Very little is known about the nature of musical Harmony. Music Theory has, with a few exceptions here and there, proven to be pretty useless. Each new theoretical system which pops up is little more than a codification of the contemporary musical style, an attempt to render the works of the great artists into a set of rules and guidelines, rather than an honest attempt at understanding what Harmony is. As a consequence, the overarching field of music theory has always been more instructive than it is informative, more practical than it is truthful. Here, at the tail end of its long history, it has failed to answer any of the big questions about music.
Most pressing of all of these questions is, without a doubt, that of the underlying nature of Consonance and Dissonance. Not only is it almost unanimously agreed upon by Western musicians that some harmonies are "consonant" and that others are "dissonant," but the entire logic of the Western style of Harmony is founded upon the polarity between the two. The entire "dance" of Western music is jumping up into unstable dissonances and landing back onto stable consonances. We cannot come close to understanding Western music without first understanding the underlying nature of these polar harmonic qualities which have been built into it.
Our understanding of Consonance suffers not, as one would expect, due to a lack of facts, but due to the persistent narrowness of those who interpret them. Music theorists, without knowing it, have all subscribed to a very specific conception of Harmony: that every harmony consists of "intervals" and that these intervals are the source of all harmonic feeling.
This conception originates with the Pythagoreans who, a couple millenia back, theorized Consonance to be the direct result of numerically simple intervals. In music, an "interval" describes the relationship between two pitches, usually in the form of a ratio. A pitch at 400 Hz and a pitch at 300 Hz, for instance, create the interval of 4/3. A pitch at 600 Hz and a pitch at 500 Hz create the interval of 9/8. According to the Pythagoreans, the former interval is more consonant than the latter because its ratio is simpler.
This "Pythagorean" conception of Harmony has dominated Western Music Theory since it conception roughly 2500 years ago. Although music theorists will sometimes deviate from the surface-level argument, that consonance derives from numerical simplicity, Pythagorean Harmony's basic underlying assumption that it is the INTERVALS which are the source of Consonance is never questioned.
We see Pythagoras' shadow cast upon all of the most famous music theorists. Zarlino, Rameau, Reimann, Schenker, Partch - all of these men begin their Treatises of Harmony by listing off a set of ratios. Only once he has finished appraising each of these ratios as either consonant or dissonant or something in-between does he then feel comfortable using them to reconstruct the classical scales and chords which he is used to.
We can also see that every one of the modern consonance-dissonance maps, constructed by modern theorists, operates by measuring consonance and dissonance along a range of intervals:
In all of these theories, the interval is assumed to be the essence of Harmony. It is never the harmonic-structure ITSELF which is viewed as Consonant, but rather the intervals which it consists of. A major triad is not viewed as consonant; the 5/4 and the 6/5 within it are. The harmonic series is not viewed as consonant; just the intervals within it. No matter how widely our conclusions vary, they are always fit into the narrow presumption that Harmony essentially only takes place at the level of the interval. Why has no one ever considered that the reverse might be true - that it is the over-arching harmonic structure which is the source of Consonance and that the intervals within this structure are merely small fragments derived from a consonant whole?
THE DOMINATION OF PYTHAGOREAN HARMONY
What has made the continual domination of Pythagorean Harmony so damaging to music theory is not just its many contradictions, not just the limitations that it has placed on our interpretation of Harmony, but that it has led us to think of Harmony as something purely numerical. Intervals are ratios, and ratios and numbers. When Pythagoras tied all of music's aesthetic qualities to its interval-ratios, he made number the sole basis of Harmony. Music became downstream from Math, and our music theory was doomed, from its very beginning, to be defined by tedious mathematical procedures.
Of all the arts, only music has been subject to such a hyper-numerical treatment. When we look at a beautiful painting, like Rembrandt's "Night Watch," are we, at the bottom of it, just perceiving some mathematically perfect object - some formula or function which rendered into visual image? Absolutely not! The qualities which define the work - its colors, shapes, light, dark, shading, depth, framing, etc. - are all qualities particular to our sense of sight, qualities which cannot be reduced to mere number. Certainly, one could measure out some set of lengths in this painting, and one could argue that their "simple proportion" contributes, in some subtle way, to the piece's beauty, but only an absolute fool would claim that the entirety of the piece's aesthetic value rests on these proportions. But this is the exact treatment we have given music! Only music is viewed as essentially just a bunch of ratios.
