Harmonic Dualism or Harmonic Polarity

Harmonic Dualism or (my preferred term) Harmonic Polarity attempts to solve two dilemmas in Western music theory: 1) the presence of the subdominant triad and 2) the relationship between major and minor triads, generally. The questions that arise around these topics are:

  1. If the harmonic series is THE central building block of tonal thought—as many profess—how do we explain the importance of the subdominant chord and, in fact, the very existence of the perfect fourth within the major scale? You can take the harmonic series up to infinity and never find the perfect fourth above the fundamental. We find (after octave reduction) the perfect fifth, the major third, the major second, the major sixth, the minor seventh and even the major seventh, high up in the series, but not the perfect fourth. How can our scales and harmony be based on the harmonic series when there is no perfect fourth within the series?

  2. Since the minor third above the fundamental is a very distant harmonic, compared with the 5th harmonic of the major third, are we then to conclude that the minor third is somehow a rare interval? If so, how are we then to understand much of the world's musical cultures, where variants of the minor third are commonplace, and in fact often more common than the "harmonically resonant" major third?

  3. Additionally, how are we to understand the preponderance of minor triads in any given classical composition, if our concept of harmony is to have its genesis in the harmonic series? Some will point to a minor triad in the harmonic series, consisting of the 8th,10th and 13th partials -- but even this is not the minor triad over the fundamental. In C, it would be an A minor triad in first inversion.

  4. Regarding this minor third, if the major third is the natural third, must we conclude then that the minor third is somehow a modified, inferior, "turbid" version of the major third? (To these "turbid" theorists, the minor triad takes on its quality by being a modification of the pure chord of nature, the major triad. Schenker, the godfather of American music theory, considered the minor scale to be "artificial")

In seeking the answers to these questions, I have come to discover a substratum of music theory known as Harmonic Dualism. At the heart of harmonic dualism (for which we will use the more modern term Harmonic Polarity) is an organizational principle that states that Western musical harmony is based, not only on the physical overtone series that results from actual vibrating bodies, but in addition to another series, mathematically and intuitively derived, which goes "in the other direction." That is, in addition to dividing the string by the mathematical ratios (producing the notes based on low prime integers as well as the full harmonic series), one also lengthens the string by the same amount. Thus, we derive two series of tones, one using simple division, and the other, simple multiplication. The result is a beautifully symmetrical series of tones, one ascending and one descending.

Here is how this could be notated, based around C. The top is the familiar overtone series, although in this context it is best to think of this purely in terms of the harmonic ratios, since this is a conceptual system of imagined, internalized tone organization, rather than a specific phenomenon of nature. The fact that the overtone series also "occurs in nature" is in a sense co-incidental—a confirmation of the correctness of integer organization.

Harmonic Series:

Reciprocal Series:

This can also be written, informatively, with the two series interleaved:

Some interested observations can be made regarding these two series:

  • Accepting the 11th member of the series as harmonically (or at least melodically) useful for the moment, we notice that in the overtone series, members 8—12 produce a strong tonal motion toward the dominant (rather like a V of the dominant), while the reciprocal series produces a strong motion toward the subdominant, with the Phrygian desending scale (in F) suggesing a Neapolitan relationship.

  • The overtone series contains the major triad constructed from below—members 4-5-6.

  • The reciprocal series contains the minor triad, but constructed from above—members 4-5-6.

  • Members 4-5-6-7 in the overtone series suggest a dominant seventh chord, which turn towards the subdominant.

  • Members 4-5-6-7 in the reciprocal series suggest a half-diminshed seventh chord, which in tonal muisc turns back towards the dominant.

  • Building from below, the overtone series suggests a major triad with an (albiet "out of tune") minor dominant.

  • In perfect contrast, the reciprocal series suggest a minor triad, built from above, with a subdominant major triad (in this case, F minor with Bb major).

The central point of this approach is that we can view any single tone as both the generator of the harmonic series and the product of the reciprocal series. (You'll be given an exercise below to grasp this at the experiential level). Combined as equal forces in tonal music, the two series seem to create a looping pattern that is reflected in the way both traditional and much modern music has been practiced:

  • The overtone series contains a major chord of rest, but also contains a tone (#7) that propels it towards the subdominant, reciprocal realm. This make sense if we consider the center tone (#1) as both being and having a generating tone.

  • The reciprocal series contains a minor chord of rest, but also a less stable "half-diminished seventh chord" that propels it back towards the dominant of the central tone.

  • The overtone series suggests the possibility of a dominant minor chord in a major key (G-Bb-D).

