The Unconservatory  

Excerpts from Harmonic Experience

by W.A.Mathieu

Copyright © 1997 by W.A. Mathieu. All Rights Reserved.

Definitions to highlighted words are can be found in the short glossary excerpts. Clicking on the word will take you directly to its definition

(Excerpts from the Introduction)

What Is Theory For?

A good thing to remember about music theory is that it typically springs up decades, often centuries, after the birth of the music it describes. A book like this one could simply not have been written during the time its primary subject matter (twelve-tone equal-tempered tonal harmony) was being discovered and developed. The rules of sixteenth-century counterpoint were not definitively codified pedagogically until the eighteenth century (by J.J. Fux, still studied today). The behavior of just intonation itself has been most confusingly and inefficiently articulated by theorists for almost three millennia, right up to the present. And only in the present generation is the code of tonal harmony being cracked, well over a century after its art came into full bloom. Theory is the scat of music, what it leaves behind. Luckily for us, we don't make music by following the rules. We don't make music at all, so much as we find our way in it. Only after the way has been found can the maps be drawn.

But the maps exist, right alongside the music. And good maps can save you centuries, soldiers, ships, fortunes. The danger: sitting at home hunched over your map table with a fixed grin thinking you've been places you haven't. The balanced way is to study these maps while you cover the territory. By the time you know your way around the territory, the maps will have become torn and faded, and you won't have to look at them anymore because the territory is teaching you more than the maps ever could.

Who Is Theory For?

The question cannot help but arise, “Don't some people get music naturally, without all of the discipline and commitment and analysis?” The answer is yes, some people do and so do you. We all do to some degree. Every action is partly intuitive and partly rational; the proportions change from action to action and from individual to individual. No one is entirely analytical in the process of learning music, and no one is entirely intuitive (although I’ve seen some serious contenders at both extremes). There is even a part of the psyche that actively seeks to not know. It wants sensual saturation and intuitive wholeness, the pure being of childhood, the animal self.

You have to work out your own recipe for learning, which includes your interaction with this book and the procedures outlined in it. A good guideline to remember is this: the rules of music - including the rules of counterpoint and harmony - were not formed in our brains but in the resonance chambers of our bodies. What feels right and good is what survives. The "rules," codified over many generations by musicians serving as teachers, arise to protect and disseminate the good vibrations. But every rule and every formulation ever made burns away at the moment of music making. Ultimately everyone makes music intuitively; individual circumstances determine the point at which the music rises above the mind and the intuition takes over.

There is a population of musicians who are both intellectually aware and musically whole; at the same time there are millions among us who achieve a high level of musical creativity in a state of musical innocence: the lyrical jazz trumpet player Chet Baker comes to mind. The musically innocent know where they are in the music without possessing a naming language; they have an internal musical map with no proper place-names on it.

So if half the folks are striving to rise above names and the other half never learned any to begin with, why go about naming everything?

People name things because transcending knowledge is different from not having any. Intellectual transcendence is not the same as intellectual innocence. A woman who has transcended her intellect and is whole in the moment of music making is a different whole woman than she was when she was musically innocent. The issue is not who has the secret, the intellectually transcendent or the intellectually innocent. The issue is how intellectual or innocent you need to be to find your musical completion.

The Sufi mystics say, “The mind is the willing slave of the heart.” The key word is willing. The intellect that wills to be in the service of the intuition is powerful and mature - the kind of mind a musician needs in order not to be paralyzed by knowledge. The great fear, especially among singers but among many instrumentalists as well, is that analytical knowledge disables us at our core in the very act of making music. Yet that need not be true. Ideally, an intellectually awake musician is not analyzing while she is performing. She is complete in the moment of pure sound; no thinking is needed. However she has thought. She has analyzed. The homework has been done, the dues paid, the mental discipline gained. The musical spirit is flying free, but it has been shaped by the work of the mind.

Music theory is not for everyone. I think each person's mind has a certain capacity for willingly serving his or her heart. Theory should be studied to the limit of that capacity but no more. When you stuff the mind with facts and rules beyond its power to yield, the mental slave rebels and fills the house with mental music, and your heart pines away.

Yet what if your heart is singing fortissimo and your mind is bored to tears?

Then you realize that your mind needs to sing too.

I think you find the balance between heart and mind by asking your heart what it really needs to become musically complete. If the question is clear, your answer will be clear too. Then your mind is finally free to offer service.

