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

Polarity

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

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

Mirror Twins

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:

200px-Unstable_equilibrium

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

Next: Polarity

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

Polarity Experiment

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.

Finally:

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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Posted by on Jul 27, 2013 in Consonance, Just Intonation, The Lattice, The Notes, Tonal Gravity | 0 comments

Consonance Experiment

In my last post I raised the idea that the consonance of an interval is not just one thing, but has two distinct parts:

  1. The sensation created by the sound of the two notes played together — smooth or rough, pleasant or unpleasant
  2. The sense of stability of the note — does it feel restful, like it has arrived, or restless, unstable, like it “wants” to move?

I propose that #1 is correlated with the harmonic distance between the two notes of the interval, that is, the length of the solid lines that connect them on the lattice.

I think #2 is determined by the polarity of those connections, whether they consist of multiplying (stable) or dividing (unstable), or a combination of the two.

This suggests some experiments.

A mostly good rule in experiments is to change one thing at a time. The prime factors used in an interval’s ratio (3, 5 or 7) strongly affect the sound, so I’ll stick to one axis (x3), and only go in one direction (multiplying, overtonal, East).

This will all happen on the backbone of the lattice, the infinite horizontal line of fifths. The notes on this line are in Pythagorean tuning, which is any tuning that uses only 3 and 2 in its ratios. Pythagorean tuning is used in music, but many of the intervals are sharply dissonant, as we shall see.

If I’m right, according to #1 the notes should get rougher or more unpleasant as I go further from the center, and according to #2 there should be a sense of stability, that gets weaker as the tonal gravity field weakens with distance.

I’ll play all the notes against a drone, several octaves of the tonic. The notes will all be played in a single octave, in between two drone notes. I will octave reduce (divide by 2) as necessary, to put all the notes in this octave.

The first interval is a perfect fifth. To get this, you multiply the frequency of the tonic by 3.

No denying it, this is a consonant interval. Only the drone sounds smoother. It sounds stable too, as it should according to #2. If a song ends like this, I’m perfectly satisfied.

Next note up is the 2. This is two perfect fifths, a factor of 9.

To my ears, this note is still quite consonant. And yes, it still sounds stable. Lots of songs end on this note — I feel wistfulness, gentle yearning, perhaps a sweet sadness, but I don’t feel a powerful need for the note to change.

Next is 3x3x3, the Pythagorean sixth or 6+. I still hear this one as consonant, but just barely. For me the pattern continues — the 6+ is right on the edge of dissonance, and stable, but more weakly. I could hear it resolving down to the 5, but the urge is not strong.

Now 3x3x3x3, the 3+, Pythagorean major third.

I don’t think there’s much disagreement that this is a dissonant note. This major third is 22 cents, an entire comma, sharper than the sweet 5/4 third that appears as “3” on the lattice. It’s has the sharpness of the equal tempered third, on steroids.

For me, the major third is the Achilles Heel of Pythagorean tuning. Any tuning with dissonant major chords is going to be of limited usefulness. I’d go nuts if all songs ended on this note.

Yep, ugly. But still feels somewhat stable. Next?

Ow! And the stability is very weak now; I’d be fine with this note resolving to the 1.

Keep going! Next up is a suitably obnoxious tritone, a version of the good old Devil’s Interval. To my ear, the stability is now lost, and actually the dissonance itself is less profound than it is with the 3+ and the 7+. Tonal gravity is very weak here in the outer solar system.

Just for fun, here are the notes in reverse order. You can hear them coming back to consonance and stability.

To my ear, this experiment supports the idea that more harmonic distance equals more dissonance. The sheer obnoxiousness does seem to fade a bit past the major third or major seventh. I’d say that both consonance and dissonance get weaker as the note gets farther from the center. The ear seems to have less signal to work with as the tonal gravity field weakens.

Next: Polarity Experiment

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Posted by on Mar 11, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 1 comment

Tonal Gravity

I believe that the great driving force in tonal music, that creates the drama and story of the music itself (independently of any lyrics), is the longing for home.

Home is the tonic. If a song is in the key of A, all the A’s in their various octaves will sound like home.

Although there are many exceptions, most music begins on the tonic, to show the ear what key the piece is in, and ends on the tonic, to bring the listener home again. In between, the music wanders, out and back again, creating tension and resolution.

One of the beauties of the lattice is that it shows a clear graphical display of this tension.

It’s as though the tonic creates a sort of gravitational field around itself. It acts a lot like real gravity. The chords and notes move in this gravitational field, like planets and moons around a sun. The gravitational field follows a few basic rules:

  1. Movement away from the center creates tension; movement toward the center gives a sense of resolution.
  2. Notes that are overtonal from the center, generated by multiplying, located to the right and up, will feel more resolved. Notes that are reciprocal, generated by dividing, to the left and down, will feel unresolved.
  3. The closer you are to the center in your journey, the stronger the sensations of tension and resolution are. The field is stronger closer in, just like real gravity.
  4. The closer together two notes are, the more consonant, or harmonious, they will be when sounded together. The farther apart they are, the more dissonant they will be, the more they will clash.

Roots generate local gravitational fields. I think of them as Jupiter to the tonic’s Sun. When the root is on the 5, for example, it shifts the gravity field to the east on the lattice, and the 2 and 7 become harmonious, consonant notes, rather than dissonant ones. The tonic still has great influence, so the entire chord feels unresolved — a 5 chord pulls very strongly toward the 1 chord, a property that is heavily relied upon in Western music. As long as the 5 is the root, though, the 2 and 7 will be consonant harmonies, because they are close to the 5 on the lattice.

Here is a movie to show how that works. The music starts with a tonic chord. Then, one at a time, the 2 and 7 are introduced. These notes are dissonant, and create a sense of tension against the tonic.

Then the root moves to the 5, and the character of the 2 and 7 changes. Now they form a major chord based on the 5, a harmonious configuration. They have become moons of Jupiter. Hear how the dissonance goes away? But there is still plenty of tension, as now there are three notes venturing away from the center, pulling the ear back toward home.

Then the root moves back to the 1, and the 2 and 7 collapse back in toward the center. There is a sense of arrival.

This movie illustrates another observation: consonance / dissonance and tension / resolution are not the same thing. They both relate to distance on the lattice, but they do not necessarily track together. When the root moves to the 5, the dissonance goes away, but there is a new tension, a drive to resolve toward the center. The ear remembers where home is, and longs for it.

These principles can be consciously used to create desired effects when writing and arranging. Resolution and consonance give the music beauty, and tension and dissonance give it teeth.

Next: Cadences

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