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Posted by 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.

Piano-keyboard

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

full lattice all-01

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.

Rosetta_stone_(photo)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.

3-01For 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

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Posted by on Nov 25, 2012 in The Lattice | 0 comments

The Major Scale

The notes 1, 2, 3, 4, 5, 6 and 7, clustered at the center of the lattice, constitute a major scale. This tuning uses the smallest ratios (the ones with the lowest numbers) available for each position in the scale. It goes back at least to Ptolemy in the 100’s AD.

I find it visually beautiful. It’s like a cat’s cradle.

Here it is again, with a drone on the tonic, to show how the notes resonate with the drone. Each one has its own flavor, its own harmonic character.

Notice how the melody never moves from a note to the note next door. It always moves two grid segments. This is a first look at the difference between harmonic space and melodic space.

Melodies “like” to move up and down on a linear scale. They want to go to a nearby note when they move — that is, near by in pitch. We hear, and sing, small movements in pitch better than we hear leaps.

Harmonies “like” to go to nearby notes too, but harmonic space is different than linear, melodic space. The 1 and the 5 are harmonic neighbors. In fact, they are as close together as notes can be, harmonically, without being the same note — a single factor of three. But they are far apart melodically — the 5 is almost at the midpoint of the scale.

1 and 2 are melodic neighbors, It’s easy to for the voice to move from one to the other. But they are far apart harmonically — two factors of three. A small move in pitch can produce a large harmonic jump.

Arranging a melody and chord progression involves interweaving the notes so they work in both spaces. The melody will tend to move up and down by small melodic steps, close together on the scale. The chords will tend to move by small harmonic steps, close together on the lattice.

It’s a bit like designing a crossword puzzle, working “up” against “down” until it all fits. The lattice is a wonderful tool for visualizing this dance.

Next: Reciprocal Thirds

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Posted by on Nov 17, 2012 in The Lattice | 0 comments

The Tonic Major Chord

The tonic is the center of the lattice. A drone note on the tonic establishes the center of that particular musical universe.

Adding a major third and a perfect fifth (5/4 and 3/2) further reinforces the center and starts to carve out some territory on the map.

This is the tonic major chord:

In my view, the tonic major helps the ear grab onto the center, by adding two notes that point directly at it. The ear has more information to work with.

The mind has amazing real time mathematical ability. Maybe a more accurate way to say this is that the mind has an amazing ability to quickly analyze and predict physical phenomena. The physical phenomena can be described by math. I don’t think the mind is working with arithmetic calculations at blinding speed, like a computer. It’s more of a massively parallel, holistic analog processor, that achieves a similar result.

Willie Mays used to catch fly balls with his back to the plate. Here’s a famous one:

Mays watches the ball start its flight, calculates the parabola it will follow (fine tuned by the conditions that day), and sets out at top speed for the spot, 400+ feet deep in center field, where he knows it’s going to land. He doesn’t (can’t!) look at the ball until it’s almost upon him. Marvelous.

So the ear hears a note, another one at 3x the frequency (remember octaves don’t count, 3/2 works like 3/1 in this regard), and another one at 5x. All three notes are direct signposts, pointing exactly at the tonic. Here we are, says the mind.

This may be why the equal-tempered major third gives me that slight queasy feeling. The tonic is the tonic, all right, but that equal-tempered third doesn’t point right at it! It’s close enough that the ear correctly identifies it, but it’s actually pointing at a note about 1% sharp of the tonic, and something sounds subtly off, like day-old sushi.

Here it is again: pure third, ET third, pure third. The middle note, the ET third, has a ratio of about 5.04/4.

JI3 vs ET3

Is it slight tonal vertigo? Where is home?

Next: Compound Notes

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Posted by on Nov 12, 2012 in Just Intonation, The Notes | 0 comments

Octave Reduction

Doubling the frequency of a note certainly changes it. The ear hears a higher-pitched note. But there is something in the essence of the note that does not change, a character that stays consistent through the octaves.

This allows a process called octave reduction. When you’re working with notes as ratios, it’s convenient to multiply or divide the raw ratio by 2, as many times as is necessary to bring it into the same octave as the tonic.

3/1 generates a perfect fifth. 3-1

This note is actually an octave plus a fifth above the tonic. Now divide by 2 and you have 3/2, one and a half times the original frequency, and just a fifth above. 3-2

The reference frequency is 1, the octave is 2, so what you want to achieve with octave reduction is a ratio, or fraction, between 1 and 2.

These are the beginnings of a scale, a collection of notes within a single octave. Such a scale can be repeated up and down the octaves to cover the whole range of hearing.

Next: The Major Third

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Posted by on Nov 11, 2012 in Just Intonation, The Lattice, The Notes | 0 comments

Notes As Ratios

Notes are pitched sounds. A given note means little by itself. It could be the tonic of a key, or some member of a scale based on a different tonic. By itself, it generates no tension, resolution or sense of place on the harmonic map.

So when I name a note in this blog, I’ll usually be referring to a ratio, the relationship between the note and a reference note — the tonic, or the root of a chord, or another note in the harmony or melody.

Ratios are fractions. The first number is divided by the second number to give the value of the ratio.

If the tonic is, say, 100 Hz, then another 100 Hz note is related to the tonic by the ratio 1/1. This is the interval of a unison, two identical notes.

Each note name on the lattice represents a unique ratio, relative to the tonic. The 1, at the center, stands for 1/1.

Next: Octave Reduction

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