tube 10 mm longer than given by Equation 8.3. Drill the embouchure hole, and
place a cork into the tube so that the cork’s inner surface sits 17 mm to the
left of the embouchure hole center. (In concert
flutes, 17.0 mm is the standard distance between the embouchure hole center
and cork.)
Test the frequency of the tube
without tone holes. Gradually shorten the tube until it gives a frequency
about 30 ¢ flat of the fundamental frequency. Stop for now. For the embouchure
hole correction use
Section 8.5
Our next task is to calculate the tone hole correction (
Before proceeding with detailed calculations, let us first distinguish between flutes with key pads, and those without key pads. A key pad has the effect of decreasing the frequency of a tone hole. Consequently, a tone hole with a key pad sits higher on the bore (or nearer the embouchure hole) than a tone hole without a key pad, which sits lower on the bore (or nearer the flute’s open end). For a tone hole with a key pad, the higher position compensates for this flattening effect and, thereby, increases the frequency of the hole. Since most modern concert flutes come equipped with key pads, we continue our analysis of the previously mentioned concert flute.
For a flute with key pads,
where
where h is the key pad distance
in an open position above the center of the tone hole, in millimeters.
Remember, however, that for a simple flute without key pads, we must rewrite
Equation 8.12 without the
Equation 8.14 suggests that the proportion expressed by Equation 8.10 applies not only to embouchure holes, but to tone holes as well. Consequently, Equation 8.15 also gives good results:
Consider now the following C#-4 tone hole dimensions, key pad and bore dimensions, and key pad correction of the same concert flute:
A substitution of these values into
Equation 8.12 gives
To determine
where x/y is either a rational or irrational interval ratio. For a scale in 12-tone equal temperament,[32] where all the intervals between adjacent tones are constant, g equals the 12th root of 2 minus one:
However, for a scale in just intonation,[33] g is not necessarily a constant. For example, if the first tone hole is a 9/8, or a “just major second,” then the interval ratio between the lowest frequency of the flute tube and the first hole is a 9/8,
and so,
But if the second tone hole is a 5/4, or a “just major third,” then the interval ratio between the first hole and the second hole is a 10/9,
and so,
Nederveen combines variables
According to Equation 8.2b, for the first C#-4 tone hole
so that,
Finally, Nederveen gives the tone hole correction[35]
so that,
For the location of the C#-4 tone hole,
substitute the values for
This corresponds to the exact location of the first hole on the concert flute.
Section 8.6
Table 8.1, Columns 3–5, list tone hole and key pad dimensions of the Armstrong
Concert Flute, and Columns 6–10 list the results of five equations required to
compute
Section 8.7
One of the most important relations to emerge from Table
8.1
is that
As the diameter of a tone hole
decreases, the distance from the embouchure hole center to a tone hole center
decreases as well; vice versa. For a given frequency, a small diameter
hole sits higher on the bore than a large diameter hole. For example, if we
decrease the diameter of the C#-4 hole in Table
8.1
to 14.3 mm, then
Most simple flutes consist of an embouchure hole and six tone holes. One plays the fundamental by closing all the holes, then six more tones by opening each hole, and finally the eighth tone, or the “octave,” by closing all the holes again and overblowing at the embouchure hole.[38] Consider such a flute tuned to G-4 at 392.0 cps. To play a just intoned diatonic scale on this instrument requires six tones holes that produce the following six frequency ratios and six interval ratios:[39]
Between 15/8 and 2/1, we do not include the last interval, ratio 16/15, because it occurs naturally while overblowing the flute.
Table 8.2(a)
lists tone hole dimensions and solutions for such a flute tuned to 392.0 cps.,
where
The embouchure hole has the following
dimensions:
We return now to Table 8.2(a)
where Column 7 gives the frequencies of the tone holes in just intonation. To
calculate these frequencies, multiply 392.0 cps by a given frequency ratio.
