Here’s a different version of my song, The Cove. The original recording is on the Flying Dream CD. I sang this one in a lower key, to leave room for two harmonies above. I found some adventurous harmonies, especially ninth intervals. I’d like to make a movie of this one; it lives deep in the southeast, minor/dominant quadrant of the lattice.
Read MoreThe V chord, the major chord based on the 5, is a powerful compositional tool. It points, very clearly and with a lot of tension, directly at the tonic. If you want to lead the ear to the I, the V chord is the top-of-the-line triad.
Why this is so is still a bit mysterious to me. It’s been discussed a lot. It seems to have both melodic and harmonic elements.
Melodies “like” to move short distances in pitch, and the move from the V to the I is elegant melodically. The 7, or major seventh, resolves up a half step to the 1. The major seventh is called a leading tone because of this very property. The 2 drops a whole step, also to the 1, and the 5 stays put.
In harmonic space, voices, especially roots, “like” to move short distances too. The shortest move of all is a fifth, and when the V goes to the I, the root moves down by a fifth. It seems natural that if the ear is anticipating the next chord, it will place its bet on the change that expends the least energy. All three notes could be seen as moving that same short distance, the easiest possible move.
I like to think of it in terms of tonal gravity. The tonic, the 1, is like a sun at the center of a solar system, and it exerts a gravitational pull. Moving away from it creates tension, collapsing into it creates resolution. Just as with gravity, the closer in you are, the stronger the force. The V is right next to the I, harmonically, so the tension is very strong.
The V chord isn’t the last word, however. It’s possible to crank it up, by adding another tense note.
The 4 and the 5 are the closest notes to the 1, in harmonic space. These two notes have the strongest tonal gravity of all. Their effect is different — 5 is the strongest overtonal note, and 4 is the strongest reciprocal note. Both point straight at the tonic.
Melodically, the 4 is two half steps below the 5. This makes it a flatted or minor seventh, added to the V chord. So the final chord is called a V7.
Of all the notes we could add to the V chord, the 4 creates the most tension, and it’s pointed directly at the tonic. I say this is the source of the power of the dominant 7th chord.
In Be Love, I add even more tension before I’m through. The melody dances around, and right before the final resolution, it lands on the 6.
I’ve added yet another tense note to the mix. It’s not as strong as the 4, but it jacks up the gravity another notch. The root is on 5, so the 6 is two half steps up from it melodically. This makes it a ninth chord — start with the basic major triad, and add a seventh and a ninth.
Now I’m set up as strongly as possible for a return to the tonic, and sure enough when the drop happens it lands with authority. I’m in major land now, and the chorus will feel entirely different from the verse.
Here’s the whole effect:
Next: To the Far Northwest
Read MoreWhen 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.
Next: Commas
Read MoreThe 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.
Next: A Reciprocal Note: The Fourth
Read MoreMultiplying 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.
Next: The Major Seventh
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