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Why Can We Hear Harmony?

Posted by on Aug 7, 2013 in Background | 0 comments



My friend Scott is an expert river rafter. I went down the American River with him once. We had about five crew. It was a big raft, and well-behaved, so he decided he’d leave the driving to us, sit in the bottom of the boat, and go through one of the rapids with his eyes closed.

He was grinning afterwards. He said that when he closed his eyes the temperature instantly got ten degrees cooler, and everything became three-dimensional.

I’ve thought about his experience a lot.

I think that Scott’s built-in orientation processor, which was already in high gear due to being in the rapids, suddenly lost an input channel (vision). The senses he had to work with were touch and hearing. They jumped up to do the job, and the model of the environment changed character according to what those senses had to offer.

We are constantly building a model of our environment in our minds, as a way of orienting ourselves in it. It’s a vital survival skill, and it takes in data from all the senses. Each sense contributes something different to the model.

I’m posting now from a restaurant, and as I walked here, I paid attention to what each sense does.

Vision gives me a really detailed view of the front surface of everything in my field of view. I have to move my eyes and head to get a wider picture. I can’t see through most things.

Touch gives me the surface texture and slope of the sidewalk, the feel of the breeze on my skin (which is a clue to my speed), the warmth of the sun which must be coming from the West at this time of day.

Balance gives me the direction of gravity, which is really important to know, in San Francisco.

I stop and close my eyes, and listen.

I have a lot less information now. A hazy picture starts to form, of what is all around me — above, below, and 360 degrees around. My ears are like dragonfly eyes, seeing in all directions at once.

The longer I listen, the more detail appears. Leaves are rustling above my head. A tree. There is a bird in it. Two birds. A couple walks by, in conversation, and I can tell exactly where they are as they come into earshot behind and above me, pass me on the right a few feet away, and fade away down the hill, below and in front of me. I can tell that the woman on the left is taller than the one on the right.

Somebody is hammering a couple of blocks away to my left, and there is sawing further away. A scooter goes by on a street behind me.

Hearing gives me three-dimensional X-ray vision, and more. I can hear far away, and through objects. I can even feel textures at a distance — the rustling of the leaves is almost tactile.

My hearing is building the big picture, the large-scale, full surround, 3D framework of the environmental model.

I think this is exactly why the organs of balance are located in the ears. If the reference frame of my world model is created by my ears, what happens when I turn my head? The balance input must be fed to the processor right along with the hearing data, or the model will spin and I will have vertigo.

Somewhere in us, we have an audio processor that is capable of doing the following, in a few milliseconds:

  • Start with two complex pressure waves, one for each ear. Each wave consists of hundreds or thousands of separate sound frequencies, made by many different objects, all mixed together into one single waveform.
  • Reverse-engineer the waves, sorting them back into individual sound sources.
  • Deduce from each sound the nature of the source.
  • Build a three-dimensional model of the environment within hearing range, including objects, their locations, movements and physical properties.

I don’t know how much computing power this represents, but I’m impressed. It’s like turning a smoothie back into peaches and strawberries. A quantum computer might be good at it — it could interpret the data in every possible way, all at once, and let the most likely, or lowest energy, scenario pop out.

Something in us, whatever is in that black box, takes in that huge mess of frequencies, sorts them into related bundles, and figures out what is making each bundle, in real time. I suggest that this processor is very, very good at recognizing and analyzing harmonic series.

Vibrating objects don’t just produce one frequency. They produce clusters of waves, with many frequencies that are small, whole number multiples of each other — 2x, 3x, 4x, 5x — the harmonic series. If the input sound contains frequencies of, say, 100, 200, 300 and 400 cycles per second, those waves almost certainly came from the same object.

And then, the harmonic content of a sound contains a huge amount of information about the object that made it. The overtones, and the way they change with time, are what tell us whether we’re hearing a barking dog, or a friend’s voice, or rustling leaves. They can give us information about what the source is made of, how big it is, its surface texture and motion.

This amazing harmonic processor must be built in at a deep level. I imagine any creature that hears can do it to some extent, and many better than we do. Imagine the hearing-model of a bat!


I think harmony feels good for the same reason sugar tastes good. Sugar is vital to our survival. The brain mostly runs on it. We usually have to work hard to get it, extracting small amounts from food, metabolizing it from starches. Our bodies are tuned to seek out sweetness, to find pleasure in it.

We humans have figured out how to concentrate and purify sugar, and the straight stuff tastes mighty good. It’s a focused shot to the pleasure center.

Same with harmony. Normally, our orientation processor gets a diet with a lot of roughage. There are dozens or hundreds of unrelated sound sources to sort out. The environment is full of noise, frequencies that aren’t in nice harmonic relationship to each other. The processor has to work hard to identify what’s making the noise.

