4.3 Bore Profiles of Brass Instruments
129
Fig. 4.23 Excerpt from the 2nd Movement of Mozart’s Horn Concerto in D Major, K.412
open tube end if it is cut to give a reasonably flat rim. A comparison of Fig. 4.23
with Fig. 1.12 shows that the notes required are all found between the 6th and 12th
members of a harmonic series based on C2; however the horn in D uses transposed
notation in which the sounding pitches are a minor 7th (10 semitones) below the
written pitches. It will be found that a length of hosepipe about 4.25 m long is
necessary for the excerpt to sound at the correct pitch. This is roughly the same
length as the bore of the D horn, which is described technically as being in ‘14 ft D’
(Sect. 7.2.7).
The hosepipe horn is an amusing musical curiosity, but it is also a very interesting
scientific demonstration that even a simple cylinder of constant diameter has a set
of acoustic resonances which seem to be reasonably close to a harmonic series.
However a musical exploration of the lower register of the hosepipe horn shows
that below the sixth natural note, the pitches are increasingly flattened relative to
the expected harmonics, the discrepancy reaching several semitones for the second
natural note.
The explanation of the apparently strange behaviour of the cylindrical tube
resonance frequencies lies in the nature of the reflections which create the standing
waves in the tube. We saw in Sect. 4.1.2 that a travelling wave is reflected with a
change of sign at the open end of the tube, creating a pressure node just outside the
end; at the input end, which is effectively closed by the player’s lips, the wave is
reflected without a change of sign, making this point a pressure antinode. These end
conditions can only be satisfied by waves for which the length L of the tube is an
odd number of quarter wavelengths. Neglecting wall losses and radiation from the
open end, the permitted wavelengths are
λ n =
4L
2n − 1
;
(4.32)
the corresponding frequencies form a series given by
f n =
c
λ n
=
(2n − 1)c
4L
= f 1 , 3f 1 , 5f 1 . . .
(4.33)
with f 1 = c/4L.
129
Fig. 4.23 Excerpt from the 2nd Movement of Mozart’s Horn Concerto in D Major, K.412
open tube end if it is cut to give a reasonably flat rim. A comparison of Fig. 4.23
with Fig. 1.12 shows that the notes required are all found between the 6th and 12th
members of a harmonic series based on C2; however the horn in D uses transposed
notation in which the sounding pitches are a minor 7th (10 semitones) below the
written pitches. It will be found that a length of hosepipe about 4.25 m long is
necessary for the excerpt to sound at the correct pitch. This is roughly the same
length as the bore of the D horn, which is described technically as being in ‘14 ft D’
(Sect. 7.2.7).
The hosepipe horn is an amusing musical curiosity, but it is also a very interesting
scientific demonstration that even a simple cylinder of constant diameter has a set
of acoustic resonances which seem to be reasonably close to a harmonic series.
However a musical exploration of the lower register of the hosepipe horn shows
that below the sixth natural note, the pitches are increasingly flattened relative to
the expected harmonics, the discrepancy reaching several semitones for the second
natural note.
The explanation of the apparently strange behaviour of the cylindrical tube
resonance frequencies lies in the nature of the reflections which create the standing
waves in the tube. We saw in Sect. 4.1.2 that a travelling wave is reflected with a
change of sign at the open end of the tube, creating a pressure node just outside the
end; at the input end, which is effectively closed by the player’s lips, the wave is
reflected without a change of sign, making this point a pressure antinode. These end
conditions can only be satisfied by waves for which the length L of the tube is an
odd number of quarter wavelengths. Neglecting wall losses and radiation from the
open end, the permitted wavelengths are
λ n =
4L
2n − 1
;
(4.32)
the corresponding frequencies form a series given by
f n =
c
λ n
=
(2n − 1)c
4L
= f 1 , 3f 1 , 5f 1 . . .
(4.33)
with f 1 = c/4L.