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Posted by on Oct 22, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 0 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:

Quadrants-1024x768

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.

Polarity central lattice-01

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.

Tonal Gravity 13-01

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.

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Posted by on Aug 26, 2013 in Consonance, Just Intonation, Recordings, The Lattice | 0 comments

100 Girlfriends, Part 2

My new song video, Real Girl, contains many examples of consonance and dissonance, tension and resolution. In my last post, I extracted a phrase from the song and slowed it way down to illustrate how the bass and melody dance, creating and resolving tension in several different ways. Here is the last half of that analysis.

When we last left our heroes, they were on the 4 and b6, quite consonant relative to each other, but still unresolved because the ear remembers where the tonic is. Here is that clip:

Now the melody moves back to the 7. This interval, against the 4, is the dreaded tritone, the devil’s interval, and it’s dissonant indeed.

Then the bass moves up to the 1, lessening the dissonance, and the melody soon joins it, and all is consonant.

But there is still a sense of incompleteness, even though both the bass and melody are smack on the tonic, the most consonant interval of all. What’s up?

The answer is that the ear remembers that the root is still the 4, and we aren’t quite home yet. Getting there requires a cadence, or final resolution. Notice that in this next clip the bass note never moves, but the harmonies and the melody signal that the root has now moved to the 1 and we are home. The bass note has magically changed character.

Here is the complete sequence, annotated.

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Posted by on Aug 15, 2013 in Consonance, The Lattice, The Notes, Tonal Gravity | 0 comments

One More Mirror Pair

I’m almost done with the next full-song video. In the meantime, here’s one more pair of mirror twins for consideration.

The 2- is a common melody note in my songs, and in the blues. It goes well with the blue note 7b3 — there is an extremely common melody that goes 7b3, 2-, 1. It’s a darker, more dissonant note than its comma sibling, the 2.

The b7 is dissonant and gorgeous — check out the sequence at the end of this post.

Each note is a compound of three legs on the lattice — two fifths, or a factor of 9, and a major third, a factor of 5. By the logic of the last post, the short leg should predominate, which would make the 2- slightly overtonal and stable, and the b7 slightly reciprocal and unstable.

This proves out when I listen to the video. Even though the 2- is distant from the center, and quite dissonant, it still feels stable. The tonal gravity field is “pulling” rather than “pushing.”

I’m setting up here for a map of the tonal gravity field. I think I can put some numbers on this stuff. Coming soon. I’ll use that new song animation as a basis — it’s full of fleeting dissonances and polarity flips.

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Posted by on Aug 14, 2013 in Consonance, The Lattice, The Notes, Tonal Gravity | 0 comments

A Mirror Quad

In the last few posts, I’ve been exploring mirror twins — notes at the same harmonic distance from the center, but of opposite polarity.

The notes explored so far are 3/1, 5/1, 7/1, 9/1, and their reciprocals, 1/3, 1/5, 1/7 and 1/9. The 9/1 and 1/9 are made up of two legs on the lattice, x3 and x3.

The next overtonal note out from the center is the major seventh, or 7. Its ratio is 15/1, or x3, x5.

The 7 has its mirror twin too, the b2-, at 112 cents. Its ratio is 1/15.

Here is how they sound:

For me, the pattern continues. The 7 is stable, but less so than the notes we’ve heard so far, and it’s getting dissonant as well, because it’s farther from the center. The b2- is both dissonant and unstable.

These notes each traverse two legs of the lattice, a 3 and a 5. The 7 is two legs “up,” or multiplying, and the b2- is two “down,” or dividing.

What if one stick goes up and the other one down?

These notes are the minor third, 3/5, and the major sixth, 5/3. They are compounds of overtonal and reciprocal energy.

How will this affect stability and instability? I’ll guess that since 3 is a shorter distance than 5 is, and closer to the center means stronger gravity, the factor of 3 will dominate the blend.

So 3/5, the minor third, should lean toward the overtonal, and 5/3, the major sixth, should lean toward the reciprocal.

This hypothesis is supported by the long tradition that the minor third is a stable note, less so than the major third but OK to end a song with.

That is indeed what I hear, although it’s less clear than it is with earlier intervals.

All four of these intervals use the same prime factors, and cover the same harmonic distance. The difference between them is polarity.

 

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

More Mirror Twins

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.

 

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