The journal of the International Society for the Interdisciplinary Study of Symmetry, 2012/1-2, Editors: Marakova/Nagy, pp. 82-85, ISSN 1447-607X

VISUALISATION OF MUSICAL INTERVALS: PATTERNS, FRACTALS AND COINCIDENCES

ANGELA LOHRI AND SUSANNE RELL

Name: Angela Lohri, violinist and PhD student. Address: International Centre of Harmonics, University of Music and Performing Arts Vienna, Lothringerstrasse 18, A-1030 Vienna (Austria). E-mail: angela.lohri@gmx.ch Fields of interest: harmonic research, mathematics in music theory, music acoustics, perception of hearing, musical intonation, combination tones, performing arts. Publications: - (2010a) Symmetrien in Partialtonstrukturen von Zweiklängen und die Entstehung von Kombinationstönen, in: Symmetry; Art & Science, The journal of the International Society for the Interdisciplinary Study of Symmetry 2010, Editors: Lugosi/Nagy, Budapest, pp. 154-157 - (2011) Combination Tones in Violins, in: Archives of Acoustics, 36, No. 4, pp. 727-740, ISSN: 0137-5075 (co-authors: Sandra Carral, Vasileios Chatziioannou). Name: Susanne Rell, Artist and Teacher (Vienna, Austria) E-mail: susannerell@hotmail.com; Home-page: www.rell-patterns.at Fields of interest: Painting, Drawing, Animation. Grants: Kunstankauf durch die Stadt Wien, 2003. Publications and exhibitions: - (2003) Solo exhibition in Artothek-Galerie, Vienna, Austria - (2006) Group exhibition 3 Generationen, 3 Regionen, Burg Schlaining, Austria - (2012) Colors and Shapes to Arvo Pärts Music, Seven Paintings out of the Series "Variations on a Theme" by Susanne Rell, 50 Jahre Promethée, Kazan 6. - 8. April 2012, 168-172 (Russian), 420-424 (English). Abstract: Harmonic sounds produce regularly occurring auditory nerve impulses in the inner ear. Based on current knowledge on hearing perception the authors have developed a model which aims at showing the mathematical nature of simultaneous musical intervals. The model represents the harmonic properties within auditory nerve stimuli and takes into account place and time of their appearance. It is programmed using the graphical framework “Processing” and based on the programming language “Java”. The program allows a continuous shifting between intervals thanks to a real time function. This tool serves as a visual approach to uncharted zones located between the “islands” of just musical intervals (intervals which are generally determined by simple frequency ratios).

Harmonic sounds generate characteristic neuronal firing patterns. In a slightly modified representation, the neuronal impulses generated by musical sound bear similarities to fractals. The model offers possibilities to study the coincidences caused by a particular vibration ratio and helps to understand the process of pitch perception. When sound waves impact the human ear they are not immediately transformed into a hearing sensation, but their frequencies continue to exist in our aural and neuronal system till the final synthesis into a pitch perception (Langner 2007, Tramo et al. 2001, Meddis/Hewitt 1991a and b). This remarkable observation has motivated the authors to design a model that represents frequencies of harmonic sounds on a neuronal level. In the case of harmonic sounds, the partial tones are multiples of the fundamental frequency and build within them vibration ratios corresponding to an arithmetic series. This principle of relations between whole numbers is maintained in the inner ear and in the brain. The information has simply been processed into electric impulses that occur at the same frequency as the original sound waves. Figure 1 exemplifies one isolated period of a harmonic sound. Thirty partial tones are taken into consideration. This kind of representation was already used by Horst-Peter Hesse in 1989.

Figure 1. Model of a neuronal firing pattern caused by a harmonic sound consisting of thirty partial tones. The time axis goes horizontally, the location of a fired nerve impulse is represented vertically. For a fundamental frequency of 200 Hz the above period would last 5 milliseconds. Pitch is a fundamental characteristic of harmonic or periodic sounds. From a biological point of view, pitch is a sign of creatures with intent to interchange signals (Langner 2007, pp. 10-11). Regarding all the simultaneous vibrations around us, it is crucial that information is being reduced and attributed to specific objects. In this sense, a pitch is a condensed information of multiples of a frequency that emanate very probably from a common source.

Figure 2. Harmonic sound composed of 60 partial tones. The representation is slightly modified compared to Figure 1. The basilar membrane location of occurring stimuli is not reported on the graph, but the amount of auditory nerve impulses generated at the same time is added up instead. The height of the lines corresponds to the amount of impulses and may therefore be associated with the intensity of a particular stimulus.

Figure 3. Auditory nerve impulses produced by two simultaneous sounds with the vibration ratio 4:5 (major third) and corresponding partial tones up to the number 60. In the upper half of the graph, one can see the stimuli originating from the higher pitch (5). In the lower half are represented the stimuli deriving from the lower pitch (4). The arrow indicates the coincidence of the two periods. This coincidence, in turn, may be perceived as an additional, lower pitch.

Figure 4. Results of the model developed by Meddis and Hewitt (1991a and b) based on measurements of the activity of auditory-nerve fibers. (b) The hair cell response and (c) the individual spike-interval histogram show parallels to the neuronal firing pattern showed in Figure 1. (Reproduced with permission of American Institute of Physics for the Acoust. Society of America in the format Journal via Copyright Clearance Center) Our newly developed model evidences that the power of the fundamental period is a mathematical consequence of the presence of harmonic overtones (Figure 2). The more overtones added, the stronger the sensation of pitch. Figure 2 and Figure 3 show that musical intervals have a geometry akin to fractals. Our model is a mathematical abstraction compared to the much more detailed model of Meddis and Hewitt (1991a and b) that takes into account several stages in the hearing process (Figure 4). However, both models lead to accordant results regarding the mathematical nature of harmonic sounds. Current knowledge in the field of auditory physiology and neurology reveals that the mechanism of pitch detection is strongly dependent on the occurring of harmonic proportions. References Hesse, Horst-Peter (1972). Die Wahrnehmung von Tonhöhe und Klangfarbe als Problem der Hörtheorie,

Arno Volk Verlag, Köln Hesse, Horst-Peter (1989). Grundlagen der Harmonik in mikrotonaler Musik, Edition Helbling, Innsbruck Langner, Gerald (2007). Die zeitliche Verarbeitung periodischer Signale im Hörsystem: Neuronale

Repräsentation von Tonhöhe, Klang und Harmonizität, Zeitschrift für Audiologie, Vol. 46, No. 1, pp. 8–21

Meddis Ray; Hewitt Michael J. (1991 a and b). Virtual pitch and phase sensitivity of a computer model of the auditory periphery, I: Pitch identification, II: Phase sensitivity, JASA 89, pp. 2866–2894

Tramo, Mark Jude et al. (2001). Neurobiological Foundations for the Theory of Harmony in Western Tonal Music, in: Zatorre et al.: The Biological Foundations of Music, Annals of the New York Academy of Sciences, Vol. 930, June 2001