General Principles of Harmony
Introduction: Why this book?
(N.B. This book is not a textbook for learning traditional, tonal harmony. Its focus is on general principles, applicable to all harmonic styles. As such, certain concepts, only relevant to tonal harmony, will not be discussed in detail. For more on this limitation, please see below.)
Of all musical disciplines, harmony is probably the most written about. Textbooks abound, from the summary to the encyclopedic. Why add to the existing plethora of resources? While we will survey some of this material below, one thing is lacking in all of them: None convincingly connects traditional harmony to contemporary practice. Although some of these books contain a chapter or two about more recent techniques, these are usually described in summary or superficial ways, and few or no connections are made with older practice.
Harmonic relationships can be divided into three categories:
- Those which are immediately audible.
- Those which become audible through attentive listening.
- Those which can never be heard, given the limits of human perception.
In this book we will explore the first two types, and systematically exclude the third. It cannot be sufficiently emphasized that not all harmonic relationships are equally important: Their location in the piece, and, especially, their relative salience must be evaluated with the limits of human perception in mind.
As in the other books in this series, our approach here will be to focus on principles rather than on styles. We contend that there are common principles in operation across various harmonic styles, and that understanding these principles – which arise more from how we hear than from stylistic conventions – can help the composer of today to find a personal harmonic language which makes audible sense.
Finally, we make no pretense here of explaining all harmonic languages; our goal here is more modest. We wish to propose some powerful concepts which are relevant to both common practice period and more recent (western) harmony.
DISCUSSION OF OTHER APPROACHES
Before starting our search for such general principles of harmony, let us list and briefly comment on the most common traditional pedagogical methods:
- Methods based on a specific style (or a limited group of styles, typically the “common practice period” from Bach to Wagner.) make no claim to universality, but simply aim to define “normal” harmonic practice within a given period. The best example of this approach is Walter Piston’s Harmony. By definition, such methods remain within relatively narrow stylistic confines, and make no attempt to generalize the principles therein. However, while not all principles of classical harmony are applicable outside the common practice period, the contrary notion – all harmonic thinking can be reduced to stylistic convention – leads to an absurdity: Can recent composers really have discovered entirely new ways of hearing? (I say “discovered” because they can hardly have invented new neurological structures and functions.) The human brain’s highly evolved, innate capacities for making sense of auditory experience have surely not changed over the past few centuries.
- Another, related method, consists of intensive drill, using harmonic formulas. Based on the notion that harmony, like language, uses many idioms, the goal here is to learn as many of them as possible, often by rote. While this approach does have some value in learning classical tonal harmony, the formulas learned are not generalizable outside of the source repertoire.
- Piston’s method and the “formula” approach both are based on Rameau’s theory of chord roots and inversions. The problem with this theory is that the root of an inverted chord is a theoretical concept, not a directly audible one. While there is some truth to the notion that all inversions of a given chord can be heard as part of the same “family”, there are important exceptions, as we shall see below. Obvious audible facts, like the actual, heard bass line, the spacing chosen for a given chord, and its linear context, can sometimes go unnoticed with this approach. It is worth remembering that Bach did not teach this way.
- Approaches based on the insights of Heinrich Schenker have the advantage that they are more directly based on hearing. Schenkerian “foreground” relationships are especially useful in understanding many harmonic situations. In particular, the notions that not all chords are of equal structural importance, and that harmonic meaning changes according to linear context, are critical insights. While the Schenkerian approach was originally intended for tonal music, certain notions of harmonic elaboration can be easily applied in other contexts; we shall do so, below. Less convincing are some of the more far-flung conclusions of Schenkerian analysis: As the connections posited become more abstract and far flung, they can become downright inaudible, in any normal mode of listening. While it is true that a full Schenkerian analysis, well done, can help refine hearing, at a certain level of abstraction, one wonders whether more emphasis on other aspects of the audible foreground would be more appropriate for most students.
- The traditional French conservatory method of teaching harmony, using mainly given basses with elaborate figures, is an outcome of continuo practice. However, whereas the latter used figures as a shortcut, some of the pedagogical extensions of this method are extremely cumbersome, with the result that the student spends a great deal of time and effort becoming familiar with an elaborate and, finally, fairly useless numeric code. This code supplies very little insight into the way harmony and form interact, and thus provides no help in teaching the student to think harmonically for himself, or to generalise what he knows outside of the realm of tonal music.
