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Posted by on Jun 23, 2013 in Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes |

The Septimal Minor Third

At the very end of the chorusBe Love showcases one of my favorite notes, the 7b3.

Most of the notes of the inner lattice can be approximated on the piano, but not the septimal minor third. It’s in between the keys. Blues pianists can evoke it by trilling between 2 and b3, but only variable-pitch instruments can actually hit the note.

Here it is in context:

It’s fun to sing this part of the song, stretching out that septimal note and tasting its flavor.

The passage illustrates the harmonic function of the note. It’s the septimal flatted seventh of the 4, also called the harmonic seventh or barbershop seventh. This is a beautifully consonant note, a great addition to a major chord. It’s generated by multiplying by seven. I use it here as a harmonic seventh over the IV.

Relative to the 1, the 7b3 is a compound note. To get there, you divide by 3, and then multiply by 7. The ratio is 7/6, octave reduced. The pitch is 267 cents, between the 2 and the b3. Here it is on the scale. The colored notes are in just intonation, the black ones are in equal temperament.

Scale with 7b3

Over a I chord, the 7b3 sounds bluesy, restless, gutsy — it’s the insistent melody note in Taking Care of Business. It’s at the heart of the guitar riff in Dizzy Miss Lizzy (George often bends it up to the major third), it’s Jagger’s haunting first “ooooh” of Gimme Shelter. Gimme Shelter

The Stones’ music is a feast of 7b3’s. So is Led Zeppelin’s. These septimal notes are found everywhere the blues has left its impression.

There’s an old question: why do the minor melody notes of the blues sound good over major chords? The web is full of discussions as to why this is so.

I think it’s because the blue minor third is not the b3, but the 7b3. The regular minor third is a reciprocal third, and harmonically it doesn’t fit with major chords — it’s in a different part of the lattice.

But the 7b3 is an overtonal 7th, built on the 4, generated by multiplication. The major notes are made by multiplying by 5. Times 5 and times 7 go together very well. The harmonic seventh chord is a thing of beauty.

There’s an implication for blues guitar. You can’t play this note in the classic minor pentatonic blues box. You can play a b3 (bend it a little to tune it up), or a 3 (bend it harder), but not a 7b3, it’s flat of the b3, and you can’t bend down.

minor pentatonic

You can play a 7b3 by grabbing the 2, one fret below, and gently bending up to it. The following box works great for septimal notes. They’re all laid out under the ring finger. Bend them by less than a half step.

major pentatonic

This box makes it easy to play the classic bit of melody, 7b3 – 2 – 1. All three songs I linked to earlier have this melody in their bones — BTO, the Stones, the Beatles. It’s everywhere. Here it is in Be Love.

Next: Mixolydian Mode

 

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Posted by on Jun 22, 2013 in Just Intonation, Recordings, The Lattice, The Notes, Tonal Gravity |

To the Far Northwest

The chorus of Be Love is solidly in major territory. The chords are I, V, IV, the classic backbone chords of the major scale.

The verse, however, is going to be back in the Northwest — I, bVII-, IV. I have to get back over there somehow, and I want to lead the ear strongly so that the move feels right.

Once again I’ll use a dominant seventh type chord. Here’s the shape again:

P1080225

The ear expects a I chord to come after this. Perhaps it’s one of those “nature abhors a vacuum” things. All the notes have tonal gravity that is pointing at the center, yet there’s nothing there. It’s as though the planets are orbiting the sun, and the sun is missing. The ear wants to put it there.

This “dominant seventh” shape is named after the dominant chord, the V7, where it shows up so often. It can be used elsewhere, and is the main tool used in classical music to change keys.

When I started to work this out, I already had a climax in mind for the chorus. In the last line, I wanted to leap all the way to the far Northwest, the ii- chord (lower case roman numerals mean it’s a minor chord), and have the melody sit still while the chords revolve around it. Here’s the ii-:

P1080430

 

It’s a long leap, but I can make it with one transition chord, the I7.

P1080421

Adding a seventh to the I chord sends a strong signal to the ear: “We’re going west!” Normally the next chord would be a IV.

This move is used all the time in music. It’s common for the bridge of a song to start on a IV chord — it’s a shift on the lattice that makes the bridge sound different from the rest of the music, much like my shift to major for the chorus of Be Love. Putting a I7 right before the bridge tells the ear to expect this shift.

I do something a little different, though. There is also a huge vacuum where the ii- ought to be — look how the two chord shapes fit into each other. Sure enough, going to the ii- after the I7 is satisfying and dramatic. Here’s how the sequence looks and sounds.

