2013 March | Untempered Music

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Posted by Gary on Mar 12, 2013 in The Lattice, The Notes, Tonal Gravity | 2 comments

Cadences

A cadence is a chord progression that gives a sense of arrival or resolution.

One particular cadence, the V-I (or V7-I) is especially powerful. In classical music, a V-I cadence is practically mandatory at the end of a piece, and it is the biggest gun in the composer’s arsenal when changing keys, or modulating.

The following movie shows a I-V-I progression. It starts on the I to establish the tonic, then there’s tension, then resolution. The V-I cadence draws the ear back to the tonic chord.

Here’s a cadence that visits the V7 first:

To me it looks like the V chord tosses out a rope, lassos the tonic and pulls.

It’s interesting to look at the notes in light of tonal gravity. In yesterday’s post, I laid out two rules of gravity on the lattice:

Movement away from the center creates tension; movement toward the center gives a sense of resolution.
The closer you are to the center in your journey, the stronger the sensations of tension and resolution are. The field is stronger closer in, just like real gravity.

There are four notes in the V7 chord.

The 5 is as close as you can get to the 1 (in harmonic space), so it creates a lot of tension. It is an overtonal note — that is, it appears in the overtone series of the 1. Pluck a string tuned to the tonic frequency, and the 5 will tend to be strongly present in the timbre of the sound. The way I see it, the ear is always searching for home. Every note gives it two clues — which direction is home, and how far away is it? The 5 gives a very strong signal, pulling the ear toward the tonic: “Home is this way, and it’s close! Come on!”
The 2 reinforces this conclusion. It’s farther out, so the signal is weaker, but it is still in the harmonic series of the 1, and it’s pretty close in. The little detective in the ear gets another clue.
Same with the 7, although now the effect is weaker. In traditional theory, the 7 is called a leading tone, and it’s thought to pull melodically toward the 1 — a sort of gravity in melodic space. It “wants” to resolve a half step upward. I feel this too, and I think the harmonic pull reinforces it further.
Then there’s the 4, which is what makes it a seventh chord (the 4 is a minor seventh of the 5). This is a reciprocal note, that is, it’s generated by division rather than multiplication. Like the 5, it points directly at the 1, from point-blank range. Reciprocal energy is different from overtonal energy. To me, it feels as though reciprocal notes are pushing toward the tonic — the message feels more like “Home is that way, now go!”

For the detective in the ear, the 4 slam dunks the case. The only reasonable conclusion is that home is located in that empty space between the 4 and the 5 on the lattice. Any other interpretation is much weaker. Every note in the V7 chord is pointing strongly to the 1, and when the notes collapse inward to the I chord, the resolution is completely satisfying.

I’ve heard a charming story about Beethoven. Apparently the composer was depressed and wouldn’t get out of bed. A friend came by and played some music, ended on a dramatic V7 chord, and sat down to wait. Beethoven finally had to get out of bed and play the tonic chord. Tough love! Don’t know if the story is true, but it certainly could be. The V7 is strong medicine.

When this particular chord shape appears somewhere else on the lattice, it can point so hard to its own center that the ear believes the tonic has moved. It’s as though the gravity of the planets is so strong that it can move the sun.

Next: Intervals

Posted by Gary on Mar 11, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 1 comment

Tonal Gravity

I believe that the great driving force in tonal music, that creates the drama and story of the music itself (independently of any lyrics), is the longing for home.

Home is the tonic. If a song is in the key of A, all the A’s in their various octaves will sound like home.

Although there are many exceptions, most music begins on the tonic, to show the ear what key the piece is in, and ends on the tonic, to bring the listener home again. In between, the music wanders, out and back again, creating tension and resolution.

One of the beauties of the lattice is that it shows a clear graphical display of this tension.

It’s as though the tonic creates a sort of gravitational field around itself. It acts a lot like real gravity. The chords and notes move in this gravitational field, like planets and moons around a sun. The gravitational field follows a few basic rules:

Movement away from the center creates tension; movement toward the center gives a sense of resolution.
Notes that are overtonal from the center, generated by multiplying, located to the right and up, will feel more resolved. Notes that are reciprocal, generated by dividing, to the left and down, will feel unresolved.
The closer you are to the center in your journey, the stronger the sensations of tension and resolution are. The field is stronger closer in, just like real gravity.
The closer together two notes are, the more consonant, or harmonious, they will be when sounded together. The farther apart they are, the more dissonant they will be, the more they will clash.

Roots generate local gravitational fields. I think of them as Jupiter to the tonic’s Sun. When the root is on the 5, for example, it shifts the gravity field to the east on the lattice, and the 2 and 7 become harmonious, consonant notes, rather than dissonant ones. The tonic still has great influence, so the entire chord feels unresolved — a 5 chord pulls very strongly toward the 1 chord, a property that is heavily relied upon in Western music. As long as the 5 is the root, though, the 2 and 7 will be consonant harmonies, because they are close to the 5 on the lattice.

