Pages Menu
Categories Menu

Posted by on Jun 19, 2013 in Consonance, Just Intonation, Resonance, The Lattice, The Notes |

The Compass Points

There are two basic directions on the lattice: multiplication and division.

If I start with a note, and then multiply it by 3, or 5, or 7, I will get a harmony note with overtonal energy. Such a note is in the natural overtone series of the original note.

Overtonal energy is stable, restful, it belongs where it is and wouldn’t mind staying there.

If I divide by 3, 5 or 7, I get a completely different kind of note. I call this division energy “reciprocal,” after W.A. Mathieu’s suggestion in his amazing book Harmonic Experience.

Reciprocal energy is restless, unstable. The note wants to move, or for the music to come to it, until it is overtonal.

On the lattice of fifths and thirds, there are two axes, fifths and thirds, and two directions, overtonal and reciprocal.

This makes four total directions one can move on this lattice. Each direction has own characteristic flavor, or energy. I use the following names for these energies, mostly after Mathieu.

  • Dominant = East = Overtonal fifths
  • Subdominant = West = Reciprocal fifths
  • Major = North = Overtonal thirds
  • Minor = South = Reciprocal thirds

Compass Points

Every interval has its own unique recipe of moves in these four directions. The perfect fifth has pure dominant energy, the major third pure major. The minor third, b3 on the lattice, is a compound note — dominant and minor.

It’s interesting to look at the minor third (b3) from the viewpoint of tonal gravity. On the horizontal axis, dominant/subdominant, the b3 is overtonal, stable, restful. On the vertical axis, major/minor, the note is reciprocal, unstable, restless.

Tonal gravity is stronger the closer you are to the center. To make a minor third, you multiply by 3 (an overtonal jump of a fifth), and divide by 5 (a reciprocal jump of a third). I know, 3 generates fifths and 5 generates thirds, a confusing coincidence.

Fifths are closer to the center, harmonically, than thirds are, so the overtonal energy is stronger than the reciprocal.

This makes the minor third a stable note, although less stable than the major third. Songs can end on a tonic minor chord and they will still sound finished.

Next: Leading the Ear

Read More

Posted by on Mar 5, 2013 in Equal Temperament, Just Intonation, The Lattice | 0 comments

Why Equal Temperament?

full lattice all-01The picture to the right shows the lattice of fifths and thirds, a map of untempered harmonic space, extending to infinity. The map shows how to venture farther and farther from home, the tonic, and stay perfectly in tune no matter how far you roam.

Every single note in this infinite matrix is tuned to a different pitch. You can go out to Mars and beyond, and you will never see the same note twice.

In pure vocal music, this is not as hard as it seems. The voices will tune up to each other, and it’s natural to sing the pure intervals. So if the piece makes its way to the far north, by small steps, the voices may be singing something like a #3, #7 and ##5, and it will be a nice pure major chord.

Want to hear this in action? Here’s a video of a piece by Guillaume de Machaut. I find this music exhilarating. Check out the astonishing note at 3:00. It’s not jazz, not classical, not blues, it’s adventurous harmony on the lattice of fifths and thirds.

In the 1300’s, before temperament started taking over in Europe, there was a flourishing of untempered music, both secular and spiritual, called Ars Nova. Machaut was one of the greatest composers of that era. In Harmonic Experience, Mathieu shows a map of a Machaut piece that wanders amazingly far on the lattice, staying in tune all the way.

The only instruments that can really play like this have infinitely variable pitch. Voice is top dog, although the fretless stringed instruments can do it too (standup bass, violin, etc).

vogel2For instruments of fixed pitch, such as pianos, organs, guitars, lutes and accordions, the tuning of the lattice in just intonation is an absolute nightmare. How do you accommodate all those pitches? The keyboard to the right gives it the old college try. (Photo is from a gallery of such keyboards at H-pi Instruments.) Yipe! ltbb_035-44k

Fixed pitch instruments work just fine if you stay in a small part of the lattice, and stay in one key. But after Ars Nova, European composers and listeners got more and more interested in wandering the map, and in changing keys, or modulation. So they started to look for compromise tunings, in which one note could represent several nearby ones, close enough in pitch that the ear would tend to interpret them as the pure note.

For example, there are two major seconds on the inner lattice. In just intonation, the 2 (ratio 9/8) is tuned to 204 cents. The other major second, 2- (ratio 10/9) is tuned to 182 cents. If the major second on your instrument is tuned to, say, 193 cents, it will be right in the middle and you can use it to play both notes, slightly but perhaps acceptably out of tune.

There are many possible ways to “temper” the scale, and each one compromises different notes. Over the next few hundred years after Ars Nova, tunings evolved through a bunch of meantone tunings, which detuned fifths and left thirds quite pure, through well temperament, which spreads out the detuning enough that it becomes possible to play in all keys. During this lattice study I discovered, to my surprise, that Bach’s Well-Tempered Clavier was not written for equal temperament. In ET, all keys sound exactly the same, but if Bach is played in the original tuning, each key sounds slightly different. Such key coloration was an integral part of the music, and composers took it into account.

Finally, by the last half of the 1700’s, equal temperament had become pretty much standard.

Twelve-tone ET completely flattens out the lattice, so that each block of twelve tones (the different colors in the top picture) is tuned exactly the same. It’s sort of like a map projection, in which the the geography is slightly distorted so that the curved surface of the earth can be represented on a flat page. In ET, the fifths are very much in tune (only off by 2 cents), and both major and minor thirds are considerably compromised (off by 14 and 16 cents respectively, quite audible). The minor seventh (Bb in the key of C) is the farthest off of the ET notes, 18 cents. Click here to hear the JI and ET minor sevenths compared.