  • The reciprocal series suggests the possibility of a subdominant major chord in a minor key (Bb-D-F).

It is important to stress that, while division of a string gives us precisely the same tones as those of the physical harmonic or overtone series, this is both a coincidence and a logical outcome of the mathematics. Since the harmonic series results from the physical fact that strings vibrate not only at their fundamental length but also at integer divisions of the string (it's just what happens, like an apple falling to earth and not floating up to the moon), it stands to reason that the same tones would be derived from a natural process that can be analyzed using integers and division. However, the series as derived from Pythagorean principles is rooted in philosophy and metaphysics, rather than in physics. It is this metaphysical principle that concerns us here.

In some old textbooks, this second series was called the "undertone" series, but this led to an unfotunate confusion, in that it suggested that the "undertone" series was a genuine physical phenomenon, like the overtone series. This is not the case, despite the fact that Hugo Reimann, one of the central proponents of this viewpoint in the 19th and early 20th Centuries, believed he could hear the undertone series when playing a tone on a very resonant piano. (Late in life, he corrected this view). The undertone series does not "exist" in the way that the overtone series can be heard in a low vibrating piano string or enduced on a trumpet. The series is a based on a mathematical and metaphysical principles, just as, in fact, the first 6 notes of the overtone series were a mathematical and metaphysical principle, from which scales and harmonies were derived (as we saw in the previous page), centuries before the scientific "discovery" or confirmation of overtone harmonics.

Rather than "undertone series" then, we shall use the term "reciprocal series," as this series is derived from taking the reciprocal of the ratio that creates the overtone series. (That is, if the higher octave is found by division, 1/2, and the next fifth is found by further division, 2/3, the next octave at 2/4 and so on, then the recipocal notes are found by multiplication. The lower octave is 2/1, the next lowest fifth is 3/2 and so on. Imagine that, instead of dividing a string, we make the string longer in length.

That said, it's important to stress that I am explaining this in terms of physical strings only by way of analogy. In fact, we are really dealing with a principle. This principle was best stated by Goethe, the great German writer and philosopher who also had a keen interest in music, and especially musical theories. Being an intuitive appreciator of music and a person for whom spirituality was a genuine personal concern (Goethe was "spiritual" without being particularly "religious"), Goethe put the concept of Harmonic Dualism in poetic terms, and in many ways explained the concept more clearly than those professional music theorists who followed him. Let's look at his language for a moment.

He made it clear that this principle, while perhaps being reflected in natural processes, is a principle that is psychological, spiritual and metaphysical in nature. "What is a string and its mechanical divisions, compared with the musician's ear?" said Goethe. In the words of two of his 20th Century followers, Levy and Levarie, the polarity of Harmonic Dualism is "one of the great princples fashioning not only the outer world of nature but also the inner world of thought and imagination." (Tone: a Study in Musical Acoustics, p. 189).

Goethe introduces a useful term, the tone-monad, meaning a central pitch around which all other tones are centered. Goethe called it a "living unit of sound." (Perhaps this concept resonates so much with me because my thinking has been deeply influenced by the philosophy behind Indian Classical music, with its similar concept of a tone-monad). "If the tone-monad expands, the result is the major mode, if it contracts, the minor mode is produced," Goethe wrote.

This then becomes a deeply perceptive and introspective approach to music, based on our own human polarity - we are drawn to both external reality (the physical, scientifically verifiable world) and also to internal experience (the world of the imagination and, if you will, the "soul."). Goethe calls it the world of concentration (also an Indian concept, since in India the subjective world of the imagination is accessed through meditative concentration). Goethe goes on, in a letter to a colleague from which I have been quoting: "The major mode is the expression of all that excites, exalts, and propels the soul toward the outer world. And, if you will, the minor is the mode of inward concentration. But concentration is in no sense synonymous with sadness. No, a thousand times, no! What is there sad about the polonaises, for example, that are in a minor key? The polonaise is a social dance and the society is drawn together into closest contact. How could this be sadness, when it is in fact the height of voluptuousness.?"

Thus, we are looking at a principle which will inform all of our work in music, reflecting a symmetrical polarity that resembles our own human experience of the inner and outer world.

Experience Polarity at the Keyboard

Now that we've looked at the full series, let's be clear about the fundamental concept, which is the discovery of the fifth above and below the tonic, based on the ratios of 3:2 and 2:3. The notated results of course are:

It is important that you have an experience with this concept. Thus, I need you to go to a keyboard and do the following:

  1. Sit down a relax.

  2. Find a note near middle C that is comfortable in your singing range. For me it's the Bb below. It could be as low as A or Ab. Use a note that you can sustain for a good long time without strain.

