Almost nothing at all is understand about the nature of Harmony. The field of music theory has consistently failed to explain the underlying meaning and causes behind various acoustical phenomena.
No subject is more emblematic of the inadequacies of our music theory than Consonance. It is almost universally agreed upon among Western musicians that there is some harmonic quality called Consonance and an opposite quality called Dissonance. And yet, there seems to be almost no useful discussion regarding the underlying nature of these two harmonic qualities.
The many developments made in acoustical science, ear anatomy, and psychology over the past hundred years have not improved our understanding of Consonance one bit. That such an accumulation of evidence fails to improve our understanding should tell us that the issue lies not in the sum of our knowledge, but in our underlying premises. I am writing this blog to address what I perceive to be the deep and central flaws in our music theory, and to establish, in their place, a totally new understanding of Harmony and of Consonance.
All music theorists have, without knowing it, subscribed to a very narrow conception of Harmony, one in which all harmonic qualities and characteristics are seen to be determined by intervals.
In music theory, an "interval" describes the relationship between two pitches, typically in the form of a ratio. A pitch at 400 Hz and a pitch at 300 Hz, for instance, are related by the interval of 4/3. A pitch at 900 Hz and a pitch at 800 Hz, by the interval of 9/8. Music theorists classify these two intervals as a "perfect fourth" and a "major second" respectively. The former interval is conventionally seen as more consonant than the latter because the ratio between its two pitches is simpler.
This wholly intervallic conception of Consonance has its origins in the Pythagoreans. Pythagoras of Samos was the first to proclaim that Consonance (in his time, Harmoniousness) was the result of simple interval-ratios. But, in doing so, he framed intervals as the essence of Harmony. Pythagoras, as the first Western thinker to really analyze Harmony, was in the privileged position to manipulate music theory in its fetal stages, to plant philosophical seeds which would, over thousands of years, harden into music theory's axiomatic roots. His most central presumption - that it is the "interval" which determines the quality of the Harmony - has become the unquestioned belief which dominates all music theory. All Western music theorists view the interval as the deepest and most fundamental unit of harmonic structure.
Read through the Harmonic Treatises of Zarlino, Rameau, Reimann, Schenker, and Partch, and you will find that each one uses the exact same method: he first lists off a set of INTERVAL-RATIOS, and only once he has finished appraising each of these intervals as either consonant or dissonant does he then feel comfortable using them to reconstruct the classical scales and chords which he is used to. It is so obvious to him that this is the correct order of operations, that the INTERVALS come first and everything else after the fact.
This belief in the supremacy of intervals has only strengthened in modern strains of theory. With the old European traditions and systems torn away, the modern theorist has to lean further into the core of the Pythagorean doctrine, which is all he has left to stand on. Harry Partch, the Grandfather of the microtonal movement, spends most of his book restating the 2500-year-old Pythagorean conception. "A Genesis of Music" begins with a section entitled "Thinking In Ratios" and ends with the "One Footed Bride," a graph which maps Consonance onto a domain of intervals. Partch rebels against the old Western doctrines of Harmony, only to hold up far older Greek doctrines in their place.
Meanwhile, the field of "psychoacoustics" (music theory with a scientific coat of paint) holds up the Plompt-Levelt curve as its grand breakthrough. Like the one-footed bride, this curve maps Consonance-level onto a domain of intervals.
The Neo-Riemannians take for granted the supremacy of 3 primary intervals - 2/1, 3/2, and 5/4 - and subject them to all kinds of mathematic and geometric manipulation in order to obtain their bloated systems of Harmony. It is still just a worship of intervals at the core.
Every one of these systems of Harmony is based on the same underlying assumption: that the interval is the source of Consonance. To music theorists, 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 consonant; the 5/4 and the 6/5 within it are. The harmonic series is not consonant; just the intervals within it. According to all of our music theorists, Consonance essentially only takes place at the level of the interval, and the broader harmonic structures we have built - the chords, the scales, the songs - are merely sets of these "consonant intervals" shoved together.
