"pls dont making fun of me"

Intro to the Harmonic Series

Numbers are first explained to children through the real-world application of counting. First, you take a simple object to be your one; your unit through which all the numbers may be represented. Anything could be the one; usually it’s an apple or a car or something. Then you multiply this thing and end up with two of your one, and then three of your one. After that you end up with four of your one, which is also just two of your twos and so on into infinity.

Through this practical game of counting apples or whatever, we learn to communicate with the divine world of NUMBERS. As we grow up, this world becomes rich and colorful in the gardens of our minds. We learn all its nuances and rules, how the numbers interact with each other, how they move around and change shapes. Every number becomes a personality with its own special properties. Out of these numbers, we build the beautiful architecture of math. School teaches us how to wield the numbers around effortlessly, intuitively projecting them onto the physical world. Every act of measurement, every equation solved, every real-world application of math is a prayer to the world numbers. The fruits of this abstract world are first tasted through the simple, childish act of counting.

I’m telling you this because harmony, at its core, is a numbers-game. Understanding harmony means learning HOW TO COUNT.

Just like with regular counting, we have to pick our 1. Our 1 can be any tone. There isn’t some special healing frequency that’s gonna make our 1 any better. That any tone can be our 1 is a reminder that the divine can exist anywhere; that every tone can act as an avatar for numbers. Let’s arbitrarily choose a tone at 100 Hz to represent our 1.

Musical "frequencies" are fundamentally just measurements of movement over time, so the only way to "count upwards" is to multiply the amount of movements per second. 100 Hz, or 100 cycles per second, is the unit for counting, so 100 MORE cycles per second will be 2; that is, 200 Hz. 3 will, of course, be 300 Hz. Counting upwards from 1 in this way produces the series of tones, (100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, ...). Here’s the crazy part: regardless of which tone we use as our 1, the infinite series of tones produced by counting upwards will sound the same, harmonically. That is, the musical intervals produced between each number are the same regardless of our starting pitch. Observe:

Counting from 100 Hz

Counting from 125 Hz

Counting from 150 Hz

When counting apples, we map numbers onto the real world, based on the unit of measurement we choose from the beginning: one apple. The moment we decide what “1 apple” is, we intuitively understand the idea of “392 apples.” Harmony is the same. No frequency has an identity until we contextualize it. By assigning a tone as our unit, the 1, we choose a specific array of tones to embody the infinite series of numbers. These tones are given numero-harmonic identities which fall in line with our 1.

So, the divine numbers map onto an unchanging series of musical intervals produced by counting. This series, my friends, is called THE HARMONIC SERIES. It is the divine tree, from which all consonance stems, and towards which dissonance forms its negative. It is the world of numbers given sound.

To understand the Harmonic Series and the personalities and idiosyncrasies of each number is to learn the rules of harmony. Everyone's relationship with "the world of numbers" should be pretty personal, but let me just kind of tell you how I conceptualize numbers. The following is, from bottom to top, my relationship with the Harmonic Series:

ONE and consonance

One is the unification of all things; the common principle. The 1 is the triumph of existence and the divine inherent in everything.

People talk about “oneness” a lot when they eat drugs or when they become buddhist. When they mention “oneness” and “being one with everything,” they’re really just trying to unify existence in their minds. The inevitable reaction to eating a shitload of mushrooms and seeing fractals in everything is attempting to resolve the entire universe into one unifying concept. Everything needs to exist in relation to THE SAME THING. The convenient thing about harmony is that the unifying concept is not just some vague abstraction; it's an audible tone!

In the harmonic series, 1 is the fundamental; it is the “tonal center” or TONIC. All tones are heard relative to the tonic. Consonance is simply a measure of a tone’s harmonic proximity to the 1. When we say a tone is “consonant,” what we mean is, “this tone is divine by its close relationship to the tonic, which is the 1.” There is no “pure consonance”, therefore, other than the tonic. Every other tone is somewhat consonant because every tone can be measured in relationship to the tonic, but only the tonic itself embodies the pureness of the 1. The tonic is stability, purity, fullness. It is existence given sound.

