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Posted by on Dec 14, 2012 in Background | 0 comments

Summary (So Far)

I messed around with electronics quite a bit as a kid. I’d put things together according to diagrams, and if they didn’t work, I’d change something and see what happened, and get a feeling for what was happening inside the black box.

When I started doing audio electronics in earnest, I found the oscilloscope. Here’s a cool handmade one by Andrew Smith.

An oscilloscope is a powerful tool, a visualizer, that lets you look right into the black box. It feels almost like cheating. All the energy that went into detective work can now be put to creative purposes. Electronics is much easier when you can directly see what’s happening in there.

I feel that hearing the notes in their untempered form, and learning their relationships on the lattice, has connected me with music in a similar way. Was blind, but now I see.

I’ve finished my first goal for this blog — to create and post the Flying Dream video, and post enough information for an interested person to understand it. I could go on for a long time about the uses of the lattice, and I imagine I will. It’s a fabulous tool.

There will be a slight pause in this blog as I write and rehearse for some upcoming shows. I intend to be back with some new subjects, especially an exploration of consonance and dissonance. If this work interests you, and you’d like to discuss it, you can reach me through the contact page.

Oh, and the matrix in which all of this is happening is love.

Next: Mozart on the Lattice

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Posted by on Dec 12, 2012 in Just Intonation, Septimal Harmony, The Notes | 2 comments


My favorite contemporary band is the Black Keys. I think Dan Auerbach is a harmonic genius. The video is funny.

Dinosaur? What dinosaur?

This song lives in the universe of 7. I just spent a half day taking apart the main riff and seeing how it works out on the lattice. It strictly uses a 5-note scale: 1, 7b3, 74+, 5, and 7b7. The notes are all in the universe of 3 and 7; there is no 5 energy at all, in this part of the song at least.

There are three new lattice notes in this video, the septimal minor third, or 7b3, the harmonic seventh, or septimal flatted seventh, labeled 7b7, and a crazy new note I’ll get to in a minute. The 7b3 is found (on the scale) between the 2 and the b3. It’s a lot flatter than the minor third. The 7b7 is pitched between the 6 and the b7, a lot flatter than the minor seventh. Neither of these can be played directly on the piano. Blues pianists can evoke them by trilling between the key above and the key below. Variable pitch instruments, notably voice and electric guitar, are capable of actually nailing these notes and delivering their full effect.

This song added a new note to my lattice! Mathieu writes that it is used in the blues, and I knew about it theoretically, but I hadn’t used it or observed it in the wild before. One source calls it the septimal narrow fourth. It is slightly flat of the 4. My name for it turns out to be 74+. (The + is a slight pitch adjustment to show exactly how it’s tuned in just intonation.) In Next Girl, it makes a harmony note with the 7b3 root — a nice interval of a ninth.

Cool, haven’t confirmed the existence of a new note in a while. The bestiary grows. Kind of like particle physics.

Septimal notes are essentially unknown in European classical music, but thanks to the blues, they thoroughly infuse the music of America and many other countries. Without them, some music just doesn’t sound the same. They are one reason the Beatles don’t translate well to elevator music. Check this out:

Listen to the signature riff, how it changes and morphs. Throughout the song, George is playing with the region between the septimal flatted third and the major third. As the chords change, the song moves around on the lattice. In response, he bends the note a little more, a little less, to evoke the septimal third, then the major, and maybe even the minor third, located between the other two notes.

By the way, this is a great little zone on guitar. It’s the second fret up from the tonic. You can play four distinct notes just by bending — the 2, 7b3, b3 and 3. George Harrison spends this whole song exploring the tension and resolution in that little melodic space.

John’s vocals are great blues, right in tune.

As I hear them, the Black Keys go even further by putting septimal notes in the roots. Great lyrics and a sense of musical history too, an excellent band.

The usable septimal notes are all close to the center. They just get too far out for me to hear, rather quickly. I personally have found three of them useful so far, and today I’ve been introduced to another.

Here are the septimal notes I have on my current lattice. I imagine I’ll add more as I explore.

7/4, the harmonic seventh, 7b7

7/6, the septimal minor third, 7b3

7/5, the septimal tritone, a staple of rock guitar and one of my personal favorite notes.

21/16, the septimal narrow fourth, or blu ma according to Mathieu. He has some great note names in his book, based on the Indian singing notes, sa – re – ga – ma and so on.