When we are presented with music, we are getting a straight shot of undiluted harmonic information — a nutrient that is vital to our survival, in an easier-to-assimilate, more concentrated form than is found in nature.

Harmony is ear candy.

Next: A Mirror Quad

More Mirror Twins

Posted by on Aug 7, 2013 in Consonance, Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes | 0 comments

Mirror twins are pairs of intervals, exactly opposite each other on the lattice.

The two intervals are reciprocals of each other, which means their ratios are flipped — if one is 5/3, the other is 3/5.

Harmonic distance is the same for each interval — the only difference is polarity. Listening to mirror twin pairs gives a good idea of what polarity sounds like.

The clearest example is the fifth/fourth pair, multiplying and dividing the tonic by 3.

Beautiful, consonant notes, one with overtonal energy, and the other with reciprocal energy.

The next closest pair is the major third / minor sixth. This has a different flavor. Now the tonic is multiplied and divided by 5.

The overtonal third feels stable and restful, though not quite as much so as the fifth. These notes are a bit farther from the center than the 5 and 4. The reciprocal sixth sounds more dissonant than the 4.

The next closest note to the center is the septimal flatted seventh, or harmonic seventh. The ratio of this note is 7/1, and its mirror twin is 1/7. I have not yet consciously used the mirror-seventh, and it’s not on my drawing of the lattice. The note is the septimal major second, at 231 cents, a dissonant interval indeed. The yellow lens shows where I would put it on the lattice.

Oy! That should put to rest the idea that just intonation is all about consonance! The septimal major second is nastier than anything equal temperament has to offer. I like the word “untempered” for this music because it better captures the wild and wooly nature of JI. “Just Intonation” sounds a bit stuffy to me, and the natural intervals of whole number ratios are anything but academic, they are burned in at a very basic level. Equal temperament is brilliant, but it’s actually the headier and less visceral of the two. IMO.

The next pair is a little further out — each note requires two moves on the lattice.

The ratios are 9/1 and 1/9. I still hear the 2 as stable, though it is less consonant than the previous notes. The b7- is suitably dissonant. It cranks up the tension in dominant-seventh-type chords, the workhorse tension-resolution chords of classical music.

I hear the effect of both tension and resolution diminishing somewhat, as tonal gravity gets weaker farther from the tonic.

These last two videos each contain a minor seventh. One is overtonal, the other reciprocal. The septimal flatted seventh, or harmonic seventh, is a stable, resolved note, the signature of barbershop harmony.

Septimal sevenths abound in this music, and they are sweet and consonant and stable.

The b7-, on the other hand, is dissonant and tense. It makes the ear want to change.

In equal temperament, these two notes are played exactly the same. ET weakens and obscures the difference, but it still can come through because of context.

The common “… and many more” tag, sung at the end of Happy Birthday, is a great example. That last note, “more,” is a harmonic seventh, 7/1, the stable, beautiful barbershop note at 969 cents. If you play “and many more” on a piano, the ear will hear the last note as a septimal seventh, only with less impact, because it is very sharp, at 1000 cents.

Next: Why Can We Hear Harmony?


Posted by on Aug 6, 2013 in Consonance, The Lattice, Tonal Gravity | 0 comments

The following video compares the perfect fifth with the perfect fourth.

These notes are the next-door neighbors of the tonic. They are equally close to the center. They are both harmonious. Yet there is a great difference in their character.

The difference between these two intervals is polarity.

I learned this term from W.A. Mathieu, in his amazing book “Harmonic Experience.”

Polarity is the main driver of tension and release in tonal music. I think it’s much more important than harmonic distance, the other component of consonance.

Here’s how I think polarity works:

Almost all sounds are actually a bundle of waves, of different frequencies. The frequencies in these bundles tend to vibrate at multiples of each other — 2x, 3x, 4x, 5x some base frequency — a harmonic series.

Our ears are highly attuned to such relationships. They help us figure out which frequencies belong together, so we can analyze them and identify the source. If we hear two frequencies in lockstep, three cycles to one, it is very likely they come from the same object.

I think this is why we can hear harmony. It’s a byproduct of our built-in orientation software. The overtone series of a sound is a powerful source of information about the object that made it. The harmonic content tells us whether it’s a barking dog or a friend or rustling leaves. Something in us sorts out and analyzes many harmonic series at once, in real time, identifying sound sources, locating them in space, and even sensing their texture at a distance. It’s a phenomenal processor.

So the processor recognizes the 3-wiggles-to-1 dance of the perfect fourth. But something is wrong — the 4 does not belong in the overtone series of the 1. The overtone series is generated by multiplication, not division.

This is a strange input for the mighty processor.

