The Harmonic Series is the basis of all harmonic structure. Long before man began clumsily shoving tones together to create chords and scales, there was, concealed within the gem-like sonority of each of these tones, a sprawling and intricate network of frequencies. This is the Harmonic Series, the only acoustical structure - that is, the only Harmony - which is produced by nature. Secretly, it has directed the flow of musical history, guiding each composer's hand to create music in its image. Triads, inversions, root, key - concepts which form the backbone of our musical system are downstream from the basic properties of the Harmonic Series. Every harmonic system has been unconsciously derived from the same fundamental structure. Every note is colored by the harmonic which it implies.
Music theorists have failed to pierce into the deeper structural aspects of the Harmonic Series. The traditional theorists - Rameau, Riemann, and Schenker - viewed the 7th harmonic as contradictory to Western systems of Harmony and so, in cowardly deference to their own culture, discarded every harmonic higher than the 6th. Modern psychoacousticians, from Helmholtz onward, rely too heavily on study-groups and ear anatomy to come to any genuinely profound conclusions. The greatest music theorist of the past century, and the only one to intuit a worthwhile model of "harmonic space," was James Tenney. Pythagorean interval-centrism has crippled the field as a whole.
At first glance, the Harmonic Series does not seem to be a "network" so much as a one-dimensional spectrum counting upwards from low pitch to high pitch, from tonic to infinity.
But a keener ear detects something lying below its ordinal surface. One begins to notice within the palette of varying harmonics, which ones are complementary and which are contrasting, which ones mirror each other and which differentiate, which ones are structurally cohesive to the series and which are decorative. In short, one who contemplates the series long enough will begin to detect the subtle relationships which bind and separate each and every harmonic, and one will quickly understand that it is not a straight line, but a network. The Harmonic Series has a FORM, a shape in acoustical space.
If we interpret the Harmonic Series superificially, in terms of a 1-dimensional pitch-axis, then it appears to just be an ordinal spectrum stretching from tonic to infinity. On the other hand, if we can interpret the Tone as what it is - a harmonic body in acoustical space - then the Harmonic Series is nothing less than this body's anatomy.
The Harmonic Series is hierarchical to its core. Each harmonic is an extension of the tonic, a into acoustical space. In this way, through this extension, the singular pitch of the tonic gains an outward form in acoustical space.
The closest we can come to understanding a thing is being able to imagine it. To imagine is to visualise. If we can conjure up a mental image of something in our head, that is the truest form of intellectual intimacy.
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.
Each successive harmonic, thus far, has presented us with a completely new acoustical feeling. The 2nd and 3rd harmonics are entirely distinct acoustical identities; each one deviates from the tonic in its own entirely distinct way. In the 4th harmonic, however, we get, for the very first time, an acoustical identity which is not new. The 4th harmonic evokes an acoustical feeling almost identical to that of the 2.
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 in it, that it is not a "new acoustical identity," but something which is derived from the 3 in the exactly same way that the acoustical feeling of 4 is derived from 2.
We should recognize that ALL of 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 true nature of "equivalence" is acknowledged, we begin to see that there is an unmistakably hierarchical character to the Series: the Harmonic Series is an acoustical hierarchy 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. On the other hand, compound-numbered harmonics such as 4, 8, 9, 10, and 12 derive their respective acoustical feelings from the prime identities of which they are multiples. 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. Compound-numbered harmonics are downstream from their parent-primes.
By charting these harmonic relationships, we arrive at the broader shape of our harmonic network. Far from a straight line extending from the tonic, the Series takes the form of an acoustical fractal in which the series of prime-numbered harmonics is generated over and over. The prime identities are the tonic's direct offshoots; they form the main "trunk" of the Harmonic Series. But from each of these prime identities sprouts a "branch" which mirrors the primary trunk. These specialized identities bare the residual feeling of whichever prime they extend from. It is as if each harmonic is, itself, the tonic of its own sub-series. The 2nd harmonic, for instance, generates the series 4, 6, 10, 14, etc.. The 3rd harmonic, likewise, generates 6, 9, 15, 21, etc..
The Harmonic Series, therefore, can be imagined as an acoustical fractal. To produce this fractal, we begin with the fundamental series of prime-identities generated by the tonic:
Then, each of these prime-identities generates a sub-series which is identical to that generated by the tonic.
Then, each harmonic within each sub-series generates its own identical sub-series, and so on, ad infinitum.
By this recursive process, we generate the entire complex network of harmonics. This network ends up resembling a tree-like structure, a branching hierarchy of identities exploding outwards from the seed of the tonic.
This Harmonic Tree is more than just a metaphor for describing the relationships between harmonics; it also functions as 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, the Harmonic Series consists of a combination of strong harmonics and weak harmonics.
Every harmonic within the series has an inherent degree of harmonic strength.
A tone consisting only of harmonics 3-8 sounds quite consonant:
While a tone consisting only of harmonics 23-28 sounds totally empty and dissonant:
From this experiment, we can infer that harmonics 3-8 are more integral to the structure of the Tone than harmonics 23-28; the latter group can be sacrificed whereas the former cannot. Harmonics 3-8, then, are inherently stronger than harmonics 23-28.
Naturally, the tonic is the strongest harmonic in the series. From there, the harmonics generally weaken in strength as their number approaches infinity. “Generally" is the operative word here. By no means does 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 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 isolate the prime-numbered harmonics, regarding them as their own prime-series, we not only notice that they weaken strictly from harmonic to harmonic, but that they do so 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. By the time we get to 11 and 13, we are at an extremely negligible degree of structural relevance. If we were to draw a graph mapping harmonic strength onto the primes, it would look like this:
Recall that the entire harmonic series is generated by the recursion of its initial series of primes. Each harmonic is the tonic of its own sub-series. Well, the harmonics of these "sub-series" weaken at an identical rate to those of the main series! 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 corresponds directly to the dilution of harmonic strength between two harmonics. 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. After adjusting in this way, 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 an actual 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, eventually, we reach 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.
