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Saturn’s Rings

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

Why are the rings of Saturn so beautiful?


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.


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



Posted by on Feb 14, 2013 in Resonance | 0 comments

Resonance, in general, is the tendency of a system to vibrate more strongly at some frequencies than at others.


A great example of this is a playground swing. Like any pendulum, the swing “wants” to swing at a particular frequency, its resonant frequency. You can make yourself go higher and higher without a lot of effort — if you swing your legs at just the right time. Move your legs a little too fast, or two slow, and you won’t go any higher. You can stop yourself cold, just by swinging your legs at a “wrong” frequency.

A similar thing happens when you sing in the shower. If you run up and down the scale, you will find that some frequencies are reinforced, and some canceled. It is much easier to make a louder sound if you are singing at the shower’s resonant frequency. Try it next time — pick your favorite shower song (One of mine is Englishman in New York — be yourself, no matter what they say) and sing up and down until you find a note that easily gets really loud. Then sing the song in that key.

Resonance, in music, makes entrainment easier, which facilitates musical joy.

Next: Saturn’s Rings


Posted by on Feb 12, 2013 in Just Intonation, Resonance | 0 comments

I recently read Mickey Hart’s beautiful book, Drumming at the Edge of Magic. Hart is best known as one of the two drummers of the Grateful Dead. His book tells the story of his lifelong fascination with percussion, and of his investigations into the ancient connection between rhythm and the human spirit.

Toward the end of the book, Hart introduced a concept that was new to me — entrainment.

It seems that if two vibrating systems are allowed to interact, and if their frequencies are already somewhat close to each other, they will become synchronized. The faster one slows down, and the slow one speeds up. Here’s a demonstration:

One of the greatest joys in my life is making music with other people. It’s great to play solo, but something different happens when two or more people make music together. I think a part of the reason is that the musicians, and often the audience, entrain with each other. They each keep their own time, and it won’t be identical at first. But as they listen to each other, and feel the common rhythms, their grooves start to adjust until they are literally on the same wavelength, and something happens inside them, one of the flavors of ecstasy. It is a bonding experience, and I feel a different connection with anyone I’ve entrained with in this way.

The beauty of this is that it’s a physical principle, which means it’s not something you have to “try” to do. Just relax and pay attention to what the other band members are doing, and if you are in the same room and have a chance to influence each other, the laws of nature will pull you toward entrainment. Do it long enough and everyone involved, including the listeners, is likely to experience a trance state. Hart should know, the Dead accomplished this thousands of times over their long career.

800px-Seattle_Center_-_Kobe_Bell_02AIn his book, Mickey Hart talked about entrainment in the context of rhythm. But I immediately connected it with harmony. I sang Christmas carols professionally for years, with the Dickens Carolers quartets in Seattle. When all four voices blend perfectly, there is a delicious sensation of being a single voice. Once, four of us were walking back to the car from a show, and we ran across the Kobe Bell, a landmark feature of Seattle Center. We all stood with our heads up inside the bell, found a key that resonated with it, and sang O Holy Night. It was one of the peak experiences of my life. The bonding that occurred was almost frightening.

I think that singing harmony is like dancing together, only very fast.

Next: Resonance

More Experimenting

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

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

An Easy Experiment To Try

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

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

The Untempered Chromatic Scale (Part 2)

Posted by on Feb 8, 2013 in The Lattice, The Notes | 0 comments

A closer look at the untempered 12-note scale reveals some interesting patterns.

Here’s the scale on the lattice again. This time, I’ve colored the notes as follows:

Red = notes from the central, Pythagorean row, or spine, of the lattice. They are generated by multiplying and dividing by 3.

Green = notes from the next row up. These notes are a major third up from the ones on the central spine — you generate them by multiplying by 5.

Blue = notes from the row below the central one. You make these by dividing by 5, which means they are a major third below.

rgb latticeNow here’s the scale again, colored the same way.

colored chromatic

Notes in red are tuned about the same as their equal tempered counterparts.

Notes in green (the majors) are all flatter than ET.

Notes in blue (the minors) are all sharper than ET.

OK, so what? I can think of several immediate ways to use this information.

If you are a singer/guitar player, remember that you can only bend notes up. You can make your guitar playing sweeter if you:

  • Avoid bending roots and fifths, play them right on the money
  • Bend minor thirds slightly, and
  • For sure don’t bend major notes. It actually helps to mute or avoid major thirds in your guitar playing, and leave those to the vocal. There are several ways to play a G chord, for example. Try singing a simple song in G (maybe Silent Night or Ring of Fire), using the classic G chord:


There are two equal-tempered major thirds in this chord.

Now try it fingering this way:

G_major_chord_for_guitar_open_position_(no_doubled_third)For even more clarity, mute the 5th string with your left hand, and there will be no major thirds in the chord at all. It is all G and D notes, roots and fifths. Can you feel the difference in your singing? There is a division of labor: the guitar plays the notes that are in tune in ET, and the voice sings the rest, including that major third. I think you will find it much easier to sing, like your voice falls into a pocket instead of fighting the intonation of the guitar.

Gotta go, I have a show tonight, but I’ll have a lot more to say about this sort of thing. I do welcome comments and questions, there’s a contact page and you are invited to email me if you’d like to discuss this stuff.

Next: An Easy Experiment To Try