Apparently it occurs to no one how narrow this conception of Harmony is. So many acoustical elements - timbre, register, noise, etc. - must be disregarded in order to maintain the purity of our Pythagorean systems. Alternatives to the Pythagorean conception are never considered. For instance, that there is no such thing as a "consonant interval." That an interval, in itself, means nothing. That the inverse of the Pythagorean conception might be true: "it is not the interval which we find consonant, but the broader harmonic structures which it is apart of."
We cannot expect there to be any developments in our understanding of Consonance as long as we continue to examine it through the same 2500-year-old lense. The lapses in our understanding of Harmony and of sound as a whole demand a broadening of our vision. Our gaze needs to pierce deeper than the mere intervals.
Of course, it's always easy to tear down first principles; to build up new ones is a Herculean task. So where does this leave us?
"If you're going through hell, keep going." So, at the risk of abusing the reader's patience, maybe it would be best to keep drilling into Pythagorean Harmony. By understanding precisely what it is, we will begin to understand precisely what it lacks. And here we will find the fertile ground upon which to begin growing a new theory: in the places where the Pythagorean weeds can't invade.
First, it needs to be understood that Pythagoras' music theory was nothing more than an extension of his own philosophy: "the essence of all things is number." Intervals are ratios, and ratios are numbers. In Pythagoras' purely intervallic music theory, number becomes the sole basis of Harmony. Music, as a whole, becomes reframed as an actualization of the principles and laws of mathematics through the vehicle of sound. Simply, it means that music is math. Even now, the idea that "music is math" is widely accepted among musicians, musicologists, and theorists alike. Even the most rebellious and individualistic philsophers submit to the ancient authority when the topic of music comes up: "music is math." Like every truly great philosopher, Pythagoras' innermost desire was not to understand the world, but to enslave it to his own philosophy. Philosophers are conquerers, and nowhere in the entire history of philosophy do we find a greater conquest than in Pythagoras' complete subjugation of music to his own philosophy.
The widespread acceptance of this "music is math" idea has imbued music theory with a cold, autistic character. Whereas other artforms like theatre, dance, and painting are viewed as sensuous, orgiastic, and intertwined with life itself, music seems to always be thought of as cold, logical, and abstract, like the numbers which supposedly form its essence.
Of all the arts, only music has been subject to this autistic, hyper-numerical treatment. Is there anyone who would actually suggest that, when looking at a beautiful painting, we are just perceiving some mathematically perfect object, some function rendered into visual image? I think most people would regard this as an absurd way of looking at things. The qualities which define a porttrait - its colors, lighting, space, depth, framing, etc. - are all qualities particular to our sense of sight. Any reduction of such qualities to a numerical framing would be totally absurd, let alone informative. Yes, one could measure out some of the proportions of the image and a reasonable argument could be made that the simplicity of these proportions contributes to the beauty of the work, 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!
Consider the words we generally use to describe sensory phenomena. Our language for describing smells and tastes is almost entirely based on analogy: "fruity," "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 color and lighting, qualities which are specific to that particular sense-world, qualities not bound by number. Even our words for colors are etymologically rooted in sensory experiences. Green shares a Proto-Indo-European root with "grass" and "growth", yellow with "glow" and "gold," red with "reed" and "ruby." From this, it seems clear to me that each color-word originated as a reference to an actual visual experience (i.e. green, red, and yellow begin as grass-like, ruby-like, and gold-like). Our language for describing Harmony lacks entirely this kind of straightforward connection to the senses. Words such as "voice-like," "buzz-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."
Here is precisely what Pythagorean music theory lacks: a connection to our basic acoustical experiences, a connection to SOUNDS. Harmonies are sounds, but they are rarely talked about as if they are. Pythagorean Harmony forces us to reduce every tone to a "pitch" and to relate these pitches to each other intervallically. But does this reduction reflect our actual acoustical experience? Do we really listen to a chord and hear "sets of pitches linked together by intervals?" No. But the interpretation is preferred simply because it reduces what we are hearing to a measureable form. This, in the eyes of academics, makes it the more valid interpretation. Academics do not love the truth; they love numbers. Numbers have this air of "objectivity" about them, this feeling of eternal validity. Therefore, every attempt is made in academia to reduce the infinitely wide and diverse array of phenomena to a mathematical form. Academics would convert the entire universe to a single all-encompassing formula, for the same reason that an ant, if given the power, would convert the universe to a single all-encompassing ant-hill.
