The picture to the right shows the lattice of fifths and thirds, a map of untempered harmonic space, extending to infinity. The map shows how to venture farther and farther from home, the tonic, and stay perfectly in tune no matter how far you roam.
Every single note in this infinite matrix is tuned to a different pitch. You can go out to Mars and beyond, and you will never see the same note twice.
In pure vocal music, this is not as hard as it seems. The voices will tune up to each other, and it’s natural to sing the pure intervals. So if the piece makes its way to the far north, by small steps, the voices may be singing something like a #3, #7 and ##5, and it will be a nice pure major chord.
Want to hear this in action? Here’s a video of a piece by Guillaume de Machaut. I find this music exhilarating. Check out the astonishing note at 3:00. It’s not jazz, not classical, not blues, it’s adventurous harmony on the lattice of fifths and thirds.
In the 1300’s, before temperament started taking over in Europe, there was a flourishing of untempered music, both secular and spiritual, called Ars Nova. Machaut was one of the greatest composers of that era. In Harmonic Experience, Mathieu shows a map of a Machaut piece that wanders amazingly far on the lattice, staying in tune all the way.
The only instruments that can really play like this have infinitely variable pitch. Voice is top dog, although the fretless stringed instruments can do it too (standup bass, violin, etc).
For instruments of fixed pitch, such as pianos, organs, guitars, lutes and accordions, the tuning of the lattice in just intonation is an absolute nightmare. How do you accommodate all those pitches? The keyboard to the right gives it the old college try. (Photo is from a gallery of such keyboards at H-pi Instruments.) Yipe! ltbb_035-44k
Fixed pitch instruments work just fine if you stay in a small part of the lattice, and stay in one key. But after Ars Nova, European composers and listeners got more and more interested in wandering the map, and in changing keys, or modulation. So they started to look for compromise tunings, in which one note could represent several nearby ones, close enough in pitch that the ear would tend to interpret them as the pure note.
For example, there are two major seconds on the inner lattice. In just intonation, the 2 (ratio 9/8) is tuned to 204 cents. The other major second, 2- (ratio 10/9) is tuned to 182 cents. If the major second on your instrument is tuned to, say, 193 cents, it will be right in the middle and you can use it to play both notes, slightly but perhaps acceptably out of tune.
There are many possible ways to “temper” the scale, and each one compromises different notes. Over the next few hundred years after Ars Nova, tunings evolved through a bunch of meantone tunings, which detuned fifths and left thirds quite pure, through well temperament, which spreads out the detuning enough that it becomes possible to play in all keys. During this lattice study I discovered, to my surprise, that Bach’s Well-Tempered Clavier was not written for equal temperament. In ET, all keys sound exactly the same, but if Bach is played in the original tuning, each key sounds slightly different. Such key coloration was an integral part of the music, and composers took it into account.
Finally, by the last half of the 1700’s, equal temperament had become pretty much standard.
Twelve-tone ET completely flattens out the lattice, so that each block of twelve tones (the different colors in the top picture) is tuned exactly the same. It’s sort of like a map projection, in which the the geography is slightly distorted so that the curved surface of the earth can be represented on a flat page. In ET, the fifths are very much in tune (only off by 2 cents), and both major and minor thirds are considerably compromised (off by 14 and 16 cents respectively, quite audible). The minor seventh (Bb in the key of C) is the farthest off of the ET notes, 18 cents. Click here to hear the JI and ET minor sevenths compared.
This is how the central portion of the lattice looks in equal temperament (in the key of C):
Whew! Familiar territory. There’s the tonic major chord, C-E-G, and the relative minor, A-C-E, and so on, and it’s easy to see how they relate to each other.
When you start expanding the ET lattice, it’s a simple repeat. Starting with the 10/9 major second:
No pesky commas, it’s just another D. Note that a new chord has appeared, the minor chord on the second degree of the scale — D-F-A, called the ii chord and very common in jazz. Here’s the whole lattice, converted to ET.
Now the blocks repeat exactly. Think of the lattice as a horizontal surface, extending to the north, south, east and west, and imagine the pitch of the notes as the vertical dimension. The untempered lattice has a tilt to it — up to the east and down to the west, by 22 cents per block, and down to the north, up to the south, by 41 cents per block. The equal tempered version is flat. You can wander at will, and play everything with just 12 pitches.
I oversimplified the ET names in order to show the repetition. For example, in the yellow block just north of the center, Ab really should be G#. In ET, these are exactly the same in pitch, but calling the yellow one G# helps in understanding where it is on the lattice and how it might be used in a composition. The following lattice shows a more informative way to name the ET notes. The pitches of the notes in the blocks are still exactly the same — 100, 200, 300 cents and so on. A C## is just the same as a D, in equal temperament. The same in pitch, but not in function.
This lattice explains why classical music has such oddities as double flats, double sharps, and weird notes such as E#. Why not just say F? E# and F are tuned the same, but they are in different places on the lattice, and if you see an E#, you know you’re in the northern zone.
