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.

Why they had to retune the whole Orchestra...

So imagine the entire Orchestra has been happily playing
Vivaldi in the key of A and now they are going to switch to
Mozart in the Key of C#. Well to switch to the Key of C, they
can't just go grabbing the middle notes of the A scales (see
above) like so:

Note

C^{#}

D

D^{#}

E

F

F^{#}

G

G^{#}

A

A^{#}

B

C

C^{#}

Hz

144

153

162

171

180

189

198

207

216

234

252

270

288

Why? Because the first half of the scale is increasing at +9 Hz
per note whereas the second half is increasing at +18Hz/note. So
the gaps are uneven, which sounds whacky - like notes are being
skipped or something. No: what they had to do was retune every
instrument to the new key of C#, whose correct tuning would be
each note increasing by ^{144}/_{12} = 12 Hz:

Note

C^{#}

D

D^{#}

E

F

F^{#}

G

G^{#}

A

A^{#}

B

C

C^{#}

Hz

144

156

168

180

192

204

216

228

240

252

264

276

288

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

How Equal Temperament Tuning is Calculated

Okay, the math is now high school level (not just simple
fractions like with Harmonic tuning), but the answer is: by
making the RATIOs of all of the intervals between all of the
successive notes the same.

He did some pretty snazzy math and realized that if each note's
frequency were = 2^(^{1}/_{12}) the previous
note's frequency, then every note would be equally
"bigger" than the previous note (geometrically
speaking) and then musicians wouldn't have to retune the whole
darn orchestra every time the composer switched keys. Nevermind
that different composers deliberately used different keys to
evoke specific emotional responses: we're after EFFICIENCY here!

Let me repeat that math: They changed the scale to make each
note's frequency = 2 to the ^{1}/_{12th} power
times the previous note's frequency.

2^(^{1}/_{12})^{th} is, by the way, not a
simple fraction: it's actually an irrational number (which means
it will never resolve to a fraction, and therefore never
be truly harmonic with any of the other notes in the scale
(except once per octave @ 2^^{(12/12)})).

Harmonic Sound Waves: Example #2

In the first example, the second wave was twice as long as the
first. But Harmony occurs whenever waves are simple fractional
multiples of eachother. So here's another example where wave A is
3/4 as long as wave B:

Sound Wave A

Sound Wave B

Tones A & B together

Notice the rhythmic Repetition between Tones A &
B... Again: the tones are harmonic

Harmonic Sound Waves: Example #3

Okay you adorable little freak! Here's a Harmonic wave generator for
you to play with... go have fun...

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:

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:

B = A#*2^{1/12} = 466.163...*2^{1/12} =
493.8833... Hz

C = B*2^{1/12} = 493.883...*2^{1/12} =
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 + 1/12^{th} the base frequency. So for example:

A = A + ^{0}/_{12}A

= 432 Hz

A^{#} = A + ^{1}/_{12}A

= 468 Hz

B = A + ^{2}/_{12}A

= 504 Hz

C = A + ^{3}/_{12}A

= 540 Hz

C^{#} = A + ^{4}/_{12}A

= 576 Hz

D = A + ^{5}/_{12}A

= 612 Hz

D^{#} = A + ^{6}/_{12}A

= 648 Hz

E = A + ^{7}/_{12}A

= 684 Hz

F = A + ^{8}/_{12}A

= 720 Hz

F^{#} = A + ^{9}/_{12}A

= 756 Hz

G = A + ^{10}/_{12}A

= 792 Hz

G^{#} = A + ^{11}/_{12}A

= 828 Hz

A = A + ^{12}/_{12}A

= 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:

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):

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.

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}/_{5}A

440 Hz

A^{#} = A + ^{1}/_{5}A

528 Hz

B = A + ^{2}/_{5}A

616 Hz

C = A + ^{3}/_{5}A

704 Hz

C^{#} = A + ^{4}/_{5}A

792 Hz

D = A + ^{5}/_{5}A

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."