It shouldn't be too difficult for the reader to accept that each sense is a world governed by a particular inner-logic, and that music, as an extension of the sense-world of sound cannot be understood without first understanding that same sense-world. Before we even BEGIN to discuss Harmony, therefore, our goal should be to first understand the nature of SOUND. Any "rule" of Harmony will necessarily be an extension of the laws which govern the sound-world.
In this blog, our goal will be to develop a theory of Consonance which has its basis, not in numbers, but in actual sounds. We reject the Pythagorean conception of Harmony which, by its reduction of Harmony to a purely numerical form, has given birth to music theories which are hyper-logical, sterile, and lifeless. Every numerical representation of Harmony, therefore, will be treated with the utmost suspicion, and every attempt will be made to understand Harmony by drawing analogies to actual sounds. The animal call, the crashing waves, the hushing wind, the crying cicadas - these basic acoustical experiences form a far greater basis for a system of aesthetics than do ratios and formulae. It is upon this basis that we will strive to create a system of Harmony which is as fiery and full of life as music itself, a music theory which dances along to the music which it interprets.
All sound is made up of frequencies, oscillations of air pressure over time. Frequencies are atomistic to the world of sound, in that they are the indivisible sound particles which by themselves mean little but out of which every sound can be produced. On the other hand, to say that the sound-world is “just a bunch of frequencies” would be a reduction as shallow and useless as the reddit-materialist motto: “we are just a bunch of atoms.” Clearly, what matters is not the frequency-atoms themselves, but the acoustical shapes which can be made by their combination.
From this infinite set of sound-shapes, I distinguish between two fundamental types: Tone and Noise. A tone is a sound which is made of many frequencies but is only heard as one pitch. Noise is a sound which is composed of many frequencies but is heard without pitch.
This is the frequency spectrum of the sound produced when a C3-note on a piano is played:
This is a tone. We have multiple frequencies playing simultaneously, but rather than hearing many pitches, as one might expect, we only hear one. Every tone that we hear, whether its source is a piano, violin, human-voice, didgeridoo, bird-call, or trumpet, is, like the tone above, a collection of many frequencies which compound together into a single pitch. This is a fact which often suprises the uninitiated into acoustics, because the pitch created by the set of frequencies is so absolute and so pure.
This is the frequency spectrum of the sound produced by a cymbal crash:
This is a noise. Once again, we have multiple frequencies playing simultaneously, but this time we do not perceive any single pitch; rather, we perceive a general range of pitchless sound.
So, we have two fundamental types of sound: sound-with-pitch and sound-without-pitch, Tone and Noise.
Too often, the terms "frequency" and "pitch" are used interchangeably. But here, already, we can see a disparity between the frequencies which make up a sound and the "pitch" that is perceived by the listener. If this disparity occurs even in the most basic of sounds, shouldn't it extrapolate further into the more complex sounds? Already we have a clear indictment of the traditional Pythagorean way of viewing things. The existence of such a disparity between a sound's "frequency" and its "pitch" demands that we distinguish clearly between the two: frequencies are the physical components of sound, but pitch is a feeling.
Frequency and pitch, as I define them, relate to each other as shape and center. Indeed, if a tone is a set of frequencies which come together to make a "shape" in acoustical space, then the "pitch" indicates the central point of this shape. The pitch is something like acoustical coordinates; it defines position and center in sound.
As we continue to discuss these two different sounds - Tone and Noise - the distinction which I have established between frequencies and pitch-feeling will be made more and more clear. It will also come to be understood that much of our sense of Harmony depends upon the disparity between the two.
Discussions of Noise are largely absent from the field of music theory. I have, in fact, never once seen Noise mentioned in any of the treatises of Harmony I have read; not even in those by acousticians, who should really know better.
But any theory of Harmony which does not deal with Noise will necessarily be incomplete. Noise is not just a kind of sound; it defines an acoustical quality which seeps its way into all harmonies. Whenever a harmony is overladen with notes, it becomes "noisy." Singular notes, when put through a distortion pedal, are sometimes said to become "noisy." "Noisiness" is everywhere.
Noise is sound-without-pitch. "Noisiness" is basically synonymous with "the deterioration of pitch-feeling."
