Almost nothing at all is known about the deeper nature of Consonance and Dissonance. That some harmonies feel "consonant" and others feel "dissonant" is a fact almost universally acknowledged by Western musicians. The entire Western style of Harmony is built around a polarity between the two. And yet we understand close to nothing about the nature of these two sound-feelings. What makes a sound consonant? What do consonances mean? Why do we experience them? - for all of Western music theory's developments over the past thousand years, it has not taken a single firm step towards answering any of these questions.
The problems with our music theory arise, I think, not due to a lack of facts but due to our deeply flawed way of interpreting these facts. Western music theorists, without knowing it, have all subscribed to a very narrow conception of Harmony, one which is framed entirely in terms of intervals.
In music, an "interval" expresses the relationship between two tones in the form of a ratio. The "interval" between a tone at 400 Hz and a tone at 300 Hz, for instance, is 4/3, and this is the interval commonly referred to as a "perfect fourth." The interval between a 400 Hz tone and a 200 Hz tone is 2/1, and this is the interval commonly referred to as an "octave."
In deeper theory, we reduce every harmonic structure - every chord, melody, and scale - as a set of intervals. A major triad is composed of a 5/4 and a 6/5, a diminished chord out of two 6/5s, the diatonic scale out of a succession of 3/2s. Whenever ethnomusicologists run into a foreign scale, they inevitably rationalize it by splitting it into intervals. That ethnomusicologists, these academics who are neurotically against "eurocentrism" still consider the intervallic interpretation to be applicable to other cultures tells us that they view intervals as objective and universally valid. To them, the chords and scales that we use are just cultural constructs, and the INTERVALS-RATIOS alone are real. Intervals, then, are treated as a first principle; they are viewed as Harmony's very ESSENCE, while all larger harmonic structures are viewed as derivatives of this essence.
All of the Classical Treatises on Harmony begin in the same way: the author lists off a series of justly tuned INTERVALS and judges each of them, using often-arbitrary methods, as either consonant or dissonant. Only once he has appraised a set intervals does he then feel comfortable using them to re-construct the classical scales and chords which he is used to. The Grand Harmonic systems constructed by Zarlino, Rameau, Reimann, and Schenker are all built upon the presupposion that there is a set of CONSONANT INTERVALS.
The modern schools of theory suppose themselves to be more "scientific" than their predecessors, but have only leaned further into their most unconscious biases. Harry Partch, the grandfather of the microtonal movement, exclusively talks about Harmony in terms of intervals-ratios. There is, in fact, no purer exclamation of the intervallic conception than Partch's Magnum Opus, "A Genesis of Music." Neo-Reimannism is, at its core, a set of systems built upon the supposed purity of the intervals, 2/1, 3/2, and 5/4. And finally, the psychoacousticians pride themselves on their famous Plomp-Levelt curve, which is just a hilariously interval-centric consonance model.
So, although countless theories have arisen to explain Consonance over the couple thousand years, all of them have been built upon the same axiom: that it is not the over-arhcing sound ITSELF which is Consonant, but the intervals which it consists of. A major triad is not consonant; the 5/4 and the 6/5 within it are. The harmonic series is not consonant; just the intervals within it. No matter how many breakthroughs we make in acoustics, music, ear-anatomy, etc. these breakthroughs have to be filtered through the same intervallic conception which has dominated our theory for thousands of years.
Apparently it has occured to no one how severely limited this conception of Harmony is. Under the intervallic conception, harmonies are never taken as a whole, but rather must be split up into a set of ratios in order to be TRUTHFULLY observed. This is, in my opinion a perversion of Harmony's spirit - an attempt not to understand Harmony in its fullness, but to render it as something numerical, as a set of quantities which can, as such, be easily constrained within the cold iron bars of logic. We are not, in actuality, observing Harmony, but dissecting it. Therefore, we only study Harmony in a lifeless state, as an unmoving set of measureable components, and our theories of Harmony reflect the spirit of number far more than they do the spirit of music.
The problems with the Intervallic conception go unnoticed only because no one has seriously considered an alternative to it. Has anyone considered, for instance, that there is no such thing as a "consonant interval?" That it is not the intervals themselves which are consonant, but some larger harmonic structure which they imply? Could it be that the booming C-major chord played by the orchestra is ITSELF the source of Consonance, and that the intervals which we can split this chord up into are, in actuality, mere fragments - the tiny derivations of this broader structure?
The problems which still confront music theory demand that we seriously consider these alternatives. The theory of Harmony which I propose, therefore, is one in which every sound is considered AS A WHOLE - where sounds do not have to be dissected into a set of numerical componenets in order to be properly. Most importantly, the banal sounds which our theory has ignored (because they cannot be split up into intervals) will be considered. Our conception of Harmony, in general, will come to encompass more than just "the proportion between two tones" as Euclid puts it; it will take into account a sound's timbre, noisiness, register, volume, and far more.
First of all, it is worth nothing that the intended meaning of the word "Consonance" is often distorted by the very people who study it. 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 refers to acoustical unity. Sounds which are “consonant with each other” naturally interlock together, becoming a tight 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, the major-triad – every sound which we have ever recognized as "consonant" shares this feeling of unity.