A long time ago, this French monk named Marin Mersenne was listening to a trumpet, and he realized that every single tone produced by the trumpet activates a series of quieter “overtones” above it. These overtones take the shape of the harmonic series! So every tone that is played will, through resonation, activate the tones in its harmonic series. This is kinda how the 1 contains every other number. Whenever you play the 1, nature will shout back the infinite span of numbers. Remember that the number “two” represents two “ones”s; the “three” represents three “ones”s; and so forth for every number. Every number is ONE of itself. So, the pot-smoking buddhist hippies have a point: oneness is all-encompassing. The 1 is simultaneously itself and the DNA for infinity.

ZERO and dissonance

If “One” is the name given to consonance, then we should also give dissonance a name. And dissonance, I name ZERO; the pure emptiness.

Although there is no tone in the harmonic series representing Zero, we must acknowledge the zero as it is heard: silence. “Zero” is the name we give to nonexistence, emptiness, non-value. Just as the tonic is fullness, silence is the emptiness which flows in between the tones of music; it is nonexistence given sound.

Silence can never be measured in proximity to the tonic; the latter and former are infinitely far from each other. Silence is the only “sound” devoid of oneness; therefore, it is the only pure dissonance.

It might be unintuitive to think of silence as dissonant at all, because it seems so unintrusive. But if we meditate on the logical extreme of dissonance, we will find our answer to be silence. When two tones are consonant, they hug each other. By virtue of their resonance, consonant tones blur into each other, achieving a warm fullness. Consonance subsumes multiple tones into oneness. This phenomenon reaches its extreme in the tonic, where the One is absolute and can only relate to itself.

In contrast, dissonant tones repel each other. When two tones are dissonant, they audibly distinguish themselves from each other, creating distance in harmonic space. So, instead of achieving a full richness, dissonant tones sound empty, like hollowed-out vessels. Seriously. Do you think I’m fucking with you? Go play the most dissonant interval you can think of right now. It sounds empty; doesn’t it? The shadow we hear in extreme dissonance is the void sitting between its tones. So, where consonance is fullness and substance, dissonance is emptiness and indifference. Dissonance is nonexistence flowing in between the gaps of being.

Dissonance reaches its extreme when existence completely cancels out: in silence. Silence is emptiness with no border.

As an experiment, let’s try to create the “purest dissonance,” by theory alone. Imagine we start with the tonic; a pure singular consonance. Adding any separate tone to this will create more dissonance, regardless of what the added tone is. The best way, therefore, to gradually approach dissonance is to add separate tones at random and see where we are approaching as our number of tones approaches infinity. This is really just a proof by induction. So we add random tones and, no matter which tones they are, our chord will only get more dissonant. As the octaves fill out, the sound becomes harsher and we begin to drown in a sea of indifference. The sound becomes more chaotic as the distance between notes becomes smaller and smaller. Eventually, something incredible happens: the sound stops being harsh and it becomes noise! When the gaps between frequencies become microscopic, we reach noise: a sea of frequencies, beneath which our tonic drowns. In noise, tones cease to exist; harmony is lost. In approaching an infinite set of tones, we have paradoxically destroyed tonality. Noise is one step away from silence. Silence, like the number zero, only exists by virtue of the imagination. We only hear “silence” when sound becomes imperceptible. Noise, by destroying any perceptible tonality, brings us closer to silence. Indeed, what we call “silence” is often unacknowledged background noise. Our gradual move towards dissonance has brought us closer to silence.


The Gnostic-Christians had this idea that the physical world was created when some god, the Demiurge, looked at his reflection in the water and realized “ohhhhhh you can have TWO of something!” Then he imagined up a world of twos: up and down, earth and sky, light and dark, self and other. He took up a sword and cut everything into separate halves. He created the universe by making things DISTINCT, cancelling out the unifying power of the one. A huge “fuck you” to the one-obsessed pot-smoking hippies.

So this is the power of “having 2 of something.” The 2 creates copies. It functions in splitting and reproducing. We find the 2 in the one-celled creature which created copies of itself indefinitely, in twins, in dichotomies. “Duality” is everywhere in the universe. You have a left-brain and a right-brain because duality was irreversibly built into you. but wat does this mean for music??

Harmonically, the 2 creates this distinct tone:

So, by the 2, we are given a new dimension: pitch! The 2 draws its sword and, for the first time, separates high and low. We have tones that are separate from the 1 now. Earth has been violently torn away from heaven. This is the creation of the harmonic universe; the first departure from oneness.

