Posted by on Oct 22, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 4 comments

## Putting Some Numbers on Tonal Gravity

I believe the sensation of tonal gravity is the most important driver of tension and resolution in tonal music, music that has a central key note.

The tonic is like a sun, creating a gravitational field around it. The lattice is a beautiful map of this gravitational field, in harmonic space.

Tonal gravity acts like real gravity, in that it’s strongest close to the center. The farther out you get, the weaker the effect.

There is a difference, though — unlike physical gravity, which only attracts, tonal gravity has two polarities — it can attract or repel. Briefly, if an interval is generated by multiplication, it will pull toward the tonic, as though to say, “You’re going the right way, you’re just not there yet.” If the interval is generated by division, the sensation is more of a push — it still points to the tonic, but now it’s saying “It’s over that way, go!”

Overtonal notes are stable, reciprocals are unstable. Reciprocal intervals create tension, overtonal ones create resolution.

The article on Polarity goes into detail, with examples.

The lattice can be divided into four quadrants, each with a characteristic tonal gravity. The northeast quadrant is entirely overtonal. This is the world of Major/Dominant: overtonal thirds, overtonal fifths.

The southwest quadrant is entirely reciprocal. Every interval is generated by division.

The northwest and southeast are zones of mixed polarity, overtonal on one axis and reciprocal on the other one. Here’s a map of the quadrants and the compass points:

Green is overtonal, stable, tonal gravity pulls.

Red is reciprocal, unstable, tonal gravity pushes.

Blue is mixed, gravity pulls on one axis and pushes on the other.

For a couple of years now, I’ve been chewing on how to represent the tonal gravity of the lattice mathematically. To describe what I experience, the equations should behave as follows:

1. Overtonal intervals have one polarity, and reciprocals have the opposite polarity,
2. Gravity gets weaker the farther one gets from the center, and
3. Multiplying or dividing by 2 does not affect tonal gravity. This is to account for the octave phenomenon — going up or down an octave does not change a note’s position on the lattice.

Here’s my latest approach. I’m not presenting this as some kind of truth — but it nicely matches my own perceptions, and it leads to some interesting graphs. Any input you may have is welcome — feel free to comment, or email me from the Contact page. Here goes:

I will call the direction and magnitude of the tonal gravity field P, for Polarity.

Intervals are expressed as a ratio of two numbers, numerator and denominator, N/D. For example, a perfect fifth is 3/1, or N=3, D=1.

1. For purely overtonal notes, of the form N/1: P = 2/N.
2. For purely reciprocal notes, of the form 1/D: P = -2/D.
3. For compound notes, with both overtonal and reciprocal components, add the overtonal and reciprocal gravities together: P = 2/N – 2/D.
4. The ratio of the tonic, the 1, is 2/1.

I’d better explain that last one, because it would seem at first glance that the ratio of the tonic would be 1/1.

But what is the actual tonic? It has no specific pitch. It is not a ratio. It is an abstraction, the anvil upon which all notes are forged, the sound of one hand clapping. If a song is in the key of A, all of the A’s from subsonic to ultrasonic are actually octaves of the tonic, created by multiplying by two. It is impossible to say that any one of these A’s is “the” tonic — the tonic is “A-ness,” that thing which connects the numbers 110, 220, 440, 880, to infinity in both directions. I submit that the “1” of the lattice, which is a real pitch (or set of pitches, an octave apart, just like all the other notes) is in fact the octave, and its ratio is 2/1.

Here’s another drawing of the inner lattice. Instead of the note names, I’ve filled in the ratios, and the value of P.

The green notes all have positive polarity, getting weaker as they get farther out. The red ones have negative polarity, also fading with distance. The blue ones have different polarities. Sometimes the overtonal part dominates, sometimes the reciprocal.

The b3, just southeast of the tonic, is a mixed-polarity note. Its ratio is 3/5, combining an overtonal fifth, P = .67, with a reciprocal third, P = -.40. If I just add the two gravities together, I get a positive net polarity of .27.

This makes sense. The minor third is considered to be a stable interval, though not as stable as the major third.

Both the major and minor triads consist entirely of stable intervals with positive P, which helps explain their special place in music.

So: now that I have some values for P, I can graph the tonal gravity of these 13 inner notes against the octave, in order of pitch.

Positive polarity is at the bottom, so that the feel is the same as real gravity. Unstable notes are up on mountain peaks, and when they resolve to stable ones they slide down into the gravity wells of the stable notes.

There is that tasty melody zone I mentioned a few posts back. The whole region from 2- to 3 is stable.

The 4 is an isolated peak, and it’s easy to imagine a 4 sliding into the pocket of the 3, or the 5. This is what happens when a Sus4 chord resolves.

The 7 is lightly stable but hanging on by its fingernails — it’s called the leading tone, because it “wants” so badly to resolve to the 1. The tonal gravity of the 7 is usually thought of in terms of melodic pull — here’s a graphic demonstration that it has harmonic pull as well.

There is an unstable region from b6 through b7, with all the mirror twins of that stable melody zone. A melody will sound unstable, unresolved as long as it stays in that region.