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.
Often, the octave - the interval of 2/1 - is held up as a “purely consonant" interval, but I would firmly disagree.
Compare an interval of 1 octave to an interval of 4 octaves:
In example 1, the two tones are almost totally united; it is even a little difficult to recognize that they are "two separate notes" if you're not looking at the sheet music as a guide. In example 2, on the other hand, the two tones are CLEARLY detached from each other. We can perceive, with utmost certainty, that the sound is comprised of 2 pitches. I would argue, therefore, that the second interval is less consonant than the first one.
But if an octave really was purely consonant, then an interval of 4 octaves should be equally so. There is no reason for a "purely consonant interval" to decrease in consonance as we multiply it. But, incredibly, this is exactly what happens:
The very fact that as we continue to stack octaves we gradually experience a weakening of the consonant feeling, a loosening of the two tones from each other, implies that the octave is not a pure Consonance. The pure consonance of the octave is another idea we have blindly accepted because of the theoretical principle of "Octave-equivalence" (The note of C is considered to be of the same harmonic quality no matter which octave it lies in. Per the principle of octave equivalence, the pitches of C1, C2, C3, etc. are all taken to be "harmonically equivalent"). Though octave-equivalence has its practical uses in our musical language, it should not be taken at face-value.
An understanding of Harmony will necessarily require us to have a full understanding of sound, which is the world Harmony exists within. So that we get the broadest possible view of this world of sound, it is important for us, first of all, to discuss Noise, a kind of sound which is COMPLETELY MISSING from all works of music theory.
Noise is sound-without-pitch. Every frequency that meets our ears is heard as one pitch, as a single "point" in harmonic space. But the more frequencies we hear simultaneously, the more difficult it becomes to identify these individual pitch-points. Multiple frequencies naturally detract from each other, cluttering our acoustical "vision." As we add more and more frequencies to a sound, the more a feeling of acoustical clutter grows.
At some point, we cease to register individual pitches at all, and we, instead, perceive the sound as a RANGE of pitches. This is roughly where a sound becomes a NOISE rather than just a set of pitches. Noise is what happens when so many pitches gather together that pitch itself is rendered arbitrary, smothered into non-existence. 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-feeling.
"Brown" noise (frequencies occupying a low range):
"White" noise (frequencies occupying the full range of human hearing):
"Purple" noise (frequencies occupying a high range):
We should note, here, the essential difference between a consonance and a 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.
Most of the sounds that we hear throughout our day are noises. The vague rumbling of crowds, the tires rolling along highways, the cry of cicadas, the rustling of leaves - all the world around us pours in a continuous stream of noises. If each of the five senses is a world of its own, then the world-of-the-ear is, for the most part, an ocean of Noise. Pitches, by comparison, are fleeting and rare. They appear briefly, like waves upon its surface, rising up and sinking back into its overwhelming mass. Every pitch that we hear is just an exception to Noise - a brief flicker of light within the dark and murky acoustical ocean that we find ourselves submerged in.
That the field of Music Theory has not uttered one word on Noise, despite it being the most rudimentary element in our acoustical universe, is embarrassing. I would go as far to say that NO conception of Harmony is complete without Noise. Noise is not just some unharmonic biproduct of rhythmic instruments; rather, it represents the entire harmonic quality of pitchlessness. Every erosion of a sound's pitch is a movement towards Noise. Every distortion-filter, every cluttered harmony, every tonal ambiguity - anything which obscures the pitches of a consonance - is a flirtation with Noise; it is a dissolving of harmony into the same frequency-soup which underlies our whole acoustical experience.
If groups of pitches tend towards Noise, how do we create meaningful acoustical structure? What is the exception to the rudimentary Noise-element? Our answer lies in the Harmonic Series.
The Harmonic Series - sometimes called the "overtone series" - is a series of frequencies which is 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 this initial frequency is at C1, then the following series of frequencies would be produced:
This naturally occuring series of frequencies is what we call the Harmonic Series. Within this series, each frequency 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, its profound implications are ignored.
First of all, no sound is ever just a pitch-class. Every tone that we hear, whether it is created by piano, violin, didgeridoo, or human-voice, has a harmonic series. Every single tone, though we regard it as a singular point in acoustical space, is actually a structure composed of many frequencies! So, when we say the word "tone," we are not referring to a single pitch-class, nor are we ever referring to a single, isolated frequency...
"Tone" refers to a sound that is composed of many frequencies, but is only heard as one pitch.
This is a single frequency:
But THIS is a tone:
The former is just an acoustical particle, while the latter is a fully-fledged acoustical structure uniting many frequencies into a single feeling of pitch.
Each tone conceals, within itself, a complex harmonic anatomy. What we hear to be a single pitch is, in actuality, a molecule of various harmonic identities. The 2nd harmonic, the 3rd, the 5th, the 7th, etc. - each is a different aspect of the tone, a different "acoustical node" within the overall tonal structure. At the center of this structure lies the tonic, the acoustical nucleus which signifies the tone's pitch. A tone is the sum-whole of these inner-harmonics which all relate to each other hierarchically. By their combination, these harmonics give the tone its breadth, its width, and its shape in harmonic space.
