Every Tone contains, within itself, a sprawling network of frequencies which we call the Harmonic Series. Contemplation of this Harmonic Series marks the beginning of an understanding of the structural laws of Harmony. The Harmonic Series is the only naturally created harmonic structure, and it lies the DNA for every harmony that has existed or will exist. In all harmony, we hear its traces. To listen to a harmonic progression is to be dizzied by many fragments, refractions, inversions, and transformations of the Series.
What is meant by the word "structure?" At a quick glance, the harmonic series does not appear to be a “structure” so much as a linear set of harmonic identities stretching from tonic to infinity. But, with a keener ear, one begins to detect something below this superficial ordinal level of the series. One begins to notice which harmonics resemble each other and which harmonics contrast, which harmonics are structurally cohesive and which are decorative, which harmonics relate to each other and which create opposing acoustical feelings. In short, one who contemplates the series long enough will begin to understand that it is a structure, an intricate network of relationships between harmonics.
The field of music theory, due to its rigid interval-centrism, has yet to conceive of the shape of this network. It has yet to even understand what a TONE is, let alone to peer into the inner-structure of this Tone and to develop a science around it. Just as Renaissance artists benefitted greatly from secretly opening up dead bodies and, for the first time, getting to understand the anatomy of men, so too will we musicians benefit from peering into the structure of the Harmonic Series and, for the first time, getting a real sense of Tonal anatomy!
The closest we can come to understanding a thing is being able to imagine it. If we can conjure up a mental image of something in our head, that is the truest form of intellectual intimacy. Therefore, let’s try to construct an image of the Harmonic Series from the bottom up.
At the core of the Harmonic Series lies the tonic, the tonal nucleus which establishes the pitch-feeling of the entire acoustical body.
Out from the seed of the tonic springs the entire branching network of harmonic identities. First comes the 2nd harmonic, the tonic’s double, the first and most rudimentary extension of its pitch-feeling into harmonic space. From the 2nd harmonic, we get the so-called “octave” interval.
Then, out from the 2nd springs the 3rd, which creates our “perfect 5th” interval.
The 2nd and 3rd harmonics are both novel acoustical identities; each one deviates from the tonic in its own entirely distinct way. So, we should expect the 4th harmonic to be something completely new as well, correct? Well, this is not the case. The 4th harmonic evokes an acoustical “color” which is almost identical to that of the 2nd.
Basically, this is because the 4 is the 2 of the 2. Do you understand? If 2 were the tonic of its own series, the 4 would function as its 2. The 4th harmonic, therefore, relates to the 2nd harmonic as the 2nd harmonic relates to the 1st. And this is, in fact, the very same way in which the 8th relates to the 4th, and 16th to the 8th, and so on. All of these harmonics - 4, 8, and 16 - are powers of 2 and, consequently, are gradually weakening reflections of the 2-feeling.
One great collective oversight of music theorists is that they acknowledge this recursive feeling in the powers of 2 - they call it “octave equivalence” - but they fail to recognize that this exact same phenomenon of “equivalence” is shared by other harmonics!
The 9th harmonic, for instance. Anyone who is listening to the 9 clearly can sense that it has the “flavor” of 3. That’s because it’s the 3 of the 3.
Indeed, the powers of 3 evoke a gradually diminishing “3-feeling” in exactly the same way as the powers of 2.
So yes. There is such a thing as “octave equivalence,” but there is also a 3-equivalence and a 5-equivalence and a 7-equivalence.
Once this fact is recognized, we begin to see that there is a logical ordering to the series which is commanded by its arithmetical inner-relations. Prime-numbered harmonics such as 2, 3, 5, and 7 are all phenomenologically distinct from each other; each one evokes an entirely unique acoustical feeling in relation to the others. On the other hand, compound-numbered harmonics such as 4, 8, 9, 10, and 12 derive their respective acoustical feelings from the prime identities which they are multiples of. When we hear the 4th harmonic, we are hearing the 2nd harmonic’s 2nd harmonic. When we hear the 6th harmonic, we are hearing the 2nd harmonic’s 3rd harmonic. When we hear the 15th harmonic, we are hearing the 3rd harmonic’s 5th harmonic.