Our language for describing smells and tastes is almost entirely based on analogy: "fruitty," "nutty," "earthy," "salty," "spicy," - every one of these descriptors invokes an association with some actual sensual experience. Sight is described almost entirely in terms of colors, shades, and light - qualities which are specific only to that sense-world and which cannot be constrained by numbers. But even our words for colors were originally inseparable from the real-world objects which they were associated, as their etymological roots suggest: green with "grass" and "grow", yellow with "glow" and "gold," red with "reed" and "ruby" and "rhubarb." Basically, it is not far-fetched to assume that our words for colors originated in the same way as our words to describe tastes and smells - that is, they began as analogies for actual things which they were associated with before gradually evolving to describe anything which shared that color. Our language for describing Harmony lacks this kind of straightforward connection to the senses. Words such as "voice-like," "howl-like," "chirp-like," "scream-like," "whisper-like," and "hum-like" are entirely absent from music theory, even though they would probably reveal far more about the meaning of a given harmony than the conventional "this chord consists of a 5/4, 5/3, 4/3, and 2/1."
In short, Sound is the only one of the fives senses that is constrained by a purely numerical language and, therefore, by purely numerical logic. Where the other sense-arts, especially those of taste and smell, are viewed as orgiastic, sensuous, and intertwined with life itself, Music, being mostly obscured in the fog of numbers we have created around it, is only ever viewed as austisticaly logical and as mysterious as number itself. Why is Harmony beautiful? "Because of numerical relations." What is beautiful about the numerical relations? "Ummm, they just ARE, okay?!!" This misguided philosophy of music has not only narrowed our understanding of Harmony, but it should be held also as partially responsible for the worst of Modern musical trends. "Serial Music" is something that could have only been produced by a generation of musicians who view their art as little more than a mathematical exercise. Indeed, the Modern stage of Western music is characterized mostly by a numerical and hyper-logical feel; it is music that is calculated, rather than composed.
So, not only the Western world's music theory, but its music too, stinks of the corpse of Pythagoras' dead number-religion, rotting and worm-infested. The purely ratio-centric language of our music theory is only the most deep-set manifestation of Pythagoreanism. Our task, then, is heavy: we must pull out his 2500-year-old philosophy by the roots, and begin to concoct some new way of viewing Harmony. Furthermore, the gap between our understaning of Harmony and our basic sensuous impressions of sound must be bridged, and the only way for us to do this is to create a music theory with a basis in actual sounds.
First of all, it would be worth setting down a definition of Consonance which is not hampered down by intervallic baggage.
It has become increasingly common to define Consonance as "pleasing/pleasurable sound." But this definition is pretty blatantly wrong, considering that dissonant harmonies can be just as pleasing and even more pleasing than consonant harmonies.
"Consonance" roughly means acoustical unity. Sounds which are “consonant with each other” naturally interlock together, becoming a singular, coherent acoustical form.
Take, for instance, the perfect-fifth:
Notice how, when played simultaneously, these 2 notes sort of "hug" each other. Rather than perceiving them as two separate sounds, we naturally feel them to be a singular acoustical form.
The octave, the perfect-fifth,and the major-triad all have this unified feeling to them.
These sounds are not "pleasing" per-se (they can actually get pretty annoying after a while); they just have this feeling of wholeness to them, of completeness. Consonance is acoustical cohesion.
The greater part of our aural experience consists of Noise: sound-without-pitch. In buildings, we hear the hush of air-conditioning; in cities, we hear the rolling of tires along pavement and the busy murmuring of large crowds; in nature, we hear the howling of wind, the crashing of waves, and the rustling of leaves. Until recently, when recorded music became so common as to be played everywhere, every moment was underlied by some background noise. Noise has been, for basically all of human history, the default of our experience of sound.
And yet, despite this, music theorists seems to think that noise is not worth discussing at all. Here is the most clear example of us treating "Harmony" and "Sound" as if they exist in two completely unbridgeable worlds. In all of the Treatises of Harmony I have read, I have not seen one single word uttered on the subject Noise; not even in those by acousticians, who should really know better.
Noise, as I said, is sound-without-pitch. If we hear a solitary frequency, we register it as a pitch, as a single "point" in acoustical space. But the more frequencies we hear simultaneously, the more difficult it becomes to identify them as individual pitch-points. Put simply: the more frequencies we hear, the more they "clutter" our range of hearing.
At some point, we cease to register individual pitches at all, and we, instead, perceive a RANGE of pitches. This is roughly where a sound becomes a NOISE. Just as we would perceive thousands of grains of sand not individually but as a single homogenous substance, so too do we experience thousands of frequencies not individually but as a single homogenous noise.
Whereas a consonance is defined by the particular placement of its frequencies, a noise is defined defined only by the general range that they occupy. The former is sound-as-structure, whereas the latter is sound-as-mass.