For example, frequency of F#-5 is 392.0 cps × 15/8 = 735.0 cps. (As before,
use c = 346,000 mm/s to calculate
Section 8.8
In conclusion to Part I, let us continue the discussion begun in Section
8.7 and examine the interaction
between flute variables in greater detail. A careful examination of the
various flute equations indicates the following proportionalities with respect
to the
In words,
Again in words, F is inversely
proportional to
[1]Schlesinger, K. (1939). The Greek Aulos, Plate 12 between pp. 74–75, and Plate 17 between pp. 420-421. Methuen & Co. Ltd., London, England.
[2]Nederveen, C.J. (1969). Acoustical Aspects of Woodwind Instruments. Frits Knuf, Amsterdam, Netherlands.
[3]See Appendix B for a complete list of conversion factors.
[4]Acoustical Aspects of Woodwind Instruments, pp. 15–17.
[5]Coltman, J.W. (1979). Acoustical analysis of the Boehm flute. Journal of the Acoustical Society of America 65, No. 2, pp. 499–506.
[6]See Chapter 7, Section 2.
[7]See Chapter 7, Section 11.
[8](a) Acoustical Aspects of Woodwind Instruments, p. 13, pp. 47–48.
(b) Nederveen, C.J. (1973). Blown, passive and calculated resonance frequencies of the flute. Acustica 28, pp. 13–14.
[9]Appendix A lists the frequencies of eight “octaves” in 12-tone equal temperament.
[10]In Figures 8.3(a), 8.3(b), and 8.3(c), the substitution tubes show displacement standing waves with nodes at the ends and an antinode in the center, while the flute tubes show pressure standing waves with nodes at the ends and an antinode in the center. Although such inconsistencies may lead to confusion, here the emphasis is on wavelength and visual imagery and not on wave physics. For a given mode of vibration, a displacement standing wave and a pressure standing wave have exactly the same length.
[11]Benade, A.H., and French, J.W. (1965). Analysis of the flute head joint. Journal of the Acoustical Society of America 37, No. 4, pp. 679–691.
[12]This value assumes no lip coverage at the embouchure hole.
[13]Acoustical Aspects of Woodwind Instruments, p. 26, 47.
With the exception of the key pad
correction (
Also on p. 63, Nederveen’s Equation 38.3 is Benade’s solution (see Note 30) for the inside correction of a tone hole. This equation states
The
effective length of a tone hole (
To use
this equation, first simplify
Now add these two corrections and simplify again:
Therefore,
the equivalent effective length of an embouchure hole (
This
latter equation is more accurate than the denominator of Equation 8.9.
Refer to Equation 8.11, and note that a substitution of
A simplification of this equation gives Equation 8.14 in Section 8.5:
And a
similar substitution of
[14]See Chapter 7.
[15]See Chapter 7, Section 5.
[16]The angular wave number is given by
[17]Richardson, E.G. (1929). The Acoustics of Orchestral Instruments and of the Organ, p. 49, 148. Edward Arnold & Co., London, England.
[18]See Chapter 7, Section 12.
[19]A flute tube three centimeters long has a terminating impedance that is too small for the production of pressure standing waves. In contrast, a tube three meters long has a terminating impedance that is too large for human beings to produce pressure standing waves. Most musicians are not able to deliver the air pressure required for sound production from very long tubes. Furthermore, with respect to tone holes, a very small diameter hole drilled into a flute tube has an extremely large terminating impedance. Pressure waves inside the tube propagate past the tiny hole as though it was not there. Consequently, the presence of such a hole does not change the frequency of the tube.
[20]Benade, A.H. (1976). Fundamentals of Musical Acoustics, p. 449, 495. Dover Publications, Inc., New York, 1990.
Also, see Equation 8.26.
[21]Acustica 28, pp. 12–23.
Nederveen states, “Regarding the outside correction for a hole . . . we observe that the main tube acts as a partial flange to the hole-end, so that we have an end correction in between those for an unflanged end and a fully flanged end.”
[22]Steinkopf, O. (1983). Zur Akustik der Blasinstrumente, p. 19. Moeck Verlag, Celle, Germany.
[23]Acustica 28, 1973, p. 16.
[24](a) Boehm, T. (1847). On the Construction of Flutes, Über den Flötenbau, p. 55. Frits Knuf Buren, Amsterdam, Netherlands, 1982.