- Schoenberg’s writings on harmony deserve special mention here. As in all his theoretical writings, Schoenberg has many provocative ideas, and his teaching is based on a profound knowledge of the repertoire. A few of his ideas have influenced our approach here – notably his notion of the structural role of harmony. The main drawbacks of Schoenberg’s writings are: his philosophy of historical necessity, his sometimes obscure formulations, and his frequent aesthetic diatribes, many of which today are simply very dated.
- Allen Forte’s set theory, like Schenkerian theory, was originally formulated with a specific repertoire in mind, in this case non-serial, so-called “atonal” music. Forte did a useful service in proposing a standard nomenclature and classification for all possible chords in the tempered system. His study of inclusion relationships is also of interest. Within limits, Forte’s ideas can provide a useful way of organising and recognizing families of pitches, which can help the composer give coherence to his harmonic language. His main weaknesses – particularly from the point of view of a composer – are:
- The lack of any serious discussion of what is or is not audible. For example, hearing the presence of a given three note harmonic cell through a short passage is a very different matter from recognizing an eight note set in two far-flung sections of a long piece.
- The lack of any discussion of issues of harmonic direction, accent, and cadence.
- Probably the best teaching work for traditional harmony is Roger Sessions’ Harmonic Practice. Written by a fine composer, it explains things in psychological terms more than in terms of convention. For example, Sessions’ notion of harmonic accent will prove very useful here. Also, his exercises are the most varied and challenging for a young composer. The limits of his book are:
- He does not discuss the important differences between vocal and instrumental harmony.
- His discussion of contemporary practice is very summary.
- Finally, Persichetti’s “Twentieth Century Harmony” is an excellent compendium of many twentieth century techniques. Written by an eminent composer/teacher, it is practical in its approach and down to earth in its explanations. However, it does not contain much in the way of general principles which are also applicable to classical harmony, and it contains little reference to long range harmonic organization – that is, to musical form.
To summarize, what is missing from all of these methods are links between tonal and other approaches. And yet such links abound. For example, while some of the specific methods of creating direction and coherence in tonal harmony cannot be transferred intact to other harmonic styles, often the principles underlying these specific solutions can be generalized. As we shall discuss below, the principles of voice-leading are clearly grounded in the way human hearing works, and therefore, appropriately formulated, remain relevant to any harmonic style.
A final inadequacy of most current approaches to harmony is that they often ignore the interaction of harmony, counterpoint, orchestration and form. However, these categories are mere pedagogical conveniences, and not realistic descriptions of the way the musical ear breaks down information. For example, voice leading cannot be separated from counterpoint, and detailed examination of the way a chord is spaced quickly leads to questions of orchestration. For this reason, in our discussions of musical examples, we will often need to refer to several different aspects of the music in order to adequately explain what is happening.
LIMITS OF OUR DISCUSSION
- This book is not a harmony textbook. Both the specifics of classical tonal harmony and of many twentieth century techniques are well treated elsewhere; there is no need to cover the same ground here. What is needed are more general, unifying principles. To the extent that we will be applying traditional principles in broader contexts, we shall assume that the reader is already familiar with their conventional applications. Where these principles are not familiar, we will explain them in more detail. To get the most out of this discussion, the reader should have a solid grounding in tonal harmony, and should also be familiar with the material in Persichetti’s “Twentieth Century Harmony”. (N.B.: Where a twentieth century technique is well covered in Persichetti, we will not supply examples of it, unless we have some extra insight to add.) Knowledge of the elementary notions of Forte’s set theory (in particular: “interval class”, “set”, “normal form”, and “interval vector”) will also be useful.
- This book is not a comprehensive method of analysis. The goals of analysis are systematic in a way ours here are not; therefore an analytical method would require a very different approach. As in the other books in this series, our aim here is practical: We are simply trying to propose some basic principles about how (harmonic) hearing works, especially those which cross stylistic boundaries, and which can therefore be useful to composers today. This is especially important given that in the recent past, systems like total serialism and aleatoric music, where the methods used to produce the music have no demonstrable relation to what any normal human can reasonably decipher by ear, were actually taken seriously. (Indeed, in some academic quarters, the preceding sentence is still “politically incorrect”.) Unfortunately, anyone who spends most of their effort during composition on what cannot be heard, risks not using audible resources to the full, and consequently producing a work whose effect on the listener can only be tepid at best. While a composer may perhaps explore such systems to break out of stale habits, if the results are not at some point severely filtered through a realistic knowledge of what can be expected from a normal listener, how can the music communicate significantly?
