The notes 1, 2, 3, 4, 5, 6 and 7, clustered at the center of the lattice, constitute a major scale. This tuning uses the smallest ratios (the ones with the lowest numbers) available for each position in the scale. It goes back at least to Ptolemy in the 100’s AD.
I find it visually beautiful. It’s like a cat’s cradle.
Here it is again, with a drone on the tonic, to show how the notes resonate with the drone. Each one has its own flavor, its own harmonic character.
Notice how the melody never moves from a note to the note next door. It always moves two grid segments. This is a first look at the difference between harmonic space and melodic space.
Melodies “like” to move up and down on a linear scale. They want to go to a nearby note when they move — that is, near by in pitch. We hear, and sing, small movements in pitch better than we hear leaps.
Harmonies “like” to go to nearby notes too, but harmonic space is different than linear, melodic space. The 1 and the 5 are harmonic neighbors. In fact, they are as close together as notes can be, harmonically, without being the same note — a single factor of three. But they are far apart melodically — the 5 is almost at the midpoint of the scale.
1 and 2 are melodic neighbors, It’s easy to for the voice to move from one to the other. But they are far apart harmonically — two factors of three. A small move in pitch can produce a large harmonic jump.
Arranging a melody and chord progression involves interweaving the notes so they work in both spaces. The melody will tend to move up and down by small melodic steps, close together on the scale. The chords will tend to move by small harmonic steps, close together on the lattice.
It’s a bit like designing a crossword puzzle, working “up” against “down” until it all fits. The lattice is a wonderful tool for visualizing this dance.
Musical nomenclature has been cobbled together over the centuries like a medieval city. Different systems leave their imprint in convention, later developments try to be compatible with accepted names, and the whole thing ends up confusing and contradictory.
Take enharmonic equivalents, for example. G# and Ab are the same note on the piano, the black key between G and A. So why do you sometimes call that note by one name, and sometimes by the other? The answer actually leads to some deep realizations about music, and it comes back to just intonation. In untempered or just music, G# and Ab are not the same note, and which one you choose becomes important. It’s important in ET too — the music establishes a context, and the ear figures out which note it’s supposed to be. But if you grew up with ET, and have no idea that there used to be two different notes there, the names can be confusing. How do you imply one note or the other? Which one is right in a given situation? Why bother? It’s a huge part of writing chord progressions that make sense, but ET by itself isn’t going to tell you what to do. You have to dig deeper for that.
I’ve slowly evolved a personal system I’m very happy with. It’s based on the lattice.
The great advantage of this approach is that it’s entirely unambiguous. Every note on the infinite lattice has a unique name, and that name tells you exactly what its pitch is, and where it is on the map.
The seven notes I’ve covered so far form the core of the system. I’ve dropped all the word names and just use numbers:
Southwest is pure reciprocal energy. You get there by dividing. All the 3’s and 5’s are on the bottom of the ratio (or fraction), and the powers of 2 are on the top.
Northwest is mixed reciprocal and overtonal energy. You multiply by fives and divide by threes.
Southeast is also mixed. Multiply by threes and divide by fives.
The major sixth is in the northwest quadrant. First divide the tonic by 3, which gives you a perfect fourth. Then multiply that by 5, and you get the major sixth. Its ratio is 5/3. You can do this in any order, of course — multiplying by 5 gets you to the major third, and dividing by 3 brings you to the sixth again.
A different flavor yet! What do you hear? How does it make you feel?
Musical sensation can be related to other senses and emotions, but really these feelings have their own quality. It’s like a new sense. I had a friend once who had never tasted a peach. She didn’t like the feel of the skin, and she hadn’t dared bite into one. I was saddened, of course, but how to describe the taste of a ripe peach to someone who’s never tasted one? You can draw comparisons forever, but nothing will prepare them for the reality.
That was a long time ago. I hope that some time later, she found herself in the perfect place and went for it. There should be a First Peach ceremony. Yum!
