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Development of Agents for Creating Melodies and Investigation of Interaction between the Agents Hidefumi Ohmura1, Takuro Shibayama2, Keiji Hirata3 and Satoshi Tojo4 1Department of Information Sciences, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba, Japan 2Department of Information Systems and Design, Tokyo Denki University, Ishizaka, Hatoyama-cho, Hikigun, Saitama, Japan 3Department of Complex and Intelligent Systems, Future University Hakodate, 116-2, Kamedanakano-cho, Hakodate-shi, Hokkaido, Japan 4Graduate School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi-shi, Ishikawa, Japan Keywords: Music, Melody, Lattice Space. Abstract: In this study, we attempted to construct computational musical theory by creating musical structure using physical features of sound without relying on the existing musical theory. Subsequently, we developed an agent system to create melodies. The agents can select the next note (a sound timing and a pitch) depending on the lattice spaces consisting of physical relationships (ratios) and probabilities. Further, we improve the agents which are interacting with each other in the system, and the system outputs various sounds such as music. We confirmed that the system could create structures of musical theory, such as mode, scale, and rhythm. The advantages and disadvantages of the lattice spaces are discovered. 1 INTRODUCTION ing the hypothesis. Next, we show the agent system creating melodies, and how agents interact with each Most human beings can hum and whistle melodies in other. Subsequently, we describe the system with an improvisational way in their daily lives. This gen- agents and the operation of the system. Finally, we eration of melodies is considered a beneficial human discuss music as the output of the system. quality for surviving in society (Jordania, 2010). In- terestingly, children can also hum melodies without music education. 2 MUSICAL FEATURES We considered how they create melodies and de- veloped an agent system creating melodies (Ohmura 2.1 Hypothesis of Creating Melodies et al., 2018). The system can provide computational musical structures such as musical scale and mode in It is typical for a musically educated person to eas- music theory because we adopted lattice spaces de- ily select the next pitch and next sound timing of ap- pending on the physical relationships of sounds to the propriateness of a present note. However, the ques- system. tion arises as to why both adults and children with- In this study, we aim to improve the agent sys- out musical education can hum or whistle melodies. tem including the three agents that interact with each We generate the following hypothesis regarding the other. We herein detail the basic elements of the pre- selection of the next note except in the case of recol- vious system in creating melodies, and demonstrate lecting a melody. “They select purely a note of physi- the improved features. Subsequently, we discuss the cal proximity to the present note.” Physical proximity outputs from the system as the music is created by the includes two elements. The first element is the rela- interactions between agents. tionship between the sound timings of notes. The sec- First, we propose a hypothesis on how humans ond element is the relationship between the pitches of create melodies such as humming and whistling, and notes. We considered how they create the melodies demonstrate the physical features of sounds underly- and developed an agent system creating melodies 307 Ohmura, H., Shibayama, T., Hirata, K. and Tojo, S. Development of Agents for Creating Melodies and Investigation of Interaction between the Agents. DOI: 10.5220/0007702303070314 In Proceedings of the 11th International Conference on Agents and Artificial Intelligence (ICAART 2019), pages 307-314 ISBN: 978-989-758-350-6 Copyright c 2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved HAMT 2019 - Special Session on Human-centric Applications of Multi-agent Technologies based on the hypothesis(Ohmura et al., 2018). In this and are defined by the relationship between the fre- section, we explain their relationships and detail the quencies. theories of musical expectation to demonstrate how A relationship of 1:2 creates an interval of a per- an agent selects a note. fect octave, and a relationship of 1:3 creates an inter- val of a perfect fifteenth, which is two octaves. The 2.2 Relationships between Pulses (Note relationship of 2:3 and 3:2 are combined duple and Values) triple frequencies (see Figure 2). This pattern is called a perfect fifth. The relationships of 3:4 and 4:3 are combined triple and quadruple (see Figure 2). This In music, an iteration is an important primitive pat- pattern is called a perfect fourth, which is a consonant tern, and a pulse is the most primitive element in a interval following a perfect fifth. rhythm. For example, when a listener hears two pulses (whose relationship is 1:2), he/she may feel a duple 880Hz 990Hz meter (see Figure 1). When a relationship is 1:3, a (A5) ! 