In my last blogpost, I proposed a system of Harmony which is less of a "system" than it is a simple allegory: Consonances are Tone-like; Dissonances are Noise-like.
The Tone is the basis of all of Harmonic structure. Every sound is a collection of frequencies, but the Tone is the only one of these in which the frequencies unify into a coherent form. Consonance is Tonality.
So it should be no surprise that the earliest and simplest form that Harmony took was the Drone, a single tone hovering in one place. Early humans produced this Drone by blowing into a conch shell or by vibrating their lips into a hollowed-out tree trunk. Imagine how shocking it was for man to discover something so alike to his own voice hiding within these husks of life, as if the . They must have held these proto-instruments up as divine vessels, as carriers of the living spirit. And so, even in its earliest incarnation, Tonality was correctly associated with the living spirit - with some divine, inner-force which animates matter. This living quality only becomes further exaggerated in the variations upon the Drone, where its timbre constantly fluctuates as if breathing, or where its pitch jumps up and down as if dancing.
Later on, we create more complex sounds by combining multiple tones together. But, because of our unconscious leaning towards the sound of Tonality, our multi-tonal structures all end up resembling a single Tone. Regardless of how greatly our harmonic progressions grow in complexity, they never fully escape from Tonality, and our harmony remains, in essence, heavily embellished-upon Drone music.
But to embellish is also to distort. Any alteration of a tone makes it less Tone-like and, consequently, more dissonant. A tone with missing harmonics, a tone with a disproportionately loud 5th harmonic, a tone with a stretched harmonic series - these are all imperfect consonances. We recognize them as Tonal, but simultaneously we recognize something off in them - something strange or asymmetrical which offsets their pitch-feeling. And so, the more we embellish upon the Tone, the more we flirt with Noise and invoke Dissonance.
So, a slight fluctuation in timbre evolves into an extreme fluctuation between Consonance and Dissonance. Tonal shapes are established before being torn apart and reconfigured in new ways. A progression from C major to a Diminished 7th to C major is fundamentally just a movement from a Tone-like sound to a Noise-like sound to a Tone-like sound:
A tonal shape is established before being distorted and then reconfigured into another tonal shape roughly resembling the first one. This formula - this fluctuation between tonal distortion and tonal purity, this cyclical destruction and restoration of a single tonal form - is the central poetic trope which all of Western music has built itself upon. This cycle, brought to its greatest extremes and variations in the hands of the great composers, is the source of Western music’s feelings of intensity and drama. There is little, here, that can be attributed to “numerical relations” or “logic” and much that can be attributed to the living-feeling inherent to the Tone, and the utter violence with which we treat it. The Tone, in the hands of the composer, becomes an acoustical ball of clay which he can play around with to produce an infinite number of harmonic shapes. He stretches, pulls, twists, presses, and cuts at the Tone to create the exact kind of sound which he wants - to create the precise harmonic shape that sounds just like how he feels.
I have already made the case, in my last blogpost, for why a consonant chord is Tone-like and, hence, consonant. But there may still be some difficulty in imagining how the greater works of Harmony could bear any resemblance to the Tone at all. In this blog entry, therefore, I will start with the presupposition that a Tone is purely consonant, and then show how it can be elaborated upon to create more complex sounds which still bear its resemblance. Through this experiment, I will not only demonstrate that every harmony is a tonal elaboration, but that the extent to which we perceive this harmony as dissonant depends entirely upon the extent to which it deviates from a pure Tone.
First of all, in order to know what qualifies as a “deviation from the Tone”, we need to establish what a purely Consonant Tone is. The Tone, you will recall, is a sound which is made up of multiple frequencies but only heard as one pitch. Any erosion of this “pitch-feeling,” amounts to a flirtation with Noise and, hence, with Dissonance.
Fundamentally, a tone has 2 qualities which determine the extent of its purity: the placement of its frequencies and the volume of its frequencies - harmonicity and timbre respectively.