This is the gravitational field in which the music moves, a sort of tonal skate park.

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

## Harmonic Distance

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).

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

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

Posted by on Jun 22, 2013 in Just Intonation, Recordings, The Lattice, The Notes, Tonal Gravity |

## To the Far Northwest

The chorus of Be Love is solidly in major territory. The chords are I, V, IV, the classic backbone chords of the major scale.

The verse, however, is going to be back in the Northwest — I, bVII-, IV. I have to get back over there somehow, and I want to lead the ear strongly so that the move feels right.

Once again I’ll use a dominant seventh type chord. Here’s the shape again:

The ear expects a I chord to come after this. Perhaps it’s one of those “nature abhors a vacuum” things. All the notes have tonal gravity that is pointing at the center, yet there’s nothing there. It’s as though the planets are orbiting the sun, and the sun is missing. The ear wants to put it there.

This “dominant seventh” shape is named after the dominant chord, the V7, where it shows up so often. It can be used elsewhere, and is the main tool used in classical music to change keys.

When I started to work this out, I already had a climax in mind for the chorus. In the last line, I wanted to leap all the way to the far Northwest, the ii- chord (lower case roman numerals mean it’s a minor chord), and have the melody sit still while the chords revolve around it. Here’s the ii-:

It’s a long leap, but I can make it with one transition chord, the I7.

Adding a seventh to the I chord sends a strong signal to the ear: “We’re going west!” Normally the next chord would be a IV.

This move is used all the time in music. It’s common for the bridge of a song to start on a IV chord — it’s a shift on the lattice that makes the bridge sound different from the rest of the music, much like my shift to major for the chorus of Be Love. Putting a I7 right before the bridge tells the ear to expect this shift.

I do something a little different, though. There is also a huge vacuum where the ii- ought to be — look how the two chord shapes fit into each other. Sure enough, going to the ii- after the I7 is satisfying and dramatic. Here’s how the sequence looks and sounds.

Notice how the melody leads the way again, by going to the 4. There’s another lovely passing chord, a stack of perfect fifths, right before the change.

Posted by on Mar 12, 2013 in The Lattice, The Notes, Tonal Gravity | 2 comments

A cadence is a chord progression that gives a sense of arrival or resolution.

One particular cadence, the V-I (or V7-I) is especially powerful. In classical music, a V-I cadence is practically mandatory at the end of a piece, and it is the biggest gun in the composer’s arsenal when changing keys, or modulating.

The following movie shows a I-V-I progression. It starts on the I to establish the tonic, then there’s tension, then resolution. The V-I cadence draws the ear back to the tonic chord.

Here’s a cadence that visits the V7 first:

To me it looks like the V chord tosses out a rope, lassos the tonic and pulls.

It’s interesting to look at the notes in light of tonal gravity. In yesterday’s post, I laid out two rules of gravity on the lattice:

1. Movement away from the center creates tension; movement toward the center gives a sense of resolution.
2. 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.

There are four notes in the V7 chord.

• The 5 is as close as you can get to the 1 (in harmonic space), so it creates a lot of tension. It is an overtonal note — that is, it appears in the overtone series of the 1. Pluck a string tuned to the tonic frequency, and the 5 will tend to be strongly present in the timbre of the sound. The way I see it, the ear is always searching for home. Every note gives it two clues — which direction is home, and how far away is it? The 5 gives a very strong signal, pulling the ear toward the tonic: “Home is this way, and it’s close! Come on!”
• The 2 reinforces this conclusion. It’s farther out, so the signal is weaker, but it is still in the harmonic series of the 1, and it’s pretty close in. The little detective in the ear gets another clue.
• Same with the 7, although now the effect is weaker. In traditional theory, the 7 is called a leading tone, and it’s thought to pull melodically toward the 1 — a sort of gravity in melodic space. It “wants” to resolve a half step upward. I feel this too, and I think the harmonic pull reinforces it further.
• Then there’s the 4, which is what makes it a seventh chord (the 4 is a minor seventh of the 5). This is a reciprocal note, that is, it’s generated by division rather than multiplication. Like the 5, it points directly at the 1, from point-blank range. Reciprocal energy is different from overtonal energy. To me, it feels as though reciprocal notes are pushing toward the tonic — the message feels more like “Home is that way, now go!”

For the detective in the ear, the 4 slam dunks the case. The only reasonable conclusion is that home is located in that empty space between the 4 and the 5 on the lattice. Any other interpretation is much weaker. Every note in the V7 chord is pointing strongly to the 1, and when the notes collapse inward to the I chord, the resolution is completely satisfying.

I’ve heard a charming story about Beethoven. Apparently the composer was depressed and wouldn’t get out of bed. A friend came by and played some music, ended on a dramatic V7 chord, and sat down to wait. Beethoven finally had to get out of bed and play the tonic chord. Tough love! Don’t know if the story is true, but it certainly could be. The V7 is strong medicine.

When this particular chord shape appears somewhere else on the lattice, it can point so hard to its own center that the ear believes the tonic has moved. It’s as though the gravity of the planets is so strong that it can move the sun.

Next: Intervals