Next: Summary (So Far)

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Posted by on Dec 11, 2012 in Just Intonation, Septimal Harmony | 0 comments

Prime Numbers and the Big Bang

Every prime number generates a new musical universe.

Prime numbers are numbers, greater than one, that can only be evenly divided by themselves and 1. All other numbers are composite — that is, they can be made by multiplying two or more primes.

1) Multiplying by 1 does nothing. 1 is a singularity, the universe before the big bang, the anvil upon which the music is forged.

2) Two starts off the explosion. Multiplying by two creates a universe of octaves, an endless, sterile line of equally spaced mile markers on the road to harmony.

Reminds me of the first chapter of Genesis, where everything is formless until the Creator starts differentiating stuff, day from night, water from sky, land from water, animals from plants, and people from animals. Start multiplying by two and before you know it you have a universe!

I’m also reminded of the current theory as to how our own universe came to be. Here’s a nice summary I found on a physics message board. It’s by Joel Novicio, an undergraduate physics student at the time.

The Big Bang singularity is a point of zero volume, but very high mass, which makes the density infinite. This singularity contained all of the matter and energy in the Universe. The initial moment of the cyclopean explosion very well remains a mystery — however, astronomers and physicists believe that after the tiniest fraction of a second, the strong nuclear force and the electromagnetic force separated, which probably caused the Universe to begin inflating. The Big Bang itself created space, time, and all of the matter and energy we know today.

OK, maybe I’m getting a little bit woo-woo here, but really I don’t think this is a trivial or accidental connection. The musical universe arises from the numbers. So does the physical one, at its deepest levels. I think that’s why we perceive music as beautiful.

I am stretching it now, but guess what is thought to have happened next after the splitting of the forces? Quarks! Quarks are the building blocks of protons and neutrons, almost all the matter we’re familiar with. And they come in threes.

3) Three makes it interesting. Keep multiplying and dividing by 3 and you can get an equivalent for every key on the keyboard, and many more. The notes never repeat, as you multiply and divide, so this universe is infinite as well.

This is the central spine of the lattice. The crucial notes 4 (perfect fourth, 4/3) and 5 (perfect fifth, 3/2) are multiples of 3. They are the backbone of music, and in my opinion, the fact that these are almost exactly in tune in equal temperament is a big reason why ET has been able to be so successful. If the 4 and the 5 were as far out of tune as the major third is, I don’t think ET could ever have been adopted.

Pythagoras based his musical scale entirely on 3 and 2. His followers expanded this, compounding it many times into what is now called Pythagorean tuning.

The first few notes generated by this tuning are beautiful. The 5 (x3) and 4 (÷3) are perfect consonances. The 2 (3×3) is really sweet. I personally like the Pythagorean sixth (3x3x3, 6+ on my map). But apparently the ear can’t follow compounds of 3 forever. By the time you get to the Pythagorean major third (3x3x3x3) you have a dissonant note. It’s on the central spine of the lattice, just off the border of my map, to the east of the 6+.

Here’s a 5/4 major third, with the tonic, and then in the context of a major chord.

just 3

Now here it is in Pythagorean tuning. It’s even sharper than the equal tempered version. Ouch!

pythagorean 3

The universe of threes is infinite, but still somewhat limited musically.

4) Four doesn’t add anything new, it’s just two octaves, every second mile marker.

5) Five, on the other hand, combines with three to create a vast and wondrous universe, the world of the lattice, and adds many more flavors of consonance, dissonance and beauty. The twelve tones I’ve just described, and virtually all of European classical music, can be found in this universe.

6) Six, like four, adds nothing fundamental. It’s 3×2, and generates only Pythagorean intervals.

7) Aha.

America doesn’t export much any more. Except culture. American music, and the movies, have spread worldwide.

Strange turn of events considering that 100 years ago, America was pretty raw. It imported much of its culture from Europe. But when it imported the music of Africa, and combined it with the music of Europe, blues and jazz and rock and roll were born, and the world’s music is still ringing like a bell. Go Johnny go!

In my opinion, the great advance in this music (harmonically, at least), is the incorporation of the prime number seven.

Next: Seven

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Posted by on Dec 10, 2012 in The Notes | 0 comments

The Augmented Fourth

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!

Next: Prime Numbers and the Big Bang

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Posted by on Dec 9, 2012 in The Notes | 2 comments

The Minor Second

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.

Next: The Augmented Fourth

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