Beethoven, who helped usher in the Romantic period, used equal temperament to roam the lattice like a wild tiger. Some of his music goes so far out on the map that quadruple flats appear. Click here for some crazy Beethoven stuff — the text is pretty dense but just look at the music notation!
Next: Rosetta Stone
The lattice goes on forever in every direction.
It starts with the tonic, the 1, the Big Bang of the musical universe.
Multiplying and dividing the tonic by 3 generates the horizontal axis. This is the familiar circle of fifths, although in just intonation, it doesn’t quite come out exact. If you multiply by 3, twelve times, you run through all of the scale degrees and land back on the tonic, 19 octaves up … almost. Three to the twelfth power, octave reduced until it’s back in the original octave, comes out to about 24 cents, not zero. Equal temperament flattens this out by subtracting two cents from every fifth. Very handy.
Multiplying and dividing by 5 generates the vertical axis. The two together create a plane, a map of harmonic space. Tonal music, that is, music that is organized around a key center or tonic, can be viewed as a journey on this map.
In the center of the map, there is a lovely pattern of twelve notes that form a chromatic scale.
Each of these notes has a cousin, four fifths down and a third up, that is tuned almost the same. It’s 22 cents flat of the original note, a distance of a Didymic comma. Thus there is another major second, the 2-, just outside the 12-note pattern.
Doing the same thing for all twelve notes creates another chromatic scale to the west. It’s the same scale, 22 cents flat of the original. It works the other way too — there is another block of notes to the east, same scale, 22 cents sharp.
The other comma I’ve discussed, the Great Diesis, shows how to extend the lattice north and south. This one shifts the pitch by 41 cents. It’s the shift that results when you go up or down by three major thirds. Equal temperament flattens this comma out too, but the adjustment is more extreme. Every major third in ET has to be sharp by about 14 cents in order for three of them to add up to an octave, a noticeable difference in pitch.
The pairs created by the Great Diesis have different note names. The b6 in the lattice above is at 814 cents, and the #5 is at 773 cents — 41 cents flat. In the key of C, these notes are Ab and G#, and they are played with the same black key on the piano, between G and A. Until I started studying just intonation and the lattice, I had no idea why one would want to think of these as different notes. It’s not an old-fashioned or obsolete distinction. It’s very useful, when writing or arranging, to know where you are on the lattice, and it’s just as useful in ET as it is in JI.
Now I can add two more blocks to the north and south.
And here’s the whole thing. The colors are arbitrary.
The chromatic scale, a block of 12 notes, has tiled the plane. The note names get pretty crazy — triple flats indeed! But they exactly describe the pitch of every note, in just intonation. Start with the major scale, 1-2-3-4-5-6-7. Every sharp (#) adds 70 cents to the original note, and each flat (b) subtracts 70 cents. Each + adds a Didymic comma, 22 cents, and every – subtracts 22 cents.
Next: Why Equal Temperament?
One more comma shows up in the central portion of the lattice.
In equal temperament, three major thirds adds up to an octave. The major third is an interval of four piano keys (out of 12) or 400 cents. Three of them is 1200 cents, exactly an octave.
In just intonation, this is not the case. The 5/4 major third is a narrower interval, 386.3 cents. Adding three of them together gives 1159 cents, 41 cents shy of an octave.
On the lattice, a stack of major thirds looks like this:
Every three places, the notes repeat — almost.
One such pair is the b6 and #5. In the key of C, they would be an Ab and a G#, and on the piano you would play them both with the black key between G and A.
In just intonation, the b6 is the ratio 8/5, which works out to 814 cents, and the #5 ratio is 25/16, which is 773 cents, almost a quarter tone flatter. (I describe cents and how to calculate them here.) The 41-cent distance between these two notes has several names. Mathieu’s is my favorite: Great Diesis (Dye-uh-sis).
Its formula is x5, x5, x5.
The reason I’m introducing these commas is to show how the lattice repeats itself. Here are the two commas on one lattice:
The lattice extends infinitely in the horizontal direction, and every time it repeats, the pitch shifts by 22 cents. It also repeats in the vertical direction, shifting by 41 cents with every repetition.
Next: The Infinite Lattice
One of the beautiful qualities of the lattice is that the patterns repeat everywhere. Notes that are in the same relationship to each other on the lattice will always have the same difference in pitch, no matter how where you go. For example, a move of one space to the right will always be a move up a perfect fifth, or 702 cents, wherever it happens.
The pitch difference between the 2 and 2-, shown in the last post, is about 22 cents (actually 21.5). This is not a big enough difference to be a different scale degree. In the key of C, the 2 is a D. The 2- is also a D, but of a slightly different flavor.
No matter where you are on the lattice, dividing by 5, then multiplying by 3 four times (the distance between the 2- and 2) will result in a pitch difference of 22 cents. The ratio, octave reduced, is 81/80. This sort of small interval is called a comma.
