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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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Posted by on Nov 7, 2012 in Just Intonation, The Lattice | 0 comments

The Tonic

The heart of the lattice is the note called 1. This note is the tonic.

Almost all the music you hear — pop, rock, classical — has one note that is at the center, a master note against which all other notes are measured. That note is the tonic. It’s the Do of Do Re Mi. When you call a scale “G Major,” or say that a song is in the key of G, the G is the tonic.

A single note means little by itself. But when it’s considered in relation to the tonic, it acquires meaning. The examples in yesterday’s post show how a note changes character when played against different tonics.

The tonic establishes the framework for the rest of the notes in a piece. It’s the anvil on which the music is forged.

The tonic can be any note. When you tune your guitar by the campfire, without a tuner, just tuning it to itself, you’ve chosen a reference frame that will make perfect sense, regardless of whether it’s the same frame as a piano or orchestra back home. You can happily play great music in the key of G-and-a-half, if you’re playing solo.

Once you’ve established the tonic, the rest of the notes are tuned, and named, relative to that note. The tonic is the center, the Big Bang of that particular musical universe. The rest of the structure comes from the interplay between the tonic and small, whole numbers — mainly 2, 3, 5 and 7.

The tonic is Home. The lattice shows how music is a journey, away from home and back again, through different lands, each with its own scenery and feeling.

Next: Harmonic Experience

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

Beauty is Truth

It’s probably Keats’ most famous pair of lines:

‘Beauty is truth, truth beauty,—that is all
Ye know on earth, and all ye need to know.’

I believe he’s right on the money. I think that when we experience beauty, it’s because we have seen a little deeper into the nature of things.

This seems especially true of mathematical beauty. I had a college friend who found math exquisitely beautiful. He bought a blackboard for his room, and stayed up until all hours, glorying in the work. Elegance, simplicity (but not too much!),  and beauty are important guidelines to the rightness of a solution or direction of research. A sense of beauty guides the scientist as well as the artist. I’m really familiar with this from my engineering career.

So there is the nugget of my own epiphany:

The beauty of music is the beauty of mathematics, perceived in real time.

We see this in its visual manifestations all the time. The curve of the cables of the Golden Gate Bridge, the pattern of seeds in the sunflower, the rings of Saturn — all clear manifestations of the way the universe works, that can be described by math, and that we find beautiful.

Music presents a pure, distilled form of this: beauty created by small, whole numbers and their relationships to each other.

Next: Notes and Intervals

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Posted by on Nov 4, 2012 in Background, Just Intonation | 0 comments

Pythagoras’ Epiphany

Pythagoras was a Greek philosopher who lived about a century before Athens’ golden age. Some time before 530 BC, he had an epiphany. He had been investigating vibrating strings, and found that when you cut the length of the string in half, the note it makes is an octave higher.

Sound is vibration. When one sound is vibrating twice as fast as another, it still sounds the same in some crucial way. The pitch is higher, or lower, but somehow we perceive it as having the same essential character. A C note, multiplied or divided by two as many times as you like, still sounds like a C.

Here are all eight C’s on the piano. They are different in pitch, but all have the same character. Eight Cs

Pythagoras also found that when you shorten the string to a third of its original length, it vibrates three times as fast. The note this creates is different in character from the C. Today it is called a perfect fifth. If you’re in the key of C (that is, if the full string sounds a C), this note will be a G.

This observation led him to what must have been a terrific epiphany — math, particularly number, is at the heart of all things. I sometimes envy those early Greek thinkers — what joy, to come across something basic for the first time!

But you know, everyone, everywhere, has lived in modern times. A thousand years after Pythagoras, Galileo was the first to find out that the Milky Way is made of stars. Can you imagine how he felt? And we are still on the cutting edge — civilization is in its infancy. Future generations will envy us our discoveries while smiling at their primitiveness. “A keyboard, how quaint!”

Pythagoras’ epiphany still has merit. Cosmologists have imagined many alternate universes, with different basic physical constants and laws, curved space, more dimensions — but it’s pretty tough to imagine a universe without number. I believe the integers — 1, 2, 3 and so on, are the most basic things we know about for sure.

Pythagoras actually founded a religion based on this insight. The inner circle were called the mathematikoi, and they lived a monastic life of study. The order had many rules, including a ban on eating beans. Perhaps they worked in close quarters. They also had a rule against picking something up when you dropped it. Cluttered, close quarters! But they found out a lot about math.

Next: Beauty is Truth

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