Connectionist Representations of Tonal Music

Michael R. W. Dawson

• Publisher: AU Press
• Year: 2018
• ISBN:
  9781771992220
• Topic: Psychology
• Subtopic: Cognitive Psychology & Cognition
• Index: Psychology

Previously, artificial neural networks have been used to capture only the informal properties of music. However, cognitive scientist Michael Dawson found that by training artificial neural networks to make basic judgments concerning tonal music, such as identifying the tonic of a scale or the quality of a musical chord, the networks revealed formal musical properties that differ dramatically from those typically presented in music theory. For example, where Western music theory identifies twelve distinct notes or pitch-classes, trained artificial neural networks treat notes as if they belong to only three or four pitch-classes, a wildly different interpretation of the components of tonal music.Intended to introduce readers to the use of artificial neural networks in the study of music, this volume contains numerous case studies and research findings that address problems related to identifying scales, keys, classifying musical chords, and learning jazz chord progressions. A detailed analysis of the internal structure of trained networks could yield important contributions to the field of music cognition.

1

Science, Music, and Cognitivism

1.1 Mechanical Philosophy, Mathematics, and Music

Natural philosophy, developed by such giants as Copernicus, Galileo, Boyle, Newton, and Descartes, reigned during the scientific revolution from 1543 (the year of Copernicus’s publication of On the Revolutions of the Heavenly Spheres) to 1687 (the year of Newton’s publication of Mathematical Principles of Natural Philosophy) (Shapin, 1996). Shapin notes that the natural philosophy that emerged during the scientific revolution could also be called mechanical philosophy because it recognized that a variety of machines could be created that appeared to be purposeful, intentional, or sentient. Early natural philosophers were inspired by the properties of clocks because during the scientific revolution a variety of clockwork automata were created (Wood, 2002).

Viewing the world as a clock, natural philosophy embraced mathematics for describing and explaining nature’s workings (Shapin, 1996). Mathematics played a key role in the theories of Galileo, Bacon, and Boyle; Newton’s discoveries revealed that physical laws expressible in mathematical form govern the universe. Music played a key role in the pursuit of the mathematical explanation of nature. Kepler sought musical harmony in the motions of the planets (Stephenson, 1994). Lagrange used musical sound to link properties of his calculus to the physical world (Dhombres, 2002). As a student, Newton explored various mathematical means for dividing the octave into smaller musical intervals (Isacoff, 2001). Many natural philosophers believed that music provided evidence of the mathematical perfection of the natural world.

Of course the relation between mathematics and music originated long before natural philosophy. Around 500 BC, Pythagoras linked perceived pitch to the frequency at which a string vibrated. The Pythagoreans also determined that the most consonant musical intervals are ratios of string lengths that involved simple whole integers: 1/1 for unison, 2/1 for the octave, 3/2 for the perfect fifth, and 4/3 for the perfect fourth. In contrast, the ratio for a very dissonant interval, the tritone, is 45/32. Pythagorean geometry led to the discovery of irrational numbers, but irrational numbers are also found in music, and are related to dissonant musical intervals (Pesic, 2010).

Pythagorean notions of consonance led to tuning discrepancies (Donahue, 2005). If one starts at one pitch, and moves to a note that is seven Pythagorean octaves higher, one does not reach exactly the same note that is produced by moving 12 Pythagorean perfect fifths higher. The two final notes differ by the so-called Pythagorean comma (which equals 24 cents, where 100 cents = 1 semitone). Thus, one cannot have a perfect musical tuning system that includes perfect ratios for both octaves and fifths. This taunted those who believed in the mathematical perfection of music or nature. If consonant ratios are mathematically perfect, then why can they not be used to tune instruments whose notes spanned multiple octaves? At the start of the scientific revolution, many scholars, motivated by this problem, attempted to develop alternative approaches to tuning (Cohen, 1984).

