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Posted by on Nov 21, 2012 in The Lattice, The Notes | 2 comments

Mixed Messages

The harmonic lattice can be divided into four quadrants.

Northeast is pure overtonal energy. All these notes are reached by multiplication alone. Powers of 3 and 5 are in the numerators, and the denominators are all powers of 2.

Southwest is pure reciprocal energy. You get there by dividing. All the 3’s and 5’s are on the bottom of the ratio (or fraction), and the powers of 2 are on the top.

Northwest is mixed reciprocal and overtonal energy. You multiply by fives and divide by threes.

Southeast is also mixed. Multiply by threes and divide by fives.

The major sixth is in the northwest quadrant. First divide the tonic by 3, which gives you a perfect fourth. Then multiply that by 5, and you get the major sixth. Its ratio is 5/3. You can do this in any order, of course — multiplying by 5 gets you to the major third, and dividing by 3 brings you to the sixth again.

A different flavor yet! What do you hear? How does it make you feel?

Musical sensation can be related to other senses and emotions, but really these feelings have their own quality. It’s like a new sense. I had a friend once who had never tasted a peach. She didn’t like the feel of the skin, and she hadn’t dared bite into one. I was saddened, of course, but how to describe the taste of a ripe peach to someone who’s never tasted one? You can draw comparisons forever, but nothing will prepare them for the reality.

That was a long time ago. I hope that some time later, she found herself in the perfect place and went for it. There should be a First Peach ceremony. Yum!

The individual qualities of these notes are easier for me to hear in just intonation than in equal temperament. They are purer flavors, a more direct experience. Later, I’ll present some notes that are very close together in pitch, yet feel different. In just intonation, there are different notes for peach, nectarine and apricot. In equal temperament, a single note, close in pitch, will represent all three. Once you’ve tasted the actual fruits, equal temperament works better than it did before. You know what it is you’re supposed to be tasting.

OK, I’m out on a quivering limb of analogy now, and way ahead of my story. We have enough notes now to start playing with them.

Next: Names

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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 13, 2012 in The Notes | 0 comments

The Major Third

Multiplying a note by 2 creates an octave, and multiplying it by 3 creates a perfect fifth.

Multiplying by 5 gives yet another new note, the pure major third.5-1

5/1 is over two octaves above the original note, so you have to reduce it twice (divide by 4) to get it down into the same octave.5-4

Now we have four notes: 1/1, 5/4, 3/2 and 2/1 — enough for a scale.1-3-5-8

This scale is contained in the chord of nature, and it pops up all over the place. A clear example is the bugle.

Bugles have no valves or keys. So how can you play more than one note on one?

A bugle is a long tube full of air, curved so it fits in a small space. The player’s lips get the air column vibrating, and by changing the tightness of her lips, the player can coax the air column into vibrating along its whole length, or get it to break up into sections, just like the jump rope in the Chord of Nature demonstration.

Here are the bugle notes: bugle scale

Two sidebars before I go on.

1) Isn’t it strange that when you multiply by 3 you get a fifth and when you multiply by 5 you get a third? The note names come from their position in a seven-tone scale. Here’s how our new scale fits with the standard do-re-mi. The notes we’ve explored are played louder to set them apart. five notes in do re mi

The 5/4 note pops up third in the scale and the 3/2 note comes up fifth. It’s just a confusing coincidence, based on our fondness for seven-tone scales.

2) Here’s a sneak preview of why I’m going to all this trouble. The equal-tempered major third that we’ve been hearing all these years is not tuned to the 5/4 ratio. It’s tuned sharp, by almost 1%. This isn’t enough to make the note sound obviously sour, but it’s certainly enough to change the feel of it.

Try listening to the following example a few times, and pay attention to how you feel while listening. JI3 vs ET3

The first note you hear is the tonic with a pure major third. The second note is with an equal tempered major third. Then it goes back to the pure 5/4 note. The pitch difference is small, but I perceive an uneasiness, almost a queasiness about the equal-tempered version. Do you hear a difference, and if so, how does it feel to you?

Next: Harmonic Space

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Posted by on Nov 9, 2012 in Just Intonation | 0 comments

Between the Keys

I grew up thinking that music was made with a particular set of twelve notes, the ones on the piano keyboard. I had a vague sense that there were other scales in the world, but I thought of them as “more primitive” or perhaps subsets of the 12-tone scale, like that pseudo-Asian music you make if you play around on the black keys of the piano. I certainly didn’t know that those 12 notes, now so unconsciously established that hardly anyone in Western culture even questions them, are a relatively recent invention. In Europe, where they first caught on, they were fought bitterly for a century or so before they became the norm. Even now, much of the world still does not tune to these notes, although they are still spreading.

But I also grew up deeply aware of blues singers, and that notes sung “blue” could not be duplicated on the piano, or on the guitar without bending strings. Something was always different about rock, country and other blues-influenced music. All my favorite music had this quality in common — somehow richer in sound, with more heart, and it wasn’t just feel. And it wasn’t just blues either — almost all vocal harmony had “it” too, regardless of genre.

When I was a teenager, I heard a tiny phrase that hit me like Sirius falling from the sky in the Truman Show. Here it is, fair use excerpt:

The Note

Hear it there, at the end? In the right channel, George Harrison plays something you absolutely cannot play on a piano, yet it is perfectly in tune. There is a wealth of information in that little phrase — it points to a whole world living there, in between the keys. That lick has stuck with me for all these years, a sign in the sky, that there was a lot more to know about music than I had been taught in textbooks.

Next: The Chord of Nature

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Posted by on Nov 8, 2012 in Background, Just Intonation, The Lattice | 2 comments

Harmonic Experience

Over the next series of posts, I’m going to explain how the lattice in the Flying Dream video works. Before I do, I want to take time to mention a terrific book.

I started investigating just intonation in earnest in early 2011. A couple of months in, my friend Kay Ashley loaned me her copy of Harmonic Experience, by W. A. Mathieu. Thank you so much, Kay!

I spent a few weeks with Kay’s copy and very soon knew I had to have my own. I devoured the book almost daily for at least a year. I still pull it out often, lug it to a cafe for browsing over breakfast, do bibliomancy with it if I’m stuck creatively, take it on vacations.

Harmonic Experience is the only music theory book I’ve read so far that actually increases my understanding of music, rather than obfuscating it. It’s huge, which could be intimidating. But I found it to be immediately accessible and entertaining. Mathieu has a great, light sense of humor. The concepts are introduced at a beautiful pace. And the ideas he presents are enlightening. “Aha” experiences abound.

Much of what I’ll present in this blog is heavily influenced and inspired by Mathieu. The lattice itself goes back to Euler in the 1700’s, but Mathieu expands on the idea enormously, arranging it so it corresponds to traditional musical staff notation, using it as a means to understand equal temperament, harmony, melody, chord progressions, world music, and much more.

Mathieu uses the term “positional analysis” to describe his system. For me, positional analysis opens the black box. It shows what’s happening in there. When my music is informed by the lattice, it makes more sense. I have more control over the effect it has on me and my audience. And it’s way more fun, because I know more about what I’m doing and why, rather than flailing around finding good sounds by instinct. And when I do compose by instinct (which is essential), I understand better why it sounds good, and can expand on my inspirations in a rewarding way.

‘Nuff said! If music theory has been frustrating for you in the past, as it has been for me, I can’t recommend this book highly enough.

Next: Between the Keys

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