The individual qualities of these notes are easier for me to hear in just intonation than in equal temperament. They are purer flavors, a more direct experience. Later, I’ll present some notes that are very close together in pitch, yet feel different. In just intonation, there are different notes for peach, nectarine and apricot. In equal temperament, a single note, close in pitch, will represent all three. Once you’ve tasted the actual fruits, equal temperament works better than it did before. You know what it is you’re supposed to be tasting.
OK, I’m out on a quivering limb of analogy now, and way ahead of my story. We have enough notes now to start playing with them.
All the notes I’ve discussed so far are found above and to the right of the tonic, in the northeast quadrant of the map. These notes are generated by multiplication alone.
What about the notes that are generated by division? These are found to the left and down on the lattice.
The closer a note is to the tonic, the smaller the numbers are, and the easier it is for the ear to tell where it is. I’ll cover this much more in later posts, but I think the character of an individual note, its unique harmonic color, is largely determined by two signals it sends to the ear and mind:
1) How far away is home (the tonic)?
2) What direction is it?
And the closer the note is to home, the clearer the signal is.
The perfect fourth is the same distance from the center as the 5 is, but in the exact opposite direction: divide-by-3 instead of multiply-by-3. So it sends a signal of equal strength and opposite direction. How does this mirror-fifth sound?
I hear beauty, and tremendous tension. Something has to happen here, and soon — it feels like a pencil balancing on its point, unstable equilibrium.
Here are two of the most powerful phenomena in music: tension and resolution.
The ratio of the perfect fourth is 1/3. This can be octave-reduced (octave-expanded) by moving it up two octaves, to 4/3.
The energy of the fourth, the division energy, has had a number of names. Harry Partch, a major composer and explorer of music in just intonation, called the quality of the right-and-up harmonies (the ones you get to by multiplication) otonality, from overtone. He called the energy of division-based harmony utonality, for undertone.
Once again I’m going with Mathieu on this one. In Harmonic Experience, he gives an excellent rationale for calling this energy reciprocal. I think he’s right. Each overtonal note has its mirror twin, and the twins are identical, just upside down from each other — reciprocals. The fourth is the reciprocal of the fifth.
The notes get more exotic as you move outward from the center. The ninth is quite consonant, but not nearly as consonant as the fifth. (Consonance and dissonance are descriptions of feelings; they are part of the flavor of an interval, and I don’t think the last word has been written on them yet. I’ll be taking my shot later in these pages.)
For very small ratios such as 3/2, the ear has no trouble perceiving where it is on the map. The signal given by 3/2 is so strong, in fact, that it’s the primary tool used in classical music to move the ear to a new key center.
As the numbers get bigger, the signal gets weaker, and the interval gets more dissonant. To get to the major second, you multiply by 3 twice. Then, using octave reduction, you can put it in any octave you want. I chose 9:4 in yesterday’s example, giving an interval of a major ninth — an octave plus a major second.
Compounding a fifth and a third gives somewhat larger numbers (3×5 = 15, or a ratio of 15/8) and, sure enough, the note is more dissonant against the tonic. Yet it has its own unique beauty. Presenting the major seventh:
Tomorrow, another kind of flavor entirely, another primary color in the crayon box, if you will.
Multiplying the tonic by 2, 3 and 5 creates the octave, fifth and third respectively. The ear hears these intervals very well. We can easily sing them. Each one has a feel, a sort of harmonic flavor, that makes a fifth a fifth and a third a third.
It turns out that the ear can also easily hear compounds, that is, combinations of these low primes. Combining 2 with anything else simply puts it in another octave. But when you combine 3 and 5, or 3 and another 3, you get entirely new flavors. Here’s an example:
The final note is an octave plus a major second above the tonic — a major ninth. Its ratio is (3/2) x (3/2), or 9/4. It has a haunting sound, to me, a different beauty certainly. A new crayon in the box.