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Posted by on Mar 4, 2013 in Just Intonation, The Lattice | 2 comments

The Infinite Lattice

The lattice goes on forever in every direction.

It starts with the tonic, the 1, the Big Bang of the musical universe.

Multiplying and dividing the tonic by 3 generates the horizontal axis. This is the familiar circle of fifths, although in just intonation, it doesn’t quite come out exact. If you multiply by 3, twelve times, you run through all of the scale degrees and land back on the tonic, 19 octaves up … almost. Three to the twelfth power, octave reduced until it’s back in the original octave, comes out to about 24 cents, not zero. Equal temperament flattens this out by subtracting two cents from every fifth. Very handy.

Multiplying and dividing by 5 generates the vertical axis. The two together create a plane, a map of harmonic space. Tonal music, that is, music that is organized around a key center or tonic, can be viewed as a journey on this map.

In the center of the map, there is a lovely pattern of twelve notes that form a chromatic scale.

full lattice 2-01

Each of these notes has a cousin, four fifths down and a third up, that is tuned almost the same. It’s 22 cents flat of the original note, a distance of a Didymic comma. Thus there is another major second, the 2-, just outside the 12-note pattern.

full lattice 2--01Doing the same thing for all twelve notes creates another chromatic scale to the west. It’s the same scale, 22 cents flat of the original. It works the other way too — there is another block of notes to the east, same scale, 22 cents sharp.

full lattice east west-01

The other comma I’ve discussed, the Great Diesis, shows how to extend the lattice north and south. This one shifts the pitch by 41 cents. It’s the shift that results when you go up or down by three major thirds. Equal temperament flattens this comma out too, but the adjustment is more extreme. Every major third in ET has to be sharp by about 14 cents in order for three of them to add up to an octave, a noticeable difference in pitch.

full lattice diesis-01

The pairs created by the Great Diesis have different note names. The b6 in the lattice above is at 814 cents, and the #5 is at 773 cents — 41 cents flat. In the key of C, these notes are Ab and G#, and they are played with the same black key on the piano, between G and A. Until I started studying just intonation and the lattice, I had no idea why one would want to think of these as different notes. It’s not an old-fashioned or obsolete distinction. It’s very useful, when writing or arranging, to know where you are on the lattice, and it’s just as useful in ET as it is in JI.

Now I can add two more blocks to the north and south.

full lattice nsew-01

And here’s the whole thing. The colors are arbitrary.

full lattice all-01

The chromatic scale, a block of 12 notes, has tiled the plane. The note names get pretty crazy — triple flats indeed! But they exactly describe the pitch of every note, in just intonation. Start with the major scale, 1-2-3-4-5-6-7. Every sharp (#) adds 70 cents to the original note, and each flat (b) subtracts 70 cents. Each + adds a Didymic comma, 22 cents, and every – subtracts 22 cents.

Next: Why Equal Temperament?

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Posted by on Mar 3, 2013 in Just Intonation, The Lattice | 0 comments

Another Comma

One more comma shows up in the central portion of the lattice.

In equal temperament, three major thirds adds up to an octave. The major third is an interval of four piano keys (out of 12) or 400 cents. Three of them is 1200 cents, exactly an octave.

In just intonation, this is not the case. The 5/4 major third is a narrower interval, 386.3 cents. Adding three of them together gives 1159 cents, 41 cents shy of an octave.

On the lattice, a stack of major thirds looks like this:

stack of thirdsEvery three places, the notes repeat — almost.

One such pair is the b6 and #5. In the key of C, they would be an Ab and a G#, and on the piano you would play them both with the black key between G and A.

diesis

In just intonation, the b6 is the ratio 8/5, which works out to 814 cents, and the #5 ratio is 25/16, which is 773 cents, almost a quarter tone flatter. (I describe cents and how to calculate them here.) The 41-cent distance between these two notes has several names. Mathieu’s is my favorite: Great Diesis (Dye-uh-sis).

Its formula is x5, x5, x5.

The reason I’m introducing these commas is to show how the lattice repeats itself. Here are the two commas on one lattice:

commas

The lattice extends infinitely in the horizontal direction, and every time it repeats, the pitch shifts by 22 cents. It also repeats in the vertical direction, shifting by 41 cents with every repetition.

