Why Harmony?

Okay. First post. Hello.

Stay tuned for Scale-of-the-day, musings about intonation and music in general. Below is an old essay that makes for a fair introduction of my thinking.

Harmony is expansive.

I first studied harmony as a young piano student. I had a facility for triadic harmony and could easily transpose, invert and otherwise pull major, minor, augmented and diminished triads off the keyboard at will. I was generally more interested in tinkering with chords than dutifully reading notes on the page so my piano studies soon stalled while my interest in music remained. (Why is early piano instruction so focused on sight-reading and otherwise divorced from listening or passion?)

Then I had the good fortune to take up jazz piano with Mary Fields. She would be the first great, and most enduring, learning influence in my life. She set me on a path of harmonic expansion that continues to spiral well beyond simple triadic harmony.

Expansive harmonic thinking began with stacking additional thirds on top of the now familiar triads to include the 7th, 9th, 11th and/or 13th degrees forming chords with more than just three notes. Some chords use every note in the scale. A full range of jazz harmonies soon followed. Suspended chords, chords with flat and/or raised 9ths (or alterations of any kind), poly-chords, tone clusters, chords respelled into new voicings, voicings that don’t include the root (leaving that role for the bass player to take up in combo settings), stacking and mixing intervals (not just thirds) and chord substitutions were just a few of the harmonic approaches I began exploring at this time. Mary Fields was also the first to introduce me to the quartertone music of Charles Ives and the “corporeal” music of Harry Partch. We even experimented with harmonies derived from phone numbers, names and other external stimuli. She instilled a sense of the importance of intervals. We studied and drilled modes until each one was second nature. She taught the value of curiosity and irreverence in the face of unthinking tradition. She gave me a copy of George Russell’s fantastic Lydian Chromatic Concept of Tonal Organization for Improvisation. These ideas formed a catalyst for my expanding sense of harmony.

I later studied serial theory with Mario Pelusi. (“Serialism” is practically a dirty word in some quarters. This is not entirely undeserved. There is a particularly unattractive strain of elitism and intolerance associated with the serialist clique of university composers of the ‘50s and ‘60s and the whole “music of the future” language is a little tough to take seriously anymore. However, many of these unattractive qualities are also present in the ongoing backlash against non-tonal music. I think serialism needs to be reconsidered within a pluralistic environment. Doctor Pelusi is a positive example of teaching atonal theory without the hyperbole and intolerance that once fermented such backlash. It is possible to study and practice serialism without turning it into dogma.) Serial theory continued to expand my concept of harmony by exposing the assumptions one develops from a fluency in tonal syntax. Serialism is a natural approach to take when the continued approximation of tonal harmony within a 12-tone equal tempered system has worn thin. Serialism is aesthetically compatible with equal temperament. This is perhaps why 12-tone music written for the piano is so compelling (in my opinion).

What I learned from studying non-tonal composition is that all harmonic combinations are possible (the so-called “liberation of the dissonance” attributed to Arnold Schoenberg). But not all harmonic combinations are probable. Set theory is a valuable discipline for developing a deeper understanding of both the possible and the probable. Serialist aesthetic generally tends toward a systematic maximization of harmonic resources just outside conventional tonal language. I learned that there are more ways to organize and conceptualize harmony than just scales and chord changes. Ordered tone rows and the kinds of operations that can be carried out on them are both fascinating (in their ability to apply continuity to a sonic landscape bereft of familiar “tonal” landmarks) and frustrating (in that ordered sets have none of the live improvisational flexibility that scales have (even if it is relatively easy to improvise something that “sounds atonal”)). Understanding intervals remained the single common thread to understanding both scales and tone rows. Intervals paint the color of harmony and the body of intervals present in a given scale or tone row determines the precise shades and hues of the larger harmonic picture.

The next addition to harmonic understanding came over the course of studying composition with James Tenney. One thing he said in his very first composition class was, “Have you ever really listened to a major third on the piano? It sounds terrible.” That statement demonstrated a deep understanding of harmony. One understands harmony through listening to the quality of intervals. There is more than one way to tune a major third and the one found on a piano is not the only one worth considering.

Tenney’s theory of harmony uses lattice structures to map out harmonic space. These structures shatter the representational adequacy of the “circle of fifths” the same way jazz harmony detonates triadic language. Intervals take on a quantifiable quality. Harmonic conception became a descriptive geography open for wild exploration. Harmonic structures became widely applicable to melody, timbre, rhythm (durations are merely temporal intervals) and seemingly all other parameters (musical or otherwise). The shaky dichotomy of ‘consonance’ versus ‘dissonance’ unfolded into a wide spectrum of perceptual relatedness. By including a larger set of intonational variants (microtonal) one opens the harmonic landscape to a rich variety of ‘consonance’ and ‘dissonance’ that are often more ‘consonant’ and more ‘dissonant’ than the familiar chain of twelfth-root-of-two/100-cent-semitones affords. By stepping into the cracks “between the keys” I found a new understanding of tonal gravity that stimulated a new compositional direction for me.

