Mathematics holds a central place in the traditional liberal arts. The four mathematical disciplines of the quadrivium-arithmetic, geometry, music, and astronomy-reveal their enduring significance in this work, which offers the first unified, textbook treatment of these four subjects. Drawing on fundamental sources including Euclid, Boethius, and Ptolemy, this presentation respects the proper character of each discipline while revealing the relations among these liberal arts, as well as their connections to later mathematical and scientific developments. This book makes the quadrivium newly accessible in a number of ways. First, the careful choice of material from ancient sources means that students receive a faithful, integral impression of the classical quadrivium without being burdened or confused by an unwieldy mass of scattered results. Second, the terminology and symbols that are used convey the real insights of older mathematical approaches without introducing needless archaism. Finally, and perhaps most importantly, the book is filled with hundreds of exercises. Mathematics must be learned actively, and the exercises structured to complement the text, and proportioned to the powers of a learner to offer a clear path by which students make quadrivial knowledge their own. Many readers can profit from this introduction to the quadrivium. Students in high school will acquire a sense of the nature of mathematical proof and become confident in using mathematical language. College students can discover that mathematics is more than procedure, while also gaining insight into an intellectual current that influenced authors they are already reading: authors such as Plato, Aristotle, Augustine, Thomas Aquinas, and Dante. All will find a practical way to grasp a body of knowledge that, if long neglected, is never out of date.

An innovative account of listening and empire-building in 17th- and 18th-century China.

The Emperor’s New Epistemology explores how listening shaped knowledge and empire-building in seventeenth- and eighteenth-century China. As the Manchu dynasty expanded deep into Inner Asian territories deemed foreign by the majority-Han population, the Qing empire’s musical culture encompassed the sounds of Mongols, Uyghurs, and Tibetans, as well as Western music brought by Jesuit and other Christian missionaries. Lester Hu reveals that this diversity transformed not only Qing court culture but the very grounds on which knowledge and power could be established.

In one revealing episode from 1692, the Kangxi Emperor (1654–1722) misinterpreted a key term of Classical Chinese music after studying European music with the Jesuits. Hu situates this anecdote within the political, social, and epistemic shifts that transformed late-imperial China. As Hu shows, listening was an instrument of Qing imperial authority: knowledge about music, language, and even the universe could only be verified through the emperor’s ears. Furthermore, listening facilitated new theories of empirical knowledge, which both lent themselves to Qing-imperial ideology and generated new epistemic tensions. By exploring their echoes in similar shifts in early modern Europe, Hu’s book brings a global perspective to the study of music in the early modern world and reconsiders the transcendental ground on which such comparative and intercultural knowledge is possible.

One of the twentieth century's most important musical thinkers, James Tenney did pioneering work in multiple fields, including computer music, tuning theory, and algorithmic and computer-assisted composition. From Scratch arranges, edits, and revises Tenney's hard-to-find writings into one indispensable collection. Selections focus on his fundamental concerns—"what the ear hears"—and include thoughts and ideas on perception and form, tuning systems and especially just intonation, information theory, theories of harmonic space, and stochastic (chance) procedures of composition.

Winner of the Society for Music Theory's Outstanding Multi-Author Collection Award

This book offers a series of essays that show the integrated role that musical structure (including harmony, melody, rhythm, meter, form, and musical association) plays in making sense of what transpires onstage in musicals. Written by a group of music analysts who care deeply about musical theater, this collection provides new understanding of how musicals are put together, how composers and lyricists structure words and music to complement one another, and how music helps us understand the human relationships and historical and social contexts. Using a wide range of musical examples, representing the history of musical theater from the 1920s to the present day, the book explores how music interacts with dramatic elements within individual shows and other pieces within and outside of the genre. These essays invite readers to consider issues that are fundamental both to our understanding of musical theater and to the multiple ways we engage with music.

Shows how musical instruments and the sounds they produce are not incidental, but an integral part of music theory.

Many musicians dread music theory: its rules and diagrams can feel stern and lifeless, far removed from the sounds that make music so thrilling. In this book, Alexander Rehding looks at music theory from a different angle. He argues that the sound of the instruments on which music theory is demonstrated—starting with the piano in the music classroom—plays a much more active role than we usually notice. They are “instruments” in a double sense: musical noisemakers, but also tools in a kind of scientific experiment. Rehding strips these devices down to their sound-generating mechanisms to show us how to read musical instruments for their theoretical content.

Instruments of Music Theory turns our attention to instruments such as multiphonic sirens, tuning forks, Chladni plates, Aeolian harps, and Wheatstone concertinas—these devices generated sounds that called for new ways of listening, measuring, and thinking about music. Bringing together music theory, critical organology, and media theory, this book can be read as a user’s manual for how to put sound back into music theory. Rehding shows us how instruments do not only produce sound—they produce knowledge about music. Through the lens of these new instruments, Instruments of Music Theory recasts fundamental musical ideas such as pitch, harmony, form, and scale from the ground up to give new meaning to the stuff we call music.

