The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician

by Kyle Gann

"Tuning is the secret lens through which the history of music falls into focus," says Kyle Gann. Yet in Western circles, no other musical issue is so ignored, so taken for granted, so shoved into the corners of musical discourse. A classroom essential and an invaluable reference, The Arithmetic of Listening offers beginners the grounding in music theory necessary to find their own way into microtonality and the places it may take them. Moving from ancient Greece to the present, Kyle Gann delves into the infinite tunings available to any musician who feels straitjacketed by obedience to standardized Western European tuning. He introduces the concept of the harmonic series and demonstrates its relationship to equal-tempered and well-tempered tuning. He also explores recent experimental tuning models that exploit smaller intervals between pitches to create new sounds and harmonies. Systematic and accessible, The Arithmetic of Listening provides a much-needed primer for the wide range of tuning systems that have informed Western music. Audio examples demonstrating the musical ideas in The Arithmetic of Listening can be found

Genres: Music

312 pages, Paperback

Published September 4, 2019

About the author

Kyle Gann is Associate Professor of Music at Bard College, a composer, and former new-music critic for the Village Voice. He lives in Germantown, NY.

Reviews

[Gexaedron · Rating 5 out of 5]

Greatest book for xenharmonic heads

[El Lector Enmascarado · Rating 4 out of 5]

Buscaba una introducción a la música microtonal y este volumen ha superado con creces mis expectativas. Me pierdo en cuanto el autor comienza a hacer malabarismos con las fracciones, y hay páginas enteras que sigo sin comprender después de haberlas leído dos o tres veces, pero salgo del libro con las ideas muy claras sobre los distintos temperamentos que se han utilizado en la historia de la música europea y sobre las principales corrientes de experimentación xenharmónica. También he podido hacerme una lista de los doce o quince compositores más relevantes de esta otra tradición, que está produciendo tanto la música más estridente como la más estremecedora. Eso sí: conviene tener claras antes de empezar algunas nociones básicas de armonía: tónica, dorminante, armónicos, intervalo, escala diatónica, escala pentatónica, modos, ciclo de quintas y cosas así.

No sé si mi ejemplar ha sido impreso on demand, pero seis o siete de las imágenes han sido reproducidas defectuosamente, con grandes franjas en negro o en gris que las dejan inservibles, como cuando a una impresora láser se le atraganta la lectura de un documento. Da rabia, porque el volumen no es precisamente barato.

[Daan Vanreybrouck · Rating 5 out of 5]

Banger, 9.5/10

[Peter · Rating 5 out of 5]

I had the opportunity to read this introduction to tuning theory and microtonal music during an independent study course concerning the music of James Tenney. The following paragraphs are excerpted from my final paper for the course.

"Kyle Gann’s The Arithmetic of Listening provided a wonderful guide through the terrain of microtonal theory and composition during my studies. I found Gann’s explanations of tuning history to be enlightening, but I was most helped by his clear explanation of separate tuning strategies based on the use of either just intonation or equal divisions of the octave. These concepts helped me to better classify some of the microtonal works with which I was familiar in terms of the tuning techniques and intellectual approaches they represent. Additionally, this dichotomy of just intonation and EDO approaches helped with the process of assimilating the subcategories of each— Gann’s progression through 5-limit, 7-limit, 11-limit, 13-limit, etc. systems made more sense given the overall context of the just intonation approach.

Perhaps the most enlightening aspect of The Arithmetic of Listening was the connection that Gann drew between the exploration of higher harmonics derived from just intonation and the use of EDO-derived temperaments to approximate these harmonics in a slightly more accessible or practical manner. Before reading Gann’s work, I was mystified as to why microtonal composers so frequently favored the use of 19edo, 31edo, and 53edo in their works. Gann’s clear presentation of the properties that these pitch systems possess (such as the useful approximation of the 3/2 perfect fifth among other pure intervals) was particularly helpful for explaining the widespread use of these pitch systems. Additionally, his presentation of Bosanquet’s theory of temperaments and Regular Temperament Theory are especially helpful for the process of generalizing and investigating similar EDO-derived systems.

I also appreciated the examples of performance practice that Gann outlined for several of the works mentioned in the book’s interludes. Some particularly useful examples included his analyses of movements from Ben Johnston’s string quartets nos. 4 and 7 and Toby Twining’s Chrysalid Requiem. Gann carefully outlined the rehearsal and performance techniques that are needed to assimilate the wide-ranging pitch relations of Johnston’s seventh quartet. Because the work is clearly at the cutting edge of present-day performance practice, it was particularly helpful to have an inside look at how the performers strategically familiarize themselves with the material. Similarly, I was struck by the ingenuity at work in the performance of Twining’s Chrysalid Requiem. While I was familiar with the work and its remarkable demands in terms of pitch accuracy, I didn’t know that the performers use headphones with a synthesized track that provides their pitches. These sorts of performance practice techniques go a long ways towards making this sometimes difficult music easier to compose and execute in a practical manner."

While the chapters on historical tunings were not as immediately applicable to my studies, they offered a necessary background to tuning systems discussed later in the work; in total, the book was a tremendously helpful, insightful, and humorous guide to the world of tuning theory and microtonal music.

[FL · Rating 3 out of 5]

I have mixed feelings on this book. On one hand, I'm glad that it exists as a readable textbook. On the other hand, I disagree with Gann at practically every turn.

He's right that tuning theory is unjustifiably neglected. But this book doesn't seem to be trying to teach it so much as it seems to want us to learn the TRUTH. Gann writes it like a missionary tract, telling us about how most people are wrong, and only a few composers in 20th-century California, pretty much, do it right. His focus becomes so narrow that he ignores most of the microtonality that's actually caught on: spectral music, barbershop, or even quarter-tones.

In making it possible to directly learn tuning theory from a textbook, this book performs a wonderful service. But in solidifying a bizarre narrative, it also does a disservice.

[RustyShack · Rating 4 out of 5]

Far more detailed than I need, but very happy to have read it even if my only takeaway is that the author constructs (what seems like) a great argument that equal temperament has come to dominate for no good reason. Looking forward to trying Young 1799!