136
4 After the Lips: Acoustic Resonances and Radiation
with f 1 = c/2L. A realistic instrument with a mouthpiece, tuning slides, etc. will of
course deviate somewhat from this idealised and simplified picture, but it remains
true that instruments with a bore profile which is mainly conical have natural notes
which correspond closely to a complete harmonic series even down to the lowest
member. The natural notes of the flugelhorn, the euphonium and the french horn,
whose bore profiles are not strictly conical but whose bores expand over most
of their length, are also close to harmonic, even if extra cylindrical sections are
introduced when valves are operated.
4.3.4 Equivalent Fundamental Pitch and Equivalent Cone
Length
The playing technique of most brass instruments is founded on the availability of a
range of pitches which can be sounded without physically modifying the instrument
and which correspond closely to the notes of a complete harmonic series. For this
reason it is useful to have a graphical method for displaying the extent to which
the pitches of the natural notes of a particular instrument deviate from the expected
harmonics. A useful quantity in this context is the equivalent fundamental frequency
(EFF) of a specific natural note, which is defined by the formula
EFF =
f n
n
,
(4.40)
for a natural note with rank number n and frequency f . As an example we can take
the frequencies of the first three impedance peaks in Fig. 4.28, which are 54.7 Hz,
109.8 Hz and 165.5 Hz. The corresponding EFF values are 54.7 Hz, 109.8/2 =
54.9 Hz and 165.5/3 = 55.2 Hz. The ‘harmonicity’ of the set of frequencies is
measured by the correspondence between the EFF values: for a perfect harmonic
series, the values would be identical.
For musical purposes it is often more useful to display the deviations of natural
notes from perfect harmonicity in terms of pitch shifts rather than frequency
changes. The equivalent fundamental pitch (EFP) is defined by the formula
EFP(n) =
1200
log(2)
log
f n
nf 0
,
(4.41)
where f 0 is the frequency of a reference pitch; EFP(n) is the difference in cents
between the pitch of the nth natural note and the nth harmonic of the reference
pitch.
Figure 4.30 illustrates EFP plots derived from the calculated impedance peak
frequencies of the cylinder and cone discussed above. Both tubes were the same
length as a B trombone, so the reference pitch was chosen to be B 1 (f 0 =
4 After the Lips: Acoustic Resonances and Radiation
with f 1 = c/2L. A realistic instrument with a mouthpiece, tuning slides, etc. will of
course deviate somewhat from this idealised and simplified picture, but it remains
true that instruments with a bore profile which is mainly conical have natural notes
which correspond closely to a complete harmonic series even down to the lowest
member. The natural notes of the flugelhorn, the euphonium and the french horn,
whose bore profiles are not strictly conical but whose bores expand over most
of their length, are also close to harmonic, even if extra cylindrical sections are
introduced when valves are operated.
4.3.4 Equivalent Fundamental Pitch and Equivalent Cone
Length
The playing technique of most brass instruments is founded on the availability of a
range of pitches which can be sounded without physically modifying the instrument
and which correspond closely to the notes of a complete harmonic series. For this
reason it is useful to have a graphical method for displaying the extent to which
the pitches of the natural notes of a particular instrument deviate from the expected
harmonics. A useful quantity in this context is the equivalent fundamental frequency
(EFF) of a specific natural note, which is defined by the formula
EFF =
f n
n
,
(4.40)
for a natural note with rank number n and frequency f . As an example we can take
the frequencies of the first three impedance peaks in Fig. 4.28, which are 54.7 Hz,
109.8 Hz and 165.5 Hz. The corresponding EFF values are 54.7 Hz, 109.8/2 =
54.9 Hz and 165.5/3 = 55.2 Hz. The ‘harmonicity’ of the set of frequencies is
measured by the correspondence between the EFF values: for a perfect harmonic
series, the values would be identical.
For musical purposes it is often more useful to display the deviations of natural
notes from perfect harmonicity in terms of pitch shifts rather than frequency
changes. The equivalent fundamental pitch (EFP) is defined by the formula
EFP(n) =
1200
log(2)
log
f n
nf 0
,
(4.41)
where f 0 is the frequency of a reference pitch; EFP(n) is the difference in cents
between the pitch of the nth natural note and the nth harmonic of the reference
pitch.
Figure 4.30 illustrates EFP plots derived from the calculated impedance peak
frequencies of the cylinder and cone discussed above. Both tubes were the same
length as a B trombone, so the reference pitch was chosen to be B 1 (f 0 =