It is not sufficient, then, for us to picture the Harmonic Series as a straight line extending from the tonic; rather, we must imagine it as a hierarchical structure which begins with the tonic and then extends into prime-numbered harmonics, and then further into compound harmonics. The prime identities are the direct offshoots of the tonic, but each of these primes gives birth to a series of compound identities. These compound identities bear the residual feeling of whichever prime they extend from. In short, every harmonic is, itself, the tonic of its own "sub-series."
The Harmonic Series, therefore, can be imagined as an acoustical fractal. It begins with the fundamental series of prime-identities generated by the tonic:
Each of these prime-identities generates a sub-series which is identical to that generated by the tonic.
And each harmonic within each sub-series generates its own identical sub-series, and so on, ad infinitum.
What we end up with is a tree-like structure, a branching hierarchy of identities exploding outwards from the seed of the tonic. The initial series of primes forms the main trunk of the structure, but each harmonic sprouts into its own "branch" of specialized identities. By this recursive process, we get the entire complex network of hamonics. Here is the exact shape our Harmonic Series takes; it is hierarchical to its core.
This Harmonic Tree is more than just a metaphor for describing the relationships between harmonics; it is also a map indicating the variance in structural importance throughout the series.
As I touched upon in my last article, certain harmonics are more structurally important than others. Like the blocks which make up a building, certain harmonics act as foundation whereas others are merely decorative. That is to say, there are strong harmonics and weak harmonics.
A tone consisting only of harmonics 3-8 sounds quite consonant, while a tone consisting of harmonics 23-28 is just a dissonant cluster of frequencies.
Clearly harmonics 3-8 are more integral to the structure of the Tone than harmonics 23-28, simply by virtue of the fact that the latter group can be sacrificed whereas the former group cannot. We can conclude that harmonics 3-8 are stronger than harmonics 23-28.
Every harmonic within the series has an inherent degree of harmonic strength. Naturally, the tonic is the strongest harmonic in the series. From there, the harmonics generally weaken in strength as their number approaches infinity. “Generally,” however, is the operative word here. By no means does the harmonic strength strictly decrease as harmonic number increases. The 4th harmonic is stronger than the 3rd harmonic. The 12th is stronger than the 7th! Harmonic strength seems, at first glance, to vary quite irregularly from harmonic to harmonic.
So the big question is: is there a logical pattern to the distribution of harmonic strength throughout the series?
Yes! And this is where our harmonic tree becomes useful. Harmonic strength correlates precisely with the acoustical hierarchy which we have mapped out in the harmonic tree!
The 4th harmonic is stronger than the 3rd, because it is structurally closer to the tonic. The 4 does not introduce a new prime-identity; rather, it is a deepening of the strong 2-identity. By contrast, the 3 introduces an entirely new structural domain. So although 3 is lower in number than the 4, it still represents a greater structural departure.
If we scutinize the fundamental series of primes generated by the tonic, it is easy to notice that these prime-harmonics weaken at an exponential rate. 1 is the strongest harmonic. 2 is only slightly weaker. 3 is still quite strong. 5 is where harmonic strength starts to noticeably diminish. 7 is much weaker. 11 is extremely weak. If we were to draw a graph mapping harmonic strength onto the primes, it would look like this:
Because the entire harmonic series is generated by the recursion of this initial series of primes, every sub-series has an identical rate of harmonic weakening! The decay of each harmonic branch mirrors the decay of the prime trunk, scaled by the strength of its original harmonic.
Therefore, harmonic strength does not depend on a harmonic’s number but on its prime-complexity. Harmonics made up of small primes - especially 2 or 3 - retain structural strength, while harmonics introducing large primes - such as 7 or 11 - represent structurally weaker harmonic departures.
To formalize it into a law: harmonic strength decays as prime factorization complexity increases.
This law explains every variance in harmonic strength throughout the series! It explains why 4 (2x2) is stronger than 3, why 9 (3x3) is stronger than 5, why 12 (3x4) is stronger than 7.
