Here is my third stop-motion animation of a full song.
Real Girl uses a custom nine-note scale. It occupies the Southeast quadrant of the lattice, the zone of the natural minor, with two added notes — the 7, which allows for a major V chord in the progression, and the 7b5, a blue note that is showcased often in the melody.
This scale contains a sharp dissonance, between the b6 and the 7. I go back and forth between those two notes a lot, with a stop on the 1 in between to help ease the transition.
Watch how the melody and bass chase each other around. In the next few blog posts, I’ll slow this dance down, and show how the polarity flips create tension and resolution. When the melody is below and to the left of the bass, the energy is reciprocal, tense. Then one or the other moves so that the melody is above and to the right, the energy becomes overtonal, and the tension resolves.
Another fun thing to watch is the alternating bass. Roots and fifths are right next to each other on the lattice. The red lens swings like a pendulum throughout the verses.
The 2- is a common melody note in my songs, and in the blues. It goes well with the blue note 7b3 — there is an extremely common melody that goes 7b3, 2-, 1. It’s a darker, more dissonant note than its comma sibling, the 2.
The b7 is dissonant and gorgeous — check out the sequence at the end of this post.
Each note is a compound of three legs on the lattice — two fifths, or a factor of 9, and a major third, a factor of 5. By the logic of the last post, the short leg should predominate, which would make the 2- slightly overtonal and stable, and the b7 slightly reciprocal and unstable.
I’m setting up here for a map of the tonal gravity field. I think I can put some numbers on this stuff. Coming soon. I’ll use that new song animation as a basis — it’s full of fleeting dissonances and polarity flips.
The 7 has its mirror twin too, the b2-, at 112 cents. Its ratio is 1/15.
Here is how they sound:
For me, the pattern continues. The 7 is stable, but less so than the notes we’ve heard so far, and it’s getting dissonant as well, because it’s farther from the center. The b2- is both dissonant and unstable.
These notes each traverse two legs of the lattice, a 3 and a 5. The 7 is two legs “up,” or multiplying, and the b2- is two “down,” or dividing.
The next closest pair is the major third / minor sixth. This has a different flavor. Now the tonic is multiplied and divided by 5.
The overtonal third feels stable and restful, though not quite as much so as the fifth. These notes are a bit farther from the center than the 5 and 4. The reciprocal sixth sounds more dissonant than the 4.
The next closest note to the center is the septimal flatted seventh, or harmonic seventh. The ratio of this note is 7/1, and its mirror twin is 1/7. I have not yet consciously used the mirror-seventh, and it’s not on my drawing of the lattice. The note is the septimal major second, at 231 cents, a dissonant interval indeed. The yellow lens shows where I would put it on the lattice.
Oy! That should put to rest the idea that just intonation is all about consonance! The septimal major second is nastier than anything equal temperament has to offer. I like the word “untempered” for this music because it better captures the wild and wooly nature of JI. “Just Intonation” sounds a bit stuffy to me, and the natural intervals of whole number ratios are anything but academic, they are burned in at a very basic level. Equal temperament is brilliant, but it’s actually the headier and less visceral of the two. IMO.
The next pair is a little further out — each note requires two moves on the lattice.
The ratios are 9/1 and 1/9. I still hear the 2 as stable, though it is less consonant than the previous notes. The b7- is suitably dissonant. It cranks up the tension in dominant-seventh-type chords, the workhorse tension-resolution chords of classical music.
I hear the effect of both tension and resolution diminishing somewhat, as tonal gravity gets weaker farther from the tonic.
These last two videos each contain a minor seventh. One is overtonal, the other reciprocal. The septimal flatted seventh, or harmonic seventh, is a stable, resolved note, the signature of barbershop harmony.
Septimal sevenths abound in this music, and they are sweet and consonant and stable.
The b7-, on the other hand, is dissonant and tense. It makes the ear want to change.
In equal temperament, these two notes are played exactly the same. ET weakens and obscures the difference, but it still can come through because of context.
The common “… and many more” tag, sung at the end of Happy Birthday, is a great example. That last note, “more,” is a harmonic seventh, 7/1, the stable, beautiful barbershop note at 969 cents. If you play “and many more” on a piano, the ear will hear the last note as a septimal seventh, only with less impact, because it is very sharp, at 1000 cents.
I’ve been quiet lately because I’ve been working on an animation of my song Real Girl. It’s a complicated one, a dance of harmonic tension and resolution. The bass and melody chase each other around the lattice like courting butterflies.
A mode is a type of scale, characterized by the pattern of intervals between its notes. The note spacings stay the same no matter what key it’s in. When we say a song is in A major, we mean the tonic is A, and the mode is major. The major mode, also called Ionian, is the familiar Do-re-mi-fa-sol-la-ti-do.
Any combination of notes, covering an octave and organized in pitch order, can be a mode. There is a particular set of modes, often called church modes, that can be played on the white keys of the piano. The different modes start on different notes. The major mode goes from C to C; Aeolian, or minor mode, runs from A to A. Stick to the white keys, and the notes will be right for that mode.
Ionian and Aeolian are the commonest modes in modern Western music, but Dorian (D to D) and Mixolydian (G to G) are popular too.
I wrote the chorus of Be Love first. It’s in major mode. I wanted the verse to have a different feel, so I decided to make it Mixolydian — a favorite for several reasons. I love the name. And, as Mathieu points out in his book Harmonic Experience, it’s particularly easy to improvise over. It’s common in rock music from the seventies on — see the BTO clip in this post. My song Driving is mostly in Mixolydian mode.
The big reason I wanted the change was to make the chorus more of an anthem, by contrast. Mixolydian has a dark, beefy quality to me, and when the chorus comes around it sounds like the sun is coming out.
In equal temperament, there is only one difference between major and Mixolydian scales. The seventh degree is minor instead of major. Starting with G, and going up the white keys, you get G-A-B-C-D-E-F-G. The G major scale goes G-A-B-C-D-E-F#-G. Only the seventh is different.
In just intonation, the situation is a bit different. There are three b7s in the inner lattice: b7, b7-, and 7b7. Which to choose?
Here is the major scale.
One possibility is to just drop the 7 to the b7:
I like the interpretation below. It uses the b7-, and also changes the 2 to a 2-.
This gives me an in-tune major flatted seventh chord, which I love. In the key of G that’s F major — lots of rock music uses this chord.
The notes of this scale (Western Mixolydian?) are in the same relationship to each other as the notes of the major scale, shifted one space to the left. The intervals are just as consonant as the major scale ones, only arranged in a different order.
It’s easy to write chord progressions in this mode. Major and minor triads form triangles on the lattice (major triads point up, and minor ones point down) and there are five in-tune triads, just like the major scale. Since the triangles are all connected, moving from one to another feels natural and is easy for the ear to follow.