A frequency, as we have said, is essentially an acoustical atom. If we hear a single frequency, that frequency will be heard as a single pitch, as a "point" in acoustical space:
But if we gradually add frequencies to the sound, they begin to "clutter" our hearing-range. As acoustical space fills up, it becomes more and more difficult to detect each frequency's particular pitch. Essentially, the sound becomes more and more "Noisy":
At some point, we are no longer able to detect individual pitches at all; instead, we can only perceive the general range which they occupy. This is the precise moment when a sound becomes a true Noise: when the placement of its frequencies seizes to be of any importance, and only the density and range:
Here, we can see the correlation between Noisiness and pitchlessness. The gradual erosion of pitch-feeling in the example above equates with its gradual transformation into a noise. Noisiness also represents a kind of acoustical uniformity. As a sound's pitches are obscured, it gradually becomes featureless. A Noise, proper, is essentially a large mass of frequencies. Just as we would register thousands of sand-particles as one, uniform powder, so too do we register thousands of frequency-particles as one, uniform Noise.
Whereas a consonance is something like a sound-structure, Noise is a sound-mass. A consonance is characterized by the placement of its frequencies, while a noise is primarily characterized by the range which they occupy. Observe how different frequency-ranges produce different "flavors" of Noise:
"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):
This acoustical mass, Noise, represents the furthest possible extreme of Dissonance. A noise is essentially a screeching mass of conflicting sounds. Noise only occurs when there is such a great number of conflicting pitches that the feeling of pitch is, itself, obliterated.
The Pythagorean conception has constrained our understanding of Harmony so tightly that no one has even considered this rather obvious relation between Noise and Dissonance. Because we can only think of Dissonance and Consonance as they apply to intervals, it never occurs to us to that the totally interval-less sound, Noise, has anything to do with either. But anyone who considers the nature of the sound-quality of dissonance, without the baggage of the Pythagorean conception weighing them down, can understand that it reflects an accumulation of acoustical contradictions. Once this is understood, it shouldn't be too difficult to further reason out that Noise, as I have defined it, represents the point where "acoustical contradiction" is taken to its furthest possible extreme - that, in Noise, acoustical contradictions become so numerous that they actually cancel each other out, leading to a total homogenization of sound, to a grey and featureless frequency-soup!
The existence of a sound like this provides a vital clue in our investigation of the Consonance-Dissonance dichotomy. If Noise is a "pure dissonance," then it will not be too difficult to take the the next logical step: that Dissonance is nothing other than Noisiness. Does this really tell us anything? Does the reader think I'm just shuffling around words, here? Be patient. Already, we've unburdened our conception of dissonance from mathematics and found its new home in a well-known and intuitively-grasped sensory experience. This was the goal all along. Further down the line, this shift will impact the whole discussion.
Noise expresses a general rule of sound: the higher the number of frequencies, the more these frequencies obscure each other's pitches. The Tone represents the singular exception to this rule. A tone's frequencies, rather than diluting each other, compound together into a single pitch.
Every Tone comes in the form of a Harmonic Series. 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, begins to vibrate at a certain frequency, that frequency will not only travel through the air but through the rest of the piano, reverberating through its body and its other strings. By these reverberations, a series of resonant frequencies, or harmonics is produced. These harmonics will all be multiples of the initial frequency, or the tonic. If the tonic is 100 Hz, for instance, a series of harmonics will be produced at 200 Hz, 300 Hz, 400 Hz, 500 Hz, and so on.
Every tone that we hear is made up of this series of harmonics: every trumpet-call, guitar-note, bird-song, animal-call, and human voice. Each one conceals, within itself, the same imperceptible structure of frequencies.
In order to begin to understand the nature of structure within the acoustical world, the reader needs to first understand how radically different a frequency and a tone are from each other. Too often, in theory, the two are treated as interchangeable. This interchangeability is justified by the assumption that the harmonic series is really not all that important - that its effects upon a harmony are negligible. To music theorist, a note at A4 is a note at A4. The harmonics don't effect how we perceive the pitch, so why would they matter?