1.2 Mechanical Philosophy and Tuning

1.2.1 Tempered Scales

The new approaches to tuning that emerged during the scientific revolution were motivated by several profound changes in music (Cohen, 1984). First, music became polyphonic: more than one voice or instrument simultaneously performed different parts. As a result, musical harmony became central to music, and the consonant combination of multiple musical parts required a proper tuning system. Second, new intervals (thirds and sixths) became accepted as being consonant. The English composer John Dunstable popularized these intervals in the early 15th century. However, Pythagorean tuning ignores these intervals. New approaches to tuning had to ensure the consonance of these new intervals. Third, fixed intonation instruments—instruments with notes tuned to specific pitches that cannot be altered during performance, like the modern piano—were more central to music. In addition, the notes of these instruments ranged over several octaves.

All of these developments created a need for a practical solution to the Pythagorean tuning discrepancy. In general, mechanical philosophers addressed this problem by developing new methods for dividing the octave into smaller intervals to define sets of available musical notes: they invented tempered scales. The primary goal of a tempered scale is to remove the Pythagorean comma (Donahue, 2005). This is accomplished by ensuring that, from some starting note with a frequency f, the note an octave higher has a frequency of 2f. In other words, the octave has primacy. Then, some other notes are added; these notes conform to the Pythagorean interval ratios. Finally, the remaining notes are included. These notes involve intervals whose ratios necessarily depart from the Pythagorean ideals. In other words, a tempered scale is a compromise: it ensures that some intervals conform to the Pythagorean ratios but deforms other intervals to eliminate the Pythagorean comma.

One example of a tempered scale is called just intonation; it was explored by mathematicians of the early 17th century (Barbour, 1972). Just intonation produces consonant harmonies involving fifths and thirds, but at the same time can produce very dissonant harmonies for other musical intervals. Another example of a tempered scale is called mean-tone temperament, which was invented by Pietro Aron in the early 16th century (Barbour, 1972). Mean-tone temperament distorts perfect fifths. This tuning favours thirds over fifths, which is the opposite of what is found in Pythagorean tuning (Donahue, 2005).

One problem with tempered scales is that they are defined with respect to a particular musical key (i.e., a particular starting frequency f). This means that the notes that define a scale in one musical key differ from those that define a scale in another. To perform the same piece in a different musical key (i.e., to musically transpose the composition), the instrument must be re-tuned. Another tempered scale, equal temperament, offered a solution to this problem.

Equal temperament, first mentioned by French philosopher, theologian, and mathematician Marin Mersenne in 1636, divides the octave into 12 equal segments, each representing the musical interval of a semitone. Equal temperament solves the problem of musical transposition because one can move a composition from key to key without having to re-tune an instrument. As a result, it became an ideal tuning for keyboard instruments (Isacoff, 2001, 2011).

However, equal temperament brought with it a new set of problems. By dividing the octave into 12 equal segments, it introduces irrational interval ratios. Consider some base pitch with frequency f. The pitch an octave higher has a frequency of 2f, or, to make an explicit link to the mathematics of equal temperament, a frequency of 212/12f. In equal temperament the tone that is a semitone higher than f will have the frequency 21/12f, the tone two semitones higher than f will have the frequency 22/12f, and so on. The appearance of irrational ratios—defined as 2 raised to some x/12 power—had two negative consequences. First, calculating the desired frequencies—a requirement for actually tuning an instrument to equal temperament—was difficult. A number of specialized tools and methods had to be invented in the 18th century to deal with this problem (Scimemi, 2002). Equal temperament existed as a theoretical notion for a long time before it became practical to use it to tune instruments in the late 19th century. Second, the presence of irrational ratios meant that many musical intervals deviated enough from the Pythagorean ideals to sound less consonant. Indeed, the distorted ratios found in any tempered scale challenge the Pythagorean notion of consonance, as does the new musical aesthetic that considers intervals like thirds to be consonant.

This latter issue raises a theoretical problem concerning music that was also a concern of mechanical philosophers. Cohen (1984) calls it the problem of consonance. While the Pythagoreans identified particular frequency ratios as defining consonant musical intervals, they had no account of why this was so. The scientific revolution developed a very popular answer to this question that related consonance to the physical properties of sound. This answer is known as coincidence theory.