Winner of the 2024 Association of American Publishers PROSE Award for Music and Performing Arts

Since its inception in the mid-twentieth century, American music theory has been framed and taught almost exclusively by white men. As a result, whiteness and maleness are woven into the fabric of the field, and BIPOC music theorists face enormous hurdles due to their racial identities. In On Music Theory, Philip Ewell brings together autobiography, music theory and history, and theory and history of race in the United States to offer a black perspective on the state of music theory and to confront the field’s white supremacist roots. Over the course of the book, Ewell undertakes a textbook analysis to unpack the mythologies of whiteness and western-ness with respect to music theory, and gives, for the first time, his perspective on the controversy surrounding the publication of volume 12 of the Journal of Schenkerian Studies. He speaks directly about the antiblackness of music theory and the antisemitism of classical music writ large and concludes by offering suggestions about how we move forward. Taking an explicitly antiracist approach to music theory, with this book Ewell begins to create a space in which those who have been marginalized in music theory can thrive.

One of the most highly acclaimed resources for the study of ear training in music education today

Solfège has become one of the most highly acclaimed resources for the study of ear training in music education today. This revised and expanded third edition includes a multitude of new musical examples and dictations covering the entire continuum of musical development from classical to modern. This edition is enhanced with a free online companion library of rich and definitive audio performances of each of the 51 musical dictations, keyed by track number to the musical notation in the text. As students progress from the simple to the complex by following these carefully interrelated lessons, the results are, according to Mel Powell, “akin to miracle-making.”
 

Reviving an educational tradition involves a double task. A new generation of students must be taught, and at the same time the teachers themselves must learn. This book addresses the teachers who seek to hand on the quadrivium-the four mathematical liberal arts of arithmetic, geometry, music, and astronomy-at the same time as they acquire it. Two components run in parallel throughout the book. The first component is practical. Weekly overviews and daily lesson plans explain how to complete the study of A Brief Quadrivium in the course of a single school year, and suggestions for weekly assessments make it easy to plan tests and monitor student progress. The second component is directed to the continuing education of the teacher. Short essays explore the history, philosophy, and practice of mathematics. The themes of these essays are coordinated with the simultaneous mathematical work being done by students, allowing the teacher to instruct more reflectively. Some users of this book are confident in their grasp of mathematics and natural science. For them, the essays will clarify the unity of mathematical activity over time and reveal the old roots of new developments. Other users of this book, including some parents who school their children at home, find mathematics intimidating. The clear structure of the lesson plans, and the support of the companion essays, give them the confidence to lead students through a demanding but doable course of study. The British mathematician John Edensor Littlewood remarked that one finds in the ancient mathematicians not “clever schoolboys” but rather “Fellows of another College.” This guide invites all teachers of the quadrivium to join the enduring mathematical culture of Littlewood and his predecessors, and to witness for themselves the significance and vitality of a tradition as old as Pythagoras.

Taken together, these comprehensive volumes offer an authoritative account of the music of Africa. One of the most prominent experts on the subject, Gerhard Kubik draws on his extensive travels and three decades of study in many parts of the continent to compare and contrast a wealth of musical traditions from a range of cultures.

In the first volume, Kubik describes and examines xylophone playing in southern Uganda and harp music from the Central African Republic; compares multi-part singing from across the continent; and explores movement and sound in eastern Angola. And in the second volume, he turns to the cognitive study of African rhythm, Yoruba chantefables, the musical Kachamba family of Malaŵi, and African conceptions of space and time.

Each volume features an extensive number of photographs and is accompanied by a compact disc of Kubik’s own recordings. Erudite and exhaustive, Theory of African Music will be an invaluable reference for years to come.

In What Music Did, experimental filmmaker and violinist Tony Conrad explores in depth the relationship between music and mathematics. A work of decades that was left unfinished at the time of his death in 2016, Conrad’s expansive history of the interrelationship of music and mathematics is published here for the first time. Editor Patrick Nickleson describes Conrad’s method as that of an anarchic, interarts haberdasher; much of the research comes from musty and out-of-print sources, giving the impression that Conrad followed paths opened up for him in used book shops and conversations, rather than seeking a direct scholarly argument. 

Throughout the book, readers encounter scenes from over two thousand years of history, mathematics, and music: Pythagoras using pebbles to articulate didactic number games to his disciples; Galileo fretting a hill and listening with a musician’s ear to calculate the rate of acceleration under gravity; Rameau trapping Western music in a five-limit tuning system, and what could have been if he were more adept with numbers. Even when drawing on classical sources to explore canonical figures like Saint Augustine, What Music Did offers idiosyncratic critical insights that highlight our ongoing cultural reverence when answers result in simple whole number ratios like 3:2, 4:3, or 5:4. What Music Did is Tony Conrad’s extended indictment of music’s role, from the Pythagoreans to the twentieth century, in upholding the use of number as a clandestine and circumscribed armature of power.