The English version of this essay does not have page numbers. A reference to the 50 mm correction in the English translation appears four pages from the end of the book.
(b) Boehm, T. (1871). The Flute and Flute-Playing, p. 42. Dover Publications, Inc., New York, 1964.
Here Boehm gives a slightly longer correction at 51.5 mm.
(c) Fundamentals of Musical Acoustics, p. 495.
[25]This cavity effect is especially significant on concert metal flutes. Here most tone holes have a small chimney that sits above the curved outer surface of the tube. These chimneys provide a straight airtight closing surface for the key pads.
[26]Acoustical Aspects of Woodwind Instruments, p. 76.
[27]Ibid., p. 48.
[28]Ibid., p. 64.
[29]See Note 25.
[30](a) Acoustical Aspects of Woodwind Instruments, p. 64.
(B) Benade, A.H. (1967). Measured end corrections for woodwind toneholes. Journal of the Acoustical Society of America 41, No. 6, p. 1609.
[31]Ibid., p. 48.
[32]See Chapter 9, Section 13.
[33]See Chapter 9, Section 14.
[34]Acoustical Aspects of Woodwind Instruments, p. 48.
[35]Ibid., p. 48, in Figure 32.3.
[36]Rossing, T.D. (1989). The Science of Sound, 2nd ed., pp. 246–247. Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1990.
[37]JASA 37, 1965, p. 689.
[38]According to Equation 7.12, a tube open at both ends generates a complete harmonic series of both even-number and odd-number harmonics; and according to Equation 7.17, a tube closed at one end generates an incomplete harmonic series of only odd-number harmonics. The act of overblowing a flute — or increasing the blowing pressure — causes these faint harmonics to sound as plainly audible tones. Therefore, the first overblown tone of a flute open at both ends produces the second harmonic, ratio 2/1; and the first overblown tone of a flute closed at the far end produces the third harmonic, ratio 3/1.
[39]See Chapter 10, Figure 15.
[40]JASA 37, 1965, p. 688.
[41]See Chapter 7, Section 5 through Section 8.
[42]Fundamentals of Musical Acoustics, pp. 473–480.
Another important fine tuning technique consists of changing the bore diameter of woodwind instruments at the location of a pressure antinode, or at the location of a displacement antinode for a given mode of vibration. If we expand the bore by lightly sanding the inner surface of the flute tube at the location of a pressure antinode, such an increase in diameter causes a local increase in volume ( V ), which in turn produces a local decrease in pressure. Refer to Equation 7.36, and note that as V increases, the springiness of the air decreases. We may interpret such a local decrease in the air spring constant as causing a decrease in the potential energy of the vibrating system. When the potential energy of a pressure standing wave decreases, the frequency decreases as well. In contrast, if we constrict the bore by applying several coats of lacquer at the location of a pressure antinode, the frequency of a given mode of vibration increases.
On the other hand, a contraction at the location of a displacement antinode decreases the frequency of a given mode of vibration. A local constriction causes an increase in the inertia — or in the resistance to motion — of the air as it flows through the constriction. However, for a given amplitude of vibration, the same volume of air must flow through the constricted portion of the bore per second of time as through the regular bore. This requires an increase in the particle velocity of the air ( u ) as it passes through the constriction. We may interpret such an increase in particle velocity as causing an increase in the kinetic energy of the vibrating system. When the kinetic energy of a displacement standing wave increases, the frequency decreases. In contrast, if we expand the bore at the location of a displacement antinode, the frequency of a given mode of vibration increases.
Reminiscent of the discussion in Chapter 6, Section 10, a pattern of
overlapping pressure antinodes and displacement antinodes means that a
constriction or expansion anywhere along the length of the bore affects
numerous modes simultaneously. Moreover, the physical length of a given
constriction or expansion also determines which mode frequencies are
affected. For example, on the Armstrong flute the constriction of the
head joint taper extends 114 mm to the right of the embouchure hole
center. Since the embouchure hole marks the location of a displacement
antinode of the first mode of vibration, the constriction flattens the
frequencies of the first “octave.” Furthermore, according to Chapter 7,
Figure 9, a pressure antinode of the second mode of vibration is located
a quarter-wavelength (0.25
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