A direct physical manifestation of the law can be observed in the frequency spectra of naturally produced tones. These spectra generally reflect the relations between harmonic strengths formalized by our Law of Harmonic Complexity:
When a tone’s timbre reflects its harmonic structure - that is, when the amplitude of each harmonic is proportional to its own inherent strength - the tone will sound purely consonant. This falls in line with what I proposed in my last article: that there is a “normative timbre” which a tone must roughly conform to in order to be heard as purely consonant. Pure timbre can now be rigidly defined: when the distribution of amplitude throughout a tone’s harmonic series reflects the natural distribution of harmonic strength.
It can be observed that tones which deviate dramatically from a natural timbre sound somewhat dissonant:
Let’s tweak our Harmonic Tree diagram to reflect the Law of Harmonic Strength. We’ll adjust the length of the lines between harmonics so that each line-length is directly proportional to the decrease in harmonic strength. Because 2 is barely weaker than 1, the line between them is short. Because 11 is far weaker than 7, the line between them is long. By this stipulation, we get the following map:
And here we have created a rough map of harmonic strength throughout the series, where any given harmonic’s strength is directly proportional to its proximity to the tonic.
Just as with a regular tree, the base is strongest, but as the tree extends outwards, it splits and branches out. As it continues to divide into thinner and thinner branches, it grows increasingly complex until, at the upper-harmonics, we have a chaotic field of overlapping harmonics which is indistinguishable from Noise. The harmonic tree is rooted in the firm root of the tonic and gradually fractalizes into Noise at its farthest edges.
This whole thing is begging to be reduced to a single, concise formula. And perhaps doing so would grant my theory some validity in the eyes of academics. In modern academia, afterall, things are never really considered “scientific” until they can be quantified. Truth must always be baptized in number.
I am not particularly gifted in math. We have, however, already basically outlined what such a formula for quantifying harmonic strength would look like.
A harmonic’s strength is inversely proportional to its prime factorization complexity, which can be defined as the magnitude and quantity of its prime-factors. So, roughly, it can be calculated this way:
Harmonic strength plays a small role in the larger image of "Consonance" which I have proposed. As a reminder, Consonance = tonal resemblance. So, am I working towards a method for quantifying Consonance? No. Not me, personally. I'm sure it can be done. Although, my guess is that there are many possible models that will all approximate "tonal resemblance" slightly differently.
In all of this mathematizing, I have maybe hinged on breaking a rule which I informally laid out in my first blogpost: "Every numerical representation of Harmony will be thoroughly distrusted." The point of my long-winded polemic against numerical representations was that, although we might use number to help us understand music, we should avoid conceiving of music as "math translated into sound." Math is a tool for comprehending ordered bodies, but the bodies, themselves, are different from number. We should be on gaurd against that insipid Pythagorean idea that the beauty of math and the beauty of music are one and the same.
But it is worth noting that I do not view this blogpost as a series of mathematical insights so much as they are insights into the structure of things. That the structure of the series is expressible in mathematical form is only downstream of the fact that everything can be expressed in a mathematical form, which is simply to say that everything is measurable. Contra Pythagoras, number can be projected onto everything, but that does not mean number IS everything.
The Harmonic Series mimics a structure which can be observed everywhere in nature: the fractal hierarchy.
All organic structures take the form of fractals. Everything which lives does so by growing, and everything which grows does so by recursion. This recursive growth process naturally produces fractal structures. By virtue of their design, these fractal structures can only operate hierarchically. A man’s fingers work in the service of his hands, his hands in the service of his body, his body in the service of his mind. Man is a hierarchical structure/ this is the hierarchical design which shapes his being: a distribution of labor beginning in the head and ending with the appendages.
Proto-Indo-Europeans envisioned the world as a tree, as Yggdrasill, because they saw reflections of the tree in everything. Animals, bacteria, LIGHTNING. Yes, even Man is a seed-head which branches out into many appendages. The Proto-Indo-Europeans intuitively understood that the fractal hierarchy was built into the very veins of nature, that the cosmos ordered itself in a tree-like fashion.
We should not be surprised, then, that order manifests itself in sound in exactly the same way that it does in every other aspect of existence. The Tone, the most perfect consonance and the epitome of acoustical wholeness, is, itself, a fractal hierarchy. It is tree-like.