4/3 ! 3/2 (B5) listener may perceive a triple meter. ! 3/4 ! 2/3 Next, we consider 2:3 and 3:2 ratios combining 660Hz duple and triple meters. These relationships provide ! 1/2 ! 2 ! 1/2 ! 2 (E5) the listener with a polyrhythm. A listener perceives ! 3/2 ! 4/3 one basic pulse as an upbeat, and another as a down- ! 2/3 ! 3/4 beat. In the cases of 3:4 and 4:3, a listener perceives a polyrhythm consisting of triple and quadruple meters. 440Hz 495Hz (A4) (B4) Figure 2: Relationships between pitch (intervals). The Pythagorean scale (and principle of the Sanfen × 2 × 1/2 Sunyi) involve only the relationships of duples and × 3 × 3 triples. However, depending on temperament, some intervals are imprecise. × 1/3 × 1/3 In the case of intervals, a quintuple is important. × 1/2 × 2 In particular, in a relationship between three pitches, a 3:4:5 creates a consonance code, called a major triad. A relationship consisting of a double, triple, and quin- tuple creates intonation. In this study, however, we Figure 1: Relationships between pulses (note values). employ only the double and triple for a simple and Actual music consists of many pulses. A listener per- easy operation. ceives the strongest or most-frequent pulse as the me- ter of the music, and less-frequent pulses as weak 2.4 Theories of Musical Expectation beats and up beats. Monophony, however, lacks beats, such that a listener at times may not perceive any me- We attempted to control the musical expectations ter. This is true in the melodies of humming. based on the theories of musical expectations (Ohmura et al., 2016). Here, we introduce these theo- ries that provide how an agent selects the next note. 2.3 Relationships between Pitches Meyer demonstrated that the deviations in ex- (Intervals) pectations arouse emotions when listening to music (Meyer, 1956). This concept is based on Dewey’s In music, patterns consisting of pitches are important. theory, according to which conflict causes emotions These patterns are explained in musical theories of (Dewey, 1894). The deviation from the listeners’ ex- temperament (how to determine the frequency of each pectation when listening to music arises from a par- note) and mode (which notes to use). The value of tial or complete disregard of rules that were accepted a pitch depends on the vibrational frequency of air. in advance. This indicates an increase in contingency Real sound consists of multiple frequencies, and we because of augmented uncertainties. These uncertain- perceive the lowest frequency as the pitch, also called ties present a commonality with complexity in the as the fundamental frequency. As with rhythm, the optimal complexity model (Berlyne, 1971) that illus- patterns of two pitches are called musical intervals, trates the relationship between complexity and hedo- 308 Development of Agents for Creating Melodies and Investigation of Interaction between the Agents nic values (Figure 3). This commonality suggests the culate the next note with probability density func- existence of a relationship between uncertainty and tions. In the next section, we describe how the agent emotion. Comparing sounds in our everyday life and selects the next note with probability density func- their relationships reveals interesting viewpoints. We tions. shall survey the points in Figure 3. At Position 1 , the complexity is relatively low. A listener can easily pre- dict the features of the sound, and a prediction event 3 AN AGENT CREATING is likely to arise in the proposed system. For example, the pure tick-tock beat of a clock not only sounds bor- MELODIES ing, but also causes displeasure in listeners. Indeed, some clock users cannot sleep with this sound. Po- To create a melody, the proposed agent selects a sound sition 2 has a higher complexity than Position 1 , timing, and subsequently selects a pitch. These ac- thus eliciting pleasure in the listener. The listener tions are based on the relationships as shown in 2.2 can predict the next sound, and recognize both real- and 2.3. The selections depend on the probabilities izations and deviations from expectation at that posi- that are built on the theory of musical expectation. In tion. Listeners may regard sounds as musical, because this section, we introduce the theory of musical ex- the sounds comprise rules as well as deviations from pectation, and explain an agent-creating melody. the rules at this position. Different levels of complex- ity exist in each musical genre. Rhythms, children’s 3.1 Lattice Spaces with Duple and songs, and folk songs have lower complexities than Triple Relationships pop music, for example. Therefore, a wide range of sounds are under this position. Position 3 exhibits an We provide a lattice space for rhythm that consists of appreciably high complexity.
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