Harmonicity refers to the actual positions of a tone’s frequencies in relation to its Harmonic Series. This is pretty straightforward. Naturally, the frequencies in a tone are positioned so that they are all multiples of the tonic. If any frequencies are out of place from these natural harmonic-positions, the tone will sound kind of off.
Compare a tone with a regular harmonic series, and one with a "stretched" harmonic series:
In the first example, the tone produced is purely consonant; every frequency is perfectly drawn into a single pitch. In the second example, we have difficulty even telling what the pitch is, as the frequencies are audibly struggling against the tonic, rather than assimilating into it.
.The sound of a bell is so easily identifiable because of the uniquely inharmonic tone which it produces. Many of its harmonics are completely offset from where they should be in the harmonic series. The result is a tone which has a clearly identifiable pitch, but which sounds "off" - almost semi-noisy.
A Youtuber, New Tonality, put out this really excellent video where he discusses the inharmonic tone produced by the Javanese Gamelan, and how the scales of Javanese musicians have naturally changed in a way which deals with this inharmonicity.
Timbre refers to the volumes of a tone’s harmonics in relation to each other. It is not enough that a tone’s frequencies fall into one harmonic series; the volumes of these frequencies also must relate to each other in such a way that none of them obscure the tonic-pitch.
Because of the way that the Harmonic Series is naturally produced, the overwhelming majority of timbres that we hear in nature follow a common pattern: the tonic is the loudest and, as the harmonics rise in number, their volume gradually tapers off.
This is the normative timbre of tones. Although timbre varies greatly from tone to tone, the variance is largely constrained to this pattern. Only recently have humans learned to artificially produce tones which deviate greatly from normative timbre. But artificial is artificial. The normative timbre is that which our ear has accustomed itself to throughout millions of years of evolution; furthermore, it is the timbre of the human-voice, that tonal sound which we inherently feel biased towards, and which therefore ought to be considered, at least in regard to human beings, the highest standard of consonance that can be reached.
So, any deviation from normative timbre is a deviation from pure consonance. Compare an oboe-tone, which has a pretty normative timbre, to an artificially produced tone which has loud upper-harmonics and soft lower harmonics:
Loud upper-harmonics and soft lower-harmonics is the total inverse of normative timbre, so these timbres are kind of "lopsided" or "top heavy." They usually have this buzzy, empty feeling to them.
Here is an even greater deviation from normative timbre, where many of the lower harmonics have been completely removed:
It's the same pitch, but now that pitch is so tenuously and bizarrely expressed that we really have to strain our ears to pick up on it. Here we see that dissonance directly correlates with the "ease" of detecting a sound's underlying pitch. The more ambiguous a tone's pitch becomes, the further the tone dissolves into a noise. I think it would be absurd not to consider these dissolved tones "dissonances" in precisely the same way that we would consider tritones "dissonances"; both have that noisy, rootless feeling that occurs whenever a sound is no longer bound by a single pitch.
Therefore, there is such a thing as "dissonant timbre." Normative Timbre and Normative harmonicity play an equal role in the purely consonant feeling of a tonal body, and any deviation from either of these two necessarily leads to dissonance. Going forward, we will only consider the overarching Timbre and Harmonicity of a sound when trying to discern the extent of its Consonance or Dissonance.
As I demonstrated in my last blogpost, many tones can come together in such a way that they impress the feeling of a single tone. This is why “consonant chords” are possible.
The following consonant chord combines 4 organ tones in such a way that they roughly resemble a tone at the pitch of C1:
The chord is, on a superficial level, composed of 4 tones - 4 separate harmonics series, each bolstering their own particular pitch. On the other hand, these 4 series all fall into one overarching “meta” series which is the source of the chord’s unity and, therefore, of a broader pitch-feeling. It is necessary, then, for us to separate the chord into two harmonic layers: the set of individual series which the chord consists of, and the broader series which they create through their union. These two layers I respectively call Sub-tonal and Meta-Tonal.
