Here is my third stop-motion animation of a full song.
Real Girl uses a custom nine-note scale. It occupies the Southeast quadrant of the lattice, the zone of the natural minor, with two added notes — the 7, which allows for a major V chord in the progression, and the 7b5, a blue note that is showcased often in the melody.
This scale contains a sharp dissonance, between the b6 and the 7. I go back and forth between those two notes a lot, with a stop on the 1 in between to help ease the transition.
Watch how the melody and bass chase each other around. In the next few blog posts, I’ll slow this dance down, and show how the polarity flips create tension and resolution. When the melody is below and to the left of the bass, the energy is reciprocal, tense. Then one or the other moves so that the melody is above and to the right, the energy becomes overtonal, and the tension resolves.
Another fun thing to watch is the alternating bass. Roots and fifths are right next to each other on the lattice. The red lens swings like a pendulum throughout the verses.
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.
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.
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.
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.
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.
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:
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.