This is how the central portion of the lattice looks in equal temperament (in the key of C):

12ET central-01

Whew! Familiar territory. There’s the tonic major chord, C-E-G, and the relative minor, A-C-E, and so on, and it’s easy to see how they relate to each other.

When you start expanding the ET lattice, it’s a simple repeat. Starting with the 10/9 major second:

12ET 2--01

No pesky commas, it’s just another D. Note that a new chord has appeared, the minor chord on the second degree of the scale — D-F-A, called the ii chord and very common in jazz. Here’s the whole lattice, converted to ET.

ET all-01Now the blocks repeat exactly. Think of the lattice as a horizontal surface, extending to the north, south, east and west, and imagine the pitch of the notes as the vertical dimension. The untempered lattice has a tilt to it — up to the east and down to the west, by 22 cents per block, and down to the north, up to the south, by 41 cents per block. The equal tempered version is flat. You can wander at will, and play everything with just 12 pitches.

I oversimplified the ET names in order to show the repetition. For example, in the yellow block just north of the center, Ab really should be G#. In ET, these are exactly the same in pitch, but calling the yellow one G# helps in understanding where it is on the lattice and how it might be used in a composition. The following lattice shows a more informative way to name the ET notes. The pitches of the notes in the blocks are still exactly the same — 100, 200, 300 cents and so on. A C## is just the same as a D, in equal temperament. The same in pitch, but not in function.

ET all real names-01

This lattice explains why classical music has such oddities as double flats, double sharps, and weird notes such as E#. Why not just say F? E# and F are tuned the same, but they are in different places on the lattice, and if you see an E#, you know you’re in the northern zone.

Beethoven, who helped usher in the Romantic period, used equal temperament to roam the lattice like a wild tiger. Some of his music goes so far out on the map that quadruple flats appear. Click here for some crazy Beethoven stuff — the text is pretty dense but just look at the music notation!

Next: Rosetta Stone

Read More

Posted by on Feb 17, 2013 in The Lattice, The Notes | 0 comments

Extending the lattice

As I’ve analyzed my songs on the lattice, and written new music using it as a tool, I have found that I have a certain palette of notes in my mind, a territory of the lattice that I can hear and think with. The notes in this portion are distinct individuals for me. Each one has its own personality, a distinct mix of attraction, repulsion, beauty and function. I’ve described and given examples of many of them.

When I wrote Flying Dream in 1981, I was consciously trying to write a song that used all twelve notes of the chromatic scale. The first part of making the Flying Dream animation was to reverse-engineer my own song, figuring out with my new tool (the lattice) what I had been instinctively hearing at the time.

220px-Crayola-64It turned out that I had been hearing about 18 notes in the song, including blue notes, and notes up in the northern part of the lattice. That made sense. I remember, as a kid, being disappointed to find out that there were only 12 total notes in music to work with. It seemed to limit the possibilities, like being stuck with the 8-color Crayola box.

The music I love to listen to, and make, has the big 64-color box with built-in sharpener. What’s up with Mick Jagger’s “Oooooh,” at the beginning of Gimme Shelter, or that guitar lick in Dizzy Miss Lizzy? These notes can’t be found on the piano, unless you have a pitch bend wheel. Check out this clip of Ray Charles bending notes in 2000 — now there’s a use of technology! The mystery of those notes, and others like them, has stuck with me, and now I feel like I’m getting to know them as friends.

Next: Another Major Second: The 10/9

Read More

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

Read More

Posted by on Nov 15, 2012 in The Lattice | 4 comments

The Lattice

In 1739, the great mathematician Leonhard Euler published something he called a Tonnetz, German for “tone network.” It looked like this:

Euler’s Tonnetz organizes the notes into a matrix, instead of a scale. Moving down and to the left represents motion by an interval of a fifth (V) in musical space. Down and to the right shows movement by a major third (III).

The lattice has been rediscovered and redrawn many times over the years. One of my favorites is the Duodenarium of Alexander Ellis, which showed up in his appendix to Helmholtz’s pioneering book, On the Sensations of Tone, in the late 1800’s.

Now we’re talkin’! C is at the center. The fifths go up and down, and thirds from left to right, leading to a square grid.

One of W. A. Mathieu’s innovations in Harmonic Experience is to slant the axes and make them line up with the musical staff:

Seriously, if this blog interests you, please get a copy of this book. I have no stake in you doing this, except that I believe the more broadly understood this man’s work is, the more great music will be made.

I’ve been messing around with the lattice for a year and a half now, and I’ve morphed it into a form that suits my own musical work.

Further slanting the thirds axis to 60 degrees makes it a hexagonal lattice, and for me the relationships between the notes become more intuitive. The major chord is now, appropriately, a stable-looking triangle. And a new axis appears, northwest to southeast: movement by minor thirds. I follow Mathieu’s example and show this one with a dotted line, because it isn’t a direct move: the minor third is a third down and a fifth up, a compound move on the lattice — a major (sorry) insight into the nature of the minor third. Much more on that one later.

Japanese mathematician Shohé Tanaka drew a hexagonal tone lattice in the 1800’s. I haven’t been able to find a picture.

Movement to the right represents multiplication by 3, that is, up a fifth. Up and to the right means you’ve multiplied by 5, up a major third. Left means division by 3, down a fifth. Down left is division by 5, down a major third. The tonic, 1, is at the center (below left of center in this portion). The grid goes out to infinity. This is the region encompassed by Flying Dream, which in fact covers most of the territory I’ve found useful so far, a major reason I chose that song for the video.

Next: The Tonic Major Chord

Read More