178
4 After the Lips: Acoustic Resonances and Radiation
When a mute with an internal tube diameter of 12 mm is inserted into the
modelled instrument, all of the green circles move to the left as shown in Fig. 4.74b,
indicating a flattening of the corresponding natural notes. The fourth peak, for
example, which was only 22 cents below the fourth harmonic of D2 without the
mute, is flattened by a further 250 cents by the introduction of the mute. Figure
4.74c shows the effect of decreasing the diameter of the internal tube to 6 mm: this
has little effect on the peak frequencies from the fifth upward, but the fourth peak
is flattened by a further 100 cents. The third and second peaks are, respectively,
591 cents and 681 cents semitones too flat and have therefore disappeared from the
diagram.
Since the flattening effect of the mute is strongly dependent on frequency, the set
of peak frequencies represented by the green circles in Fig. 4.74c is very far from a
harmonic series. Looking to the right of these green circles, however, we see a set of
red circles which lie very close to the vertical line with an EFP value of +100 cents.
This suggests that a well-supported harmonic series of natural notes is available
based on the pitch E 2, a semitone higher than that found without the mute. On
the Extended EFP diagram, all circles on a given magenta line represent the same
impedance peaks, but different circles have different assigned peak numbers. Thus
the frequency of the fourth impedance peak is 237 Hz when the mute with a 6 mm
diameter channel is inserted; this is 371 cents too flat to be the fourth harmonic of D2
(73.4 Hz), but only 26 cents too sharp to be the third harmonic of E 2 (77.8 Hz). The
red circles on the Extended EFP plot show the effects of reassigning peak numbers;
for the simplified mute modelled here, the nth natural note of the E 2 harmonic
series is supported by the (n − 1)th impedance peak.
The trumpet mute whose extended EFP values are plotted in Fig. 4.74c behaves
like a transposing mute, raising the pitches of the playable notes by a semitone.
Experiments with original and reproduction baroque mutes have shown that they
also transpose upwards by approximately a semitone (Klaus and Pyle 2015).
No original mute has been found which transposes by a tone, as described by
Monteverdi and many other seventeenth- and eighteenth-century authors. It has been
suggested that, in order to avoid transposition by a semitone into keys which could
be difficult for other woodwind instruments, trumpet players may have routinely
removed a short section of tubing when asked to play with mutes, raising the pitch of
the unmuted instrument by a semitone; inserting the mute would have then resulted
in a total transposition of a whole tone (Stradner 2015).
4.5.6 Hand Technique on the Horn
Although mutes similar to those used on trumpets and trombones are employed
on the french horn, the standard method of playing from the eighteenth century
has involved the insertion of the hand of the player into the bell of the instrument
(Fig. 4.75a). The pitch of each of the natural notes of the instrument can be modified
by adjusting the position of the hand; increasing the degree of closure of the bell
4 After the Lips: Acoustic Resonances and Radiation
When a mute with an internal tube diameter of 12 mm is inserted into the
modelled instrument, all of the green circles move to the left as shown in Fig. 4.74b,
indicating a flattening of the corresponding natural notes. The fourth peak, for
example, which was only 22 cents below the fourth harmonic of D2 without the
mute, is flattened by a further 250 cents by the introduction of the mute. Figure
4.74c shows the effect of decreasing the diameter of the internal tube to 6 mm: this
has little effect on the peak frequencies from the fifth upward, but the fourth peak
is flattened by a further 100 cents. The third and second peaks are, respectively,
591 cents and 681 cents semitones too flat and have therefore disappeared from the
diagram.
Since the flattening effect of the mute is strongly dependent on frequency, the set
of peak frequencies represented by the green circles in Fig. 4.74c is very far from a
harmonic series. Looking to the right of these green circles, however, we see a set of
red circles which lie very close to the vertical line with an EFP value of +100 cents.
This suggests that a well-supported harmonic series of natural notes is available
based on the pitch E 2, a semitone higher than that found without the mute. On
the Extended EFP diagram, all circles on a given magenta line represent the same
impedance peaks, but different circles have different assigned peak numbers. Thus
the frequency of the fourth impedance peak is 237 Hz when the mute with a 6 mm
diameter channel is inserted; this is 371 cents too flat to be the fourth harmonic of D2
(73.4 Hz), but only 26 cents too sharp to be the third harmonic of E 2 (77.8 Hz). The
red circles on the Extended EFP plot show the effects of reassigning peak numbers;
for the simplified mute modelled here, the nth natural note of the E 2 harmonic
series is supported by the (n − 1)th impedance peak.
The trumpet mute whose extended EFP values are plotted in Fig. 4.74c behaves
like a transposing mute, raising the pitches of the playable notes by a semitone.
Experiments with original and reproduction baroque mutes have shown that they
also transpose upwards by approximately a semitone (Klaus and Pyle 2015).
No original mute has been found which transposes by a tone, as described by
Monteverdi and many other seventeenth- and eighteenth-century authors. It has been
suggested that, in order to avoid transposition by a semitone into keys which could
be difficult for other woodwind instruments, trumpet players may have routinely
removed a short section of tubing when asked to play with mutes, raising the pitch of
the unmuted instrument by a semitone; inserting the mute would have then resulted
in a total transposition of a whole tone (Stradner 2015).
4.5.6 Hand Technique on the Horn
Although mutes similar to those used on trumpets and trombones are employed
on the french horn, the standard method of playing from the eighteenth century
has involved the insertion of the hand of the player into the bell of the instrument
(Fig. 4.75a). The pitch of each of the natural notes of the instrument can be modified
by adjusting the position of the hand; increasing the degree of closure of the bell