(Doctoral Thesis Abstract)
by Jason Martineau


Theories suggesting that the overtone series is a powerful, generative force in Western musical thought have existed from Rameau to the present. One manifestation offers that the basic musical element is the chord, specifically the major triad, which is a naturally occurring sound in the series. The movement of harmonic progressions in tonal music, as well as the structure of the diatonic scale itself, are also said to be influenced by the series. To extend this notion, it is suggested by this writer that the actual unfolding of harmony through known history, considered as the gradual addition of intervals, provides a mirror of the overtone progression. Whether the phenomenon is physiological involving the inner ear structures, or spiritual involving consciousness itself, the perception of musical sound as a process of growth and change can be clearly observed as evolving through time. The musical composition accompanying this paper will serve as an historical microcosm and unfold harmonically in this manner; in accordance with the overtone series.

~diagram~ (pdf)


This paper and composition are intended to illustrate a theory about the history of Western harmonic thought. Due to the abstract and metaphorical nature of this inspiration, some preliminary, cautionary notes will be necessary. The reader is encouraged and invited to free himself of any preconceptions or adherence to other strict, academic interpretations of the history or nature of Western musical expression. The ideas presented and developed are based largely upon observations and conclusions. These ideas are primarily those of the writer, but often they will be drawn from formulations and concepts presented by many people throughout written history. The acceptance of the validity of these notions will be dependent in part upon the reader's ability to entertain them without seeking to find exclusions. All rules have exceptions, and all a priori theories have fallibility. No attempts at comprehensive explanations of history or theory can be entirely inclusive. The approximation of trends, patterns, and probabilities are often the closest we can come to understanding the true nature of a thing.

The fundamental intervallic content of music from (at least) the 9th century to the present reflects a development, when viewed loosely as a timeline, which closely resembles the overtone series. The intervals inherent to the acoustic properties of any pitch when sounded contain a microcosm of musical collective consciousness. It can be seen that, beginning with Chant, the process begins as man is concerned mainly with the fundamental of the series; i.e. the first, generative force of vibration. With the development of organum, the first two overtones of the octave and fifth make their appearance. With the gradual development of polyphony in the 12th and 13th centuries came the inclusion of major and minor thirds and their inversions, sixths. As these sounds created triads, leading tones, key relationships and tonal sensibilities also formed. When the triad as a basic musical harmonic unit was firmly in place by the late 16th and early 17th centuries, the first through fifth overtones were represented (remember that the third overtone is two octaves above the fundamental, so it is not a different pitch). With the use of dominant harmonies and their substitutes such as augmented and diminished chords, the seventh overtone (or seventh member of the set) is the somewhat mistuned interval of a minor seventh. It is mistuned because it does not directly correspond to any sounds in the equal-tempered scale. In fact, the intervals of a fifth, fourth, third also do not precisely equal their acoustic derivations; but remember that pitch is fluid, and any attempt at temperament will involve compromise (the problem of the Pythagorean comma). The point here is that the minor seventh shares a more distant relationship with its natural, acoustic derivation than do most other intervals. With a pitch collection of major triad plus minor-seventh, the great emancipation of tonal centers was firmly in place, in part by virtue of the tritone created between the third and seventh of the chord. The equal exploration of all twelve notes as tonal centers, transported by modulation, accounts for much of the repertory from Bach to Beethoven. Also remember that when tonal centers first began to move, it was most often to the dominant or mediant, both of which are the first differentiated pitches in the series. Again, the overtones are apparent, underlying the harmonic and tonal impetus of music from this time period. From the 18th century through the early 19th century, modulations to more distantly related keys began to push the envelope of harmonic functionality, but only in a small way. This inevitably led to the extension of harmonic thought (also involving the prolongation of non-chord tones) so obviously exploited in the 19th century. If we look at the overtone series, we see that the interval of a second makes an appearance for the seventh through ninth harmonics. Dominant-seventh harmonies, as active agents by their nature, due in particular to the presence of a tritone, easily accommodated appoggiaturas and anticipations of unusual modulations by the addition of harmonic extensions. Ninths and thirteenths become chord members, and slowly their need to resolve is countered by their sonorous qualities. Some have compared the derivation of the thirteenth as another presentation of the more flatted version of the mistuned minor seventh of the series. Additionally, the extension of chromatic participation in time weakens the tonal implications of a given harmony, and this contributes to one of the most controversial developments in music history: the dissolution of functional tonality based on the acoustic nature of pitch. The eleventh overtone, the tritone or raised eleventh, created another fascinating gateway into the worlds of symmetrical pitch constructs and harmonic ambiguity. This partial, like the seventh, is also mistuned in the same sense as the former. These intervals, when considered as a whole in a fully extended dominant-seventh chord, represent the saturation of the diatonic system. A chord of C-E-G-Bb-D-F#-A, contains all the notes necessary to create a seven-note scale, often termed the acoustic mode, or the melodic minor constructed upon the dominant. Of particular popularity were the octatonic and whole-tone scales which began to make their appearance in the late 19th century. These symmetrical designs and their harmonic derivatives were the companion to twelve-tone music as deteriorating forces of tonal perception. At this point in the series, we begin to depart from the range of audibility, as each partial also diminishes in amplitude as well as in interval size; vibration fading further upward. Despite this, the intervals present there, minor seconds and microtones, have also found their resonance in music of the twentieth century. In the second half of the twentieth century, the crisis may have truly begun, as the series may have been exhausted and art (as well as mankind) frantically tries to rediscover itself.

