Elektronika 2009-11.pdf - Instytut SystemÃ³w Elektronicznych

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References [1] Russel S., Norvig P.: (2003) Artificial Intelligence. A Modern Approach. Prentice Hall, New Jersej. [2] Zielona Księga dla Rady Parlamentu Europejskiego, Europejskiego Komitetu Ekonomiczno-Społecznego i Komitetu regionów. Adaptacja do zmian klimatycznych w europie-warianty działań na szczeblu UE. Komisja Wspólnot Eropejskich (KWE), Bruksela, 29.06.2007 (in Polish). [3] Fiuk G., Budziński R.: (1995) Model Systemu Informatycznej Obsługi Miasta Szczecin. W ks.: Systemy Informatyczne w Zarządzaniu Aglomeracjami Miejskimi. PAN IBS, Oddział w Szczecinie, Warszawa-Szczecin (in Polish). [4] Tretyakova T.: (2009) Wiedza i modele dla inteligentnych lokalnych składników systemu wspomagania decyzji. Studia i materiały Polskiego Stowarzyszenia Zarządzania Wiedzą nr 18. PSZW, Bydgoszcz (in Polish). [5] Tretyakova T., Zair A.: (2008) The Structure and Knowledge of the Intelligent System of Warning and Decision’s Support that Includes Local Systems. Polish Journal of Environment Studies, vol. 17, no 4C. [6] Tretyakova T. (2005). Knowledge base of expert system “HMDecision “ in information system of class DSS - the objective approach to designing. Proceeding of 6-th International Conference “ the Analysis, forecasting and management in complex systems”, St Petersburg (in Russian). [7] Zadeh L. (1965) Fuzzy sets. Information and Control, vol. 8. [8] Zadeh L. (1988) Fuzzy logic. IEEE Transactions on Computers, vol. 21, no 4. [9] Kacprzyk J. (2006) Komputerowe systemy wspomagania decyzji dla potrzeb zarządzania wiedzą. W ks. Pod red. R.Kulikowski, Z.Bubnicki, J.Kacprzyk. Systemowe-komputerowe wspomaganie zarządzania wiedzą. ELIT, Warszawa (in Polish). [10] Yu X., Kacprzyk J. (2008) Applied Decision Support with Soft Computing, Springer. [11] Piegat A. (2001) Fuzzy Modeling and Control, Physica-Verlag Heidelberg, New York. [12] Leonenkov A. (2003) Fuzzy modeling in the environment MAT- LAB and fuzzyTECH. BHV- -Petersburg (in Russian). Generation of harmonic sequences in accordance to tonal harmony rules with artificial neural networks (Generacja sekwencji harmonicznych w zgodzie z zasadami harmonii tonalnej przy użyciu sieci neuronowych) dr inż. WOJCIECH ZABIEROWSKI, prof. dr hab. inż. ANDRZEJ NAPIERALSKI Technical University of Lodz, Department of Microelectronics and Computer Science There has been a continuous research throughout the world. in the field of computer generated music. Many implementations has appeared as well as a lot of disputes on this topic, and it is now perceived as a branch of science. The computer participation in the process of music creation is not only restricted to being an aid for the composer. In fact, a machine is able to replace human in the creation process. There are some realizations that are purely machinegenerated and which were able to reach the top of pop music lists, successfully competing with human-made works. The elements of the branch of science mentioned above include musicological analysis of musical opus and automatic harmonization of the melody line. The research presented below can be applied in particular in these areas. Problem specification A musical piece can be described by a number of various parameters. These parameters determine, what is later called, a musical work - an art. Music is not a sole matter of putting together notes and resulting sounds (and vice versa). Moreover, even if these sounds are combined in chords, they still remain just a set of more or less ordered tones. One of the most basic parameters, determining whether what we obtain can really be called music, is the harmonic meaning of mentioned elements. More precisely, this meaning is, of course, a harmonic meaning in musical sense - an accordance of tones. Basic harmonic constructions [2-4], also known as functions, form together a harmonic triad and, depending on the musical scale degree on which one builds them, they are called: tonic (T), subdominant (S) and dominant (D) correspondingly. Only by combining these harmonic functions into sequence, we obtain music. Secondary functions may be distinguished as well. These secondary functions retain similarity to the main functions, and hence, they received similar names, such as T II - tonic of second degree. By using secondary functions, a harmonic sequence can be greatly enriched. There is, of course, a strict (depending on a chosen harmonic) set of rules for progression of chords that have specific harmonic functions. Table 1 contains the basic rules for harmonic function progressions. Tabl. 1. Basic functional progressions Tab. 1. Podstawowe następstwa funkcyjne Harmonic function Tonic Subdominant Dominant Allowed consequent function S, D, T T, (D) T 22 ELEKTRONIKA 11/2009
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 ELEKTRONIKA 11/2009 23
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