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Posted by on Jan 16, 2013 in Just Intonation, The Lattice | 2 comments

Melodic Space, Harmonic Space

Throughout my musical education, I’ve been taught that music happens in a linear space. This is the space so beautifully laid out on the piano keyboard.

Piano-keyboard

Music teaching is organized around scales. In most Western music, the full scale consists of twelve notes, equally spaced. Other scales, such as the seven-note minor and major scales, are subsets of this full, “chromatic” scale. Due to octave magic, a mysterious and crucial aspect of our inner perception, when we get to the thirteenth note, we have multiplied the original note by two, and the sequence starts over again.

So, fortunately for musical analysis, melodic space can be described in one octave. It takes about ten of these octaves to cover the range of human hearing.

On the piano keyboard, melodies look the way they sound. When the pitch goes up, you move up the scale, and when the pitch goes down, you move down the scale. Short distances (the shortest is from one key to the next, a half step), feel short. Long distances (more than about three half steps) feel long. This is a good and useful space for visualizing melody.

Harmony, not so much.

Musical nomenclature, as I’ve pointed out before, has grown like an old city over the years. As music theory changes, bits and pieces of the old terminology are appropriated and redefined by new thinkers. The result is a cobbled-together mass that has a lot of weird contradictions and misleading names.

I think one of the most regrettable bits of confusion comes from the word interval.

The distance between two notes on the keyboard is called an interval. When my melody moves by an interval of a minor third, it has covered a distance of three half steps. When I move by a major third, I’ve covered four half steps. The major interval is bigger than the minor one — that’s why it’s called “major.” No problem! The move feels bigger when you sing it.

The problem comes when you start to think about harmony — two or more notes sounding simultaneously. The word “interval,” with the same connotation of pitch difference, is also used to describe the distance between harmony notes. Yet in the world of harmony, the interval, or pitch distances don’t make any intuitive sense at all.

For example, two notes a fifth apart (seven half steps) sound wonderful when played together. C and G are two such notes. They are closely related to each other, harmonically. So are C and F, which are a fourth apart (five half steps). These are the best consonances there are, except for unisons and octaves.

So what about the note in between them, an interval of six half steps?

Yep, none other than the dreaded tritone, the devil’s interval, definitely a dissonant note.

If the linear scale were the best way to think about harmony, wouldn’t the tritone be between the fourth and fifth in consonance? Why would three notes in a row, next-door neighbors on the scale, be so drastically different from each other harmonically? The scale gives no clue. You just have to remember.

Perhaps there is a more intuitive way to visualize harmony, one that puts harmonically related notes closer to each other, and puts the notes that are harmonically farther apart … farther apart?

I think there is indeed a harmonic space as distinct from a melodic space. This space can be illustrated on the lattice. It’s not a good model for melody — scales do a much better job. But it’s a great model for visualizing harmony — what you see corresponds intuitively to what you hear.

The interplay between these two spaces creates the beautiful dance that is harmonized music.

Next: Cents

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Posted by on Dec 10, 2012 in The Notes | 0 comments

The Augmented Fourth

I’ve described eleven notes now, and each one has a piano key to go with it, an equal tempered equivalent.

The one remaining black key has a lot of names. It’s the note between the 4 and 5, right in the middle of the octave — the tritone, devil’s interval, flatted fifth, augmented fourth.

In ET there’s only one tritone, and it precisely splits the octave in half. In JI, there are several tritones, with different tunings, that sound and function differently from each other.

One tritone, that nicely fills out the set of 12 notes, is the augmented fourth:

This note is not like the other black keys. It’s completely overtonal, that is, it is generated entirely by multiplication — x3, x3, x5, or 45/1. It does appear in the Chord of Nature, but so far up that it wouldn’t be audible in the harmonics of a vibrating string. I think the fact that we can hear any harmony at all with this note shows that we can hear compounds of simple ratios, even when the numbers are getting pretty big. If pure ratios were all that mattered, 13/1 would be far more harmonious than 45/1 — the numbers are smaller. But 13/1 is almost nonexistent in the musics of the world, and even 11/1 is very rare.

So the harmonic connection with the tonic is tenuous, but it’s there. I hear a different kind of dissonance than the b6 or b2-, more harmonically distant, but without as much of that urgency-to-move that the reciprocal notes have.

It’s natural to resolve it melodically to the 5:

Or once again we can travel through harmonic space to get back home.

Can you hear yourself getting closer to home with each step?

We now have a set of 12 notes, one for each key of the keyboard. Next, the prime number 7, and then some notes between the keys. Oh, the places we’ll go!

Next: Prime Numbers and the Big Bang

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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 19, 2012 in The Notes | 0 comments

The Major Seventh

The notes get more exotic as you move outward from the center. The ninth is quite consonant, but not nearly as consonant as the fifth. (Consonance and dissonance are descriptions of feelings; they are part of the flavor of an interval, and I don’t think the last word has been written on them yet. I’ll be taking my shot later in these pages.)

For very small ratios such as 3/2, the ear has no trouble perceiving where it is on the map. The signal given by 3/2 is so strong, in fact, that it’s the primary tool used in classical music to move the ear to a new key center.

As the numbers get bigger, the signal gets weaker, and the interval gets more dissonant. To get to the major second, you multiply by 3 twice. Then, using octave reduction, you can put it in any octave you want.  I chose 9:4 in yesterday’s example, giving an interval of a major ninth — an octave plus a major second.

Compounding a fifth and a third gives somewhat larger numbers (3×5 = 15, or a ratio of 15/8) and, sure enough, the note is more dissonant against the tonic. Yet it has its own unique beauty. Presenting the major seventh:

Tomorrow, another kind of flavor entirely, another primary color in the crayon box, if you will.

Next: A Reciprocal Note: The Fourth

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Posted by on Nov 6, 2012 in Just Intonation, The Notes | 0 comments

Notes and Intervals

A note, in music, is a sound with a particular pitch. Pitch is frequency, measured in cycles per second, or Hertz (Hz). The faster the vibration, the higher the pitch.

A vibration, at, say, 220 Hz, all by itself is a note by that general definition. But the note doesn’t acquire its distinct personality until it’s considered in relation to some other note. That relationship is called an interval.

Here is that 220 Hz note, played on a cello, all by itself: 220 Hz

Here it is in relation to a note an octave below, vibrating half as fast, at 110 Hz: 220 and 110

It still sounds like the same note. But now play it with a note vibrating at 1/3 of its frequency, or 73.33 Hz. The 220 Hz note acquires a very different character: 220 and 73

And now with a note at 1/5 its frequency, 44 Hz: 220 and 44

Even though the 220 Hz note always has the same pitch, in a different context it has a different personality and function.

The lattice of the Flying Dream video does not show absolute pitch. Each intersection, or node, represents a note, named according to its relationship to one special note: the Tonic.

Next: The Tonic

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