The apparent ascension of the harmonic series may also mirror the Humanism that began in the Renaissance. The idea that man is to grow and transcend always upwards is reflected in this ascension of the series through history. There also exists a suggested correlation between historical time and the relative strengths or amplitudes of each partial. Because of the lack of explicit music notation prior to the ninth century, it is difficult to incorporate prehistory into a chronologically precise timeline. However, it can be seen that as the new intervals became standardized, the rate of ascension accelerated. This theory of harmony can be reconciled in part by the notion that Western perception and collective consciousness has in some measure expanded or heightened over time (all spirituality aside), embracing each new overtone according to its respective harmonic resonance.

Also contained in the basic aspects of the series are some other historical curiosities. For the last three centuries the pitch C, regardless of its actual Hz value at the time, has always carried with it a symbol of purity; the most elemental, primordial state. This is due in part to the fact that most tuning systems prior to equal temperament used C as the first tuned note. This gave C a practicality and assurance in the use of a scale built upon it. Some of the "mystic" composers used the chord of C major in an otherwise chromatic framework to express an etiological state or state of absolute simplicity, such as Messiaen (the last chord of his opera Saint François d'Assise ), Scriabin (the last chord of Le Poème de l'Extase ), Holst (last chord of The Hymn of Jesus ), and Berg (Wozzeck, Act II, scene I, mm. 116-123). Indeed, perhaps it is no less an accident that in the key of C major, F# and Bb each carry the tonality to either the dominant or subdominant, respectively. Likewise, the acoustic mode can be directly transformed to the whole-tone and octatonic scales by the contraction of the fifth and sixth degrees by a half-step or the splitting of the second degree into two half-steps, respectively. In these cases, the acoustic collection as a realized scale or chord in tempered music plays an embryonic role: a primigenial entity. My composition, therefore, will attempt to serve as a microcosm or illustration of this theory.

The first sound heard in the work is an all-note, all-interval chord built upon C in consecutive thirds. This can be thought of like the big bang; the birth of creative musical force. This chord will then pervade the opening section, at times only in shadows or in pieces; providing a stage for the primordial chaos. As there is presently much dispute about the actual nature and sound of music from prehistory, this indefinite time span up to the ninth century will be handled as a metaphorical sea of ideas. The lack of explicit written record or repertoire will be symbolized by surges of simultaneous events and gestures, violin glissandi, momentary wind flurries, bubbling string tremolos; the textural equivalent of white noise. These will be a seemingly random series of events, each momentary and disconnected from the other. The reader is invited to imagine scenes of the world in drama; a meteor shower, volcanoes, tumultuous rivers of lava, lightning storms, or the aurora borealis as visuals to accompany these sounds. It is the dancing and playing of creative energy seeking form, structure, and pattern.

As all audible pitch is indeed actually rhythm or periodicity, rhythm and percussion will underpin the pitch material. The particular rhythms to be presented are some of the Pythagorean ratios; (1:1, 2:1, 3:2, 4:3, and 5:4). Musically, they are the unison, octave, fifth, fourth, and major third, respectively. These are chosen because their discovery is considered among the most important in all of music history. The ratios are expressed by means of polyrhythms. These rhythms are articulated by various non-pitched percussion instruments while the remainder of the orchestra will represent the musical chaos. It is hoped that the polyrhythms will offer a more direct, visceral experience of intervallic relationships. This portion of the piece constitutes an introduction, and lasts approximately one minute, thirty seconds.

What then follows is the first stable pitch, representing both early Chant and the fundamental of the series. Because of the nature of the design of music theory, and in particular the keyboard, C is the first sound heard; the "Ur-note" if you will. To remind the listener of the prevailing concepts, the note undergoes a shifting of timbre. This is accomplished by a small variety of instruments swelling and subsiding at the unison. This shifting of timbre is in fact tantamount to the heightening or lowering of certain partials in the series. The tempo here is rather slow, to intensify the monolithic quality of this section. The first motion of the theme is a descending minor second, then ascending, then a descending major second. Slowly, each interval will be introduced in order of expansion. The theme will vary slightly with each repetition, gradually adding to its length each interval of the twelve-note set. This could be visualized by thinking of a gnomonic spiral such as a nautilus shell. At the conclusion of this is a transposition of the theme at the fifth, the first natural departure. This variation is followed by a presentation in the subdominant, completing the cycle of primary root functions I, IV, and V. These can be thought of as the Earth (stable) I, the Moon (passive) IV, and the Sun (active) V.