Commas in general are little intervals that pop up again and again in just intonation. There are three or four of them that are important enough to have names. This one is called the syntonic comma, or comma of Didymus, or just plain “comma.” It has its own (very good, I think) Wikipedia article.
There are three such pairs in my home territory of the lattice, the part I currently feel comfortable roaming:
Each of the notes in the lower right portion is 22 cents sharp of its namesake in the upper left.
Mathieu calls these pairs of notes Didymic pairs, or comma siblings.
In my naming system, I use a minus sign to show that the note is a Didymic comma flat of its sibling, and a plus sign to show that it’s a Didymic comma sharp.
These siblings start to show how the lattice repeats (almost) as it expands. The almost-duplication goes out forever in all directions.
Next: Another Comma
When I started exploring the extended lattice beyond the central 12 notes, the first note that was really new to me was the 10/9 major second, also called the minor or lesser whole tone. Now I call it the 2-.
The lattice extends forever in all directions. When you continue multiplying and dividing, generating new notes beyond the boundaries of the central zone, the notes start to repeat, but not quite. The notes in red, the 2 and the 2-, are very close in pitch. They are different flavors, if you will, of the interval of a major second, or whole tone — a distance of two half steps, two keys on the piano.
Even though they are so close in pitch (204 cents for the 2, 182 cents for the 2-, only 22 cents apart), the two major seconds are generated in different ways and have very different functions and characters.
The 2 is an entirely overtonal note, that is, generated by multiplying alone. Such notes can be found in the chord of nature, the harmonics of a vibrating string. The character of notes is somewhat subjective, but for me, overtonal notes have a stable, sort of upbeat or positive character, and even though the 2 is somewhat dissonant, it has a kind of peaceful sound, that shows up well in ninth chords. Its recipe is x3, x3, or x9, octave reduced to 9/8.
The 2- is a combination of reciprocal and overtonal energy. It’s farther from the center than the 2, and more dissonant. Its recipe is /3, /3, x5, or 5/9, which octave reduces (or expands, really) to 10/9. It is darker, bluesier perhaps, and functions differently in chord progressions.
These very similar ratios, 10/9 and 9/8, 182 and 204 cents, are in fact entirely different beasts. Equal temperament has obscured this difference over the years. In ET, both notes are played at the compromise pitch of 200 cents, but that does not change the functional difference. It is extremely useful when writing or arranging to know whether you are playing a 2 or a 2-.
I tried making a demo of how they sound, as with other notes, but I think that played by themselves, out of context, the 2 and 2- are hard to tell apart. To get the difference, I think you have to sing them against a drone (scroll down the linked page a bit and there’s a list of Indian drones to play around with, it’s really fun to improvise melodies over these) and feel them in your own body. Mathieu shows you how to sing the 10/9 note in Harmonic Experience.
The functional differences really show up when you’re designing chord progressions that make sense. A chord progression is a journey on the lattice, and if you’re roaming in western territory, that is, to the left of the center, you want to use the 2- in your chords and melodies, and if you’re in overtonal, eastern lands, to the right of center, the 2 is going to sound better. It’s a crucial distinction in just intonation. Not so much in ET, since the notes are tuned the same — but awareness of where you are on the lattice really helps when you’re writing ET chord progressions.
It’s an old puzzle. Why do some progressions feel “right,” and others “wrong”? Knowing the map of harmony, the lattice, helps a lot. Much more to come in later posts.
As I’ve analyzed my songs on the lattice, and written new music using it as a tool, I have found that I have a certain palette of notes in my mind, a territory of the lattice that I can hear and think with. The notes in this portion are distinct individuals for me. Each one has its own personality, a distinct mix of attraction, repulsion, beauty and function. I’ve described and given examples of many of them.
When I wrote Flying Dream in 1981, I was consciously trying to write a song that used all twelve notes of the chromatic scale. The first part of making the Flying Dream animation was to reverse-engineer my own song, figuring out with my new tool (the lattice) what I had been instinctively hearing at the time.
It turned out that I had been hearing about 18 notes in the song, including blue notes, and notes up in the northern part of the lattice. That made sense. I remember, as a kid, being disappointed to find out that there were only 12 total notes in music to work with. It seemed to limit the possibilities, like being stuck with the 8-color Crayola box.
The music I love to listen to, and make, has the big 64-color box with built-in sharpener. What’s up with Mick Jagger’s “Oooooh,” at the beginning of Gimme Shelter, or that guitar lick in Dizzy Miss Lizzy? These notes can’t be found on the piano, unless you have a pitch bend wheel. Check out this clip of Ray Charles bending notes in 2000 — now there’s a use of technology! The mystery of those notes, and others like them, has stuck with me, and now I feel like I’m getting to know them as friends.
Next: Another Major Second: The 10/9