Chapter 12 Further Reading: Tuning Systems — The Mathematics of Consonance and Compromise

Primary Sources and Historical Documents

Werckmeister, Andreas. Musicalische Temperatur (1691). Reprinted in various anthologies. The foundational document of well temperament. Werckmeister describes three temperament systems, of which Werckmeister III became most influential. Available in translation in various music theory anthologies. Essential primary source for understanding what "well temperament" meant in Bach's time.

Partch, Harry. Genesis of a Music (1949; 2nd ed. 1974, Da Capo Press) Part autobiography, part manifesto, part physics treatise. Partch explains the mathematics of 11-limit just intonation, describes his 43-tone scale in detail, explains his instrument designs, and articulates his aesthetic philosophy of "corporeal music." The most complete alternative to the Western tuning tradition produced by a single mind in the 20th century. Chapter 4 (on the harmonic series) and Chapter 8 (on his scale) are directly relevant to this chapter.

Barbour, J. Murray. Tuning and Temperament: A Historical Survey (1951; reprint 1972, Da Capo Press) The standard historical survey of Western tuning systems from ancient Greece to the 19th century. Dense, systematic, and authoritative. A valuable reference for the historical dimension of Chapter 12.

Acoustics and Physics

Helmholtz, Hermann von. On the Sensations of Tone (1863; English trans. 1954, Dover) Chapters 10-14 cover consonance, dissonance, and the relationship between interval ratios and beating. Helmholtz's theory of consonance (simple ratios minimize beating) remains the foundation of modern accounts. His analysis of different historical tuning systems is thorough and physically grounded.

Jorgensen, Owen. Tuning (1991, Michigan State University Press) Comprehensive technical reference covering the history and mathematics of Western tuning systems. Provides precise instructions for tuning historical temperaments by ear using beats. Invaluable for anyone who needs to actually tune instruments in historical systems.

Sethares, William A. Tuning, Timbre, Spectrum, Scale (2nd ed., 2005, Springer) The most sophisticated modern treatment of the relationship between instrument timbre and optimal tuning system. Sethares demonstrates that consonance depends on the specific overtone structure of the instruments, not just on abstract ratios. His work implies that Indonesian gamelan instruments might actually sound most consonant in the gamelan's own non-standard tuning — the physics of consonance is relative to the timbre, not universal. Mathematically demanding but essential.

Historical and Musicological Studies

Lindley, Mark. "Temperament," in The New Grove Dictionary of Music and Musicians (6th ed., 2001, Oxford University Press) The authoritative reference article on the history of tuning systems in Western music. Technically precise and historically comprehensive. Available through most university library databases.

Lehman, Bradley. "Bach's Extraordinary Temperament: Our Rosetta Stone" (2005). Early Music 33(1): 3-23 and 33(2): 211-231. The most provocative recent contribution to the WTC tuning debate. Lehman argues that the spiral decoration on Bach's title page encodes his tuning system. Published in a peer-reviewed musicology journal and subject to extensive subsequent debate. Read alongside the responses published in Early Music in the same year.

Duffin, Ross W. How Equal Temperament Ruined Harmony (and Why You Should Care) (2007, W.W. Norton) An accessible polemic against equal temperament, arguing that historical tuning systems produced better-sounding harmony for the repertoire of their respective eras. Technically informed, historically grounded, and written for a general audience. An excellent companion to Chapter 12's historical sections.

Microtonal Music and Alternative Tuning Systems

Sims, Ezra. "Yet Another 72-noter" (1988). Computer Music Journal 12(4): 28-45. A composer-theorist argues for 72-tone equal temperament (a system that divides each 12-TET semitone into 6 equal parts) as a practical compromise between 12-TET and just intonation. Technical but accessible.

Tenney, James. A History of 'Consonance' and 'Dissonance' (1988, Excelsior Music Publishing) Traces the shifting meaning of consonance and dissonance through Western music history, from ancient Greek theory through the 20th century. Shows how the concepts are not fixed acoustic facts but culturally variable aesthetic judgments. Essential intellectual context for the tuning debates.

Xenharmonic Wiki and the Xenharmonic Alliance (various online resources) The online community of microtonalists maintains extensive resources on alternative equal temperaments, just intonation systems, historical tuning, and microtonal composition. The Scala scale archive (freely available online) contains thousands of historical and theoretical tuning systems. Both resources are accessible to non-specialists.

Recordings

Bach, J.S. The Well-Tempered Clavier, played by various pianists in various temperaments: - Gustav Leonhardt (harpsichord, various well temperaments) — essential historical recording - Kimiko Ishizaka (piano, various temperaments — free download under Creative Commons license) - Angelo Invernizzi (harpsichord, Vallotti temperament) — demonstrates the effect of a specific historical temperament

Partch, Harry. Delusion of the Fury (Columbia/Odyssey recording, 1969, conducted by Danlee Mitchell) The essential Partch recording — a complete theatrical work demonstrating the full sonic world of 43-tone just intonation. The 2012 reissue by Bridge Records also includes extensive liner notes.

Partch, Harry. The Harry Partch Collection, Volumes 1-4 (CRI/New World Records) Comprehensive collection of Partch's major works, including "Barstow," "Revelation in the Courthouse Park," "And on the Seventh Day Petals Fell in Petaluma," and others.

Blackwood, Easley. Microtonal Études (1980, Cedille Records) Systematic exploration of equal temperaments from 13-TET through 24-TET, composed and performed by a professional composer. The most accessible introduction to what different equal temperament systems actually sound like.

Various artists. Just Intonation: Exploring the Untamed Spaces of Music (2010, various labels) A compilation exploring just intonation by composers and performers including La Monte Young, Phill Niblock, and others. Demonstrates the range of aesthetic possibilities within just intonation.

Online Resources

"The Tuning Spreadsheet" (various authors, freely available) Multiple spreadsheets are available online (search: "tuning systems comparison spreadsheet") that calculate exact frequencies for various historical temperaments. Useful for the mathematical exercises in this chapter.

Xenharmonic.net / Xenharmonic Wiki Community-maintained resource for alternative tuning systems, with listening examples, theoretical explanations, and composition guides. Essential for anyone who wants to explore beyond 12-TET.

"Why the Circle of Fifths Doesn't Actually Circle Back: The Pythagorean Comma Explained" (multiple YouTube videos) Several high-quality explanations of the Pythagorean comma problem are available on YouTube; search for the specific term and select videos from musicians or music educators with substantial technical depth. Adam Neely's YouTube channel is particularly recommended for clear, rigorous explanations.