Every consonant chord is, in essence, a “meta-tone” (MT), a tonal image constructed through the union of a set of lesser “sub-tones” (ST).
Each sub-tone acts as an extension of a particular meta-tonal harmonic. The pitch of the E3-tone, for instance, is the same as the meta-tone's 5th harmonic; so, although this E3-tone has its own harmonic series, this entire series accentuates the 5th harmonic of C1. If we view the harmonic series as a network of interrelated frequencies, then it is as if C1's series forms the main body of this network, and the series of G2, C3, and E3 are blossoming out from different harmonic-points.
Sub-tonally, C1, G2, C3, and E3 are individuated tones, but meta-tonally they are accentuations of different parts of the broader structure set in place by C1. We would notate this chord as C1:[ST 1 3 4 5] to indicate that our meta-tonal pitch is C1, and that we have sub-tones accentuating its 1st, 3rd, 4th, and 5th harmonics.
In our harmonic language, C1 is both the MT and ST1. This may seem a little confusing, but we have to distinguish between the overarching tonal image resembling a C1-tone (MT), and the sub-tonal C1 (ST1) which contributes to fleshing out this image. This distinction becomes more apparent when we actually withhold from playing the ST1 and the larger meta-tonal feel remains:
When judging the extent of a chord’s consonance, we examine the whole meta-tone. In this case, we compare our meta-tone C1 to a pure C1-tone. Upon doing so, we find that, although its harmonicity is pure, its timbre is slightly abnormal, because ST2, ST3, and ST5 disproportionately raise the volumes of the 2nd, 3rd, and 5th meta-harmonics. So, although the chord is very consonant, it is not purely so.
You can always expect a meta-timbre to be somewhat “lopsided” or “top-heavy” if we have sub-tones anywhere but the tonic. This lopsidedness is what makes all chords, by necessity, a little bit dissonant.
Observe how moving the sub-tones onto higher and higher meta-harmonics gradually makes the meta-timbre more and more abnormal, intensifying the feeling of dissonance:
Until, as with every Tonal distortion, there is some breaking point where the image being distorted is no longer recognizable. For instance, if we only have sub-tones at 7 9 13 and 15, for instance, the timbre of the meta-tone thus produced is so abnormal that we no longer perceive any tonality in the chord at all. We have stretched the meta-timbre to its breaking point, and all we are left with is a bunch of tones - a mere noise:
So, by changing our sub-tonal "voicings" we create different meta-timbres. The expressiveness of "chord-voicing" as a whole is downstream from the expressiveness of timbre.
At last, we have the context and tools to create an actual harmonic progression. By constantly changing the sub-tonal voicing of a meta-tone, we fluctuate through various meta-timbres, resulting in a very safe and inoffensive harmonic progression. If you've ever listened to 1-hour ambient meditation god-frequency music on youtube, this is exactly what they’re doing:
What we call an "arpeggio" is really just playing one sub-tone at a time:
A suprising number of famous musical progressions rely upon this technique of "tracing" through a single meta-series. For instance, we have the famous opening to the William Tell overture:
The influence of timbre upon consonance can be taken advantage of here to create a kind of resolution. Moving from an abnormal meta-timbre (sub-tones at higher harmonics) to a more normative meta-timbre (more sub-tones at lower harmonics added) constitues a movement from Consonance to Dissonance:
The first chord, in a purely meta-tonal sense, is an incomplete tone - a tone with the entire lower part of its series missing. The second chord fills in all of these missing parts. This is why the resolution sounds so satisfying: the restoration of an incomplete tonal structure. Moreover, because the first chord implies a tonal structure, and the second chord blatantly affirms it, we are validating an assumption made by the ear.