Following my studies with Tenney I pursued a decidedly anti-scale aesthetic toward composition. The ideal approach seemed to be tuning each interval relative to what preceded it in a free harmony reminiscent of Lou Harrison’s “free style.” Tone rows seemed like a logical target for independently shaping specific intervals over time and composing a music that roams freely into the outer reaches of harmonic space without the opacity that equal temperament brings to serial music. I began speculating on the “liberation of the tonic (1/1)” and working toward a realization of vast, rootless harmonic landscapes. Then I took another look at George Russell’s Lydian Chromatic Theory and decided to re-think the scale. I tried to systematically examine every possible scale and ended up altering my own concept of scale. Scales regained a high value for my harmonic sensibilities. I began to develop a more inclusive theory of harmony that builds upon the cross-fertilization of progressive jazz harmonic theories, microtonal theories and adaptive set theories.

Harmony seems to expand further outward the more I study it. There is something fundamental about frequency and periodicity that just seems to resonate with my personal sensibilities. There is something compositionally suggestive about scales, tone rows and intervals.

Harmony as color.

When the Pythagorean chain of just perfect fifths is bent into a circle, a measure of infinity is lost. The F-sharp and G-flat are coerced into sharing the same note on the piano keyboard as “otonal” and “utonal” qualities are blurred into a shade resembling the vibrancy of mud. Picasso’s “Blue Period” would have been much less compelling if he had only a single shade of “blue” to work with. The current equal tempered system affords twelve identical harmonic modulation points at the expense of such shading. Modes turn into mere scales as the sonic qualities of a “Lydian” or “Aeolian” are snapped into an opaque reordering of a singular sequence of five major and two minor seconds.

Modes are singularly important to understanding harmony. Yet current practice obscures a clear understanding of what modes are and why they paint the kinds of colors they do. A scale is said to be a “mode” of another scale if the sequential order of intervals varies only by starting point (a shifting of the tonic degree). In current practice the Lydian mode is said to be a mode of the “Major” scale (Ionian = Major) that starts from the fourth degree (resulting in the major seventh degree of the Ionian serving as the augmented fourth degree of the Lydian). The relative content of “otonal” versus “utonal” member pitch classes determines the overall “brightness” of harmonic color shading. These shades are dimmed when interval classes are folded into closed, flat-earth, cyclical approximations of more vibrant realities.

“Otonality” is sonically similar to “major tonality.” In traditional just intonation an otonal pitch has a frequency ratio with the largest prime factor in the numerator (i.e. the 3/2 just perfect fifth is otonal because the largest prime factor is three). In a lattice structure this would be any pitch whose coordinate point along its largest prime axis is a positive integer. In a Pythagorean system this would be any pitch that has a power of three in its numerator and a power of two (or whatever the scale replicating interval is if it is not the octave) in the denominator. This happens to be any pitch that is derived from tuning upward by just perfect fifths from the tonic. “Utonal” pitches are the inversion of “otonal” and sonically similar to “minor tonality.” As a coordinate point in a Pythagorean system the “utonal” pitch is negative (and derived from tuning upward by just perfect fourths from the tonic). A member pitch of any given scale will be otonal, utonal or a tonic in much the same manner integers are positive, negative or zero. The balance of otonal versus utonal pitches determines the modal identity of a scale.

A scale that is in Lydian mode is any scale that consists of a tonic and all otonal pitches (that is, zero utonal member pitches). A Lydian scale, however, refers to a common-practice relative of the Pythagorean Lydian that is typically made up of a tonic, major second, major third, augmented fourth, perfect fifth, major sixth and major seventh degree. It is possible to tune a Lydian scale to a non-Lydian mode, and as long as the tonic and augmented fourth degrees are left intact the other member pitches may be altered (i.e. minor or augmented seconds, augmented fifths, etc.) without changing its Lydian scale-ness. The key point is to differentiate between the Lydian scale and the Lydian mode.

The Ionian mode, therefore, is any the mode of any given scale that contains one utonal member. The Ionian mode is a single shade darker than the Lydian mode of that same scale. The following sequence then follows: Mixolydian mode = two utonal members, Dorian mode = three utonal members, Aeolian mode = four utonal members, Phrygian mode = five utonal members, Locrian mode = six utonal members, etc. Beyond six utonal members the ancient Greek names are abandoned in favor of the clarity of 7-utonal mode, 8-utonal mode, 9-utonal mode, etc. Given a seven note scale, one can regard the “brightness” spectrum as spanning from the Lydian mode as the “brightest” mode to the Locrian mode as the “darkest.”

When thinking of composition allegorically as painting the color implications of modes are compelling. Shifting between dark and light is an effective method for affecting contrast without falling into the clichéd “major = happy/minor = sad” dichotomy. (I generally find painting allegories like “color” and “brightness” to be more useful than the vague emotion/mood similes. While it is admittedly possible to calculate a specific emotional response range within a hypothetical listener, that task that lies well beyond the scope of harmonic theory.)

Harmony as an expression of the inner workings of tone.