  3. While singing this note (without sounding it simulataneously on the piano) , play on the piano the note an octave a fifth higher. If you're singing middle C, you'd play the G on top of the staff. In my case, I play the F.

  4. Keep pounding the high fifth until your singing voice is perfectly in tune.

  5. Now, while still singing the note, play the reciprocal tone, an octave and a fifth below. If you're singing middle C, you'd play the F at the bottom of the staff, as in the diagram above. In my case, I play the Eb.

  6. Again, keep playing this note until you here the pure perfect fifth resonance betwen your voice and the piano.

  7. Repeat this process a number of times until you hear the polar relationship.

What you are discovering here is the fundamental concept of "fifth above, fifth below," in beautiful symmetry. When you play or sing the high fifth, you are tuning to the upper partial of the tone. When you play or sing the low fifth, you are tuning to the note which is the generating tone of the note you are singing. This is the intuitive, metaphysical experience. A note both generates a fifth and is itself the product of a tone that is silent, but present within the silence. The central tone is the creator of the higher fifth, but the central tone itself is the product of another creative principle, which we can find reciprocally an octave and a fifth below. (I am speaking poetically, of course, but it is a poetry rooted in an ancient tradition).

What you might notice in this exercise, if you really hold to your sung note, is the very subtle shift you have to make as you play the higher fifth and then the lower fifth, back and forth. This is of course due to the fact that the piano is not tuned to perfect fifths, so you're having to make that little adjustment in your voice to retune to the genearating tone.

Now, you can add the final steps, which are to sing the low tone that is the generating tone of a central pitch, and then to sing the upper partial of the central tone.

First, you are going to sing the high fifth, which is already a common experience for you if you've sung at all in a capella groups. Play the low tone that you sounded before, your F or Eb or whater, and then sing the 3/2 interval an octave and a fith above. Tune it.

Finally, sing the reciprocal tone. For this, you'll probably need to change key a little. Consider what your lowest comfortable note may be. For men, it's usually F or G. Once you're clear on this note, first sit at the piano and play what will be the note an octave and a fifth above. It will probably be middle C or D. Then sing the low tone, tuning your voice to create a perfect fifth, 2/3, below the central tone. Tune it

Alaudin Mathieu, in his book Harmonic Experience (the work that began my explorations along these lines), calls this the mother-father principle. Here is how he puts it, referring to how the F below seems to "create" the C above:

Of all the mysterious events in music, we have come to perhaps the most myserious of all—mysterious because of how the tables have turned: instead of resonating with a note that has already been placed into the musical space by a generating tone, we have invented—created—a note that contains the generating tone in its harmony. It is one thing to produce a vibration that is already given, that is "already there." It is quite another to become that vibration which produces, as one of its children, the very tone you started with.

Within the boundaries of music, the generating tone does behave somewhat like a creation principle. It is the god of its tonal world. C is the god of the realm of all C modes, of all the music "in C." But when you sing F, you create C. How can you create the creative principle? How does one go about giving birth to a musical god? That is the work of the Musical Mother. Hello, Mother. You who dare to sing F in the C world beome the embodiment of the creative and the sacred.

This is only a way of talking, of course, but it is an old and useful way...

Having established the basical principle of dualism, let's make some further observations on the two series and explore a fundamental dichotomy.

The Harmonic Series

The Harmonic Series should be known by every musician. It is closely related to the Pythagorean principle, since it is based on simple integer divisions, although of course the integers involved are greater than those of the 3-limit system. In this case, the intervals are not derived from the monochord but from the actual, physical attributes of a vibrating body, such as a string or tube of air.

The existence of the harmonic series has both a practical and a theoretical aspect. From a practical aspect, it informs how instruments are invented and created, such as the valved brass instruments and chromatic woodwinds. But from a theoretical, compositional point of view, it brings up many reactions. Theorists in historical times, such as the godfather of modern theory Rameau, who were excited about the confirmation and codification of the harmonic series drew some conclusions:

  1. Beginning with Rameau, who was writing his Treatise on Harmony at the same time that scientific experiments were being conducted on vibrating bodies, the discovery that the 4th, 5th and 6th degrees of the harmonic series form a perfectly tuned (in pure 5-limit just intonation) major triad created a tremendous amount of practical and philosophical speculation regarding this "chord of nature." It no doubt fueled the developing centrality of the major triad during the common practice period.