Chopin absolutely LOVED this type of resolution. He ended a lot of his songs by playing sub-tones in the middle, then sub-tones way up high, and then finally landing onto ST1 and ST2 (this interval he ends on being almost indistinguishable from a pure tone):
This is such a common way of ending a piece of Western music that it is a wonder music theorists still consider consonance to be entirely dependent upon intervals and scale. No system of theory other than mine can explain how these resolutions sound like resolutions. How can we “resolve” from a C-major chord to a C-major chord, Hugo Reimann? How can we “resolve” from less tones onto more tones, Harry Partch? The answer is: intervals don’t matter, function doesn’t matter, only resemblance to the Tone matters!
The second way we can distort a meta-tone is harmonically - that is by detuning its harmonics from their natural places in the harmonic series, thereby corrupting the structure.
For instance, take a fairly strong consonance, C1:[ST 2 3 4 5]
Now, let’s move the E up to an F:
Really, the F3 should be regarded, not as a new harmonic identity, but as a detuned version of the E3. We feel a progression from Consonance to Dissonance because the ST5 is pitch-shifted away from its natural position in the series and wants to snap back into place much like a rubber band which has been pulled away from its ball.
Whereas, at the beginning of the progression, ST5 is hidden away in the consonant embrace of the meta-tone, when we displace it from its position in the harmonic series, it protrudes like a jagged edge. When ST5 is retuned back to its original place, it sinks back into the warm body of sound from which it had departed. Overall, this slight push and pull upon the ST5 is quite charming. The dissonance is not so intense as to be "painful"; it is more like a subtle tonal "massage."
But what about a not-so-subtle dissonance? What if we displace ALL of the sub-tones?
Here, we really blur the line between what is "Tone-like" and what is "Noise-like." Whereas, in the previous example, most of the tonal structure is left alone, here it is utterly mangled. Here is where we get that feeling of pain or displeasure which is often ascribed to harsh dissonances, where “tonal massage” crosses over to “tonal torture.” One can almost hear the Tone screaming out in pain as its harmonic-limbs are stretched and pulled into the most unnatural positions, forced into an excruciatingly Noise-like sound. But the intensity of this dissonance only amplifies the relief we feel when the harmony resolves:
Now, in a way that sort of combines harmonic-distortion and timbre-distortion, we can displace a sub-tone far enough that it actually travels to a new place in the harmonic series. For example, we can detune ST5 all the way up to ST6:
Or, we can shift ST6 downward to ST5 and ST5 to ST4 simultaneously:
And you can just keep doing this - traveling from meta-harmonic to meta-harmonic over and over to create a decent chord progression:
Through this technique, Beethoven opens up his Piano Sonata in F#:
Throughout the progression, SUB1 and SUB2 remain steady; they are the base of our drone, the solid foundation upon which the rest of the harmony lies. Because these two sub-tones are active, there is little ambiguity in the progression; we always know exactly which harmonic identities are being activated and which are being deviated from. Meanwhile, the upper-tones are slowly crawling upwards, occaisonally locking into the meta-tonal structure at various point. Therefore, the meta-tone gradually expands outwards, ultimately growing from one consonance - SUB 1, 2, 4, 5, 6 - to a broader one - SUB 1, 2, 8, 12.
If we were to actually remove the ST1 and ST2 from this progression, we still feel the underlying meta-tonal structure; it’s just that this meta-tonal structure is less blatant:
It is worth noting that, fundamentally, these progressions are all still just elaboration upon the Drone. The harmony still only consists of a single tone changing in timbre. The only difference, now, is that our timbre is far more variable, because it consists of many sub-tonal parts which can be moved around independently.
This is kind of a side note, but remember how I mentioned that we hear all tones in relation to a normative timbre which we are used to? Well, if our ear is used to a harmonic series which gradually decreases in volume as the harmonics count up, then wouldn’t an upper-harmonic that is the same volume as the lower-harmonics necessarily stand out?
[That’s why, in all of the chord progressions we’ve done, the pitch of the highest tone is always the most clearly audible. Our ear is always drawn to the tonic of the highest sub-tone, because that is always the most disproportionately loud frequency of the entire structure. This, my friends, explains the melody, and its ability to draw the ear. If any other theoretical system has some explanation for melody, I have missed it.
