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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 10, 2012 in Just Intonation | 0 comments

The Chord of Nature

When I first learned to play guitar, I would sit on the couch late at night and pluck the low E string, again and again, and just listen to the beauty of the sound as it died away.

That low E note is not just a simple vibration. The full length of the string is vibrating at about 82 Hz. But a pure 82 Hz note doesn’t sound like a guitar string at all. It sounds like this:

82 sine

The guitar sound is much more complex:

guitar low e

The difference comes from the fact that a string doesn’t just vibrate along its whole length — it also vibrates at twice the frequency, three times, four, and so on — all at the same time!

Maybe you did this as a kid. I did. When you get a jump rope going, you are essentially vibrating a big string. It has a characteristic frequency, maybe two cycles per second, set by the length of the rope and the amount of tension, just like a guitar string. This frequency, the natural vibration speed of the whole string, is called the fundamental.

But if one person holds their end still, and the person on the other end moves the rope twice as fast as usual, a funny thing happens. The rope divides in two, and the center point stays still, while each half does its own circle. Again, the length and tension determine the natural speed. Go three times as fast, and three sections will appear. These higher-mode vibrations are called harmonics.

Here are a couple of guys in lab coats to demonstrate:

Video

This only works when you hit the right frequencies. Spin the rope at, say, 2 1/2 times the natural frequency and everything falls apart. The stable frequencies are the fundamental, 2x, 3x, 4x, 5x and so on. This video shows a string getting stable at 6x, 5x, 3x, and the chaos that happens in between.

Video2

When you pluck a real string, it will vibrate in all these modes, generating a complex sound. The particular recipe of added harmonics creates the timbre, or tone, of the note.

Here’s that same pure 82 Hz tone, with the harmonics 2x, 3x, 4x and 5x added successively:

Chord of Nature 2

This is the Chord of Nature. It is a sonic manifestation of number, and of the laws of the universe, and it’s very simple. If the fundamental frequency is 1, then the frequencies of the harmonics are 2, 3, 4, 5, 6 and so on. And somehow, our perception of sound is designed so that this sounds beautiful.

Here, in contrast, is the same demonstration but with the harmonics detuned randomly by less than two percent:

Chord from Hell 2

Yipe! Now go back and listen to the first one as a palate cleanser.

There is something deep inside us that recognizes the series of harmonics, and, for most of us, labels it “beautiful.” There is some connection between those small, whole numbers and musical beauty.

Next: Notes As Ratios

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