"Brown" noise (frequencies occupying the lower range of human hearing):
"White" noise (frequencies occupying the full range of human hearing):
"Purple" noise (frequencies occupying the higher range of human hearing):
The Harmonic Series - sometimes called the "overtone series" - is a series of frequencies naturally created by the resonations of a physical body. If a piano-string, for instance, were to vibrate at a certain frequency, this frequency would resonate with the rest of the piano's body, and these subsequent resonations would produce a series of higher frequencies which would all be multiples of the initial frequency. If a string vibrates at 100 Hz, for instance, it will create a series of frequencies at 200 Hz, 300 Hz, 400 Hz, 500 Hz, etc.
This naturally occuring frequency spectrum is the Harmonic Series. Each frequency in this Series is called a "harmonic," and the lowest of these harmonics, the initial frequency, is called the "tonic."
Though the existence of the Harmonic Series is tacitly accepted by theorists, the deeper implications of this natural sound-structure and the way we hear it remain largely ignored. Every tone that we hear, whether it comes from a piano, trumpet, didgeridoo, bird, or human voice, has a harmonic series. Music theorists generally still treat every tone as a single "pitch-class," but the existence of the Harmonic Series should dissuade us from reducing the Tone to just a pitch. Every tone is composed of many frequencies. The Tone is, itself, a harmonic structure!
This is a single frequency in isolation:
But THIS is a tone:
There is a world of difference between the two! Both are heard as a single pitch, but, while the frequency is just an acoustical "atom," the Tone is an elaborate acoustical molecule - one in which many interrelated frequencies contribute to a single over-arching feeling of sound.
Although the Tone is a structure made up of several frequencies, we only hear this collection of frequencies as a single pitch. No other harmony can boast this level of unity. Every sound is really just a collection of frequencies, but only in the Tone do these frequencies interlock together so seamlessly that we fail to distinguish between them entirely.
These facts lead us to a conclusion which should have been obvious from the beginning: The Tone is the only pure consonance.
This is an idea which goes overlooked only because of the persistence of Pythagorean theory, which has led music theorists to only look at the intervals between tones, rather than the tones themselves. But, once we ditch the narrowness of the Pythagorean theory, it becomes obvious that the Tone epitomizes every quality which belongs with Consonance. In the Tone acoustical unity is absolute. Every harmonic is perfectly assimilated into a single feeling of pitch. The Tone lacks any of that desire to "resolve" which is so indicative of dissonances. The Tone is a perfect harmony.
With the examination of these two types of sound, Tone and Noise, we can finally begin to envision the acoustical universe and to wrestle some meaning from it.
Frequencies should be regarded as atomistic to the World of Sound, in that they are the acoustical particles which, by themselves, mean very little but, out of which, every possible sound can be constructed.
A noise is merely a large quantity of these acoustical particles. Much like a cloud is just a temporary almagam of various water-particles, a noise is really just cluster of disjointed frequencies which have happened to float into one another. Wherever we delineate the beginnings and ends of this cluster will be totally arbitrary. Two noise-clouds will not be heard as separate; they will bleed into each other, combining into a larger frequency-mass.
A tone, on the contrary, is a self-contained body of sound. Every tone has a pitch, which is its definite center in harmonic space. Every frequency which falls along the tone's harmonic series is assimilated into its pitch. Consequently, each tonal frequency is unambiguously attached to the Tone it belongs to. Whereas noise-frequencies are homeless sound-particles flung to the wind, each tonal frequency is a unique part of a tonal whole. the Tone, therefore, is more than just a haphazard collection of frequencies; it is a coherent sound-form cleaved off from the entire rest of the sound-world.
These two rudimentary sounds, Tone and Noise, represent the furthest extremes of our Harmonic experience. In the Tone, sound is completely drawn into one pitch; in Noise, pitch is obliterated. Pitch-structure versus frequency-mass, hierarchy versus homogeneity, form versus unform: our entire sense of Harmony rests upon the distinction between these two extremes.
At every moment, our ear distinguishes between Tone and Noise. One only has to walk outside, away from the intrusion of ambient music and public speakers - always growing more and more inescapable by the day - to hear the profound importance that the Tone/Noise dichotomy has in the world of sound. The buzzing of cicadas, the blowing of wind, the rustling of leaves, the rolling of tires along pavement - these are all noises; they are blurry waves of pitchless sound. The various bird-songs, the animal-calls, the mindless chatter of neighbors - these are all tones; each is a single fluctuating pitch, given an outward body in the form of a harmonic series. The world of sound, at its core, is a vague and boundless landscape of noises upon which many diverse tonal-voices blossom outwards, always appearing briefly and then dissolving back into Noise's ever-shifting mass. THIS is the world of sound in its purest, most untouched state. This is the acoustical environment within which our ear developed and according to which it transformed itself. There were no intervals, no perfect-fourths, no perfect-fifths, no major triads; only Tone and Noise poised in violent contrast to one another.