Harmony is all auditory magic tricks. It’s about taking advantage of the listener's sense of Tonality to create a bunch of Tonal hallucinations. There is a way to take this "sleight of hand" further. It involves a simple variation on our harmonic displacement trick.
Just like before, let’s take a C1 Meta-tone and displace its sub-tones and then resolve them back into place:
This should just be harmonic displacement. But, wait. What is different about this progression? Just like before, we distort the Meta-tone of C1, but this time the second chord doesn’t JUST sound like a distortion; it actually sounds like a completely new tonal shape. Wait, is that the harmonic series of F0?
Yes, this is a kind of harmonic “pun”: the second chord is a distortion of the C1 chord, but this “distortion” also resembles an entirely new tonal image, a meta-tone at the pitch of F0. These two separate tonal meanings weigh upon our ear simultaneously: have we distorted the C1 or resolved onto the F0? Which is the consonance and which is the dissonance? It is only when we resolve back to a C1-chord that this ambiguity is answered for. It is as if the composer says “Nah just kidding. It was a C1-drone the whole time.”
Because of this tonal pun, the second chord has a kind. Tone-A is heard as both S2 of MTC1 and S3 of MTF0; Tone-C as both the detuned S5 of C1 and S8 of F0; Tone-D, as both the detuned S6 of C1, and the S10 of F0. Each of these tones is colored by the shades of TWO separate harmonic identities.
Good Harmony is rich with these double- triple- quadruple meanings. Composers are constantly juggling around various meta-tonal feelings, deciding on a whim whether they should resolve onto a certain one or continue to string the listener along. I should not have to tell the reader how much depth this adds to the art of Harmony - how many new shades of harmonic color are added to our palette in the form of these hybrid-tones, and how much excruciating uncertainty now becomes imbued with our progression.
But, with that added depth comes an extra layer of tonal embellishment to be acknowledged. The progression is still just an elaborate C1-drone; every single frequency which makes up the progression is heard in direct relation to the pitch of C1; every harmonic color derives from the C1’s harmonic series! The meta-tone of F0 which is briefly touched upon in the second chord is only secondary to this primary meta-tonal feel established by C1. In addition to our Meta-tonal/Sub-tonal distinction, then, we must also distinguish between our primary meta-tonal structure which is heard throughout the whole progression and our secondary meta-tonal structure which pops up briefly, before disappearing. Let’s say that the latter, F0, is just a regular meta-tone (MT) and that the former, C1, is a meta-meta-tone (MMT). We get the following analysis:
The MMT-layer is still where we retain our drone. Observe the following chorale, in which many MTs are touched upon, but the MTT is always C1:
On the MMT level, we hear a single C1-chord warping over and over before finally resolving to a stable shape. But, every time the C1-chord warps, it does so onto a new meta-tonal structure: first onto G, then A, then. The harmony is constantly transforming and renewing into different shapes, but these shapes are always heard against the backdrop of C1’s Harmonic Series. Every tone played simultaneously stands in relation to its own microcosmic tonal cell, and to the broader C1, who’s pitch envelops the whole of the Harmony. We are still embellishing upon the Drone!
Now, there is probably a reader or two who, at this point, is absolutely sure that I am jumping through mental hoops in order to justify the axioms which I established in the last blogpost, and that this “meta-meta-tonal” pitch I’ve described is as purely theoretical as it is arbitrary. But, I assure you that we feel this pitch with as much clarity as we do the center of a circle. To demonstrate, observe how it sounds when the tonic of the MMT layer is sounded throughout the whole progression:
Notice that this MMT doesn’t clash with the progression as you might expect it to; rather, it seems to form the perfect foundation upon which these upper-chords dance. By playing the pedal tone, we are just blatantly affirming what was already implied by the upper-harmony; we are playing what was already imagined by the listener: a single, unmoving structural throughline.