I hate to repeat myself, but this is, once again, the result of the Pythagorean conception which reduces Harmony to "pitches held together by intervals." It is a fatal mistake for a theorist to conflate a frequency, an infinitesimally small acoustical point, and a tone with its fully-fleshed out series of harmonics. We only have to listen to these two sounds side-by-side to recognize how far apart they are acoustically.
This is a single frequency in isolation:
But THIS is a tone:
Both are heard at the same pitch, but whereas the former is a simplistic expression of that pitch, the latter adds structure, depth, and weight. A series of harmonics, when added along to a simple tonic-pitch, gives it an acoustical body. Harmonics flesh out pitch. While the frequency is just an acoustical "atom," the Tone is an elaborate acoustical molecule, one in which many interrelated harmonics contribute to a single over-arching feeling of sound.
And, so, Pythagorean music theory, through its over-simplification of Harmony, has obscured another brutally obvious truth: that the Tone is, itself, a harmony. The Tone unifies many sounds into a single coherent, acoustical form. Isn't that exactly what "harmony" is? Isn't that exactly what... Consonance is?
Every time we hear something "consonant" we are really just hearing a bunch of frequencies that seem to blend together. The major triad, the perfect fifth, the octave - each of these feels like ONE acoustical object. The only difference, really, between all of these consonances and the Tone, is that, whereas the we can still detect some traces of acoustical separateness in the former, we detect none in the latter. So seamlessly do the Tone's frequencies combine together that we do not even register them as separate. It is a totally faultless acoustical body. Not only, then, is the Tone a consonant harmony; it is the purest possible consonant harmony, a perfect unification of many frequencies into a singular feeling of pitch. The Tone is the one and only pure Consonance!
Tone and Noise represent opposite relationships between frequencies and pitch. Whereas the frequencies within a noise all detract from each other, a tone's frequencies all assimilate into the pitch of its tonic. By forgoing their own pitches, the harmonics actually intensify the pitch of the tonic. The Tone, then, is an anti-Noise. It is the only way for frequencies to fit together in such a way that pitch-feeling is intensified, rather than obscured.
So, we have two fundamental sounds: Tone and Noise. The Tone is a pure consonance in which many frequencies assimilate into a common pitch-feeling. Noise, on the contrary, is a pure dissonance in which acoustical contradictions are so numerous that pitch-feeling is nullfied. Sound-with-pitch and sound-without-pitch, acoustical structure and acoustical mass, 1 and 0, form and un-form, unity and homogeneity. Our entire sense of Harmony rests upon the duality between these two extremes of sound.
In order to understand how fundamental Tone and Noise are to the sense-world of sound, I encourage the reader to go outside, away from the intrusion of public speakers (background music needs to be avoided. We are trying to observe the sound-world in its most natural state) and to simply take note of the sounds which reach the ear. The reader will quickly find that nearly every sound can be neatly categorized into either a tone or a noise. The crashing of ocean waves, the blowing of wind, the cry of cicadas, and the roar of planes are all noises, waves of pitchless sound which bleed into one another. Meanwhile, the bird-song, the baby crying, the chatting neighbor, and the car horn are all tones; each is a single fluctuating pitch buffered by a harmonic series. The world of sound is by-and-large a constant interplay between tones and noises.
Noise forms the eternal backdrop of our acoustical experience. Every moment is marked by a background noise. There is no silence in our world; only a constant undercurrent of pitchless sound. Right now, as I write this, I am listening to the hush of air conditioning and the rolling of tires along the nearby highway. The world of sound, at its lowest level, is a boundless landscape of various noises, against which pitch is merely an exception.
A tone is a fleeting acoustical form juxtaposed against the constant landscape of noises. Whereas noises naturally dissolve into each other, becoming one undulating acoustical mass, tones are distinct acoustical objects. Each one, by virtue of the absolution of its frequencies into a particular pitch-feeling, cleaves itself off from the rest of the sound-world. Timbre is nothing less than acoustical shape, the outward expression of a pitch-center. Tones appear briefly against the backdrop of Noise, flexing these brilliant acoustical shapes, before fading away, dissolving back into the Noise-mass. Each tone is a brief spark of light, a flickering star which temporarily draws the ear from the overwhelming Noise-scape.
Everywhere we look, therefore, we find the same image of the World of Sound: a timeless landscape of noises upon which various tones sprout and decay. THIS is the world of sound in its purest, most untouched state, before background music was inserted into every public space. 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 contrasted against each other like figure and landscape.
