Biomedical Engineering Reference
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(a)
female time interval
between notes
male time interval
between notes
female note
6000
1/2
1/2
1/3
1/3
1/4
male note
0.0
1.0
2.0
time (s)
(b)
6000
1/2
1/2
1/2
2/5
1/3
1/3
1/4
0.0
1.0
time (s)
2.0
Fig. 9.1.
Duet sonograms. (
a
) Sonogram of a typical duet of the suboscine hornero
(
Furnarius rufus
). Male and female voices are easily distinguished, owing to the
differences in their spectral and temporal features. The male begins the vocalization,
monotonically increasing the note production rate as the duet develops. The female,
on the other hand, may respond with an increasing, decreasing or nonmonotonically
varying note production rate, but their notes lock in a systematic way. The locking
sequence of this duet is shown in the sonogram. (
b
) Sonogram of a synthetic duet.
Female syringeal and respiratory gestures were simulated by a nonlinear oscillator
subjected to a periodic forcing of increasing frequency. Both a periodic forcing and
a forced nonlinear oscillator were used to drive physical models of the syrinx to
generate male and female notes. Adapted from [Laje and Mindlin 2003]
9.2.2 A Devil's Staircase
In Fig. 9.3a, we plot
r
ap
as a function of the normalized average time interval
between male notes
T
male
for
n
= 11 duets (from at least 10 hornero couples),
in order to inspect the female response. The average time interval between
male notes has been normalized to the average time interval between notes
of the corresponding female when locked with
r
ap
=1
/
3 (every female locked
with
r
ap
=1
/
3 at least once). A clear stair-like structure emerges, with
steps at several values of
p/q
. Notice that some values of
p
are different
from 1. These are located at the regions predicted by the Farey sum rule
[Gonzalez and Piro 1983]. For example, we found a number
r
ap
=2
/
7; it fell
between
r
ap
=1
/
3and
r
ap
=1
/
4. A segment with
r
ap
=3
/
10 was also found,
located between
r
ap
=1
/
3and
r
ap
=2
/
7.
