BEYOND PYTHAGORAS Paulina Sundin Monty Adkins Adrian Gierakowski Music Department University of Huddersfield Music Department University of Huddersfield Electronic Engineering and Computer Science Department Queen Mary University ABSTRACT This paper discusses the new harmonic possibilities enabled through the implementation of Sethares’ theory of the dissonance curve in Max/MSP and its use in a live electronic composition by two of the authors. 1. INTRODUCTION Much of the research informing the authors’ composition Shards (2012) has been informed by the notions of consonance and dissonance outlined by William A. Sethares in his book ‘Tuning, Timbre, Spectrum, Scale’. Sethares writes that the notion of sensory consonance and dissonance has two implications. Firstly, individual complex tones will have an intrinsic or inherent dissonance: Since dissonance is caused by interacting partials, any tone with more than one partial inevitably has some dissonance. This is a stark contrast to all the previous notions, in which consonance and dissonance were properties of relationship between tones. [1] Secondly, that consonance and dissonance will depend not only on the interval between tones, but also on the spectrum of the tones used, Since intervals are dissonant when the partials interact, the exact placement of these partials is crucial. [2] The latter is something that Pierce was already aware of more than thirty years earlier in the 1960s when working with his arbitrary scales and correlating sounds [3]. Based on these experiments, our research examines how sounds with other kinds of spectral relationships work together to derive a new sense of harmonic consonance and dissonance in our electroacoustic compositional practice. 2. SETHARES DISSONANCE CURVES Composer and theorist Harry Partch begins chapter nine of his ‘Genesis of a Music’ with the following, According to Galileo, “agreeable consonances are pairs of tones which strikes the ear with a certain regularity; this irregularity consists in the fact that the pulses delivered by the two tones, in the same interval of time, shall be commensurable in number, so as not to keep the eardrum in perpetual torment, bending in two different directions in order to yield to the ever-discordant impulses.” The fairly “perpetual” torment which is our heritage in Equal Temperament has long obscured this aural axiom. [4] Partch’s work and research is based on a tradition dating back to the ancient Greeks, the Pythagoreans and Ptolemy in particular, through music theorists and mathematicians such as Zarlino, Rameau, Galileo, Kepler, Helmholtz until the early 1900s. Partch was interested in creating music based on scales with more than 12 notes per octave. He built his own instruments, such as the Chromelodeon, a reed organ, in order to play the music he had composed with a scale with 43 scale steps per octave. He tuned his reed organ with “no other aid than the ability of the ear to distinguish pulsations ‘commensurable in number’ and those which bend its tympanum ‘in two different directions’” [5] in this 43 tone per octave scale with the focus on Just Intonation. By doing this he, as summarised by Sethares, “classified and categorised all the 43 intervals in terms of their comparative consonance”. [6] A consonance curve portrays the perceived consonance and dissonance versus musical intervals. Helmholtz’s roughness curve [7], Plomp and Levelt’s consonance curve [8] as well as Partch’s ‘One Footed Bride’ [9] are examples of dissonance curves. All of these dissonance curves show how the ear perceives sounds with harmonic or no (pure sine tones) spectra as sensory consonant at certain traditionally “consonant” scale steps, if the scale is tuned in Just Intonation (rather than the equally tempered tuning). The points of maximum sensory consonance occur on these scale steps, which shows the correspondence between spectrum and scale. Sethares’ dissonance curve is however, mathematically constructed to portray the perceived consonance and dissonance versus musical intervals with sounds containing any spectra. A comparison with Sethares’ dissonance curve (Figure 2) and an experiment carried out by Kameoka and Kuriyagawa [10] show that Sethares’ calculations are related to the results of their experiment (Figure 1). In Kameoka and Kuriyagawa’s third experiment presented in 1969, chords of two identical complex tones were used. One of the tones containing eight partials was fixed at 440 Hz and the other tone was played together with the first from 440 Hz to 880 Hz (an octave) divided into fifteen steps. The degree of dissonance was calculated for each step according to the circles seen in Figure 1. The results from the experiment showed that the degree of consonance and dissonance seemed to occur on the same minima and maxima steps that were calculated in advance.