1.2.2 Coincidence Theory

In 1558, Gioseffo Zarlino explained the consonance of certain Pythagorean ratios with what is known as his scenario (Cohen, 1984). According to Zarlino, the consonant Pythagorean ratios all involve the first six integers; the set of integers from 1 to 6 defined the scenario. Zarlino proposed that any ratio of numbers that belonged to the scenario would be consonant, and supported this proposal with a variety of mystical arguments about why the members of the scenario were perfect numbers. In spite of this appeal to mysticism, coincidence theory was the most popular theory of consonance that emerged during this period (Cohen, 1984). It was then adopted, extended, and popularized by many leading figures of the scientific revolution including Galileo, Mersenne, Descartes, and Euler.

Coincidence theory attempts to establish the physical basis of consonance (Cohen, 1984). It begins with the observation that a plucked string produces sound by striking or percussing the surrounding air. We hear sound when these percussions reach our ears. Coincidence theory then recognizes that strings vibrating at different frequencies generate percussions at different rates. In some instances, the percussions generated by two different sound sources will reach the ears at the same time; these coincident percussions will be pleasing to the ear, or consonant. When the coincidence of percussions diminishes, so too will consonance. In other words, coincidence theory linked the purely mathematical view of consonance developed by the Pythagoreans to physical properties of sound vibrations, as they were understood in the 16th and 17th centuries.

1.3 Psychophysics of Music

The study of music during the scientific revolution produced new approaches to tuning and developed a theory that attempted to explain consonance in terms of coincident patterns of vibrations (Cohen, 1984). From this perspective, musical properties were physical properties of the world that could be studied scientifically and described mathematically. For instance, consonance was related to physical vibrations, and was not a construction of the mind.

1.3.1 Psychophysics

The systematic psychological study of consonance did not begin until the latter half of the 19th century; the golden age of this research occurred between 1840 and 1910 (Hui, 2013). It was during this period that psychophysical explorations of music began.

Psychophysics was invented by Gustav Fechner, who began by searching for relationships between physical properties of stimuli and the properties of the experiences that they produced (Fechner, 1966/1860). The psychophysical study of music began with attempts to identify universal mathematical laws that related physical properties of sound and music to mental properties of musical experience (Hui, 2013).

To Hui (2013), the psychophysical study of sound sensation is in turn related to an understanding of musical aesthetics. Her general argument is that the psychophysical study of music involved a constant tension between universal psychophysical laws and individual aesthetic responses, a conflict that could only be resolved by acknowledging that psychological processes contribute to the experience of music. Crucially, this meant that the physical properties of sound were not the only determinants of such phenomena as consonance.

At the same time, new theories of tuning became practical realities. When psychophysicists began to study music, equal temperament was rising in popularity (Isacoff, 2001). In short, psychophysicists studied music at a time when radically new notions of consonance and dissonance were emerging. How could one reconcile a natural science of music or consonance with the many changes in music and musical preference arising in the latter half of the 19th century?

1.3.2 On the Sensations of Tone

One of the most influential accounts of musical psychophysics (Hiebert, 2014) was Hermann Helmholtz’s book On the Sensations of Tone as a Physiological Basis for the Theory of Music (Helmholtz & Ellis, 1863/1954). Helmholtz wove three different threads together. One concerns the physics of sound, musical sound in particular, and includes details about various devices for sound production and measurement. A second concerns the physiology of hearing, and includes a detailed account of the structure and function of the cochlea and the basilar membrane. A third concerns the implications of the properties of sound, and the physiology of hearing, for the perception of music, and provides detailed discussions of different tunings and musical aesthetics.

The core idea that Helmholtz uses is a reinvention of coincidence theory. Coincidence theory focused on the relationship between two pure tones. Helmholtz recognized that in almost every case a musical instrument will not generate a pure sine wave with frequency f. Sympathetic vibrations add to this fundamental frequency additional sine waves at various frequencies defined by the octave (e.g., 2f, 3f, 4f, and so on). These additional frequencies, called partials or harmonics, occur at weaker intensities than the fundamental, and tend to be weaker and weaker as the partial frequency becomes higher and higher. Helmholtz’s theory of consonance emphasized harmonic interference.

