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Posted by Gary on Mar 14, 2014 in Equal Temperament, Just Intonation, The Lattice | 3 comments
The Harmonic Lattice can be viewed as a map of harmonic space. Music moves in harmonic space, just as it moves in melodic space (the world of scales and keyboards). The two spaces are very different from each other.
In melodic space, such as a piano keyboard, when two notes are close together, it means they are close in pitch.
In a harmonic space, such as the lattice, when two notes are close together, it means they are harmonically related.
"Harmonically related" means that one note can be converted into the other note by multiplying and dividing by small whole numbers. A note vibrating at 100 cycles per second is closely related to a note at 300 cycles per second. In melodic space, these two notes are far apart, but in harmonic terms they are right next door to each other — they harmonize.
In my video, Flying Dream, I animated the movement of one of my songs on the lattice. Now I've animated a composition of W. A. Mathieu's.
Mathieu is the author of Harmonic Experience, an astonishing book that takes music back to its origins in resonance and pure harmony, and then uses the lattice concept to bring that harmonic understanding forward into the world of equal temperament. For me, the book opened the study of music like a flower.
The lattice, and my stop-motion animations, have given me a sort of musical oscilloscope. Instead of the music being some sort of black box, I can see inside it, get a visual image of what is going on harmonically. The new tool has made songwriting, improvising and arranging much easier.
I've animated Example 22.10 from the book. It's intended to be an illustration of unambiguous harmony — the chord progression moves by short distances on the lattice, so it is clear to the eye and ear where you are. I think it's a beautiful piece of music in its own right, a one-minute tour of a huge area of the lattice. It uses 28 different notes!
There are two versions of the video. The first one, in ET, has a soundtrack of Allaudin Mathieu playing the piece on his beautifully tuned piano. This is perfect equal temperament. It uses twelve notes to approximate the twenty-eight notes that the piece visits.
For the second one, in JI, I retuned the piano to the actual pitches of the lattice notes. Now, magically, the piano has all 28 notes. There is a whole new dimension to the music. In the JI version, I feel:
Slight vertigo when the music moves quickly
Satisfaction when a spread-out (tense) pattern collapses to a compact (resolved) one
A great sense of homecoming at the end
Stronger consonance and dissonance than in the ET version.
The four voices, from lowest to highest, are red, green, orange and yellow. It's fun to follow one voice at a time.
This lattice is notated differently. It's my usual system, but with letters instead of numbers. C is the tonic.
Posted by Gary on Nov 1, 2013 in Off Topic | 0 comments
A Theory of Everything
Hey, shouldn't everyone have one?
I've suggested many times on this blog that number is at the heart of all things. Here's one article: Pythagoras' Epiphany. I think the beauty of music, especially harmony, is similar to the beauty of math, but happening in real time. It slightly parts the veil, deepening the view of what is most basic and true about our universe. I believe that this connection with the deeper reality gives us a sensation of beauty. Here are a few more articles that go into detail, with examples and illustrations.
Saturn's Rings
Beauty Is Truth
Prime Numbers and the Big Bang
If the Big Bang actually happened, then our universe blossomed outward from a point of infinite density, a singularity. What existed before that?
I assert that it was number. The Big Bang was a mathematical event.
What space did it occur in?
One of the gnarliest problems in physics is the issue of reconciling gravity with the rest of the basic forces. There are four: gravity, electromagnetism, and the strong and weak nuclear forces. Gravity is the odd one. It is far weaker than the others. It appears to operate in a continuum, a smooth field, while the others seem to be quantized — that is, they come in little discrete packets. Even space and time themselves may be quantized.
Math is like gravity, in that it works in a continuum. In the world of mathematics, there is no smallest anything. You can zoom in forever.
Here's an example. If I measure the circumference and diameter of a circle very accurately, and divide one by the other, I always get the same number, pi, about 3.14. If I measure more and more accurately, I can get the value of pi to quite a few decimal places, but even with the best equipment imaginable, I will eventually run into the sizes of atoms themselves, and the calculation can't get any finer.
But in the ideal world of math itself, pi goes out to as many digits as you please. There is no point where it runs into the grainy nature of reality.
It's as though in the real world, there are pixels, and when you blow things up enough they start to show, but math itself has no such trouble. Things are infinitely divisible.
Einstein's relativity and quantum mechanics are famously incompatible. Each describes the universe beautifully within our ability to measure so far. But it's really hard to come up with a theory of reality that allows both to be true. Part of the problem is that relativity assumes a smooth continuum, and quantum mechanics assumes that on a tiny, tiny scale, everything happens in jumps, rather than smoothly. A theory of everything would have to explain how both relativity and quantum mechanics can be so true.
So how about this for a TOE:
The deepest reality, the substrate, upon which our universe is based, is simply the world of number.
The universe we live in, with its stars and planets and galaxies and people, is a mathematical object, like the Mandelbrot Set or a cellular automaton, that grew from this substrate.
Here's a tiny, tiny piece of the Mandelbrot Set. Click to enlarge for full glory! This is a mathematical object, at least as big as our universe, and if our universe is one too, then maybe the beauty of this object is related once again to the beauty of music, or of Keats' Grecian Urn — it shows us, a little bit, the nature of Creation.
If the deep reality is a continuum, and the immediate reality of stars and planets sprang from this continuum, then maybe gravity is different from the other three forces because it is a feature of the deep reality. It is a manifestation of the shape of space-time. It is working in the substrate.
Relativity and gravity are happening in the basic reality, the continuum, the world of number.
