(Sensory) Dissonance is a collection of research materials on the relationship between room acoustics, musical tuning systems and different perspectives on “consonance“ and “dissonance“. I wrote a short essay and software tools to outline a possible method of deriving musical tuning systems from the acoustic properties of specific enclosed spaces (for the production of highly site-specific music).

Essay

Last year, I started to develop a strong interest in the topic of musical tuning systems. A musical tuning system is, by definition, a system used to determine which pitches to use when playing a musical instrument, which in turn determines the frequencies produced by that instrument. My interest in tuning systems grew out of a number of different, intersecting concerns in my practice.
For one, I have a longstanding practice of working with digital audio as an artistic material. Most of the conventional software “tools of the trade” used by musicians and artists for audio production work only offer the possibility of working with the musical tuning system predominant in contemporary Western classical music – namely, the division of the octave into twelve equally spaced intervals (I will provide a more exact description of this system later in this text). This (in some sense arbitrary) built-in assumption struck me as a significant restriction: After all, different musical traditions have used highly varied approaches to tuning in order to fulfill very different musical purposes, with some of them mathematically subdividing the octave in very different ways than the conventional division into twelve equal parts and others tuning instruments purely according to ear (taking into account the natural acoustical properties of the instruments in use).
While I knew of this blind spot in my musical practice, it made sense considering my instrumental background. The only acoustic instrument I can play at least somewhat competently is the piano. I learned to play the instrument by taking classical piano lessons during my teenage years, which consisted of rote memorization and recitation of Western classical repertoire pieces. Little importance was placed on two acts I now appreciate most about instrumental practice: Firstly, opening up the instrument to see what actually happens “under the hood” - both literally and metaphorically speaking – and being able to modify the instrument according to my needs or interests, and secondly, taking the time to listen closely to the actual, physical sounds created as a result of playing the instrument (as opposed to listening for mere musical “note events”). I eventually found myself feeling alienated by the restricted timbral and rhythmic possibilities of the instrument and my role as an instrumental performer: In the (common practice era) Western classical arrangement, the pianist is only a “user” controlling the instrument via the keyboard and pedals. The pianist is generally not responsible for the tuning or technical maintenance of the instrument, which both require expertise, and they usually lack the knowledge to do so. The task of tuning the instrument is therefore relegated to being a specialist activity from which the actual performers are excluded. I felt that this had a big effect on the relationship of pianists to their instruments: the act of tuning an instrument is one of intense listening, requiring a lot of focused attention on simple intervallic relationships between tones. Slight changes in pitch correspond to significant changes to in sound, as the listener is made aware of a physical phenomenon called beating, wherein the presence of two closely spaced tones produces a third oscillation at the speed of their difference in frequency.
At the time, I also took an interest in acoustics. I was especially interested in the idea of matching music to the acoustical properties of the surrounding space it is played in, as opposed to doing it the other way around (think large concert halls being purposely built to best accommodate the sound of an orchestra).
This brings me to the concept of dissonance, which in my estimation is the primary reason behind the diversity of tuning systems encountered in different musics. Dissonance is a property of musical intervals, which are “distances” between two different pitches. More specifically, dissonance is the degree to which an interval sounds unpleasant or rough. Historically, tuning systems in the Western classical tradition include both dissonant and consonant intervals, which server different musical functions, but differ in the exact construction of the systems. In the early 20th century, German physicist Hermann von Helmholtz made a hypothesis on how the perception of consonance and dissonance is produced in the human auditory system: Informed by Fourier theory, he regarded complex tones as superpositions of sinusoidal components, and suggested that the perception of dissonance was produced by rapid beating between adjacent, closely-spaced components within the tone. This was experimentally investigated by various studies, most famously by Dutch Psycholinguists Reinier Plomp and Willem Levelt, who presented an experimental dissonance measure based on the intervallic distance of two pure sinusoidal components. Since instrumental tones are more complex than pure sine tones, this measure is not enough to determine the dissonance of an interval played on an actual instrument, even in theory. During my research, I encountered the work of music theorist William Sethares, who titled categorized the Helmholtz measure of dissonance as a measure of “sensory” dissonance and developed the idea further: by considering all of the pairwise interactions (i.e. dissonance amounts) between the sine components of two tones, he was able to compute a dissonance curve for any two instrumental timbres. This dissonance curves describes the amount of sensory dissonance of every interval between the two instrumental tones. I want to emphasize that my interest in this theoretical framework did not come about because of its rather universalizing explanatory power, but it’s specific applicability to my situation of wanting to derive tuning systems from acoustics.

During the process of researching these topics, I wrote a number of classes in SuperCollider, my audio programming language of choice, to implement the individual steps outlined above: A simple program for synthesizing exponentially swept sine chirps and capturing the room impulse response, classes for calculating dissonance curves for arbitrary timbres and a code snippet for spectral peak-finding. The resulting code is available online as free software under the link below.

In more technical terms, a procedure for deriving a tuning system for a specific room might consist of the following steps:

1. Record the impulse response of a room. To do this, “excite” the room by playing an impulse signal through loudspeakers (or, more ideally, an exponentially swept sine chirp, which is equal to an impulse “smeared” over time) and record the resulting reverberant tail until it disappears into the room’s noise floor. This recording encodes the acoustic character of the space relatively well: the reflected source signal (impulse) is sensitive to the details of the environment’s geometry, physical dimension and material properties.
2. Compute the spectrum of the impulse response by computing the Fourier transform of the recorded room response.
3. Find the room modes: that is, find the set of existing resonant frequencies that a room produces when excited by an acoustic impulse. These are the N highest peaks in the computed spectrum.
4. Using the N loudest room modes with their corresponding amplitude, compute a sensory dissonance curve.
5. A choice of particular frequencies that make up the final tuning can now be made according to the desired amount of dissonant/consonant intervals and number of pitches in the final tuning. The extent of the sensory dissonance of a particular interval is directly related to its position on the dissonance curve: minima of the curve correspond to intervals of minimal dissonance, and vice versa.

Note that this procedure hinges on some simplifying assumptions. For example, the room is assumed to be an LTI (linear, time-invariant) system – this means that the room’s acoustical properties are assumed to not significantly change over the short time it takes for the response’s reverberant tail to decay (time invariance). Additionally, the summed room responses of two separate given “inputs” (sonic events happening within the space) is assumed to be roughly equal to the room response of the sum of those inputs (linearity).

Downloads

Code (14KB .zip)