Elektronika 2009-11.pdf - Instytut Systemów Elektronicznych

In addition, one has to remember that a musical work
always begins with a tonic (T), and each stress build up by
a dominant is resolved to a tonic (T). In short, creating music
is a matter of choosing such progressions.
Input data preparation
An idea was born to try, in accordance with Computer Generated
Music trends, to implement (simplified) tonal harmony
rules in generation of harmonic sequences as an initial phase
of further research in the area of Computer Generated Music.
To achieve this, neural networks have been used. After performing
extensive evaluation and testing, a three layer perceptron
model with 32 neurons in hidden layer has been
chosen. Each of 16 input neurons and 16 output neurons has
been assigned a corresponding value from the learning sets.
An example of such mapping is presented in Table 2.
Tabl. 2. Exemplary values of corresponding input and output neurons
used in the network architecture
Tab. 2. Przykładowe wartości odpowiednich neuronów wejściowych
i wyjściowych w użytej architekturze sieci
Neurons In
In1 In2 In3 In4 In5 … In14 In15 In16
1 0 0 0 1 0 1 0 0
Neurons Out
Ou1 Ou2 ... Ou8 Ou9 … Ou14 Ou15 Ou16
0 0 0 1 0 0 1 0 0
To represent the corresponding tones in chords of given
harmonic meaning, a MIDI notation was used (see tabl. 2),
more precisely - the part of information stored in the voice
messages. Occurence of a certain value in a voice message
has led to a ‘1’ as an input for a given neuron. No tone was
equivalent to ‘0’ as an input.
Tabl. 3. Values corresponding to sounds in MIDI notation
Tab. 3. Wartości MIDI odpowiadające wartościom dźwiękowym
Note MIDI value (hex) Decimal value (dec)
C 3c 60
D 3e 62
E 40 64
F 41 65
G 43 67
A 45 69
H 47 71
Data was prepared in accordance with tonal harmony
rules using a synthesizer, which was utilized to obtain the desired
groups of dependencies as sets for teaching the neural
network. The sets contained from 50 sequences, in a simplest
case, up to 300 in a case that considered dependencies between
triads in I and II position of the secondary functions.
Input data was tested according to the rule that three tones
making a chord have specified harmonic meaning. In the course
of using a taught network to generate harmonic sequences,
the input consisted of the two tones plus the harmonic
meaning of the previous tonal sequence, in a hope that the
network would point the correct tone for filling in the specified
harmonic value. The tonal range was limited to a single octave
to simplify the occurring dependencies.
The research
The learning process, depending on the method used,
consisted of up to 40 000 iterations. In the process [6,7] the
following methods were taken into consideration:
• gradient descent back-propagation, (with and without momentum),
• resilient back-propagation,
In addition, the following transfer functions were used there [5]:
• log-sigmoid transfer function - Matlab - logsig(n);
• hyperbolic tangent sigmoid transfer function - Matlab
tansig(n).
As a result of teaching the neural network, a good match,
measured with the mean squared error (MSE) and the number
of erratic classifications, was obtained (see tabl. 4). The
magnitude of the classification error Er means the number of
direct error matches with respect to the expected precise
answers and can be defined in the following manner:
where: er - number of erratic classifications, L - test set size
Tabl. 4. The results of learning processes and their Er error values
Tab. 4. Wyniki uczenia wraz z wartościami błędu Er
Network
architecture
Learning
time
(iterations)
16/32/16 40 000
16/32/16 40 000
16/32/16 500
Transfer
Function
tansig/
tansig/tansig
tansig/
tansig/tansig
tansig/
tansig/tansig
Teaching
Function
MSE
The most interesting results for an architecture with one
hidden layer and 32 neurons in the layer, as well as obtained
adjustment and mean squared error (MSE), is shown in
tabl. 4. The architecture of presented solution was chosen
empirically. All of the sets were prepared with default network
parameters taken into account. Various network configurations
with different numbers of neurons per hidden layer were
tried out. Many more setups were investigated , but not all of
them are included because of the unsatisfactory results they
produced.
Concluding from the tabl. 4, the reached level of mean
squared error of learning process for this issue is, in fact, the
same in every case. The same can be observed for the Er
classification error, which shows similar values in different network
configurations. Due to this, more attention has to be paid
to the configurations that allow to obtain similar level of adjustment
with less calculation overhead.
(1)
(2)
(3)
Er
traingd 0,0214148 8
taingdm 0,021857 8
trainrp 0,0182302 8
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