Almost nothing at all is understood about the nature of Harmony.
The various systems of Harmony which music theorists have constructed, from the Ars Noveau of the Middle ages to the modern Neo-Reimannian theories, have been far more devoted to practicality than truth. Music Theory should, at its core, be a kind of acoustical phenomenology, a study of our perception of Harmonic qualities and why we perceive them the way we do. But when we look out across the whole body of works on music theory, we see nothing of the sort; we only see a long list of shallow categorization-systems: harmony categorized into modes, Harmony categorized into Reimannian functions, Harmony categorized into Lewinian transformations. None of these harmonic taxonomies reveal anything even close to the underlying nature of Harmony; they only succeed in reducing the harmonic styles of the great composers into rigid and communicable systems, thereby making them teachable.
Consonance is the central problem of Western Music Theory. It is almost universally agreed upon among Western musicians that there is some harmonic quality called Consonance and an inverse quality to that called Dissonance. The whole Western style of music is built upon the perceived polarity between these two. Every well-trained musician is aware of the polarity, and some of the more gifted ones even know how to weild it properly. And yet there is no music theory which has really pierced into the deeper nature of these sound-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 has utterly failed to improve our understanding should tell us that our issue is not due to a lack of evidence, but due to some flaw 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 intervals are seen to be the essence of Harmony.
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. Conventionally, we would identify the former interval as more consonant than the latter, due to its greater numerical simplicity.
In Western music theory, the interval is viewed as the deepest and most fundamental unit of harmonic structure. A given harmony's qualities and characteristics are seen as determined by the intervals which it consists of. This, of course, includes the extent of the its Consonance. Music theorists view the phenomenon of Consonance as something which occurs exclusively within the interval, or "between two pitches."
This interval-centric conception of Consonance has origins as far back as the Pythagoreans. Everywhere in Pythagorean literature, we see a sharp duality constructed between consonant (symphonos) intervals - the fourth, the fifth, and the octave - and dissonant intervals - the rest of them. These consonant intervals, once discerned, are used to construct the Greek scales. Thus, the interval is framed as the essence of Harmony. The Pythagoreans, as the first thinkers to dabble in something which could be called "music theory", were in the privileged position to manipulate music theory in its fetal stages, to plant philosophical seeds which would, over thousands of years, spread out and harden into music theory's axiomatic roots. The idea central to Pythagorean music theory - that it is the "interval" which determines the quality of the Harmony - was adopted by Catholic monks in the middle ages, and has, since then, crystallized into an unquestioned belief which dominates all Western music theory.
Read any Harmonic Treatise from Zarlino all the way up to Schenker and you will find a method quite similiar to that of the Pythagoreans. First the theorist 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 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 semi-religious faith in the primacy of intervals has only become 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. To a modern musicologist, the scales and chords and tuning-systems that we use are nothing more than cultural constructions, but INTERVALS are still eternally valid. 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," which, in practice, appears to be little more than music theory touched up with an empirical coat of paint, is willing to analyze the harmonic spectrum within tones, but then still frames its conclusions intervallically. The Plompt-Levelt curve is no different from Harry Partch's One Footed Bride: it is a crude mapping of degrees of Consonance onto all the intervals within an octave. The only difference is that, where Harry Partch relies on his musical intuition, the Plompt-Levelt Curve relies on a synthesis of Helmholtz's ideas and ear anatomy.
Lastly, the Neo-Riemannians base all of their work upon the supposed primacy of 3 intervals - 2/1, 3/2, and 5/4. All of their systems are achieved by mathematic and geometric manipulation of the these 3 intervals. 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 as-a-whole 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. No matter how greatly our conclusions about Consonance have varied, they have always been constrained at the very outset, forced to fit into a narrow and suffocating Pythagorean view.
Apparently it occurs to no one how narrow this conception is. So many acoustical elements - timbre, register, noise, etc. - must be disregarded in order to maintain the purity of the Pythagorean view. 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."
The problems of music theory demand a severe broadening of our vision. 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. Intervals only represent Harmony at its most superficial level, Harmony as a set of measureable lengths. We should try to look beneath this intervallic layer and apprehend the more subtle structural aspects of Harmony.
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? What is a suitable alternative to Pythagorean music theory?
"If you're going through hell, keep going." So 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. 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 else 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 his philosophy - "music is math" - has imbued music theory with a cold, autistic character. Treatises of Harmony bore even the most curious students away from the study with their painful, mechanical analyses of the "mathematical relations between pitches." Conclusions always come in the form of mathematical models. Charts, diagrams, formulae: these are the greatest possible achievements of a music theory which regards ratios as the essence of Harmony. In these, music is degraded to its lowest possible form; it is crucified upon a y-axis, tortured, entrapped in a steel cage of numbers. 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. Even the most abstract of paintings are defined by qualities which are particular to our sense of sight: color, lighting, framing, space, and depth. Any reduction of such qualities to a numerical framing would be totally absurd, let alone informative. Yes, there are mathematical aspects to certain works of visual art, and their are even pieces of art who's aesthetic value cannot be decoupled from these mathematical domain, but only an idiot would claim that the entirety of EVERY piece's aesthetic value rests on numbers. 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 treated as such. Pythagorean Harmony forces us to reduce every chord to a set of "pitches" and to relate these pitches to each other intervallically. But does this reduction reflect our actual perception of the sound? Do we really listen to a chord and hear "sets of pitches linked together by intervals?" I really don't think we do. But this interpretation is eagerly accepted because it converts Harmony 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.