Here is a movie to show how that works. The music starts with a tonic chord. Then, one at a time, the 2 and 7 are introduced. These notes are dissonant, and create a sense of tension against the tonic.

Then the root moves to the 5, and the character of the 2 and 7 changes. Now they form a major chord based on the 5, a harmonious configuration. They have become moons of Jupiter. Hear how the dissonance goes away? But there is still plenty of tension, as now there are three notes venturing away from the center, pulling the ear back toward home.

Then the root moves back to the 1, and the 2 and 7 collapse back in toward the center. There is a sense of arrival.

This movie illustrates another observation: consonance / dissonance and tension / resolution are not the same thing. They both relate to distance on the lattice, but they do not necessarily track together. When the root moves to the 5, the dissonance goes away, but there is a new tension, a drive to resolve toward the center. The ear remembers where home is, and longs for it.

These principles can be consciously used to create desired effects when writing and arranging. Resolution and consonance give the music beauty, and tension and dissonance give it teeth.

Next: Cadences

Posted by Gary on Mar 6, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

Rosetta Stone

Almost all Western music, including my own, lives in the world of tonal harmony. This means:

There can be, and usually are, multiple notes playing at the same time.
There is a key center, or tonic, around which the notes are arranged. The tonic doesn’t always sound — it’s an intangible presence, the home from which you leave on your harmonic journey, and to which you will hopefully return.

The multiple notes can have different functions:

Roots are the fundamental notes of chords. A G chord has its root on G. Roots are local centers that move the ear around the lattice as they change.
Harmonies flesh out the chord. In a G major chord, the harmony notes are B and D. They stake out more lattice territory and add definition to the chord. Is it a G major, minor, seventh? The harmonies establish this.
Melodies dance in the harmonic field set up by the tonic, roots and harmonies. They have more freedom than the others. Melodies travel fast and light, and though they can sing the same notes as the others, they can also travel farther afield, further embellishing the chord, or leading the ear toward the next chord in the progression, or lingering on the last one after it has changed.

All this action is happening in two musical spaces at once.

Melodic space is the world of scales. It’s organized in order of pitch. The piano keyboard is a perfect representation of melodic space.

Harmonic space is the world of ratios. Multiply a note by a small whole number ratio, and you have moved a small distance in harmonic space. Multiply by large numbers, and you have moved a large distance. The lattice is a map of harmonic space.

The two worlds are not the same. Often, they are opposites. The perfect fifth is a small move harmonically but it’s a mile in the melody — bass singers have to jump all over the place in pitch. Small melodic moves tend to be big harmonic ones. A chromatic half step, the distance between the 3 and b3, is only 70 cents, less than the distance between neighboring keys on the piano. But on the lattice, it’s a long haul — down a third, down another third, and up a fifth.

Writing and arranging a song is sort of like designing (rather than solving) a crossword puzzle. There are two intersecting, independent universes, Up and Down. To design the puzzle, you work back and forth between the two, massaging them until they don’t conflict, and each one makes sense on its own.

All of the notes live in both harmonic and melodic space. They may have a foot in one more than the other — the roots tend to move small distances on the lattice, the melodies usually move small distances in pitch, and the harmonies tend to bridge the two, moving melodically while staking out the form of the music on the lattice. But every note moves in both spaces, all the time.

A great advantage of the lattice is that it serves as a sort of Rosetta Stone, a bridge or translator between the two worlds.

The Rosetta Stone was carved in 196 BC and rediscovered in 1799. It immediately became famous because it repeats the same text three times, in three different languages. It was the key that allowed scholars to decipher Egyptian hieroglyphs.

The lattice bridges the two musical spaces by means of the patterns it presents to the eye.

When two or more notes are plotted on the lattice, they will form a particular visual pattern. Any time you see this pattern, no matter where on the lattice it is, the relationship between the notes of the pattern will be exactly the same, in both harmonic and melodic space.

For example, this pattern shows an interval of a major third. The ratio of the frequencies of these two notes is 5/4 (or 5/2, or 5/1 — twos don’t count, they just shift the note by an octave). Any time you see two notes in this formation, no matter where they are, you know they have the following relationship to each other:

Harmonic space: When the notes are sounded simultaneously, they will have the characteristic sound of a pure major third.
Melodic space: When you move from one note to the other, you are traveling a distance of 386 cents, or about four semitones on the piano.

Getting familiar with these patterns, and learning to recognize them wherever they are, has made it easier for me to think in harmonic and melodic space at the same time, which makes writing and arranging music much easier.

Next: Tonal Gravity

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

Why Equal Temperament?

full lattice all-01 The 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).

For 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):

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:

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.

Now 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.

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

Posted by Gary 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.

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.

Doing 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.

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.

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.

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

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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Tonal Gravity

Rosetta Stone

Why Equal Temperament?

The Infinite Lattice

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