At the completion of the presentation of the opening theme and its modulations, octaves will begin to appear; and the rhythm accelerates slightly. In fact, during the course of each new interval presented throughout this work, either the tempo or the rhythmic subdivisions also increase. This idea stems from the fact that historical time compresses between each new harmonic unit; and that as the pitches presented are higher in sound, their frequencies (rhythms) are also faster. Some contrasting thematic material is used here to assist in articulating the entrance of a new interval, but it is also drawn in part from the first theme.

The section of the piece containing thirds and sixths will probably seem a bit more familiar in its sound. Open intervals and triads (and their inversions) will abound in both homophonic and polyphonic settings, but they will not behave functionally. Both the first and second themes will be united here as the tempo increases. The cumulative effect of theme combination, instrumental addition, and tempo increase hope to express the culmination of the first musical unit; the triad iself. The third in musical history contains within it the will of man, singing with his own voice with the architectural support of the timeless and stable fifths, fourths, and octaves. The seeds of leading-tone behavior are also planted by the proliferation of thirds, and with them the beginnings of modulation.

With the arrival of the seventh, tonal implications around C fall away rather quickly. The thematic structures also begin to crumble in order to enhance this sensation. There is implied here the beginning of dissolution. As modulation into other keys becomes possible in music, so does the weakening of each key relative to the other. This ultimately leads to symmetrical constructs and equal-interval tonality (dodecaphony). Harmonic expansion of the raised eleventh completely removes stability from the piece; and as microtones are introduced expressed as small groups of violin glissandi, the listener is reminded of the beginning of the work. The Pythagorean relations presented as polyrhythms in the introduction also recapitulate. It is in fact, a cycle. The hypothesis is that the cycle of harmonic exploration of the acoustic series resets after the highest, presently inaudible partials are reached; and so the piece ends as it began, and begins again.

~hear the entire piece~


Blavatsky, H. P. The Secret Doctrine: The Synthesis of Science, Religion, and Philosophy. Volume 2, Anthropogenesis. California: Theosophical University Press, centennial edition, 1988.

Bowers, Faubion. Scriabin: A Biography. Second, revised edition. New York: Dover Publications, Inc., 1996.

Cooke, Deryck. The Language of Music. Oxford: Oxford University Press, 1959.

Dallin, Leon. Techniques of Twentieth-century Composition. Dubuque, Iowa: Wm. C. Brown Company, 1974.

Fleming, William. Art, Music & Ideas. New York: Holt, Rinehart and Winston Inc., 1968.

Ghyka, Matila. The Geometry of Art and Life. New York: Dover Publications Inc., 1977.

Grout, Donald Jay. A History of Western Music. New York: W. W. Norton & Company, 1973.

Hall, Donald E. Musical Acoustics. Pacific Grove, California: Brooks/Cole Publishing Company, 1991.

Hargreaves, David J. The Developmental Psychology of Music. Cambridge: Cambridge University Press, 1986.

Hoppin, Richard H. Medieval Music. New York: W. W. Norton & Company, Inc., 1978.

Huntley, H. E. The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications Inc., 1970.

James, Jamie. The Music of the Spheres. New York: Copernicus, Springer-Verlag New York Inc., 1993.

Jarman, Douglas. The Music of Alban Berg. Berkeley: University of California Press, 1979.

Kandinsky, Wassily. Concerning the Spiritual in Art. New York: Dover Publications Inc., 1977.

Lawlor, Robert. Sacred Geometry: Philosophy and Practice. New York: Thames and Hudson Inc., 1989.

Morgan, Robert P. Twentieth-century Music. New York: W. W. Norton & Company, 1991.

Nectoux, Jean-Michel, Roger Nichols, Robert Orledge, Patrick Gowers, Nigel Wilkins, G. W. Hopkins, Paul Griffiths. The New Grove Twentieth-century French Masters. New York: W. W. Norton & Company, 1986.

Ratner, Leonard G. Music, the Listener's Art. United States: McGraw- Hill Inc., 1977.

Schimmel, Annemarie. The Mystery of Numbers. Oxford: Oxford University Press, 1993.

Scriabin, Alexander. Poem of Ecstasy and Prometheus. New York: Dover Publications Inc., 1995.

Stolba, K Marie. The Development of Western Music. Dubuque, Iowa: Wm. C. Brown Communications Inc., 1994.

Thompson, D'Arcy Wentworth. On Growth and Form: The Complete Revised Edition. New York: Dover Publications Inc., 1992.

Wiess, Piero and Richard Taruskin. Music in the Western World: A History in Documents. New York: Schirmer, 1984.



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