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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 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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Posted by on Feb 10, 2013 in Just Intonation, The Notes | 0 comments

An Easy Experiment To Try

It would be natural to read these posts and wonder why I’m so passionate about intonation, and why I’m going to so much trouble to explore it in this blog and in real life. After all, we’re talking about tiny differences in tuning here, why be so picky when it’s the heart that counts?

It’s true, the tuning differences are small, and hard to hear. Thing is, it’s not actually about the pitch. It’s about the way it feels, and in that realm, the difference is not subtle at all. It is profound, and once you hear (no, feel) it, I think you may be hooked, or at least understand more of why I’m so interested in this subject. I think it opens the door to music that truly moves both the performer and the listener, a recipe for audio joy. You bet it’s about the heart. This is not just an intellectual pursuit.

Here is an experiment you can do, to feel that difference in yourself. It uses a chord, and a melody, that you probably already know.

The open G chord is one of the most common chords in guitar music. It looks like this:

Open G

The notes, from left to right, are: G–B–D–G–B–G. If G is the tonic, these notes are the 1, 3, 5, 1, 3 and 1.

One of the best-known melodies in the world is Frère Jacques, or in English, Are You Sleeping (Brother John), a round that is hundreds of years old. It could be harmonized in several ways, but the melody is such that it sounds fine sung over just the tonic chord, over and over again.

Here’s the experiment. First, tune your guitar carefully. A tuner is best. When the open strings are in tune, double check the notes of the open G chord. I think Jody showed me this — it often sounds better if you tune to the tonic chord of the song instead of to the open strings.

Now play a full, open G chord as above. Make sure all the strings sound clearly. You are playing a chord with an unusual property: It has two equal-tempered major thirds in it. This chord is highly equal-tempered in character.

Strum away, and sing Frère Jacques over it, several times through. You may wish to capo and tune again, if this is not a comfortable key for you.

When you have a good sense of what this feels like, try fingering the G chord as follows:

Open G5

I’m a thumb-wrapper, so I finger the low G with my thumb, and mute the A string with more of my thumb. (This is heretical to some, but it’s a wonderfully useful technique when used at the right time. Here’s a beautiful explanation by guitar teacher Jim Bowley.) Then I finger the two high notes with my index. Any fingering will work as long as it mutes the A string.

Now you have a chord with no major thirds at all. It goes G—D–G–D–G, or 1—5–1–5–1.

Sing Frère Jacques over this chord, several times through and check out what happens.

I won’t tell you what to feel. Don’t worry about trying to hear or sing subtle tuning differences. Just pay attention to your singing, and to your body’s reaction.

Seriously, go do this now, or the next time you’re near a guitar. It works great with piano too, and in any key. First play a major chord, with a couple of thirds in it to really make the point. Then play only roots and fifths. Sing the song over each version of the chord, back and forth. The difference may surprise you.

I’ll check in tomorrow with my own conclusions.

Next: More Experimenting

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Posted by on Feb 5, 2013 in Just Intonation | 0 comments

The Major Scale in Cents

The simplest untempered tuning of the major scale is:

1 — 0 cents

2 — 204

3 — 386

4 — 498

5 — 702

6 — 884

7 — 1088

Here’s how that tuning compares with the equal tempered scale:

Cents

 

The black numbers show the pitches of 12-tone equal temperament. They are equally spaced, like inches on a ruler.

The red numbers show the tuning of the untempered major scale. They are spaced in the way they naturally turn out when you generate them with small whole number ratios. As is so often the case with the natural world, they don’t line up too well with the nice human grid lines we love.

The way I see it, when you play in equal temperament, you’re playing the grid lines on the map.

When you sing or play the untempered notes, you are visiting the actual territory.

I’ve read that it’s not possible to combine the two, but I disagree. It’s a matter of showing the ET instrument who’s the boss. My favorite example is Ray Charles. Here’s a video from 1976. He’s playing the piano, laying down those grid lines, and the rest of the band is too, but when he sings, his voice owns the sound, and the sound becomes him. A great, dominant singer will infuse the whole combo with that soul.

Another great example is Ella Fitzgerald. Want some goosebumps? Check this video out.

Next: The Untempered Chromatic Scale (Part 1)

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