Speech is proto-Harmony; it is "harmonic progression" stripped down to its most rudimentary form. Nowhere else is the duality between Tone and Noise more clearly expressed than in our speech. No matter what language we are listening to, every sound that the speaker produces is either a tone or a noise. "Ah," "Eh," "Ee," "Oo" and "Oh" are all tonal sounds which we distinguish according to their different timbres. "Ch" "Ss" "Tuh" and "Puh" are all noises which we distinguish according to their different ranges. Speech, then, consists of the quick fluctuation between different Tone-timbres and different Noise-ranges; and this, I would argue, is Harmony in its purest form. In this mechanism of speaking lies the most rudimentary Harmonic elements, the basic tone-shapes and noises which our ear began to acclimate to many thousands of years ago. Wouldn't any evolution of our language also necessarily propel the evolution of our sense of Harmony? In the face of more sophisticated languages, wouldn't we be forced to develop a more sophisticated ear?
The dichotomy between Tone and Noise is as fundamental to the ear as the dichotomy between Light and Dark is to the eye, and equally rife with spiritual meaning. Our ear is not, as theorists are so eager to believe, a cold and mechanical interval-calculator; our ear simply recognizes the basic sound-forms that it evolved alongside. It does not measure out;it recollects.
My answer to the Consonance problem is simple: Consonance is Tonality, and Dissonance is Noisiness.
A Consonance is nothing other than a Tone-like sound. A sound may not be a pure tone, but it might be similar enough to a tone to at least rouse our Tone-seeking ear - to make it turn its head and go "huh? wuh?" Even a chord is just a set of tones which come together to create the impression of a single tone.
The reader might ask how it is possible for many tones to "create the impression" of a single tone. Say that a church-organist is having an off-day, and he accidentally starts the hymn by playing a pretty harsh dissonance:
This chord is composed of 6 separate tones - 6 ENTIRELY DIFFERENT PITCHES all fighting for the ear's attention. The result is a sound which is acoustically cluttered - that is to say, Noisy.
So, quickly fixing his mistake, he voice-leads to a strong consonance, a towering C-major chord:
Just by slightly shifting each of these pitches, the organist has eliminated that unsettling feeling of 6 pitches fighting against each other. But why? Why do these 6 tones feel unified whereas the previous 6 tones do not? It is because, although the organist is technically still playing 6 tones, all of these tones conform to a single Harmonic Series!
When multiple tones fall along a single Harmonic Series the resulting sound tricks the ear. Our ear recognizes not the individual tones but the over-arching Series which they fall into. As a consequence, the overall sound is perceived as a single Tone-like structure. This is why, in the strongest consonances, tones seemingly fuse together; it's because they literally become one Tone!
The most widely accepted psychoacoustical explanation for Consonance is Helmholtz's theory that Consonance is the direct result of overlapping harmonics. But this is only half-true. It doesn't get to the core of what's really happening. What Helmholtz misses is the fact that whenever 2 separate harmonic series overlap with each other, they necessarily create a larger kind of meta series through their union:
It is THIS series which is the actual source of their consonance. It is the meta-series which holds them together! So thoroughly does our ear scan the environment for tones, that even a series with missing harmonics (like the one above) is scutinized as a possibly Tonal sound. The phenomenon of "virtual pitch" proves that the ear recognizes incomplete harmonic series, and it absolutely astounds me that I have never seen anyone expand the phenomenon of virtual pitch to intervals and chords.
So, consonant chords and intervals are all Tone-like sounds. The more closely a sound resembles the Tone, the more Consonant it is; the less closely it resembles the Tone, the more dissonant it is. The relative Consonance and Dissonance sounds should only be judged on this basis.And the fluctuation between Consonance and Dissonance which is so definitive of Western music should be regarded as nothing less than a fluctuation between Tonality - sound defined by form, hierarchy, and pitch - and Noisiness - sound defined by formlessness, homogeneity, and pitchlessness.
Tonality, then, is not something peculiar to Western Harmony, but a principle which is common to all styles of Harmony. Wherever one finds a Tradition of Harmony, whether it is Islamo-Indian Raga, American Jazz, or any of the thousands of locally-bound folk musics lying scattered across the globe, one will always find some elaboration upon the Tone. All sound-structure is Tonal, and the properties inherent to this structure are something which every composer must deal with, just as every architect must deal with the laws of gravity. This is a fact not consciously realized but instinctually felt by every composer, every tuner, and every performing musician. Everyone who has ever contributed to the progression of their own musical culture has, without realizing it, felt the pull of the Tone upon his ear. The sense for Tone-like sound, hammered into his psyche by millions of years of evolution, unconsiously influences every musical decision he makes, driving him to create scales, chords, progression, melodies, and songs which ground themselves in the firm structure of the Tone. For the moment he strays too greatly from this structure, he invokes Noise, and, with it, the grey and banal feeling of acoustical homogeneity.