  • The 4 has to belong to the 1, because the two are in step with each other at three beats for one. In nature, that’s a dead giveaway.
  • But the 4 can’t belong to the 1, because the ratio is the wrong way around — natural sounds do not contain 1/3 in their overtone series.
  • Maybe the 1 actually belongs to the 4. The 1 is three times the 4, so if the 4 were the base, all would make sense.
  • But every other clue, all the harmonics in the drone (and, importantly, the listener’s memory), are pointing to the 1, hollering “This is the basic frequency!”

What’s a supercomputer to do?

I feel this sensation as unrest or instability, a need for something to change. Either the 4 needs to resolve to an overtonal note such as the 5 or major 3, or the root needs to change. Moving the root to the 4 will resolve the dissonance, introducing a new reciprocal tension — now the root “wants” to go to the 1, because the ear remembers. There are videos of this here.

On the lattice, there are two basic polarities, overtonal and reciprocal. Some notes combine both energies.

Overtonal notes are created by multiplying the base frequency. They appear in the natural harmonic content of sounds. They sound stable, restful, resolved — more so the closer they are to the center of the lattice, where tonal gravity is stronger. Pure overtonal notes are in the Northeast quadrant of the lattice.

Reciprocal notes are created by dividing the base frequency. The mind recognizes them as being related to the base frequency, but they do not appear in its natural overtone series. They sound unstable, restless, tense — and, like the overtonal ones, the effect is stronger closer to the center.

Polarity is our sense of the tonal gravity field. It is how we orient ourselves in harmonic space.

Next: More Mirror Twins

Mirror Twins

Posted by on Aug 3, 2013 in Consonance, Just Intonation, The Lattice, The Notes, Tonal Gravity | 0 comments

For every note on the lattice (except the 1), there is another note, the same distance away from the center and exactly opposite it. The harmonic moves for the two notes are the same, but the directions are opposite.

Mirror twins are reciprocals of each other. Flipping a note’s ratio upside down will produce its twin.

Listening to these mirror twins helps demonstrate polarity.

The simplest ratios on the lattice, the ones with the smallest numbers in them, are 3:1 and 1:3, the perfect fifth and perfect fourth. Here they are:

Tonal gravity is strong here close to the sun. The fifth sounds remarkably resolved, like it’s part of the drone. In fact, this note shows up so strongly in the natural overtone series that it is part of the drone. If you listen carefully to that tonic drone by itself, you can hear it. The following video shows a 5 by itself, followed by the straight drone on the 1. I hear the fifth appear again, quietly, after the drone has a few seconds to settle in and bloom.

The fifth says, “You’re home, relax.”

The fourth also shows where home is, but in a very different way. Instead of saying, in effect, “Home is here, come on,” It is saying “Home is over there, now go.”

Our built-in audio processor is always looking for mathematical relationships between notes so we can tell which frequencies belong together, identify different sound sources and orient ourselves in our surroundings. I think this is why we hear harmony and why it sounds like a journey — it’s a part of our built-in orientation software.

Three cycles for every one is a “right” ratio. It strongly says, “These frequencies belong together, they are being made by the same thing.” But the ratio is upside down, 1/3, the exact opposite of the “right” sound, 3/1. It’s the shadow version of the 5, yin to its yang.

The 5 feels like this:

200px-Stable_equilibrium.svg copy

The 4 feels like this:


The beautiful stability of the 5, contrasted with the equally beautiful instability of the 4, is what I mean by polarity.

Next: Polarity

Harmonic Distance

Posted by on Jul 30, 2013 in Consonance, The Lattice, The Notes, Tonal Gravity | 2 comments

Harmonic distance is the total length of the connection between two notes on the lattice, as measured on the solid lines. The more tinkertoy sticks you traverse to get from one note to the other, the greater the harmonic distance.

It’s not the same thing as melodic distance, which is a difference in pitch. Two notes can be far apart in harmonic space, but close together in melodic space, or vice versa. This post has a demonstration.

Each solid line on the lattice is a prime factor — 3, 5 or 7. A simple way to put a number on harmonic distance is to multiply together all the prime factors used in the ratio of the interval. Doesn’t matter if you’re multiplying or dividing by the factor, the distance is the same. Twos don’t count; these are octaves and they don’t add distance on the lattice.

The closest intervals on the lattice are the perfect fifth and perfect fourth. To get these intervals, you multiply or divide the original note by 3. The ratio of the fifth is 3/1, and the ratio of the fourth is 1/3. The harmonic distance is 3, in both cases.

The major seventh, or 7, is a more distant interval. Its formula is x3, x5, or 15/1, so its harmonic distance is 15.

The b2- is the reciprocal of the 7. Its formula is ÷3, ÷5, or 1/15, and it is equally distant. The polarity is opposite, but it’s the same distance away from the center.

There are two other notes at this same distance of 15 — the 6 and the b3. Their ratios are 5/3 and 3/5 respectively. They are reciprocals of each other, and have opposite polarities.

