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Posted by on Jun 19, 2013 in Consonance, Just Intonation, Resonance, The Lattice, The Notes |

The Compass Points

There are two basic directions on the lattice: multiplication and division.

If I start with a note, and then multiply it by 3, or 5, or 7, I will get a harmony note with overtonal energy. Such a note is in the natural overtone series of the original note.

Overtonal energy is stable, restful, it belongs where it is and wouldn’t mind staying there.

If I divide by 3, 5 or 7, I get a completely different kind of note. I call this division energy “reciprocal,” after W.A. Mathieu’s suggestion in his amazing book Harmonic Experience.

Reciprocal energy is restless, unstable. The note wants to move, or for the music to come to it, until it is overtonal.

On the lattice of fifths and thirds, there are two axes, fifths and thirds, and two directions, overtonal and reciprocal.

This makes four total directions one can move on this lattice. Each direction has own characteristic flavor, or energy. I use the following names for these energies, mostly after Mathieu.

  • Dominant = East = Overtonal fifths
  • Subdominant = West = Reciprocal fifths
  • Major = North = Overtonal thirds
  • Minor = South = Reciprocal thirds

Compass Points

Every interval has its own unique recipe of moves in these four directions. The perfect fifth has pure dominant energy, the major third pure major. The minor third, b3 on the lattice, is a compound note — dominant and minor.

It’s interesting to look at the minor third (b3) from the viewpoint of tonal gravity. On the horizontal axis, dominant/subdominant, the b3 is overtonal, stable, restful. On the vertical axis, major/minor, the note is reciprocal, unstable, restless.

Tonal gravity is stronger the closer you are to the center. To make a minor third, you multiply by 3 (an overtonal jump of a fifth), and divide by 5 (a reciprocal jump of a third). I know, 3 generates fifths and 5 generates thirds, a confusing coincidence.

Fifths are closer to the center, harmonically, than thirds are, so the overtonal energy is stronger than the reciprocal.

This makes the minor third a stable note, although less stable than the major third. Songs can end on a tonic minor chord and they will still sound finished.



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Posted by on Dec 3, 2012 in The Notes | 2 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.



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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.

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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?



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Posted by on Nov 15, 2012 in The Lattice | 4 comments

The Lattice

In 1739, the great mathematician Leonhard Euler published something he called a Tonnetz, German for “tone network.” It looked like this:

Euler’s Tonnetz organizes the notes into a matrix, instead of a scale. Moving down and to the left represents motion by an interval of a fifth (V) in musical space. Down and to the right shows movement by a major third (III).

The lattice has been rediscovered and redrawn many times over the years. One of my favorites is the Duodenarium of Alexander Ellis, which showed up in his appendix to Helmholtz’s pioneering book, On the Sensations of Tone, in the late 1800’s.

Now we’re talkin’! C is at the center. The fifths go up and down, and thirds from left to right, leading to a square grid.

One of W. A. Mathieu’s innovations in Harmonic Experience is to slant the axes and make them line up with the musical staff:

Seriously, if this blog interests you, please get a copy of this book. I have no stake in you doing this, except that I believe the more broadly understood this man’s work is, the more great music will be made.

I’ve been messing around with the lattice for a year and a half now, and I’ve morphed it into a form that suits my own musical work.

Further slanting the thirds axis to 60 degrees makes it a hexagonal lattice, and for me the relationships between the notes become more intuitive. The major chord is now, appropriately, a stable-looking triangle. And a new axis appears, northwest to southeast: movement by minor thirds. I follow Mathieu’s example and show this one with a dotted line, because it isn’t a direct move: the minor third is a third down and a fifth up, a compound move on the lattice — a major (sorry) insight into the nature of the minor third. Much more on that one later.

Japanese mathematician Shohé Tanaka drew a hexagonal tone lattice in the 1800’s. I haven’t been able to find a picture.

Movement to the right represents multiplication by 3, that is, up a fifth. Up and to the right means you’ve multiplied by 5, up a major third. Left means division by 3, down a fifth. Down left is division by 5, down a major third. The tonic, 1, is at the center (below left of center in this portion). The grid goes out to infinity. This is the region encompassed by Flying Dream, which in fact covers most of the territory I’ve found useful so far, a major reason I chose that song for the video.


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