Philosophy behind Harmonic Polarity
This is a page that I wish to add to in the future, but for now it serves to summarize a few key points and bring balance to the more technical details on the surrounding pages.
We saw in chapter 2 that music theory has often maintained an aspect of philosophical speculation. It was seen as an inquiry into meaning, and to be based on more than just musical intervals or mathematical ratios.
This is especially true of the concept of Harmonic Polarity.
Harmonic Polarity, or Harmonic Dualism has a long history, and is usually taken up by persons like myself, who enjoy looking at music from a philosophical or (vaguely defined) spiritual standpoint. For me, it helps to give music theory a deeper level of meaning and connection to our personal experience.
Briefly put, the notion that we discover tones both upwards through the harmonic series and downwards through the reciprocal series corresponds to the polarity of our inner and outer experience as human beings. Major harmony seems to reach upward and outward, while minor harmony reaches downward and inward.
A fundamental principle of Harmonic Polarity is the acceptance of the fundamental equality between major and minor harmonies, and between major and minor keys. On one level this should seem obvious: all tonal music is made of a some fairly equal combination of major and minor chords. Compositions in a major key nevertheless emply minor chords for variety. Minor key compositions move frequently to the major tonality for the effect of brightening the sound. But in terms of music practice, they are essentially equal.
It can be said that major harmony is outward looking—it takes us out into the world, and that the the harmonic series seems to penetrate us from outside. This is because the harmonic series itself is a materially demonstrative principle, and the major triad is certainly linked to the existence and sound of the harmonic series (forming steps 4, 5 and 6, as you know). The harmonic series exists "in nature" — we hear it within the vibrations of a sounding string or a blown pipe.
When we think of music that is in a major key—especially classical music—it is outward looking, goal based, often affirmitive in a way that is heroic, goal oriented. Music that is strongly major in quality is almost patently optimistic or simplistic. The author Deryl Cooke in his book The Language of Music points out that major key music came to dominante the musical landscape during the time of the European Enlightenment, when there was a strong belief in material progress and the possibiity of an enlightened society.
When we think of minor music, it seems (does it not?), to draw us more inward. The cliche is to call minor music "sad" but this is not exactly the word. Certainly it is a bit withdrawn—minor music allows us to reflect on our experience, to contemplate deeper realities. How could a funeral march be major? How could Beethoven's struggle in the 5th symphony be anything but a struggle in a minor key?
The point here, in this rambling first attempt at expressing these principles, is that major chords are clearly built from the bottom up, as reflected in the harmonic series. But if polarity theory has any validity at all, it would then seem to show that minor chords are built from the top down. Even though we conventionally label minor triads by their lowest note—A minor, E minor and so forth—I have found it deeply instructional to think of these chords, and to hear these chords, as being born from the top note down.
In other words, a C major triad is built upwards, C-E-G, whereas the C minor triad is built downwards, G-Eb-C. This is most evident in Beethoven's 5th Symphony, which we will be studying. It begins "Ba-ba-ba-BUM", with the notes G-G-G-Eb. Some commentators say that this is ambiguous, since the opening could either be "Mi-Mi-Mi-Do" in Eb major or "Sol-Sol-Sol-Mi" in C minor. But to the Harmonic Dualist, it is clearly a perfecly expressed C minor triad, spoken painfully and inwardly from the top down!
Goethe, the great German philosopher, poet and playwright, was keenly interested in the question of harmonic dualism. He had studied music theory in great detail and was a contemporary of Beethoven. It is useful to explore his understanding of Harmonic Polarity.
Goethe explained the basic principle as follows: "The fundamental C yields in an upward direction the C major harmony, and in the downwards direction the F minor harmony. Major and minor are the polarity of harmony, the principle of both.The major arises through ascending, through the upward tendency, through an extension of all intervals upwards; the minor through the extension of all intervals downwards...The carrying out of such opposition is the basis for all music."
First off, it is important to understand that we are really dealing with a principle. This principle was best stated by Goethe. 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.
Opponents of the Harmonic Dualism consider the minor third, and thus the minor triad, to be only a false or "turbid" version of the "true" major third, as evidenced by the harmonic series. But to this Goethe also has a reply:
"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 be a major third, and yet still be a third? And when does it cease to be a third altogether?"
The crux of harmonic dualism has to do with this next principle:
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, but I will give you all a handout which contains the Lamdoma. It basically presents a graph, dating back millennia, that lays out a series of interlocked overtone and reciprocal series.
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 natrual 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, or=f 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 ciolating the postulates of musicianship. The hypothesis is based on a given condition into which we are botn—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 fros in opposite directions—the stem upward, the root downward—the flow os 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 i leterature? 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:
There is more to say on Harmonic Polarity, but this is an introductory summary on the principles.
NEXT: 5H More on Dominant Seventh Chords
ADDENDUM
(For those of you, especially music majors, who might be interested in exploring this topic, I am adding here an exercise that I gave my third semester theory students. Some of you might wish to investigate this, so I paste it here. If not, just let it go.)
It is important that you have an experience with this concept. Thus, I need you to go to a keyboard and do the following:
- Sit down a relax.
- 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.
- 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.
- Keep pounding the high fifth until your singing voice is perfectly in tune.
- 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.
- Again, keep playing this note until you here the pure perfect fifth resonance betwen your voice and the piano.
- 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...
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