Pages Menu
Categories Menu

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.

Read More

Posted by on Dec 2, 2012 in The Lattice | 0 comments

Reciprocal Thirds

There are four basic moves available on the lattice of fifths and thirds. They are:

Up a fifth (x3)

Down a fifth (÷3)

Up a major third (x5)

Down a major third (÷5)

Each of these moves has its own harmonic flavor, and they can be combined to create new flavors.

The major scale only uses the first three building blocks. What about the fourth one?

The land of reciprocal thirds is where most of the black keys reside. It’s the world of minor tonality. Here is the sound of a pure reciprocal third:

The new note is a mirror of the 3, an upside-down 3. Its ratio is 1/5, which can be octave-shifted to 8/5. That ratio puts it a little over halfway up the scale, between the 5 and the 6. It’s called the minor sixth or flatted sixth; I use the symbol b6.

There is a beautiful shift of feeling when you move from overtonal energy to reciprocal and back again. To me it feels like breathing in and out. Maybe that’s because I play harmonica. When you blow on a few holes of a Marine Band, you get the 1 chord. When you draw, you get the 4. Breathing in and out takes you back and forth between reciprocal and overtonal territory.

The same action can happen on the 5-axis, with a more exotic flavor:

Hear the shift? Overtonal, reciprocal, back again. Every note on the lattice except the 1 has its mirror twin.

Read More

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

Read More

Posted by on Nov 20, 2012 in The Notes | 2 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.

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

Read More