What is Harmonic Tuning?

Harmonic tuning is when you tune your musical instrument to notes that are Harmonically Resonant with each other. Resonance is when sounds (which are basically waves of energy traveling through the air) have wave-lengths that reinforce each other at regular intervals to create pleasing overtones.

How is that different from regular Tuning?

Ever since the Industrial Revolution (when the world got really excited about efficiency), Western Orchestras have been using "Equal Temperament" tuning, which - oddly enough - is intrinsically DISHARMONIC.

We have become so used to it that most people don't even notice it at this point. After all, most of us grew up with it and have never known anything different.

However, because it is essentially disharmonic, the notes do not reinforce each other, which means that much of the power that music has to vibrate physical objects resonantly (and thereby physically change/alter/organize them) is lacking from Western-tuned instruments. There's still power to music, but it's mostly the power of emotion and lyrics; The physical power of true harmony is dissipated by the chaos intrinsic to the disharmony of the notes that make up the current Western scale.

We have become so used to it that most people don't even notice it at this point. After all, most of us grew up with it and have never known anything different.

Explain Harmony again...

Okay, from a Math & Physics perspective, Harmony is when sound waves reinforce eachother at regular intervals, like this:

Tone 1
Tone 2
Tones 1 and 2 together

Notice the rhythmic Repetition between Tones 1 & 2... This is the definition of Harmony.

So that's pretty simple, right?

So what are Disharmonic notes?

Pretty simple: non-harmonic notes are notes where the wavelengths do not line up at regular intervals, so no pattern of reinforcement (aka pleasing overtones) ever emerges between the notes (well, not within a reasonable amount of time). It turns out that human ears are actually pretty picky about that: if the sound waves do not harmonize with each other within about 13-17 wavelengths, average human ears will not register the sound as being harmonic. In fact that's one of the reasons scales typically have 12 notes (there are some cultural variations on that, but more on that later...)

I thought you said Western music is Disharmonic

Yeah, that's right, it currently is.

It didn't used to be though: Until the late 1800's, Western notes were harmoniously tuned & many cultures worldwide still use harmonic tuning. So every note in an octave was 1/12(A) higher than the previous one where A was the root tone of the octave. So for example, if A was 108Hz, and there were 12 notes in the scale, each note in the Octave would be 108/12 = 9Hz higher than the previous one. Thus, a typical octave would look like this:

Note A A# B C C# D D# E F F# G G# A
Hz 108 117 126 135 144 153 162 171 180 189 198 207 216

See? Each note was 9Hz higher than the previous one (in that octave). Harmonious and easy right? Well, the only problem is that the same rules apply to the next octave: The new A is 216 Hz, so each note in its octave is 216/12 = 18 Hz higher than the previous note. so the new octave looks like this:

Note A A# B C C# D D# E F F# G G# A
Hz 216 234 252 270 288 306 324 342 360 378 396 414 432

This is still all very well and good except for one little problem: In the 1800s, music was the biggest, most powerful form of mass entertainment (remember, television hadn't been invented yet) and there were all these fantastic composers writing epic music in different keys (to evoke different moods). There were also a lot of big-production opera houses performing these pieces. The problem came when you had to switch keys. If you wanted to switch from one key to another, you had to retune the entire orchestra.

At some point some misguided genius said "Hey! This is a real pain in the butt. Let's just 'fix' it

Before you think I'm making this up, here's the table of frequencies that result from his math, using 440Hz as the base frequency:

Note A A# B C C# D
Hz 440 466.163... 493.883... 523.251... 554.365... 587.329...
Note D# E F F# G G# A
Hz 622.253... 659.255... 698.456... 739.988... 783.990... 830.609... 880

If you compare this to an actual table of current Western tuning, you will unfortunately find that this is really how modern day "equal temperament" tuning is accomplished, which is the standard tuning for all Western instruments today.

Harmonic, Shmarmonic, Why Should I Care?

We should care because those notes are intrinsically disharmonic to each other!

Harmony is determined by wavelengths being simple ratios of each other. These numbers are not simple fractional ratios of each other. In fact, since they are based on 2^(1/12), which is an irrational number, the wavelengths will never realign precisely / harmonize with each other. This is SHOCKING! And it means that all of modern Western Tuning is fundamentally disharmonic.

Ready for more Math?

When European orchestras changed the scale from a harmonic scale to the current Western scale, they did so by retuning notes so that every note was 2^(1/12th) times the frequency of the prior note on the scale. Feel free to check this:

  • A = 440 Hz
  • A# = A*21/12 = 440*21/12 = 466.1637615... Hz
  • B = A#*21/12 = 466.163...*21/12 = 493.8833... Hz
  • C = B*21/12 = 493.883...*21/12 = 523.25113... Hz
  • etc...

Unfortunately, that means that NONE of the notes in the Western scale are truly harmonic with each other (except for one octave intervals, where the note is harmonious with itself on the new octave). This is really very unfortunate & musicians wind up learning all kinds of tricks & rules (thirds, fifths, etc) to find notes that sound close to (but are not actually) harmonious with each other.