Here is the inner lattice, showing the ratios (without any factors of 2), and harmonic distances instead of the note names. The ratio of an interval defines it completely; it would make perfect sense to name the notes by their ratios alone (it’s been done).Harmonic Distance central lattice


In the consonance experiment from a few posts ago, I played intervals in order of harmonic distance, and sure enough, as they got further out, they got more dissonant. I used the Pythagorean axis (multiples of 3) to keep it simple. Pythagorean tuning is somewhat limited musically; harmonic distance increases so fast that there are very few consonant notes.

On the lattice of thirds and fifths, there are more consonant notes to play with. How would that same experiment sound, when you add in these new intervals?

I’ll stick with the overtonal, Northeast quadrant of the lattice. Every ratio involves multiplication only, so there is no reciprocal energy, and I’m not comparing apples to oranges. My intention is to test only one ingredient of consonance, the harmonic distance. The intervals travel away from the center, and back again. Listen and watch a couple of times, and hear what happens.

I think the pattern holds very nicely. At the very end, the #4+ with its distance of 45, I think the dissonance has lost some of its obnoxiousness. It does appear that as the distance gets big enough, both consonance and dissonance start to weaken. The ear has less to go on, the signal is weaker.

Also note how the other component of consonance, stability/instability, changes as we roam farther out and come home again. All these intervals are stable, since they are all overtonal. This sense of stability gets stronger the closer we are to home, as though the ear is receiving a stronger signal and is more and more sure of itself. I start to clearly hear the stability at the major seventh (15/1), and it quickly gets stronger from there on in.

Next: Mirror Twins

Polarity Experiment

Posted by on Jul 29, 2013 in Consonance, Just Intonation, The Lattice, The Notes, Tonal Gravity | 0 comments

In the last post I did a consonance experiment, listening to intervals with wider and wider spacing.

In that experiment, I kept the axis (3) and direction (multiplication, overtonal) the same, and increased the distance.

This time I’ll keep the axis and the distance the same, and switch direction. Each illustration will compare a note with its mirror twin, its reciprocal.

First up is the strongest polarity flip there is, the perfect fourth and fifth. One divides the tonic by 3, the other multiplies it by 3.

The 4 is clearly unstable, it wants to move. The 5 is clearly stable. If a song ends with this interval, I will feel completely satisfied.

The next matchup is the b7- and the 2. The b7- is the crucial note that provides the tension in dominant-type seventh chords and makes their resolution so satisfying. Here it is in undiluted form.

The 2 is fairly stable. Quite a few songs end on this note, and there is a pretty good sense of resolution, maybe with some wistfulness mixed in.

The two notes are about equally harmonious, and of opposite polarity. This is the same pattern as the 4 and 5, only weaker.

Moving outward, we get the b3- and 6+ pair:

The pattern continues — now both notes are rather dissonant, with the b3- weakly unstable and the 6+ weakly stable. It would be rather unsettling to end a song on the 6+, but maybe you could get away with it.

Here are the next two:

These are interesting. They are dissonant, all right, and the b6- is unstable and the 3+ is stable. But I actually hear the polarity a little more strongly than the last pair.

I think my ear is trying to interpret these notes as out-of-tune versions of the b6 (a strongly unstable note) and the 3 (strongly stable).

How is my ear to interpret this 3+ note, the Pythagorean major third? Can I even hear a ratio of 81/64? Maybe not well enough to really recognize it.

Perhaps the ear “decides” that it’s simpler to read this strange note as a badly tuned version of a simpler interval, one I am familiar with. So I hear it as an out-of-tune 5/4 instead of an in-tune 81/64.

This is why equal temperament works, as Mathieu demonstrates so well in Harmonic Experience. A painting doesn’t have to be exactly straight on the wall for the eye to interpret it as straight. Thank goodness! In the same way, a note doesn’t have to be exactly in tune to be heard as that note. The ear is willing to accept “close enough” and hear it as the real thing, though the consonance will not be as good.

Maybe the part of the mind that processes this stuff is like a quantum computer, taking in the sound, trying out all possibilities at once, and spitting out the “most likely” interpretation, which would be the solution with the lowest “potential energy,” the one that is closest to the center, just like real gravity.

We’re probably too far out now to really recognize these intervals as what they are, but for the heck of it:

Suitably nasty, and now the sense of polarity is pretty much gone, I can’t hear it.


The Pythagorean spine, the sequence of fifths, has come full circle — almost. The two notes are 24 cents apart, a Pythagorean Comma. All that remains of tonal harmony at this distance is a generic sort of dissonance. I hear no polarity at all. The tonal gravity field is too weak to detect.

Here’s one more video to bring it all back home. I start to smell the stables at about the b3-/6+, and the sense of direction gets rapidly stronger from there.

Next: Harmonic Distance

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