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Posted by on Dec 5, 2012 in Just Intonation | 0 comments

Untempered Vs. Tempered

I’ve been listening to yesterday’s chord progression showing off the b7.

I think it offers an excellent opportunity to hear the difference between equal temperament and just intonation.

Equal temperament works by implying or evoking a note rather than playing it exactly. There are dozens of singable notes per octave; ET represents them all with just twelve tones.

Some ET notes are close to their just counterparts; the 4 and 5 are close enough to be essentially right on. The major third is not so great. It’s 0.8% sharp, enough to change the feeling it produces.

The ET b7 is even further off, a full 1% flat of the untempered note. For me, this is enough to change its flavor entirely, and dilute its resonance to the point where it’s just not the same note. I would contend that the real experience of the b7 is not actually available in equal temperament.

Here it is again:

And in ET:

To me, the real b7 sounds triumphant, like its arms are outstretched to the sky after a great victory.

The ET one sounds very different. It’s not unpleasant, but it sure is different. It it a little sad? The leaping dance is gone. The b3 is flat too. Poor minor, no wonder she’s sad! A mortal has seized the hem of her garment and made her earthbound, in order to put her in his power and make her a little better behaved.

Now go back and listen to the JI version. My experience is that I hear it a little sharp for a second, and then it settles in and wow. This is all subjective; you may hear entirely different things. But this example makes it pretty clear, I think, that JI and ET do not sound the same.

So here we have a note, with a distinct (and unique) personality, that produces a physiological sensation that just isn’t quite available in equal temperament. There are a lot more of these to come, with strange and beautiful colors. Really getting into JI and the lattice is like getting the 64-color Crayola box for Christmas. Orange-yellow and yellow-orange, what riches!

One of my favorite phrases in any song comes from the great Greg Brown. In Eugene, from The Evening Call, he says,

The Northwest is good, once you get off I-5 and wander up and down the Willamette dammit, on the back back roads. I know a few people who’d let me park in their drive, plug in for a night or two, stay up late, and talk about these crazy times — the blandification of our whole situation. And then back to the woods. A dog is bound to find me sooner or later. Sometimes you gotta not look too hard — just let the dog find you.

The blandification of our whole situation. Nice one, Mr. Brown. I recommend going back and forth between the last two vids a few times. Deblandification!

By the way, The Evening Call is packed with great lyrics and music. Top notch.

Next: The Minor Second

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Posted by on Dec 4, 2012 in The Notes | 0 comments

The Minor Seventh

The farther we get from the center, the less consonant the notes are, when played against the tonic. Consonance is a whole subject. It’s generally spoken of as though it could be plotted on a scale, from consonance to dissonance. I think this is a big mistake. Consonance has more than one dimension. Trying to force these independent dimensions of consonance onto a one-dimensional scale leads to unnecessary confusion.

Anywayy … The minor seventh is a pretty dissonant note in all dimensions. It’s three moves away from the tonic, down a third and up two fifths:

The ratio is 9/5.

This is some beautiful, exotic harmony.

Here’s a progression that shows off the flavor of the b7, in just intonation:

Hey, that’s beautiful! I worked it out as an illustration, with the idea of showcasing the minor seventh, and it turned out to be really nice music.

See why I’m in love with the lattice? It’s a beauty engine.

Next: Untempered Vs. Tempered

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Posted by on Dec 3, 2012 in The Notes | 4 comments

The Minor Third

Here’s an interesting and perhaps misunderstood note.

It’s a compound move on the lattice: down a third and up a fifth. Or up a fifth and down a third, it doesn’t matter what order. So the ratio is 3/5, or 6/5, octave reduced. The note is the minor third. I call it b3.

It lives a little bit flat of the major third — much less than an equal-tempered half step.

The closeness of major and minor, the small size of this particular half step, is one of the revelations I’ve had in the past couple of years. Major and minor are only about 2/3 of a semitone apart.

The difference between major and minor third is not so much one of pitch, but of polarity. The minor third contains reciprocal third energy and the major is overtonal third energy. A smile is just a frown turned upside down … Here’s an example that shows the reversal in polarity between major and minor third. This is untempered tuning. The pitch is moving by less than a piano key while dramatically shifting the harmonic ground.

I hear that same sort of “breathing” as in yesterday’s post — in, out, in, out.

I say “misunderstood,” because equal temperament changes the character of this note. Mathieu has a nice passage in Harmonic Experience:

When I first found my own voice inside a minor triad, I couldn’t believe it was so — well, so (arggh! I can scarcely say the dreaded word, but here goes) — so … happy. There. We are told from the beginning that minor is sad, the designated mode for angst and funerals. Well, to be honest, the equal-tempered version of the minor third is rather sad. [It] is too narrow, or flat. So piano minor is flat and sounds dull — the fire is out of it. But minor thirds in just intonation, and the minor triads they support, are swift and burning. They have the gypsy left in them, and do some leaping kind of dance.

– W.A. Mathieu, Harmonic Experience, p. 55

The gypsy really comes out to dance when it’s actual music, but to get an idea, here’s that same seesaw between minor and major. This time it’s tuned to equal temperament.

Is it my imagination, or do I hear a little melodrama here? Is the minor overly sad, the major a little over-the-top happy?

You may hear something entirely different. It is very interesting to go back and forth between these last two videos.

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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 22, 2012 in The Lattice, The Notes | 0 comments

Names

Musical nomenclature has been cobbled together over the centuries like a medieval city. Different systems leave their imprint in convention, later developments try to be compatible with accepted names, and the whole thing ends up confusing and contradictory.

Take enharmonic equivalents, for example. G# and Ab are the same note on the piano, the black key between G and A. So why do you sometimes call that note by one name, and sometimes by the other? The answer actually leads to some deep realizations about music, and it comes back to just intonation. In untempered or just music, G# and Ab are not the same note, and which one you choose becomes important. It’s important in ET too — the music establishes a context, and the ear figures out which note it’s supposed to be. But if you grew up with ET, and have no idea that there used to be two different notes there, the names can be confusing. How do you imply one note or the other? Which one is right in a given situation? Why bother? It’s a huge part of writing chord progressions that make sense, but ET by itself isn’t going to tell you what to do. You have to dig deeper for that.

I’ve slowly evolved a personal system I’m very happy with. It’s based on the lattice.

The great advantage of this approach is that it’s entirely unambiguous. Every note on the infinite lattice has a unique name, and that name tells you exactly what its pitch is, and where it is on the map.

The seven notes I’ve covered so far form the core of the system. I’ve dropped all the word names and just use numbers:

1 — the tonic, 1/1

2 — the major second, 9/8

3 — the major third, 5/4

4 — the perfect fourth, 4/3

5 — the perfect fifth, 3/2

6 — the major sixth, 5/3

7 — the major seventh, 15/8

The rest of the notes are named by adding accidentals to modify the pitches. I’ll quantify these later, and explain how they work, but approximately they are:

b — flat by about 2/3 of an equal-tempered semitone

# — sharp by about 2/3 of a semitone

— flat by about 1/5 of a semitone

+ — sharp by 1/5 semitone

7 — flat by 1/2 semitone.

The basic notes occupy the center of the lattice. These seven notes form the major scale.

Next: The Major Scale

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