Pages Menu
Categories Menu

Posted by on Jan 28, 2013 in Just Intonation, The Lattice, The Notes | 0 comments


Musical notes can be mapped onto many different spaces. The two I find useful so far are:

— Harmonic space, the space of the lattice, organized by harmonic connections (ratios of whole numbers).

— Melodic space, the space of the scale, organized by pitch, or frequency.

Both maps show the location of a note relative to a reference tone, the Tonic, the “do” of do-re-mi.


Distance on the lattice could be measured by the number and length of the connections to the tonic, sort of “how many Tinkertoy sticks away are we?”

How to measure distance in melodic space?

One of my favorite music theorists is Alexander Ellis. Ellis was an interesting character, a researcher in phonetics, and the prototype for Professor Henry Higgins of George Bernard Shaw’s Pygmalion (My Fair Lady). He wrote a huge appendix for Helmholz’s foundational book about psychoacoustics, On the Sensations of Tone, in which he laid out a version of the harmonic lattice that is very much like the one I’m using. The appendix was published in 1885.

Ellis proposed dividing each equal-tempered semitone into 100 equal parts, called cents. This gives 1200 cents to the octave. Cents have caught on almost universally as a way to describe and compare pitches of tones.

Cents are a logarithmic unit. Logarithms form a bridge between addition and multiplication. When you add logarithms, you are multiplying in the real world. Adding 1200 cents is the same as multiplying by 2. When you add one cent, you are multiplying by a small number, the same number each time. It’s the 1200th root of two, in fact, a very small number, about 1.0006. Multiply by 1.0006, 1200 times, and you get 2.

The ratios themselves show what the pitch of a note will be, and there’s a formula for translating from harmonic space (ratios, the lattice) to melodic space (cents, pitch). It is great fun, if you’re a geek like me, to plug this formula into a spreadsheet and start exploring the musical spectrum.

For any ratio, b/a, the pitch in cents is:

1200 x log2(b/a)

That’s log to the base 2. A good straightforward explanation of logarithms can be found here. They are a handy concept in the study of perception, since many human senses, including visual brightness, loudness and pitch, work in a logarithmic way. A 100-watt amplifier sounds louder than a 10-watt amp, but it’s nowhere near 10 times as loud. Maybe three times as loud, subjectively? A 10-watt amp is louder than a 1-watt by about the same amount. I have a 1-watt Vox tube amp that the neighbors have yelled at me about. For something to sound “twice as loud,” it has to be moving something like 4 or 5 times as much air.

So let’s run that formula. The untempered major third is a ratio of 5/4.

log2(5/4) = 0.32

x 1200 = 386.3 cents

The ET major third is at exactly 400 cents, 14 cents sharper. This is a clearly audible difference — the ear can distinguish a difference of about 5-10 cents.

Cents give us a language for comparing pitches, and quantifying the differences between them.

Next: Untempered vs. Just Intonation

Read More

Posted by on Jan 16, 2013 in Just Intonation, The Lattice | 0 comments

Melodic Space, Harmonic Space

Throughout my musical education, I’ve been taught that music happens in a linear space. This is the space so beautifully laid out on the piano keyboard.


Music teaching is organized around scales. In most Western music, the full scale consists of twelve notes, equally spaced. Other scales, such as the seven-note minor and major scales, are subsets of this full, “chromatic” scale. Due to octave magic, a mysterious and crucial aspect of our inner perception, when we get to the thirteenth note, we have multiplied the original note by two, and the sequence starts over again.

So, fortunately for musical analysis, melodic space can be described in one octave. It takes about ten of these octaves to cover the range of human hearing.

On the piano keyboard, melodies look the way they sound. When the pitch goes up, you move up the scale, and when the pitch goes down, you move down the scale. Short distances (the shortest is from one key to the next, a half step), feel short. Long distances (more than about three half steps) feel long. This is a good and useful space for visualizing melody.

Harmony, not so much.

Musical nomenclature, as I’ve pointed out before, has grown like an old city over the years. As music theory changes, bits and pieces of the old terminology are appropriated and redefined by new thinkers. The result is a cobbled-together mass that has a lot of weird contradictions and misleading names.

I think one of the most regrettable bits of confusion comes from the word interval.

The distance between two notes on the keyboard is called an interval. When my melody moves by an interval of a minor third, it has covered a distance of three half steps. When I move by a major third, I’ve covered four half steps. The major interval is bigger than the minor one — that’s why it’s called “major.” No problem! The move feels bigger when you sing it.

The problem comes when you start to think about harmony — two or more notes sounding simultaneously. The word “interval,” with the same connotation of pitch difference, is also used to describe the distance between harmony notes. Yet in the world of harmony, the interval, or pitch distances don’t make any intuitive sense at all.

For example, two notes a fifth apart (seven half steps) sound wonderful when played together. C and G are two such notes. They are closely related to each other, harmonically. So are C and F, which are a fourth apart (five half steps). These are the best consonances there are, except for unisons and octaves.

So what about the note in between them, an interval of six half steps?

Yep, none other than the dreaded tritone, the devil’s interval, definitely a dissonant note.

If the linear scale were the best way to think about harmony, wouldn’t the tritone be between the fourth and fifth in consonance? Why would three notes in a row, next-door neighbors on the scale, be so drastically different from each other harmonically? The scale gives no clue. You just have to remember.

Perhaps there is a more intuitive way to visualize harmony, one that puts harmonically related notes closer to each other, and puts the notes that are harmonically farther apart … farther apart?

I think there is indeed a harmonic space as distinct from a melodic space. This space can be illustrated on the lattice. It’s not a good model for melody — scales do a much better job. But it’s a great model for visualizing harmony — what you see corresponds intuitively to what you hear.

The interplay between these two spaces creates the beautiful dance that is harmonized music.

Next: Cents

Read More

Posted by on Dec 12, 2012 in Just Intonation, Septimal Harmony, The Notes | 0 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)

Read More

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

Read More

Posted by on Dec 5, 2012 in Just Intonation | 0 comments

Untempered Vs. Tempered

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!

One of my favorite phrases in any song comes from the great Greg Brown. In Eugene, from The Evening Call, he says,

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!

By the way, The Evening Call is packed with great lyrics and music. Top notch.

Next: The Minor Second

Read More

Posted by on Nov 12, 2012 in Just Intonation, The Notes | 0 comments

Octave Reduction

Doubling the frequency of a note certainly changes it. The ear hears a higher-pitched note. But there is something in the essence of the note that does not change, a character that stays consistent through the octaves.

This allows a process called octave reduction. When you’re working with notes as ratios, it’s convenient to multiply or divide the raw ratio by 2, as many times as is necessary to bring it into the same octave as the tonic.

3/1 generates a perfect fifth. 3-1

This note is actually an octave plus a fifth above the tonic. Now divide by 2 and you have 3/2, one and a half times the original frequency, and just a fifth above. 3-2

The reference frequency is 1, the octave is 2, so what you want to achieve with octave reduction is a ratio, or fraction, between 1 and 2.

These are the beginnings of a scale, a collection of notes within a single octave. Such a scale can be repeated up and down the octaves to cover the whole range of hearing.

Next: The Major Third

Read More
Page 7 of 9« First...56789