In order to begin to understand Harmony beyond its superficial mathematical aspects, we need to re-establish a connection between Harmony and the sensory world. Harmony is an extension of Sound and cannot be understood without reference to that particular sense-world. Any claim we make about the nature of Harmony, therefore, should be rooted in some principle of how we perceive sounds generally. Music theory should, at its core, be acoustical phenomenology.
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In this blog, our goal will be to develop a theory of Consonance which has its basis, not in numbers, but in 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 will be treated with the utmost suspicion, and every attempt will be made to understand Harmony by drawing analogies to our most basic acoustical impressions. The animal call, the crashing waves, the hushing wind, the crying cicadas - we regard these sounds as a far greater basis for understanding Harmony than ratios and formulae. It is upon this basis that we will strive to create a new theory of Harmony which is as robust, strange, and alive 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 possible 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 clear. It will also come to be understood that much of our sense of Harmony largely depends upon the relationship between the two.
Discussions of Noise are largely absent from the field of music theory. I have, in fact, never seen the phenomenon of Noise mentioned in any of the treatises of Harmony that I have read; not even in those by acousticians, who should really know better. But any theory of Harmony which refuses to 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. A chord over-laden with notes, or a band with too many parts, or a guitar tone run through a distortion pedal: all of these are recognized as "noisy." The sound-quality, "noisy" - Noise-like - is intuitively recognized by many listeners without musical training.
A Noise is a pitchless range of frequencies. A frequency, as we have said, is essentially an acoustical atom. A single frequency, in isolation, we always be perceived 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, individual pitches become completely obscured. The space between them disappears. This is the precise moment when a sound becomes a true Noise: when the feeling of separate, identifiable "pitches" disappears, and the sound becomes a mere range of unidentifiable frequencies.
Here is the primary difference between a consonance (a sound-structure) and a Noise (a sound-mass). A consonance is characterized by the placement of its frequencies, while a noise is characterized only by the range which they occupy:
"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 definitive quality of Noise is acoustical uniformity. 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. The dissolution of pitch-feeling means the flattening of all acoustical distinctions; it means that each frequency is hidden away, obscured within the overwhelming mass of frequencies surrounding it.
Noise is the greatest possible Dissonance. A noise is essentially a screeching mass of conflicting sounds. 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 think of Consonance and Dissonance as qualities which strictly 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 reflects on the nature of the sound-quality of dissonance, without the baggage of the Pythagorean conception weighing them down, will find that it always points to 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 conceals, within itself, this same imperceptible frequency-structure: every trumpet-call, guitar-note, bird-song, animal-call, and human voice. Every tone is made up of harmonics. "made up of" - this is the decisive break. A tone does not have a harmonic series; it is a harmonic series. Too often, music theorists view only a tone's tonic to be of any consequence, and the rest of the harmonic series to be tacked on. Even the term "overtone" implies something separate and above the actual tone. This misleads us into thinking that the harmonics are arbitrary, when, in fact, they form the very body of the acoustical form that we call the Tone. The Tone is not a pitch plus a series of overtones; rather, what we call a tone is, itself, an acoustical form, a structure composed of many frequencies.
The structural aspect of the Tone is undermined precisely because we only perceive the whole structure as one pitch. It is easy to assume that a frequency at 440 Hz and a Tone at 440 Hz are really not that different, that, because they lie at the same pitch, there's no reason to differentiate between the two. But this is a fatally unwarranted assumption. In reality, there is a world of difference between a frequency, an infinitesimally small acoustical point, and a tone with its fully-fleshed out harmonic structure. 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 as a singular pitch, but whereas the frequency is a simplistic expression of that pitch, the latter is that same pitch-feeling buffered by volume and depth. 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.
The Tone is, itself, a harmony. The Tone consists of many acoustical elements unified together into a single coherent, form. Isn't that exactly what a "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 basic sound-form of the Tone. Consonant harmonies, for instance, tend to mirror the structure of the Harmonic Series. In consonances 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. This root-feeling is nothing other than the hallucination of a pitch-feeling in a Tone-like sound. Just as, when we look at a cloud which resembles a human face, we instinctually project facial qualities onto it such as eyes and a mouth and hair, so too do we project the tonal quality of pitch-feeling onto a sound which is sufficiently Tone-like. This pitch-feeling in every consonance has been labeled "root" for centuries but never quite understood for what it is: a tonal hallucination!
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 Holistic view of Harmony. To view harmony Holistically 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.