For Whom This Book Is Written

These pages are looking for musicians within a very wide range of training, including near-beginners who really want to know how harmony works, as well as advanced composers and performers who feel the pinch of traditional concepts and need to redefine music in a whole-making way. Since most people are somewhere in between, let me say that the perfect reader is simply one who wants to know music intimately from the inside out. New paradigms of integrative experience operate at all levels of development: a new way of perceiving old ways can be useful at every stage of learning.

To benefit most fully from this work, however, you must bring to it certain skills. You need to be able to read music. For the last three parts you need to be able to play a keyboard instrument well enough to navigate at least most of the musical examples. It helps to have previously investigated the overtone series, or be ready to (the text helps you). Also, the more you are in the habit of listening to the music of the world’s various cultures, the more insight you will bring to this study.

Harmonic Experience Copyright © 1997 by W.A. Mathieu. All Rights Reserved.

(Excerpt from Chapter 13)

Singing Practice

You can’t learn everything right away, but you can expect to form good habits, which means, basically, practicing regularly and staying awake. However long your practice sessions, I suggest you spend most of the time using sargam and the rest of the time using open vowels - ah mostly - or any syllables that bear repetition or accrue power through repetition.

It is good to have favorite modes that you learn well; you don’t have to learn more than a few to cover most of the ground. For instance, Lydian and Phrygian between them take in all twelve notes. Try composing melodies in your favorite modes, learning them with appropriate meend, and then using these as departure points for improvisation.

The results of singing practice come on quickly. You can expect your ear to improve, with better pitch and greater certainty. You gain musical authority. And you learn, too, about your own musical behavior, including both the denial and acceptance of your innate gifts. And last but not least, you set the stage for what is to come: an understanding of the entire range of harmonic experience in twelve-tone equal temperament.

Harmonic Experience Copyright © 1997 by W.A. Mathieu. All Rights Reserved.

(Excerpt from Chapter 19)

Approximation versus the Real Thing

The primary concern of this book is the behavior of twelve-tone equal temperament, and we are raising a question central to it: Physiologically and psychologically, how can we accept a musical approximation for the real thing? Any answer to this question, at least at the present state of our knowledge, must be subjective, evanescent, and nonquantifiable. Even the terms of the question cannot be rigorously defined. Scientific method cannot save us.

We do, however, have ourselves, not only individually but also collectively. By examining deeply our own responses to musical phenomena, we can shed light on our musical behavior. And by examining the music that people leave in their wake we can understand human response on the scale of human populations.

To better explain what I mean by “the real thing,” let’s step outside the musical frame of reference for a moment and use, as an example, the sense we have of what is "vertical" or "perpendicular" or "a ninety-degree angle with the horizon" or "straight up and down." We all have an inborn sense of the plumb line of gravity; it is part of the nature that made us. This inborn sense functions as an internal norm from which we can discern more or less deviation: we know how to “zero in” on a center we can feel.

The internal norm is further formed and nurtured by experience: we learn to walk; to balance our way across logs; we practice ballet; we study geometry; we build houses, internalizing what is "vertical" from the hard world outside. We might call this sense of center that is both given by nature and internalized by experience an inner/internalized norm. Both inborn and amplified by learning, it is an active interface between the inside and the outside world.

What is fascinating, and relevant to our musical study, is how things can appear slightly off the vertical yet be accepted by us as if they were vertical anyway: telephone poles and streetlamp standards are good examples. Even skyscrapers - those perfect symbols of straight-upwardness - lean a little. On the other hand, consider our response when the norm is stretched beyond our tolerance: Aunt Edith's painting that never seems to hang quite straight, or the ever-defiant Leaning Tower of Pisa.

Things that are approximately vertical still refer to the vertical norm up to a certain point, but where is that point? The same goes for things that are approximately horizontal. How tilted does your bed have to be before you can't sleep? How flat is Kansas? Given the intrinsic quantum lumpiness of atomic energy, how flat is anything? Do you know what I mean when I say “flat” on account of a norm that neither one of us has ever actually experienced? Is there such a thing as a straight line? If not, how crooked can it get and still be straight? Can a field be square? Can a cookie be round?

These are the kinds of questions that will become crucial to our study as we step back inside the musical frame of reference.

Harmonic Experience Copyright © 1997 by W.A. Mathieu. All Rights Reserved.

(Excerpt from Chapter 34)

The Dark Side

The ear is deep; it is difficult to overestimate the collective profundity of the ear. And the depth of the ear's musical response is cumulatively enhanced and nurtured by culture, passed on from one person to the next, from generation to generation. I am alive in the first generation of ears that can listen to the greatest master musicians of virtually every culture and subculture. For a few dollars I can buy recordings of the best musicians in the world, some of whom are long gone from us. The more I listen the more I can hear the wit, pathos, and intellectual complexity that connects the Australian aborigine’s didjeridoo and the European conductor’s baton, and the more I realize there is only one pair of ears.