Finding a universe of harmony within the overtone series of a single sounding tone is a bit like finding a preponderance of “solar systems” embedded within the atomic structure of matter. It makes for interesting speculation, but tends toward overly mystical interpretation. There is an undeniable suggestiveness to finding so much intervallic content at the “microscopic” level of acoustic phenomena. Vibrating objects almost seem to carry a sonic DNA that favors the 386-cent major third found between the fifth and fourth partial over the 400-cent approximation forced into place by frets and piano tuners. Unaccompanied choirs and violins certainly gravitate to “nature’s” version of the major third. Even more disconcerting is the 31-cent deviation between the 7/4 minor seventh and the 1000-cent 12TET version that is often sited as the acoustic origin of “blue” notes. Just intonation allows the overtone series’ of multiple notes to mesh more cleanly (and conversely “beat” against each other with more ferocity (wolf-tones)) than tempered systems allow. (This is a generalization, one must remember that unisons and octaves are every bit as clean in either system (because they are identical in just and equal temperaments) and that the human ear is willing to accept a 700-cent perfect fifth as “close enough” to stand in for the 702-cent just perfect fifth).

The overtone series as a compositional resource has inspired a number of interesting pieces. But as a larger, more encompassing harmonic system it tends to lack the darkness that inversion/utonality brings to harmonic language. The idea of constructing scales derived from the FFT analysis of complex timbres (or vice versa; constructing timbres derived from scales) is a fascinating compositional possibility. There is an undeniable aesthetic appeal to using a singular source to organize disparately related parameters. I would argue that a good FFT analysis will document harmonic content over time, exposing the “ordered” onset and fade of frequency energies, which has more in common with tone rows than the non-linear scale.

The larger conclusion to be drawn from studying the overtone series and the phenomenon of additive frequency sums forming complex tones is there is an abundance of harmonic content being effortlessly processed by our auditory systems all the time. The sonic world is rich with harmony. This is, by itself, a definitive answer to the question: why harmony? Studying and experimenting with harmony offers incredible and sometimes startling insights into the process of human auditory perception.

Harmony as an outward expression of natural phenomena.

When one numerically measures the size of intervals one can’t help but notice parallels to all the things quantifiable. One can take the interval of time required for the earth to orbit the sun (approximately one beat per 365.25 days) and transpose it up several octaves to conclude that the orbit is tuned to a G-natural. Like the overtone series, it is suggestive, but risks toppling into a miasma of mystical allusion (especially if one never asks the question “why transpose by octaves? Is the universe constructed of multiple powers of two?”).

The larger conclusion is that we can learn any number of things (and notice any number of coincidences) when we quantify any phenomena. I tend to regard music inspired by the overtone series as exploring internally derived logic and music structured by the orbits of celestial bodies (or the Earth’s magnetic field) as exploring externally derived logic. Life is rich. The grounding principle here is to recognize potential compositional resource (inspiration) without turning “meta-physical” or building a “grand unified theory of everything.” One does not knock the earth out of its orbit by playing in F-sharp minor.

Harmony bends well to structural machinations.

The real marvel of harmony is that our auditory system can process so much information so effortlessly and involuntarily. One doesn’t need to “understand” detailed harmonic theory to hear it. The octave sounds naturally consonant. Triadic qualities are communicated without needing proper names. Ordered tone rows reveal a consistency over time that an alert mind can process as structurally unifying. An open and curious mind is adaptable to harmonic ideas that are familiar or surprising.

Often when the question of “why harmony?” is posed the implied query is “why not just rely on intuition to determine what ‘sounds good?’” or the near-rhetorical and insidious question of “what is the point of over-thinking such things?” I am aesthetically opposed to the anti-intellectual attitude toward music (or any other endeavor) lurking behind these implied questions. All the calculating with numbers and experimenting with intonation is directed at achieving a sound that stimulates the auditory system irregardless of whether one is conscious of the machinations behind it. Wallowing in the familiar comforts of unthinking tradition leads to a mental atrophy that closes minds and dulls the senses.

There is also the pull of a harmonic traditions spanning across continents, cultures and centuries. Harry Partch drew inspiration from ancient Greece, the music of Chinese theatre and the rhythm of spoken word to develop a theory (and practice) of harmony that reflects the integrity of these influences. Partch contributed to an understanding of harmony that draws upon a sweeping view of music history coupled with acoustic principles. Charles Ives painted sonic portraits of the New England of his youth that creatively expanded the harmonic fabric of his own time. He speculated on a future when children would sing songs in quartertones as a natural evolution of harmonic development. Ivan Wyschnegradsky dreamt of angels visiting him and delivering “divine revelation” to write microtonal music for string quartet. The pulsating rhythms of Indonesian Gamelan music (with the unique pentatonic Slendro scale or 7-tone Pelog scale) have inspired both shallow imitation and fully formed performing ensembles worldwide. The story of Western “classical” music spanning from Gregorian chant to a developed tonal language and beyond to the total serialism of Pierre Boulez or aeleatoric practices of John Cage is itself a harmonic story of the gradual addition and acceptance of increasingly complicated intervallic relationships. Given a progressive view of harmonic evolution it is hard to resist the multiplicity of ideas available to forward-thinking composers of the early 21st century looking to build upon the examples of Partch, Ives, Wyschnegradsky and countless others.