Next: The Infinite Lattice

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Posted by on Feb 26, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

Another Major Second: The 10/9

When I started exploring the extended lattice beyond the central 12 notes, the first note that was really new to me was the 10/9 major second, also called the minor or lesser whole tone. Now I call it the 2-.
Other-major-2-latticeThe lattice extends forever in all directions. When you continue multiplying and dividing, generating new notes beyond the boundaries of the central zone, the notes start to repeat, but not quite. The notes in red, the 2 and the 2-, are very close in pitch. They are different flavors, if you will, of the interval of a major second, or whole tone — a distance of two half steps, two keys on the piano.

Even though they are so close in pitch (204 cents for the 2, 182 cents for the 2-, only 22 cents apart), the two major seconds are generated in different ways and have very different functions and characters.

The 2 is an entirely overtonal note, that is, generated by multiplying alone. Such notes can be found in the chord of nature, the harmonics of a vibrating string. The character of notes is somewhat subjective, but for me, overtonal notes have a stable, sort of upbeat or positive character, and even though the 2 is somewhat dissonant, it has a kind of peaceful sound, that shows up well in ninth chords. Its recipe is x3, x3, or x9, octave reduced to 9/8.

The 2- is a combination of reciprocal and overtonal energy. It’s farther from the center than the 2, and more dissonant. Its recipe is /3, /3, x5, or 5/9, which octave reduces (or expands, really) to 10/9. It is darker, bluesier perhaps, and functions differently in chord progressions.

These very similar ratios, 10/9 and 9/8, 182 and 204 cents, are in fact entirely different beasts. Equal temperament has obscured this difference over the years. In ET, both notes are played at the compromise pitch of 200 cents, but that does not change the functional difference. It is extremely useful when writing or arranging to know whether you are playing a 2 or a 2-.

I tried making a demo of how they sound, as with other notes, but I think that played by themselves, out of context, the 2 and 2- are hard to tell apart. To get the difference, I think you have to sing them against a drone (scroll down the linked page a bit and there’s a list of Indian drones to play around with, it’s really fun to improvise melodies over these) and feel them in your own body. Mathieu shows you how to sing the 10/9 note in Harmonic Experience.

The functional differences really show up when you’re designing chord progressions that make sense. A chord progression is a journey on the lattice, and if you’re roaming in western territory, that is, to the left of the center, you want to use the 2- in your chords and melodies, and if you’re in overtonal, eastern lands, to the right of center, the 2 is going to sound better. It’s a crucial distinction in just intonation. Not so much in ET, since the notes are tuned the same — but awareness of where you are on the lattice really helps when you’re writing ET chord progressions.

It’s an old puzzle. Why do some progressions feel “right,” and others “wrong”? Knowing the map of harmony, the lattice, helps a lot. Much more to come in later posts.

Next: Commas

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Posted by on Feb 15, 2013 in Resonance | 0 comments

Saturn’s Rings

Why are the rings of Saturn so beautiful?

400px-Saturn_eclipse_exaggerated

The photo to the right was taken by the Cassini spacecraft when the sun was behind the planet, and backlighting the rings and the edge of the atmosphere. A type of solar eclipse never before seen by humans!

The rings are composed of millions of small particles, mostly ice, orbiting Saturn. They are neither arranged in a uniform disc, nor are they evenly spaced like the grooves on a record. Instead, they have an exquisite natural pattern, not quite like anything I know of on Earth. Click the image below for a full size, zoomable version. The bright bands are higher density, dark ones lower density.

2200px-Saturn's_rings_dark_side_mosaic

As with so many patterns in nature, this one is generated by simple rules. The main generator of the ring patterns is orbital resonance. The chunks that form the rings come in all sizes, from dust grains to small moons. When the orbital periods of two bodies are related to each other by a ratio of small whole numbers (sound familiar?), they will have a lot more gravitational influence on each other, just like the playground swing example from the last post. They give each other a little kick every time they come around, and either the relationship is unstable (one or both get booted out of their orbit) or stable (they settle in to a pattern and their resonance locks them into, um, harmony).

There are other examples in the Solar System. Pluto and Neptune are in a 3:2 resonance. Pluto orbits the sun twice for every three times Neptune goes around, and the relationship has persisted for a long time. They are playing a very slow perfect fifth. Orbital resonance draws them into this pattern. The legs are kicking at just the right time.

I think there is a very real connection between the beauty of the rings and the beauty of harmony. Stand close to someone, and sing a note while the other person sings a perfect fifth above. I think you will feel the resonance in your vocal cords, as it draws you into entrainment. Resonance influences and creates physical structures on every scale from subatomic particles to spiral galaxies.