By comparison, this is how notes were originally determined:

There was a Base note (For example A = 432Hz)... hold on! We'll get to why it's not 440 Hz in a minute... & every note was equal to that note + 1/12th the base frequency. So for example:

A = A + 0/12A = 432 Hz
A# = A + 1/12A = 468 Hz
B = A + 2/12A = 504 Hz
C = A + 3/12A = 540 Hz
C# = A + 4/12A = 576 Hz
D = A + 5/12A = 612 Hz
D# = A + 6/12A = 648 Hz
E = A + 7/12A = 684 Hz
F = A + 8/12A = 720 Hz
F# = A + 9/12A = 756 Hz
G = A + 10/12A = 792 Hz
G# = A + 11/12A = 828 Hz
A = A + 12/12A = 864 Hz

But why 12 notes? Why not 13 or 5 or 17? Well it turns out in fact that there are at least two very good reasons to have 12 notes in a scale:

  1. 12 has lots of prime factors, so any notes that involve those prime factors will harmonize within that many wavelengths and sound good together. For example: 0, 3, 6, 9 & 12 (or any subset thereof) should all sound good together because their waves will match up harmonically in < 4 cycles (wavelengths):

  2. Similarly, 0, 4, 8 & 12 (or any subset thereof) will sound good together, harmonizing within 3 wavelengths. 0, 2, 3, 6 & 12 will also sound good together, although a slight dynamic tension will exist between 2 & 3 who will require a full 12 cycles to match up. You could probably create a pretty amazing instrument just using the 0, 2, 3, 4, 5, 6, 8, 9, 10, 12 tones from this scale, which would create the following potential harmonics: {0, 2, 4, 6, 8, 10, 12}, {0, 3, 6, 9, 12}, {0, 4, 8, 12}, {0, 5, 10}. For added tension potentials add #1, 7, & 11 back in: they'll still all harmonize with each other within 12 cycles.
  3. Why 12? Oh yeah! Well as it turns out, most average human ears only pick up as harmonic tones that synchronize within 13 cycles of eachother. So of all the numbers < 13, 12 is the one with the most prime factors and hence the most potential harmonic arrangements within a single scale. But you could just as easily do scales with 5, 6, 7 or 24 notes (& historically many cultures have): Just adjust the Denominator in your multiplier to get the right frequencies. For example, for a 5 note scale:
    A = A + 0/5A 440 Hz
    A# = A + 1/5A 528 Hz
    B = A + 2/5A 616 Hz
    C = A + 3/5A 704 Hz
    C# = A + 4/5A 792 Hz
    D = A + 5/5A 880 Hz
    All of these frequencies harmonize (realign) with eachother within 5 wavelengths of the dominant tone, (intrinsically: because of how they were constructed) and so will all sound good together...

Okay, so wait a second: I just switched back to 440 Hz. Why? Well, because Harmonics is all about waves overlapping within a reasonable number of wavelengths, which is (as we have seen above) all about Prime Factors. And now we get to talk about the fundamental problem with using 440 Hz as the Base Tone upon which your whole scale is constructed(!)

440 = 11*4*10 = 11*2*2*2*5

Okay, so right off the bat, you can see that the prime factors are not harmonic. In fact, this is probably one of the least harmonic numbers that the ear & mind could withstand before screaming "Foul!" and rejecting it right off the bat as unpleasant. There is something positively eerie about it, so would it surprise you to learn that it was established by Hitler's Minister of Propaganda? (Now there was a man with a knack for manipulating the populace to do all sorts of crazy sh*t)

There is a much, much better way, which is to return to the worldwide historical base tuning of 432 Hz. First of all, check out the prime factorization:

432 = 2 * 2 * 2 * 2 * 3 * 3 * 3

Wow! How many Harmonic Subsets can be derived from those prime factors?

1 3 9 27
2 6 18 54
4 12 36 108
8 24 72 216
16 48 144 432

That's a pretty rich field! And probably explains why our ancestors intuitively (through the power of their hearts and ears versus the power of electronic devices and deceived brains) tuned all sorts of instruments worldwide ranging from Native American Flutes to Tibetan singing Bowls to this base frequency, which we now measure as 432 Hz.

And Hz is not an arbitrary scale... It stands for Cycles per Second, where there are (by definition of a second), 60 seconds in a minute, 60 minutes in an hour, & 24 hours in a day, so Hz are intrinsically calibrated and related to earth time within the solar system.

The prime factorization of 60*60*24 is also harmonically resonant with 432, so there is a validity to this number within the context of human time on this planet.

I could get into all sorts of theory about Base Tones here, including using it to resonate with systems in the body, energy centers, Kung, or other planets in the solar system. For that, I recommend a book called The Cosmic Octave.

But meanwhile, I have a gift for you to play with in your personal quest for harmonic resonance in a disharmonious society. This is a Harmonic Calculator: simply input the frequency that you want to use as your base tone, how many notes you want to define in an "octave" (perhaps better called an Interval), and it will spit out the exact frequencies in Hz that you want to tune your instruments to.

I am also going to post a SongTree style blog where you can upload your musical results. Please be sure to tag them with your Base Tone & notes per interval so we can propagate beauty as it is discovered. As Elon Musk said when he publicized all of Tesla's engineering plans: "If we're all in a [sinking] ship together... and we have a great design for a bucket, then... we should probably... share the bucket design."

Have fun!