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The Arithmetic of Listening: Tuning Theory & History for the Impractical Musician – Kyle Gann

Thoughts: I read The Arithmetic of Listening for a seminar I sat in on in Winter 2021 on Tuning and Temperament. Its subject matter is quite niche, but for those interested in alternate/extended musical tuning systems, Gann’s book is both mind- and ear-opening.

(The notes below are not a summary of the book, but rather raw notes - whatever I thought, at the time, might be worth remembering.)

Gann, Kyle. 2019. The Arithmetic of Listening: Tuning Theory & History for the Impractical Musician. University of Illinois Press.

You can listen to audio examples for the book at


1. The Cosmic Joke

2. The Harmonic Series

3. Generating Scales

Interlude A: Ptolemy and Ancient Greek “Parts”

4. The Pythagorean Scale

Interlude B: Guillaume de Machaut’s Notre Dame Mass

5. The Five Limit, the Second Dimension

Interlude C: Some Modern Five-Limit Notions

6. Meantone Temperament and the Primacy of Thirds

Interlude D: Meantone Examples

7. Well Temperament and Key Color

Interlude E: Bach, Beethoven and Temperament

8: Twelve-Step Equal Temperament

9: The Seven Limit and Johnston Notation

Interlude F: La Monte Young’s The Well-Tempered Piano

Interlude G: Ben Johnston’s String Quartet No. 4

10: The Eleven Limit and the Fourth Dimension

Interlude H: Harry Partch

The Thirteen Limit and Beyond

Interlude I: Ben Johnston’s String Quartet No. 7, Movement 3

Interlude J: Kyle Gann’s Hyperchromatica

Interlude K: Toby Twining’s Chrysalid Requiem

Chapter 12: Non-Twelve-Divisible Equal Temperaments

Interlude L: Nicola Vicentino’s Archicembalo

Chapter 13: Twelve-Based Equal Temperaments

Interlude M: Some Quarter-Tone Impressions (Hába, Ives, Wyschnegradsky)

Interlude N: Ezra Sims’s String Quartet No. 5

Chapter 14: A Few Numbers Drawn from Non-Western Musics

Chapter 15: Brief Miscellaneous Thoughts

Posted: Dec 21, 2021. Last updated: Dec 21, 2021.