Appendix
1. Partial Correlations
In all
behavioural experiments reported in this dissertation, probe
tone ratings were initially correlated with the overall
exposure frequencies of each pitch. This results in a
correlation strength r for each participant's ratings
before and after exposure. While the value of r was
generally larger for post-exposure ratings than for
pre-exposure ratings, it was observed that the value of
r for pre-exposure ratings was often significantly above
the chance level of zero. While the observation of a
significant correlation between exposure frequencies and
pre-exposure ratings might suggest some pre-existing knowledge
of tone frequencies prior to hearing the exposure set of
melodies, a possible alternative hypothesis is that the
correlation is boosted due to the use of the probe tone
paradigm (after Krumhansl, 1990). In each trial of the probe
tone ratings, a melody is presented, followed by a probe tone,
and the participant's task is to rate how well the tone fit the
preceding melody. Importantly, the melody presented during the
trial (which is the same for trials) is one instance of the new
musical system. If participants rated any tones that occurred
in the melody as good fits, then the resulting correlation
could be uniquely associated with the melody, rather than
sensitivity to exposure frequencies of tones overall. Figure 1
shows how the melody used during the probe tone ratings is
correlated with the exposure set.
Figure 1. Profiles of the exposure set and the melody
used in probe tone ratings trials. Partial correlations involve
partialling out the effects of the probe melody (dashed line)
from the probe tone ratings, leaving only the correlations
uniquely associated with the exposure set (solid
line).
To
assess sensitivity to tone frequencies due to the true
contribution of the set of exposure melodies, rather than the
single melody used in probe tone ratings, we conducted a
partial correlations analysis for all probe tone ratings data.
Effects of the melody were partialled out using the partial
correlation equation:
rXY•Z= (r
XY- rXZrYZ)
_
√(1 -
r2XZ)(1 -
r2YZ)
where
X was the ratings profile, Y was the exposure
profile, and Z was the profile of the melody used to
obtain probe tone ratings. When X was the pre-exposure
ratings profile, rXY•Z dropped to zero, but when X was
post-exposure ratings, rXY•Z was significantly above chance (see figure.
1). This suggests
that participants did not have any knowledge of tone
frequencies prior to the start of the experiment; any knowledge
they initially demonstrated was a result of the experimental
paradigm used. Thus, we use partial correlation as a powerful
statistical tool for assessing sensitivity to frequencies,
while removing the effects of the experimental
paradigm.
2. Melodies
used in Chapter 3
Here we list the melodies used in all experiments from
Chapter 3. Twenty melodies from each grammar are listed. Each
melody is written as a string of numbers, with each number
representing an increment n in the Bohlen-Pierce scale
formula,
F = 220*3^(n/13),
which derives tone frequency F from the number of steps
(or increments) along the scale.
Grammar I
melody
1
6
6
4
3
6
melody
2
0
7
4
3
3
melody
3
6
7
3
7
10
melody
4
0
0
7
6
melody
5
6
4
7
7
7
6
10 10
melody
6
10
10
7
10
3
6
melody
7
6
6
10
4
7
10
9
10
melody
8
0
4
3
0
melody
9
0
0
0
7
6
10
melody 10
6
7
7
4
4
3
6
melody 11
10
4
7
6
10
melody 12
6
4
10
10
6
melody 13
0
4
3
6
0
melody 14
6
4
7
7
3
10
6
melody 15
6
7
7
0
6
melody 16
10
4
7
7
6
10
melody 17
6
6
7
3
10
6
melody 18
0
0
3
7
6
melody 19
6
0
3
10
6
melody 20
10
4
7
3
6
10 10
Grammar II
melody
1
0
7
4
6
6
melody
2
10
7
4
4
6
melody
3
6
3
0
0
6
6
melody
4
6
10
4
7
6
10
melody
5
0
3
7
6
melody
6
6
10
4
6
10
melody
7
0
7
7
6
6
10
melody
8
10
10
4
6
melody
9
6
7
0
6
melody 10
0
7
7
3
4
6
melody 11
10
7
4
6
melody 12
6
7
4
7
6
melody 13
0
3
4
0
0
melody 14
0
3
0
4
7
6
6
melody 15
10
10
7
6
6
melody 16
6
3
4
4
6
melody 17
0
0
3
0
4
7
melody 18
10
7
7
4
7
melody 19
6
6
3
4
6
melody
20
6
7
7
4
6
10