  2. Important theorists such as Hemholtz in the 19th Century (in his On the Sensation of Tone) and Hindemith in the 20th (The Craft of Musical Composition) have concluded that the major triad is the fundamental structure of harmonic thought, invoilable due to its existence within the naturally vibrating system.

  3. From this, it seems clear, as it was to Schenker, that the major mode is the natural mode, and all others are in some ways inferior or "unnatural."

Other theorists, however, have been less sanguine about the musical importance of the harmonic series from a theoretical or compositional point of view. They (Levy and Levarie among them) note that:

  1. The "chord of nature" concept is over-rated, since in fact most musical cultures, while having full access to hearing this chord within any naturally vibrating system, do not immediately gravitate towards the major third or the major triad (to the extent that they have any triadic thinking at all). Indeed, the less "outwardly directed" the culture, the more is there a preponderance of non-major thirds (variants on the minor third).

  2. Even within western music, the minor triad remains a consonance, not a dissonance. The concept, to the critics, that the minor third is somehow a "turbid" or altered major third led Goethe to offer this perceptive comment: "If the third is an interval provided by nature, how can it be flatted without being destroyed? How much or how little may one flat or sharp it in order that it may no more by a major third, and yet still be a third? And when does it cease being a third altogether?

  3. Among believers in the use of the harmonic series as a basis for music theory, there is a curious lack of attention paid to the seventh harmonic. The decision to "stop at the sixth harmonic", since the seventh is out of tune and "not part of our system" seems arbitrary. If the centrality of the major triad is based on our "listening in" to the harmonic series, why is the "out of tune" seventh harmonic not brought into the system? It's energy (loudness) is not so far from the energy of the 7th harmonic (the second fifth in the series). This fact suggests that our harmonic system does not derive from the "natural" harmonic series, but from a philosophical approach that is, ultimately, a choice made within a myriad of harmonic possibilities that give order to the chaos of sound. Nature reflects this order, but does not create it.

The Reciprocal Series

The Reciprocal Series is derived from continuing the process that you experienced in finding the 2/3 fifth below your tone-monad. When the multiplications (which, parodoxically, correspond with the inward direction of the tone-mandala) continue, a beautiful minor triad below the reciprocal fifth is formed, as we saw above.

I don't about you, but I find this both logical and liberating. The polarity of major-minor, which after all is central to our experience as musicians, is accurately reflected in this system. When I sit at the piano and play , especially when I improvise, the minor triad and minor tonality takes on a great depth of meaning and connection, due to this new awareness of it being derived from the polarity of major (outward) with minor (inward). It speaks to me musically, logically, spirtually and creatively.

Listening from above

The most radical aspect of the theory of harmonic polarity lies in the fact that, in this system, the minor triad is "created from above" even though in our practical experience we seem to hear harmonies as built up from below, as from a bassline. These needs some discussion.

This discussion comes to us from a kind of teaching lineage that has roots in Pythagorean thought, but finds its way to us from some of the Renaissance writings of Zarlino, through to the 19th century theorists Hauptman and Riemann and through to the American professors Siegmund Levarie and Ernst Levy, who in turn were influenced by another German named Hanz Kayser. Kayser, in his book The Theory of World Harmonics, takes the idea of harmonic polarity into vast realms of thought and application, including botany and architecture.

Kayser, for one, bases his theories on the Lamdoma, which can be traced back to Pythagorean thought. I couldn't find a good link to this, so I'll also give you a clearer handout with this reprinted.

The Wikipedia entry on the undertone series is worth a read, as it presents a dis-passionate summary. I was amused by the entry that "few bother about these things now," when in fact "bothering about them" is exactly what I am doing! (Wiki doesn't know everything; perhaps I should edit the article).


You will notice here references to Harry Partch, which helps connect what we are doing to contemporary composition, the point of this semester's work. Also referenced are Ernest McClain and Hugo Reimann, as well as the book The Spiritual Basis for Harmony, which also is having an influence on my thinking about this subject.

To address the question of "listening from above," I provide an extended quote from Siegmund Levarie and Ernst Levy's Tone, A Study in Musical Acoustics. Please read this carefully and have a reaction, for or against (or in between).

Although the polarity theory seems to offer a more satisfactory explanation of the major-minor problem than the turbidity theory, at least two difficulties remain.