The dichotomy between Tone and Noise is as as fundamental to the ear as 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, then, takes the form of a simple analogy: Consonances are Tone-like. Dissonances are Noise-like.
Every consonance has Tonal qualities, because it is always, in some way, an elaboration upon the Tone. It has often been observed, for instance, that consonant harmonies mirror the structure of the Harmonic Series. In consonances, as well, we tend to recognize a "root-note" or a "tonal center," whereas, in dissonances (like the diminished 7th or the tritone), this feeling of a root is entirely lacking. To me, it is clear that this feeling of a "root" is just a pitch-feeling. Just as
Our sense of Harmony originates in our ability to distinguish tonal voices from a sea of noises out in nature. This basic sense evolves greatly in response to the ever-increasing complexity of human language, and even further in response to man-made harmony, in which the line between Tone and Noise blurs. A consonant chord is something in between Tone and Noise, something which is tonally structured but which is, itself, made up of many tones. In Harmony, then, the sharp duality between Tone and Noise transforms into a messy polarity between Tone-like and Noise-like, a polarity which we have, up until this point, never recognized for what it is, and have filled in its blanks with "Consonance" and "Dissonance."
To understand the terms "Tone-like" and "Noise-like" requires that we resist the Pythagorean urge to view harmonies as groups of notes bound together interval-by-interval. Instead, the harmonic picture must be viewed in its entirety. In opposition to the traditional Pythagorean, or hyper-numerical, view of Harmony, we can call this The Organic viewing of Harmony. To view harmony organically means to view each harmony as a whole and complete SOUND, and to place a higher value on the over-arching impression of this sound than its individual parts. The moment we begin to think of harmonies in this way, it becomes clear how certain harmonies could resemble the Tone while others could more closely resemble Noise.
For instance, a tone at the pitch of C1 and an octave spanning from C1 to C2 are nearly identical!
Here is a C1-tone:
And here is a C1-tone and a C2-tone played simultaneously:
The addition of the C2-tone BARELY changes the sound at all. In fact, if we hadn't heard the initial C1-tone, we might even confuse the latter sound for one tone. THAT is Consonance. THAT is what it means for a sound to be Tone-like. It means that, although a "single tone" and an "octave" are easily distinguishable on paper, in terms of ACTUAL SOUND, there is barely any distinction between the two. This is the ACTUAL reason why two tones at an octave distance from each other form such a strong consonance: not because of the 2/1 ratio between them, but because, together, they form a sound which is barely distinguishable from a single tone.
All consonant harmonies, like the octave, are groups of tones which fit together in such a way that they create the impression of a singular tone. Many tones can be "Tone-like" in combination.
Say that a church-organist is having an off-day, and he accidentally starts the hymn by playing this chord:
The chord consists of 6 separate tones - 6 ENTIRELY DIFFERENT PITCHES which, in their fight for the ear's attention, end up obscuring each other. The chord, therefore, is a quasi-noise. A dissonance.
So, quickly fixing his mistake, the organist 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, rather than hearing 6 separate tones, we perceive a single Tone-like sound! A consonance!
In the consonant chord, the pitches of the 5 upper-tones are each incorporated into the harmonic series of C1, thereby, assimilating into its pitch-feeling. The tonic of the C2-tone is the same as the 2nd harmonic of the C1-tone. C2, therefore, is heard as an extension of C1's 2nd harmonic. G2, likewise, extends from the 3rd harmonic; C3 from the 4th; E3 from the 5th; and G3 from the 6th. What we get, as a result, is a single network of frequencies stemming from the pitch of C1.
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 a half-explanation. 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 by virtue of this "meta" series that the tones unify. The source of consonance is not in the two actual tones but in the overarching Tone-like sound which is created by their union. It is not the interval, but the fact that they literally become one tone!
So, Consonance is not a matter of number, but of semblance. 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?".
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 Indian Raga, American Jazz, or any of the thousands of locally-bound folk musics lying scattered across the globe, one always finds 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. The moment that he strays too greatly from this structure, he invokes Noise, and, with it, the grey and banal feeling of acoustical homogeneity.