Because the 2 is a reflection of the 1, it’s still REALLY consonant. The 1 and 2 are so consonant with each other that, sometimes, it’s difficult to tell them apart. That’s because the 1 and 2 have the exact same tone-quality, which just means that if our tonic is C, the 2 of the series will be a higher C. If we multiply any frequency by 2, we produce that same tone one octave up. If we continue to multiply C by 2, we will only ever produce a range of Cs. If 2 is a reflection of the tonic, then the powers of 2 (4, 8, 16, etc.) are reflections of reflections of reflections of the tonic:

The 2 can only create copies for infinity! So although the 2 offers a beautiful gift, varying pitch, we are still trapped. It is easy to get trapped in the octave-world of duality. If our only tool is splitting things in half, then we will only create copies of copies of copies forever. This is why the Gnostics actually hated their creator-god, by the way. The Gnostics had a lot of anxiety about “being trapped.” They found this fleshy reality of infinitely reproducing, infinitely self-eating organisms to be gross and evil. They wanted to GO BACK. “Take us back to the 1!” they shouted. “It was nice there!” Unfortunately, there is no going back to the 1. This train has no brakes. But, good news! There is a way forward and through duality! We simply need to count higher.


There’s this religion called “Catholicism,” and a central idea in this religion is that their One God is also THREE GODS. I don’t really know what that means, but these “Catholics” really liked the number 3. Whereas the Jews loved weird numbers like 7 and 12, the Catholics only really cared for 3. They were obsessed with this number, which is important for musicians, because the Catholics controlled Western music theory for the longest time. Most of recorded musical history is the Catholic churches trying to decide which musical scales were usable, and they figured the safest way to do that was to exclusively use their God’s favorite number. So the Catholics built their scales by multiplying 3 over and over.

One reason this is a good thing is that the 3, for the first time, produces different tone qualities. By that, I mean if our tonic is C, the 3 will produce a G!

We have transcended the world of octave-pitch and added a dimension of NUANCE to harmony!! This is a really beautiful thing! Whereas the 2 could only create reflections of the tonic, the 3 creates COLORS. But wait. There’s a down-side to this: different tone qualities means we are now digging farther into dissonance. With nuance comes individuality and freedom but also instability and chaos.

All the same, the Catholics couldn’t get enough of this harmonic. Hundreds of years ago, the Catholics were creating scales just by multiplying by 3 and dividing by 2 (dividing by two brings the tone down an octave, so the distance between the former and latter tones decreases, but the tone-quality still changes). If you do this five times, you get a scale like this:

All of these tones sound completely different but still contain an aura of 3-ness. It sounds really nice. But, as I said, the 3 changes tone quality; so, as the connection between these tones and the tonic becomes more indirect, the tones rapidly become less consonant. 3 is pretty consonant, but if you continue to stack 3s for too long, they lose their purity. So, a 3 of a 3 of a 3 of a 3 of a 3 of a 3 of a 3 (roughly a C#) is so many layers removed from the 1, that it's actually gonna sound pretty harsh:

Now you might be thinking "OW how the fuck did THAT sound come out of a CHURCH????" Well, Catholics are experts at hiding their perverted desires in plain view. In art, they did this through beautiful paintings of half-nude Jesus being BDSM’d to death. In music, they did this by taking advantage of a loophole in harmony. They were allowed to push the 3 to such an extreme because it’s technically God’s favorite number. “It’s such a pure harmonic, right?” said the church. “It’s so innocent, there’s no way it could be perverted, right?” So they begrudgingly gave their musicians the green-light to compose the most ear-grating, fucked-up dissonances because “muh 3rd harmonic.” This is how the Catholics hid their gravest sins behind a wall of divinity and how they got away with abusing the shit out of the 3. This abuse culminated in the invention of a scale that stacks 3/2 TWELVE TIMES. This way, the last note ALMOST sounds like if you stacked the 2 seven times.