First, undertones do not exist as a spontaneous phenomenon of physical nature. This objection is irrelevant to a polarity theorist, who develops a harmonic theory not from physical phenomena but from spiritual principles. The results of number operations, such as division and multiplication, applied to the string are independent of the existence of nonexistence of parallel natural phenomena. Goethe expresses his amusement at the observation that nature produces only one half of the acoustical polarity: "It is asking too much for an experiment to perform everything ... One should devise an experiment which could demonstrate the minor mode as being equally original." The experiment requested by Goethe consists in the application, as we have shown, of the reciprocal series of integers—acoustically not a phenomenon of physics but rather the projection of a principle.

The second difficulty is more serious. It concerns our inability to hear a chord from above. The minor triad, generated downward from C, will not be heard by us as C minor but as F minor...We offer a hypothesis which might explain this contradiction without violating the postulates of musicianship. The hypothesis is based on a given condition into which we are born—that of tellurian (i.e. earth-bound) gravity. Gravity, whatever its physical explanation, permeates our whole being—not only our body but certainly our total imagination. Ideally, as in the Pythagorean table, the major and minor triads spring from the same generator as a pair of identical chords in opposite directions. This absolute conception suffers as soon as the concept of high and low pitches, of altitude, in short, of gravity, enters the system. The influence of gravity does not affect the major triad, for the generator C is also the fundamental of the chord. But in the other member of the pair, in the minor triad, the generator and the fundamental become divorced. "Absolutely," we ought to hear the minor chord generated by C as C minor, but "tellurically" we do hear it as F minor. This inner schism between structure and apperception is based on polarity. This situation in music is not much different from the geotropism of plants (again, see Kayser's work). Although the plant grows in opposite directions—the stem upward, the root downward—the flow of the sap is unidirectional, that is, "tellurically adapted."

<<They then suggests that this be tested out with examples in music literature, which we are doing with the Beethoven Waldstein and also, a bit later, with some Mahler. They go on:>>

What accounts for the eminent role of the subdominant in literature? Here the overtones cannot help at all, for F is not present in the overtone series of C...This inconsistency speaks strongly against the turbidity theory. No such difficulty exists in the polarity theory,where the subdominant directly emerges as having equal force with, through opposite character to, the dominant.

The harmonic behavior of chords is well explained by the recognition that two triads spring from one generator—major triad upward, and a minor triad downward. This unfolding of a tone in both directions forms a stable whole:

Critiques of Harmonic Dualism

Of course, a great many theorists in the last hundred years have rejected this viewpoint, and that must be acknowledged. I would say it quite likely that I am one of the few music theory teachers in the US who makes this a central tenet of his teaching, and it's important that you understand this as well! Although much of my research has been based on professors from opposite sides of the country—Levy and Levarie, who taught at Brooklyn College, and Allaudin Mathieu, who taught at Mills College in California and was, incidentally one of John Adam's harmony teachers—it remains the case that this is a minority viewpoint. Nevertheless, it is how I experience harmony, so it's important that you at least know where I am coming from and how I hear music.

The arguments against Harmonic Dualism and the reciprocal series can be summed up as follows. The rebuttals of these viewpoints have essentially already been made.

  1. There is no undertone series in nature. Well, this argument has already be dealt with by basically agreeing with it. If you base your musical theories on the centrality of the harmonic series and the "chord of nature," then you will naturally reject this view, since there is nothing in the physical, outer world which corresponds with the reciprocal series. Indeen, Heinrich Schenker, whose views form the basis of most of the music theory taught at colleges, universities and conservatories in the US, actually considered the minor mode to be an "artificial system." (His book on Harmony actually calls it that: "articifial.") If you subscribe to this view, as a great many theorists in the US have been trained to do (Schenker is a religion in US music theory departments), then you will instinctively reject his approach.

  2. Continuing with the first point, Hugo Riemann, who codified Harmonic Dualism and is the name most associated with it, didn't help matters any by insisting for decades that there "really was" an undertone series. He was wrapped up (at the end of the 19th century) with a felt need to make music theory "scientific", and thus tried to make a metaphysical system physical. Today, followers of Dualism are more comfortable with understanding it purely as a philosophical and psychological orientation.

  3. More serious is the second major argument against dualism, which is that we don't, in fact, hear a minor triad from the "top-down", but as musicians, we still hear all harmonies from the ground up, as growing up out of the roots of a bass note. Again, the answer here is psychological and intuitive, rather than scientific.

My final "rebuttal" to this approach is simply a practical one. I hear harmony in terms of harmonic polarity. It was only recently that I learned that it had a name, but this is always how harmony has presented itself to my ear and my psyche and my soul. It is possible to begin to hear minor chords from the top down while understanding that they function from the bottom up. It's all a matter of orientation. And since I was hearing this way before the theory for it came to my attention, this all has served to reinforce my personal experience.

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