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Posted by on Nov 20, 2012 in The Notes | 0 comments

A Reciprocal Note: The Fourth

All the notes I’ve discussed so far are found above and to the right of the tonic, in the northeast quadrant of the map. These notes are generated by multiplication alone.

What about the notes that are generated by division? These are found to the left and down on the lattice.

The closer a note is to the tonic, the smaller the numbers are, and the easier it is for the ear to tell where it is. I’ll cover this much more in later posts, but I think the character of an individual note, its unique harmonic color, is largely determined by two signals it sends to the ear and mind:

1) How far away is home (the tonic)?

2) What direction is it?

And the closer the note is to home, the clearer the signal is.

The perfect fourth is the same distance from the center as the 5 is, but in the exact opposite direction: divide-by-3 instead of multiply-by-3. So it sends a signal of equal strength and opposite direction. How does this mirror-fifth sound?

I hear beauty, and tremendous tension. Something has to happen here, and soon — it feels like a pencil balancing on its point, unstable equilibrium.

Here are two of the most powerful phenomena in music: tension and resolution.

One resolution is right next door: the major third. It’s only a half step lower, and it is a point of stable equilibrium.

Aaaaahhhh.

There’s another way to resolve the tension, and that is to move the playing field so that the fourth is in a stable spot. This is done by changing the root. It’s a chord change. Check this out:

The mountain has come to Muhammad.

The ratio of the perfect fourth is 1/3. This can be octave-reduced (octave-expanded) by moving it up two octaves, to 4/3.

The energy of the fourth, the division energy, has had a number of names. Harry Partch, a major composer and explorer of music in just intonation, called the quality of the right-and-up harmonies (the ones you get to by multiplication) otonality, from overtone. He called the energy of division-based harmony utonality, for undertone.

Once again I’m going with Mathieu on this one. In Harmonic Experience, he gives an excellent rationale for calling this energy reciprocal. I think he’s right. Each overtonal note has its mirror twin, and the twins are identical, just upside down from each other — reciprocals. The fourth is the reciprocal of the fifth.

Next: Mixed Messages

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Posted by on Nov 18, 2012 in The Lattice, The Notes | 0 comments

Compound Notes

Multiplying the tonic by 2, 3 and 5 creates the octave, fifth and third respectively. The ear hears these intervals very well. We can easily sing them. Each one has a feel, a sort of harmonic flavor, that makes a fifth a fifth and a third a third.

It turns out that the ear can also easily hear compounds, that is, combinations of these low primes. Combining 2 with anything else simply puts it in another octave. But when you combine 3 and 5, or 3 and another 3, you get entirely new flavors. Here’s an example:

The final note is an octave plus a major second above the tonic — a major ninth. Its ratio is (3/2) x (3/2), or 9/4. It has a haunting sound, to me, a different beauty certainly. A new crayon in the box.

Next: The Major Seventh

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Posted by on Nov 11, 2012 in Just Intonation, The Lattice, The Notes | 0 comments

Notes As Ratios

Notes are pitched sounds. A given note means little by itself. It could be the tonic of a key, or some member of a scale based on a different tonic. By itself, it generates no tension, resolution or sense of place on the harmonic map.

So when I name a note in this blog, I’ll usually be referring to a ratio, the relationship between the note and a reference note — the tonic, or the root of a chord, or another note in the harmony or melody.

Ratios are fractions. The first number is divided by the second number to give the value of the ratio.

If the tonic is, say, 100 Hz, then another 100 Hz note is related to the tonic by the ratio 1/1. This is the interval of a unison, two identical notes.

Each note name on the lattice represents a unique ratio, relative to the tonic. The 1, at the center, stands for 1/1.

Next: Octave Reduction

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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 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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