- One final caveat: In this work we will limit our discussion to the tempered scale. This is not to deny the interest or the musical potential of non-tempered and micro-tonal systems. Quite possibly, some of the principles mentioned here also apply to non-tempered harmony, but a thorough discussion of such harmony would require expertise I do not possess. In addition, the tempered scale is so ingrained in our notation, performance practice, and instrumental construction, that serious attempts to work outside of it require groundwork which goes far beyond the scope of this book. Similarly, we will not address harmony which makes significant use of portamento effects.
A NEW APPROACH TO UNDERSTANDING HARMONY
Since mankind’s evolutionary capacities and limitations for hearing and understanding relationships between tones have clearly not changed in a very long time, it follows that there must be connections in the way we hear “old” and “new” music. Recent works by Bergman (Auditory Scene Analysis), Deutsch (Ear and Brain), and Snyder (Music and Memory), shed significant, new light on these basic auditory/cognitive systems. Combined with what musicians already know and intuit about how music works, they provide a useful starting point for a more general understanding of harmony and other musical disciplines.
The main theories which will prove useful to us are those which refer to the most easily heard phenomena. (Incidentally, the aspects of cognition we take for granted are often the most complex.) The disdain with which “salience” is referred to in some current (music) theoretical literature is entirely at odds with the practical needs of the composer.
For example, some of the assumptions behind current ideas of pitch structure need to be reexamined. Recent psycho-acoustical research, as well as practical experience, lead to the conclusion that some of these notions are conventions with only limited usefulness, focusing on connections that are often quite obscure to even the most trained and attentive listener. Worse, they often do not explain what actually is heard, and thus can lead the analyst or the aspiring composer to ignore factors much more relevant to the sonic result.
We may draw a parallel here to the exaggerated importance given to chord roots in harmonic theory before Schenker. Schenker`s thinking, by contrast, emphasized the fact that the sounding bass line usually has more effect on the sense of harmonic direction than any theoretical root. Here a widely accepted theory (that of chord roots and inversions) often led to ignoring or undervaluing direct musical experience.
Similarly, the vast literature about pitch class sets and series often veers into the musical equivalent of numerology. Again, overemphasis on extremely subtle intervallic relationships, especially over long stretches of time, where their aural perception is often impossible, easily leads to inadequate emphasis on relationships that are audible even to the uninitiated. This in turn can lead to serious misjudgments about a work’s effect. Clearly salient events are always the best pillars supporting musical architecture. (See my article, On Salience, for more on this issue.)
An example of a common basic assumption needing qualification is that of octave equivalence. While in the middle register C3 and C4 are clearly in some sense equivalent, comparing C1 and C7 is quite another matter. In the extreme registers, pitch discrimination is very inexact and dependant on many factors, including orchestration, duration, etc.
The two chords in this example include the same four pitch classes. However, in terms of perception, what does it mean to speak of them as being “identical”? The exact pitches in the first chord are quite difficult to distinguish due to their extreme register, and their short duration makes this harder still. Most important, the differences in register and spacing between the two chords have the effect of thrusting the common pitches into the perceptual background. Even if we concede that, played one right after the other, careful listening might recognize these common pitches, what if the chords are separated by several bars of other music? In this case the similarity between the two chords is surely at best a refinement, compared to their surface contrast. Except in the case where the two are placed side by side for comparison or otherwise “pointed out” to the listener, the pitch similarity between them is relatively unimportant. Note that to make the listener’s job easier I have used the same pitch classes. Imagine if I had also transposed the chords (at an interval other than the octave), requiring the listener to compare intervals rather than just pitch classes.
In this example, the first chord is the same as the second one from the previous example. The second chord here contains two new pitches, and different intervals as well. And yet the two chords seem much more similar than the pair in the previous example, because: They are in the same register, share two common tones, and each contains a sharp dissonance in the middle, with richer intervals surrounding it.
These examples raise two critical questions:
- How can a composer make pitch identity relationships clear to the listener?
- When should other relationships (as in the second example) be considered more important?
These questions are largely ignored in the literature, despite their vital importance for understanding musical form – which, after all, works mainly through association and memory. Both of these cognitive capacities depend on surface salience. A good deal of our discussion here will therefore focus on the ways composers can create, and realistically differentiate, audible harmonic relationships, to fulfill various formal functions.
- Discussion of other approaches
- Limits of our discussion
- A new approach to understanding harmony
- A definition of harmony
- Pitch and interval limitations
- Linear aspects: melody and bass lines; voice leading
- An aside: open vs. closed harmonic systems
- Hierarchy, landmarks, cadences
- General aspects of harmonic accent
- Creating momentum and renewing interest on various structural levels
- Higher Levels
- Harmonic rhythm
- Modulation and harmonic transition
- Spacing and register
- Harmony with multiple planes of tone