All this is to say that the tones that we HEAR are far more complex than the notes which represent them. A note only signifies a certain pitch-class, but a TONE is a complex sound-structure. And the more we meditate on this, the more the Intervallic conception which has dominated Harmony seems absurd. It makes no sense to treat Harmony as something which occurs "between two tones" - because a tone is, itself, a work of harmony.
The Tone is a sound which is composed of many frequencies but only heard as one pitch. No other harmony can boast this. Every consonance is a synthesis of many frequencies into one, but only in the Tone do these frequencies interlock together so seamlessly that we fail to distinguish between them entirely - that the entirety of the sound seems compressed into a single acoustical center, into a single, absolute feeling of pitch.
And, with these obversations, we reach a truth which should have been obvious: The Tone is the only pure consonance.
In the Tone acoustical unity is absolute. There is, in the Tone, no acoustical separation, no tension, none of that desire to "resolve" which is so indicative of dissonances. A single pitch envelops the entirety of the sound, assimilating every harmonic perfectly into itself. In this consolidation of many frequencies into a single pitch, we experience not only the epitome of Consonance, but the epitome of what is meant by the word "Harmony": the resolution of the many into one.
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
Tone and Noise represent the furthest extremes of our Harmonic experience. The Tone is sound completely bound up into a single pitch; Noise is sound completely deprived of pitch. Our entire "sense of Harmony" is based upon the distinction between these two extremes.
At every waking 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 calls of mammals, the mindless chatter of neighbors - these are all tones; each is a single fluctuating pitch, fleshed out by 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 most primal state. This is the acoustical environment in which our ear developed and which it transformed itself according to. There were no "just 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. Speech is the most basic form of "musical progression," and that out of which all of our musical styles and idiosyncracies have evolved. 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 qualify according to their differing timbres. "Ch" "Ss" "Tuh" and "Puh" are all noises which we qualify according to their differing 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. If humanity's languages were to evolve from then until now, then wouldn't this 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 sophistocated 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. To the former belongs pitch, identity, and hierarchy; to the latter, pitchlessness, density, and homogeneity.
Every "consonance" is Tone-like. 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?"
It has long been observed that Consonant harmonies tend to conform to a Harmonic Series and that divergence from this Series (i. e. "inharmonicity") results in Dissonance.
Every consonance has a "tonic" of sorts. The more consonant a sound is, the more clearly we apprehend this tonic. Perfect-fifths, perfect fourths, and major triads all have very definite roots, while intense dissonances, such as the tritone or the diminished 7th, cannot be assigned a root until we "resolve" them onto some consonance. This "root" implied by every strong consonance, and which dissipates whenever the sound becomes dissonant, is nothing other than the tonic-feeling which naturally arises with every Tone-like sound. A tone is incomprehensible without its tonic, which represents its pitch and which every harmonic-identity is heard in relation to. Consequently, any sound which mimics the Tone also mimics its tonic. Consonance and Tonicism are bound at the hip.
In Western music, we have the "key-center," the kind of "meta-root" which every scale degree is heard in relation to. Likewise, in the Indian Classical Tradition, there is the Shadja, the one svara which cannot be ommitted from the musical performance and which must accompany the singer at all times. Further Eastward, in the Chinese Classical tradition, there is the Huangzong or Gong which, in the Huainanzi, was associated with the element Earth and with Centrality (where every other primordial tone is associated with a direction stemming from this center). Each of these Grand Harmonic cultures, once it began to critically examine its own style (i.e. music theory), recognized that every harmonic structure was grounded in a single fundamental pitch - a TONIC.
So every consonance conforms to a Harmonic Series and has a pitch-feeling corresponding to this Series' tonic. From this, it should be clear that every consonance has a tonal essence to it. All Consonances, regardless of their complexity, are just elaborations upon a single Tone!
A group of tones which orients itself along a single Harmonic Series will trick the ear. We will not hear "multiple tones" as much as we will hear the single Harmonic Series which they fall into; therefore, our ear will comprehend the whole chord as a single Tone-like sound - a Consonance. But the moment these tones diverge from their positions in a Harmonic Series, the illusion is ruined; we go back to hearing the set of tones as a mere set of tones, unbound by any singular pitch-feeling. Such a sound will be Noise-like - a Dissonance.
Tones falling within a harmonic series:
Tones falling outside of a harmonic series:
Tonality, then, is not something peculiar to Western Harmony, but a principle which is common to all styles of Harmony across the world. All sound-structure comes in a Tonal form. The properties inherent to this form 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 tonal feeling on 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. The threat of Noisiness has always kept Harmonic Styles within certain boundaries. To keep Harmony firmly rooted within the stable earth of Tonality and to prevent it, even in spite of its wild transformations and embellishments, from being flung into the chaotic wind of Noise has always been the underlying goal of every Harmonic Tradition.