I’ve described eleven notes now, and each one has a piano key to go with it, an equal tempered equivalent.
The one remaining black key has a lot of names. It’s the note between the 4 and 5, right in the middle of the octave — the tritone, devil’s interval, flatted fifth, augmented fourth.
In ET there’s only one tritone, and it precisely splits the octave in half. In JI, there are several tritones, with different tunings, that sound and function differently from each other.
One tritone, that nicely fills out the set of 12 notes, is the augmented fourth:
This note is not like the other black keys. It’s completely overtonal, that is, it is generated entirely by multiplication — x3, x3, x5, or 45/1. It does appear in the Chord of Nature, but so far up that it wouldn’t be audible in the harmonics of a vibrating string. I think the fact that we can hear any harmony at all with this note shows that we can hear compounds of simple ratios, even when the numbers are getting pretty big. If pure ratios were all that mattered, 13/1 would be far more harmonious than 45/1 — the numbers are smaller. But 13/1 is almost nonexistent in the musics of the world, and even 11/1 is very rare.
So the harmonic connection with the tonic is tenuous, but it’s there. I hear a different kind of dissonance than the b6 or b2-, more harmonically distant, but without as much of that urgency-to-move that the reciprocal notes have.
It’s natural to resolve it melodically to the 5:
Or once again we can travel through harmonic space to get back home.
Can you hear yourself getting closer to home with each step?
We now have a set of 12 notes, one for each key of the keyboard. Next, the prime number 7, and then some notes between the keys. Oh, the places we’ll go!
The last three notes (b6, b3 and b7) are related to each other. They all contain a reciprocal third. There is a family resemblance of sound and function. (They also all happen to be a little flat in equal temperament. On a guitar it’s a nice trick to bend them a little to sweeten them.)
Here is another note in the family, farther out harmonically, the minor second:
That’s a dissonant interval. The b6 is already tense with reciprocal third energy. Now this b2- (The minus is an accidental to show its exact pitch; more later) is another reciprocal fifth beyond (below?) that note. Its ratio is 1/15, which expands to 16/15 — just above 1. See how the ratios show where the pitch of the note is? 1/1 is the tonic, 2/1 is the octave. 16/15 is just a little bit greater than 1, so it’s just a little sharper than the tonic.
It’s not pitch so much that makes consonance and dissonance. It’s harmonic relationship.
Music is all about tension and resolution. Here’s a very tense note. How to resolve it?
One answer is just a half step away, a drop to the tonic.
That’s a move in melodic space. The tonic is right next door and it’s an easy drop.
On the lattice, the 1 is not a next door neighbor. How about going home through harmonic space instead?
Going to the 4 is an interesting experience for me. There’s still reciprocal tension, but I’m much closer to home — I can smell the stables. It’s as though I felt a bit lost at the b2-, the harmonic distance was too great to really get my bearings. But moving to the 4 allows me to figure out where I am, and where the tonic is, so that the final move home sounds really right. The 4 says to me, “There is home, now go.”
Then the melody moves to the 5, and there is resolution. The 5 sends just as strong a signal as the 4, but of opposite polarity. The 5 says, “Here is home. Now stay.”
It’s a little story, a journey on a microcosmic landscape of attraction, repulsion and beauty.
I’ve been listening to yesterday’s chord progression showing off the b7.
I think it offers an excellent opportunity to hear the difference between equal temperament and just intonation.
Equal temperament works by implying or evoking a note rather than playing it exactly. There are dozens of singable notes per octave; ET represents them all with just twelve tones.
Some ET notes are close to their just counterparts; the 4 and 5 are close enough to be essentially right on. The major third is not so great. It’s 0.8% sharp, enough to change the feeling it produces.
The ET b7 is even further off, a full 1% flat of the untempered note. For me, this is enough to change its flavor entirely, and dilute its resonance to the point where it’s just not the same note. I would contend that the real experience of the b7 is not actually available in equal temperament.
Here it is again:
And in ET:
To me, the real b7 sounds triumphant, like its arms are outstretched to the sky after a great victory.
The ET one sounds very different. It’s not unpleasant, but it sure is different. It it a little sad? The leaping dance is gone. The b3 is flat too. Poor minor, no wonder she’s sad! A mortal has seized the hem of her garment and made her earthbound, in order to put her in his power and make her a little better behaved.
Now go back and listen to the JI version. My experience is that I hear it a little sharp for a second, and then it settles in and wow. This is all subjective; you may hear entirely different things. But this example makes it pretty clear, I think, that JI and ET do not sound the same.
So here we have a note, with a distinct (and unique) personality, that produces a physiological sensation that just isn’t quite available in equal temperament. There are a lot more of these to come, with strange and beautiful colors. Really getting into JI and the lattice is like getting the 64-color Crayola box for Christmas. Orange-yellow and yellow-orange, what riches!
The Northwest is good, once you get off I-5 and wander up and down the Willamette dammit, on the back back roads. I know a few people who’d let me park in their drive, plug in for a night or two, stay up late, and talk about these crazy times — the blandification of our whole situation. And then back to the woods. A dog is bound to find me sooner or later. Sometimes you gotta not look too hard — just let the dog find you.
The blandification of our whole situation. Nice one, Mr. Brown. I recommend going back and forth between the last two vids a few times. Deblandification!
The farther we get from the center, the less consonant the notes are, when played against the tonic. Consonance is a whole subject. It’s generally spoken of as though it could be plotted on a scale, from consonance to dissonance. I think this is a big mistake. Consonance has more than one dimension. Trying to force these independent dimensions of consonance onto a one-dimensional scale leads to unnecessary confusion.
Anywayy … The minor seventh is a pretty dissonant note in all dimensions. It’s three moves away from the tonic, down a third and up two fifths:
The ratio is 9/5.
This is some beautiful, exotic harmony.
Here’s a progression that shows off the flavor of the b7, in just intonation:
Hey, that’s beautiful! I worked it out as an illustration, with the idea of showcasing the minor seventh, and it turned out to be really nice music.
See why I’m in love with the lattice? It’s a beauty engine.
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 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.
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.