Once again, I propose that when we experience the joy of musical harmony, we are seeing (and hearing and feeling) a little more deeply into the nature of the universe. The window is resonance. Here’s an interesting site with lots more about the connections between physics, sound and resonance.

And, to ride my hobby horse for just a second, I believe the dominance of equal temperament has obscured this deep insight and feeling. For many notes, the legs just don’t kick at quite the right time. No worries, I do think equal temperament is extremely useful, and it’s been used to make a whole lot of gorgeous music. I use it myself. But it has distanced us somewhat from the shot of pure joy that the resonances of music, in tune, can deliver. I’m hooked on the straight stuff, and the reason I’m writing this blog is the desire to share that joy.

Actually, I do know of an Earthly structure that resembles Saturn’s rings. It’s the scale, in just intonation.

colored chromatic

 

Next: Extending the lattice

 

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Posted by on Feb 11, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

More Experimenting

Yesterday, I described a simple way to hear, and more importantly, feel, the difference between equal temperament and just intonation, by singing Frère Jacques over an open G chord with ET major thirds, and then over a G chord that has only roots and fifths in it.

The second half of the experiment is called singing over a drone, and it’s a great way to get acquainted with the resonance of the pure notes. The 1–5 drone is a bedrock foundation of North Indian (Hindustani) classical music. Check out this gorgeous song with Ravi Shankar’s two daughters, Norah Jones and Anoushka Shankar. It’s not drone music, but it’s a beautiful blend of East and West, and I feel the purity of the notes down to my bones. This is untempered music.

At this point I must pause again to acknowledge my enormous debt to W.A. Mathieu. I had been studying just intonation for several months when my friend Kay Ashley, a fine singer, guitarist and student of Hindustani music, loaned me her copy of Mathieu’s book Harmonic Experience. She did not get it back until I had my own copy.

The first part of the book introduces the pure notes by showing the reader how to sing them over a drone. I have found no better way to understand the notes of just intonation than to sing them. It isn’t just about hearing them, even though that can be beautiful and illuminating. It is about feeling the resonance in your body.

200px-Stable_equilibrium.svg copy

My own experience in singing over a drone of a root and perfect fifth is that it is much easier to sing in tune. It’s as though the drone sets up a sort of sonic field that has grooves or troughs in it, points of stable equilibrium into which my voice falls, and wants to stay.

The equal-tempered chord is not so friendly. The tempered major third is not in tune, that is, its natural resonance does not point exactly to the tonic. It’s as though it points to a different tonic, a little sharper than the one the root and fifth are pointing to. The groove is obscured, and there is a fight between the two worlds that makes it harder to know exactly what to sing. The reference is shaky.

I confess, yesterday’s experiment was a little unfair to equal temperament. I changed two things at once, which is not a good way to investigate nature. It’s much better to change only one thing at a time, so you know what causes what. When you sing over the straight G–D drone, you’re hearing two changes — the simpler chord (1–5 instead of 1–3–5), and the effect of removing the equal tempered third.

In the interest of scientific honesty, here’s one more exercise that shows only the effect of ET.

I’ve recorded some synthesized strings to sing along with. These are the same notes as the open G chord: G-B-D-G-B-G. Once again, sing Frère Jacques. Row, Row, Row Your Boat and Three Blind Mice are also excellent, I recommend trying them too.

Here is a G major chord in just intonation:

G chord JI

And in equal temperament.

G chord ET

If this is not in your most comfortable vocal range, here are some six-note chords in the key of C. I find these better for my own voice. These are note-for-note the same as the first position C chord on guitar, another common chord with two equal tempered thirds in it.

C major in just intonation:

C chord JI

And in ET.

C chord ET

I invite you to go back and forth between the JI and ET versions of the chord that is most comfortable for you, singing over each.

While you’re singing, pay attention to how the notes feel, in your body.

Also notice how easy, or how difficult, it is to hold your notes, to jump straight to the next note, to not waver when you hold a long one.

And perhaps most importantly, pay close attention to the emotion you feel while singing.

I have long experienced flashes of musical ecstasy — it’s why I make music, to experience and share that transcendence. But such experiences have been sporadic, and somewhat mysterious. Encountering, studying and internalizing the pure notes, and their relationships to each other (the lattice), has thrown open the double doors, and I am now in the long process of walking through them.

Next: Entrainment

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