You ALMOST get an octave-equivalent to C on that last note there. It's only a little sharp. So the Catholics, proud of having almost resolved 2 and 3 with each other, took this 12-tone scale and made it their ONLY SCALE FOR HUNDREDS OF YEARS. This kind of shit sends shockwaves through musical history. The Catholics were so influential that even today most people only work within this scale. The Catholic-scale is so thoroughly integrated into the Western canon that anytime something escapes from this scale, we call it "microtonality." I know I'm making the this scale sound like a prison right now, but I think its popularity is actually a good thing! The Catholic-scale fucking rocks! The "microtonal" people bitch and moan about it all the time because they hate being marginalized or whatever, but there is incredible poetry to be derived from the 12 tones and how they relate. Anyways, that's a discussion for later.

FOUR and other non-primes

The thing about 4 is it’s really just an extension of the 2, as are all the powers of 2. The 16 is just the 2 of the 2 of the 2 of the 2, so you will only ever end up extending in octaves. That's why 16, despite being way farther along in the series, is more consonant than 3: because it is an extension of the extremely consonant tone, 2.

The 9 (3x3) and 27 (3x3x3) are just extensions of the 3, but because the 3 changes note-qualities, the extensions of 3 progressively get more dissonant.

What about a number like 6? It’s just 3x2. Here, we see the reflective power of the 2 going to work at the 3. if our 1 is C, the 3 is G and the 6 is a G one octave higher. Consider the 6 to be a synthesis of two prime numbers and, therefore, a synthesis of the harmonics they represent. The same goes for 15, which is a synthesis of 3 and 5.

The primes are the only numbers which truly have their own solidified identities. Any other number is just an extension of one or many primes. The primes are each powerful heads of their infinitely-extending non-prime families. This is why mathematicians are so obsessed with the primes and why WE as MUSICIANS should be so obsessed with them. Each prime member of the harmonic series produces a completely unique tone-quality.


The 5 is a seriously beautiful tone! It makes this sound:

Pretty cool, huh? It's only slightly less stable than the 3 and it feels like a warm hug.

So here's where we get to a really common trick in Western Music: if you put the 1, 3, and 5 together, you get what people like to call the "Triad." If you look around, triads are everywhere. Modern musicians will rarely write a 1 and 3 without adding in the 5. I think this is because the 5 really affirms our tonic. Like, a C and G played in the upper-range of the piano is probably the 4 and 6 of C, but it COULD ALSO BE the 6 and 9 of F! But if we add an E in there(the 5 of C), it's much more clear that our tonic is C. All doubt has been erased.


For most of music history, people thought of this harmonic as necessarily dissonant. Although we are getting farther down the consonance-hierarchy, the 7 can still be pretty stable when you give it context:

Oooh yeah. To me, the 7 sounds really sexy. That's probably why the Catholics treated it as a dissonance for so long. The Catholics hate sex! They had no room in their music for coomer-harmonics. But these French guys, Satie and Debussy, were super ironic and LOVED cooming because they were French. So they were using the 7 all over the place while the tight-ass Catholic musicians cried and pissed their pants. Then JAZZ happened and the coom ramped up to an inferno... a coom inferno... I'm not even gonna talk about jazz yet holy shit this is getting long.


Okay. NOW we're going pretty deep down the rabbit hole.

The eleven is very rarely used. EXTREMELY hard to find in the wild. The connection to the tonic is getting pretty tenuous at this point. Do you notice that as the list goes on, I have less and less to talk about? That's by design. As we count upwards, the primes have less and less prominence in the universe. Primes which are more inherently dissonant are also more difficult to talk about.

What's important about the 11, though, is it creates an interval called the "TRITONE." If I started talking about the tritone now I would probably go on infinite segues trying to explain it, so I'll just save the rant for a later time.


Here's 13:

okay cool.

Tonal Hierarchy

Okay I think that’s enough for now. Obviously there are infinite primes and infinite solidified harmonic identities, but you don’t need me to explain them to you. The exploration of the harmonic series is a life-long journey. A rich understanding of the Harmonic Series will only be achieved by a deeply personal connection. Dwell upon the fundamental, inner harmonics. Experiment with the obscure, outer ones. In meditating upon the series, you will gain an intuitive understanding of harmony at its most universal.

The Harmonic Series is a tonal hierarchy. At its core is the tonic, the prime consonance. And from this unifying consonance, the series branches outwards into infinity, brushing against the far expanses of dissonance.

The Harmonic Series is the divine structure from which all harmony derives.