Quantum mechanics, electromagnetism, light, nuclear forces, matter and energy are all happening in the particular reality that came about when the singularity happened.
Relativity and quantum mechanics can't be reconciled because they actually operate in different realities — gravity in the basic reality, and the other forces in the immediate universe.
If this is true, it may offer insight into the nature of dark matter.
Dark matter has not been observed directly within our quantized universe. Its existence has been deduced, or conjectured, because galaxies move and rotate as though there is a lot of mass there that we cannot see. The idea is that dark matter interacts with "our" universe only through gravity, and not through the other forces. That is why we can't see it, because seeing requires light.
What if dark matter is something that exists in the basic reality, rather than in our particular Big-Bang-generated one? The only link between the realities would be gravity.
There is no reason why our particular singularity should be the only one.
Perhaps what we call dark matter is just the gravitational shadow of other universes.
Here's a scenario:
Ours is one of many universes, each one starting with a different set of "seed" values.
Ours is of course perfectly designed for us to exist, and the other universes are also "coincidentally" perfect for whatever exists in them.
The universes all exist in a space-time continuum, in which gravity is the only "force," being actually a distortion of space-time as Einstein described.
The universes can attract each other through gravity, and so they tend to clump in the same places.
The ratio of dark matter to ordinary matter (about 5:1) may turn out to be an important number. Maybe "nearby" universes (those with similar seed values) attract each other more strongly than more "distant" ones (those with more different seed values), and the 5:1 ratio is the result of an infinite sum — the total pull of all those other universes, fading off into the "distance."
Here's a terrific article on dark matter from April 2013 if you'd like to explore further.
Next: A Harmonic Journey: ET and JI Compared
Posted by Gary on Oct 27, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 4 comments
Tonal Gravity and the Major Scale
In my last post, I proposed a simple way to graph tonal gravity against the octave.
Overtonal notes, generated by multiplying, are restful, stable — they have positive polarity, pulling toward the center. Reciprocal notes, generated by division, are restless, unstable — they push. I call this negative polarity. Mixed-polarity notes have both, and I've chosen to simply add their overtonal and reciprocal components together to get the total polarity.
Here again is the graph of the 13 most central notes of the lattice.
The stable notes are gravity wells, and the unstable ones are peaks. Melodies and harmonies dance in this gravity field. Higher points represent tension, lower ones resolution, and the lower they are, the more resolved and stable. The tonic major triad, most stable of all, occupies the lowest spots — 1, 3 and 5.
The polarity map of the major scale looks like this:
The notes are all overtonal except the 4, which is strongly reciprocal, and the 6, which is mixed and slightly unstable.
Here's a split screen video showing the major scale, against a tonic drone, on both the lattice and the octave. This is an example of how the lattice serves as a Rosetta Stone, a translator between harmonic and melodic space.
Can you hear the push/pull quality of the notes? Each note has its own feeling against the steady 1.
Next: A Theory of Everything
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:
Overtonal intervals have one polarity, and reciprocals have the opposite polarity,
Gravity gets weaker the farther one gets from the center, and
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.
For purely overtonal notes, of the form N/1: P = 2/N.
For purely reciprocal notes, of the form 1/D: P = -2/D.
For compound notes, with both overtonal and reciprocal components, add the overtonal and reciprocal gravities together: P = 2/N – 2/D.
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.
Next: Tonal Gravity and the Major Scale
Posted by Gary on Oct 14, 2013 in The Lattice, The Notes | 0 comments
Straight Line Chords: Aug, Dim, Sus2, Sus4
Major and minor chords look like triangles on the lattice. They are closed loops, which I think contributes to their stable feeling. Up a major third, up a minor third, and down a fifth (due West on the lattice) brings you right back where you started. The intervals interlock, and they are all consonant, reinforcing the sense of harmonic rest.
Other combinations of notes are more open. Many chords look like straight lines on the lattice.
Augmented chords are stacks of major thirds:
Diminished chords are stacks of minor thirds:
These chords often show up in my animations as fleeting transitional harmonies. In Flying Dream, for an instant, there's even a stack of three major thirds:
To my ear, this stack of thirds has a distinctive sound, almost like cloth ripping.
A Suspended Second chord is a straight line, like an augmented chord, but on the horizontal axis. It's a stack of two fifths. At the time the Sus2 and Sus4 chords were named, the 2 and 4 were usually "suspended," or held over from the previous chord, as a tension to be resolved. Now they are often used as full chords.
The Police used these stacks of fifths a lot. Andy Summers' guitar part for Every Breath You Take is full of Sus2 chords, alternating with major and minor thirds.
To me, Sus2 feels lightly stable, and wistful. All the intervals are overtonal, and quite consonant, but unlike the major and minor triads, the Sus2 doesn't come full circle. If it keeps going, it will never return home, but climb on up the endless spiral of fifths.
It's a great way to end a certain sort of song, finished but with a sense of longing.
The Suspended Fourth consists of a root, a perfect fifth, and a perfect fourth. The Sus4 looks the same on the lattice as the Sus2. The difference is the root.
This chord has a wonderful tension. The root establishes home. The 5 is overtonal, stable, with strong tonal gravity that attracts. The 4 is reciprocal, unstable, with a strong tonal gravity that repels.
Resolving to the 3 is satisfying indeed, as the unstable 4 slides from its unstable peak into the stable gravity well of the major third.
Sus4 chords are all over rock music. Pinball Wizard is a study, here it is by Townshend on acoustic guitar:
Next: Putting Some Numbers on Tonal Gravity