When two musical sounds occur together, not only do their fundamental frequencies interact but their partials interact as well. For Helmholtz, interference patterns among all of the frequencies that are present cause the consonance of combined musical tones. Interference between partials would result in beats that produced an effect of roughness; consonant tones had little roughness; dissonant tones had more roughness because of more interference among partial frequencies. Importantly, these interference patterns affected the perception of music because they were detectable by the physiological mechanisms of hearing (Hui, 2013).

Helmholtz provided a detailed analysis of beat patterns related to a variety of musical intervals, and used this analysis to make fine-grained predictions about consonance. His quantitative account of consonance coincides with a variety of experimental studies of consonance conducted in the 20th century (Krumhansl, 1990a). Helmholtz used his new theory of consonance to inform musical aesthetics and to defend his personal views on the beautiful in music. Helmholtz was a champion of just intonation (Hiebert, 2014; Hui, 2013) and was highly critical of the rising popularity of equal temperament. He believed that just intonation produced music that was far more consonant than was possible in equal temperament. Hui points out that Helmholtz makes the physiological argument that just intonation produces music more consistent with the physiology of hearing than is produced using equal temperament. However, Hui also notes that Helmholtz recognized that there were individual differences in musical aesthetics. How was this to be reconciled with his detailed psychophysical theory?

Helmholtz argued that while his psychophysical account of the sensations of tone could inform the elementary rules of musical composition, these rules of composition are not natural laws. Different listeners were capable of enduring (and appreciating!) different degrees of roughness. Individual differences in musical tastes were a function of experience, culture, and education. One’s musical knowledge, culture, or experience could influence the aesthetic experience of consonance and dissonance.

1.3.3 Individual Contributions

Helmholtz’s influential studies suggest that musical perception is only based in part on universal laws involving the physics of sound and the physiology of hearing. Music perception is also affected by aesthetic considerations that stem from an individual’s own culture, experience, and expertise. This position coincided with the tenets of musical Romanticism, which arose during the same era as musical psychophysics. Musical Romanticism emphasized individuality. That is, in Romanticism individual composers aimed to communicate their personal emotions and imaginations; Romanticism also heralded the virtuoso instrumentalist (Longyear, 1988). Informed by the theories of music critics like Eduard Hanslick (Hanslick, 1854/1957), audiences began to carefully listen to music—not just to enjoy it but also to understand it (Hui, 2013). Such intellectual listening depends heavily upon an individual’s musical tastes and knowledge.

One consequence of psychophysicists accepting that an individual’s mind or knowledge greatly affects musical perception is the need to discover the psychological laws that govern these influences. This research goal is central to the cognitive study of music that began in the middle of the 20th century. The cognitive approach recognized that there could exist abstract laws governing perception and thinking, but these laws are in turn linked to physical mechanisms. In the next section, we briefly introduce this cognitive approach and then describe the kinds of theories it produced to account for musical perception.

1.4 From Rationalism to Classical Cognitive Science

1.4.1 Rationalism

Mechanical philosophy was not only interested in explaining the natural world but also in explaining the mind, as exemplified in the 17th century philosophy of René Descartes (Descartes, 1637/2006, 1641/1996). To establish a rigorous philosophy, Descartes adopted the mathematical approach of derivation from axioms. The basis o...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• List of Figures
• List of Tables
• Acknowledgements
• Overture: Alien Music
• Chapter 1: Science, Music, and Cognitivism
• Chapter 2: Artificial Neural Networks and Music
• Chapter 3: The Scale Tonic Perceptron
• Chapter 4: The Scale Mode Network
• Chapter 5: Networks for Key-Finding
• Chapter 6: Classifying Chords with Strange Circles
• Chapter 7: Classifying Extended Tetrachords
• Chapter 8: Jazz Progression Networks
• Chapter 9: Connectionist Reflections
• References
• Index