What continually amazes me is the fierce love of harmonic resonance that is evident everywhere. Not all music has the complex tonal system or the counterpoint of the European legacy, or the modal variety of India, or the rhythmic complexity of Ghana, but everywhere one listens one hears the same passionate intelligence. The most subtle conceivable nuances of intonation are shared worldwide. It becomes apparent that our one pair of ears recognizes how wide harmonic territories can be traversed by small melodic intervals. Even in music that does not require commas, micro-intervals that radically affect the harmonic flow show up again and again. This is true for the best musicians regardless of culture or style - the hip pop singer in the recording booth no less than the muezzin in the tower or cellist in the concert hall.

Worldwide listening reveals that master musicians sing and play the gamut running from the dead-in-tuneness of low-prime norms, through harmonic complexities of every sort, all the way to momentary, deliberate out-of-tuneness. Playing with simple pure harmonies is like playing with light, and the fascination with that cusp between complexity and out-of-tuneness is like playing with the dark side. The flirtation between the light and the dark is a kind of confrontation with destiny itself, a reenactment of the battle between survival and annihilation, harmony and noise. We cherish this in our music.

As long as we are singing about life and death we know we are still alive. The various commas, whether they are used as expressive shading or as a functional way of zapping through harmonic territory, far from being the pesky problems that many theorists have assumed them to be, are in fact windows to the affective dark side of the psyche, and they enable music to fill up to the brim with the full range of human feeling.

Even the commas implicit in twelve-tone equal temperament imbue equal-tempered music with a tremendous range of human response. We need to elevate the status of the equal-tempered comma from that of a pun, where a single thing stands for two incidentally related things, to that of matchmaker, where complementary energies are conjoined and synthesized into a higher meaning.

Harmonic Experience Copyright © 1997 by W.A. Mathieu. All Rights Reserved.

(Excerpts from Chapter 44)

The Value of the Theory

Assuming that you find the resonances of just intonation real, and that they serve as the norms of equal temperament, and that the inherent ambiguities of equal temperament result in a network of commas that are functional and affective for you, the theory and its consequent practices may be valuable in a variety of ways.

It draws unique distinctions. The theory allows you to draw subtle but real distinctions between harmonic values - especially between Pythagorean and pentamerous values - that have not heretofore been theoretically provided for.

It organizes tonal harmony. On the basis of these distinctions, all available tonal harmony in twelve-tone equal temperament can be organized and thus made more accessible for hands-on use.

It defines harmony as form. The affective qualities of perfect fifths and major thirds operate in tonal music over every musical order of magnitude, from tones to symphonies, from Medieval modality to twentieth-century cyclic harmony verging on atonality. The theory gives a handle on how the harmonic drama of centering and dispersing is played out at every level. It demonstrates how elemental harmonic principles are architectonic - that is, how their structure controls or directs other structures - and how this active, forming energy cuts both ways across the orders of magnitude, from tone lattices to key lattices. By this method, one begins to grasp the dialectic of extremes, that is, how the smallest elements and the largest formal shapes are mutually nourishing.

It promotes self clarity. Because the work involved in learning these principles is so highly experiential, the theory throws you back onto your own perceptions, into your own hearing. Ultimately, one senses how the lattice is an expression of an inner working, and how harmony is the geography of the mind.

It gets you unstuck. Even though we learn music by imitation and are beholden to our teachers and our legacies, we don’t want anything or anybody to stand between us and the source of sound. We want direct access. The theory allows you to track your stuck places to the source, both conceptually and inspirationally, so your music can move forward.

It reveals a hidden gift. Although we began our work with a search for what is innate and ancient in human hearing, and were thereby led through an interior world of harmonic resonance, we eventually found ourselves in a forest of temperament, only to emerge with a realization that equal temperament, with its inherent lack of resonance, is nonetheless a radiantly beautiful gift, one of many from the musical gods.

Harmonic Experience Copyright © 1997 by W.A. Mathieu. All Rights Reserved.

Innner Traditions International can be contacted at:

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Please join our Harmonic Experience Forum for discussions related to Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression, by W.A. Mathieu (Rochester, Vermont: Inner Traditions International, 1997)

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Updated: January 25, 2001 (KB)

Copyright 2001 The Unconservatory, All Rights Reserved.