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100 Girlfriends

Posted by on Aug 22, 2013 in Consonance, Recordings, The Lattice, Tonal Gravity | 0 comments

There is a passage, in my song Real Girl, that clearly showcases both kinds of dissonance — the kind that comes from harmonic distance, and the kind that comes from reverse polarity.

This melodic passage occurs many times in the song, and it contains a rather dizzying series of tensions and resolutions. My friend Jody Mulgrew, who has an exquisite sense of pitch, experienced actual nausea the first time he heard the song. He told me, “I was wondering how to tell my friend Gary that I didn’t like his new song. Then, before the chorus, it started to sweeten up, and when the song was over I immediately hit the ‘replay’ button. I realized it was just tension and resolution.”

I think my friend was experiencing what I call tonal vertigo. His comment spurred some of my thinking on the nature of harmony, how it may be a byproduct of our orientation software. The “100 girlfriends” section is a roller coaster ride in the tonal gravity field. Here it is in its original form:

Now to slow it way down and take it apart.

The first dissonant melody move is to the 7. The interval is a major seventh, down a half step in pitch, and the harmonic distance is great enough (3×5=15) that the note is quite dissonant. But the bass, alternating between 1 and 5 as so many bass lines do, quickly moves to resolve the dissonance.

Note that there is still an unresolved, unfinished feeling. Even though everything you can hear is beautifully consonant, the ear still remembers that the real root of the chord is the 1. This memory is crucial to tonal music.

The next move creates a different kind of dissonance. This is the tension of reverse polarity.

First the melody moves to the 1. This note is right next to that 5 in the bass, and beautifully harmonious. But there is tension, because it’s a reciprocal note. The way to get from a 5 to a 1 is to divide by 3 — it’s one move to the left on the lattice.

Then it makes a crazy move, to the b6, that gives me vertigo. Not only is this note distant from the bass note (a factor of 15), but it’s the reciprocal version of the major seventh, its mirror twin, the minor second. You’re dividing by 15, rather than multiplying. Here’s the article that explains why this is such an important difference.

If this weren’t enough, the b6 is also a reciprocal of the root. Remember, even though the bass is the 5, the root is still the 1. The b6 is the mirror twin of the 3, an intensely reciprocal note. So the tension is very high.

And, in two moves, the melody has covered a lot of harmonic territory, all in the reciprocal, Southwest direction. No wonder Jody felt nausea! It’s an E-ticket ride.


Once again, the bass moves to save the day. The chord changes too — that 4 in the bass is the new root. The melody note magically becomes a minor third, not fully consonant, not fully resolved, but a lot better.

In the next post, the famous tritone! Then full resolution.

Next: 100 Girlfriends, Part 2

Real Girl, Animated

Posted by on Aug 20, 2013 in Just Intonation, Recordings, The Lattice | 1 comment

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.

One More Mirror Pair

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

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.

Next: Real Girl, Animated

A Mirror Quad

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

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.

Next: One More Mirror Pair

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?

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