Notice how the melody leads the way again, by going to the 4. There’s another lovely passing chord, a stack of perfect fifths, right before the change.

P1080425

 

Next: The Septimal Minor Third

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Posted by on Jun 21, 2013 in The Lattice, The Notes, Tonal Gravity |

The Power of the Seventh Chord

The 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.

P1080225

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.

P1080237

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

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Posted by on Jun 19, 2013 in Just Intonation, The Lattice, The Notes |

Leading the Ear

Be Love was written and arranged on the lattice. I consciously used the lattice as a tool to make the music do what I wanted it to do.

Working with this song has taught me a lot about leading the ear.

Different parts of the lattice have different sounds. The upper right, the northeast, is major scale territory. Music in this zone sounds major, you know, that uplifting, stable, “happy” majorness. The northwest region, up and to the left, has a darker, dramatic sound, not like minor, but with its own flavor. It shows up a lot in rock. A great example is BTO’s Taking Care of Business. The progression is I, bVII-, IV, I. (I use numbers for notes and roman numerals for chords.)

I wanted the song to start in the northwest for the verse, and then move eastward for the chorus, and then go back again, and I wanted to choose notes that would lead the ear on the journey.

Here’s the beginning. The chords plant a flag in the Northwest.

The music stays there for a while, and then it starts to move. The chord progression changes, and the guitar melody reaches out to the east and starts to rope in more territory.

Finally, right before the chorus, the V chord takes the song firmly into dominant territory.

Notice how the melody leads the way into the far east. When the melody goes to the 2, in advance of the chord progression, it sets up tension. The tension is resolved when the root moves up to the 5 and creates a more consonant interval.

One of the pleasures of these lattice movies is watching the fleeting, exotic harmonies that are formed as the melody dances around the basic chords. This chord is a type of sixth chord.

P1080176

When 4 is the root, 2 is its sixth degree. I call the interval between 4 and 2 a Pythagorean sixth, because it is generated entirely by multiples of 3 — a characteristic of Pythagorean tuning. The ratio, octave reduced, is 27/16. It sounds different than the 5/3 sixth, and is tuned sharper — 906 cents instead of 884.

The Pythagorean sixth chord leads the ear to the east. The tension of the 2 in the melody is resolved by moving all the music up to meet it.

Now there’s a new tension, against the tonic, which is in the back of the listener’s mind all the time. I will want to resolve this tension by collapsing to the center, but first I want to increase it as much as possible. I want to dive into the chorus from a great height.

Next: The Power of the Seventh Chord

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Posted by on Jun 19, 2013 in Consonance, Just Intonation, Resonance, The Lattice, The Notes |

The Compass Points

There are two basic directions on the lattice: multiplication and division.

If I start with a note, and then multiply it by 3, or 5, or 7, I will get a harmony note with overtonal energy. Such a note is in the natural overtone series of the original note.

Overtonal energy is stable, restful, it belongs where it is and wouldn’t mind staying there.

If I divide by 3, 5 or 7, I get a completely different kind of note. I call this division energy “reciprocal,” after W.A. Mathieu’s suggestion in his amazing book Harmonic Experience.

Reciprocal energy is restless, unstable. The note wants to move, or for the music to come to it, until it is overtonal.

On the lattice of fifths and thirds, there are two axes, fifths and thirds, and two directions, overtonal and reciprocal.

This makes four total directions one can move on this lattice. Each direction has own characteristic flavor, or energy. I use the following names for these energies, mostly after Mathieu.

  • Dominant = East = Overtonal fifths
  • Subdominant = West = Reciprocal fifths
  • Major = North = Overtonal thirds
  • Minor = South = Reciprocal thirds

Compass Points

Every interval has its own unique recipe of moves in these four directions. The perfect fifth has pure dominant energy, the major third pure major. The minor third, b3 on the lattice, is a compound note — dominant and minor.

It’s interesting to look at the minor third (b3) from the viewpoint of tonal gravity. On the horizontal axis, dominant/subdominant, the b3 is overtonal, stable, restful. On the vertical axis, major/minor, the note is reciprocal, unstable, restless.

Tonal gravity is stronger the closer you are to the center. To make a minor third, you multiply by 3 (an overtonal jump of a fifth), and divide by 5 (a reciprocal jump of a third). I know, 3 generates fifths and 5 generates thirds, a confusing coincidence.

Fifths are closer to the center, harmonically, than thirds are, so the overtonal energy is stronger than the reciprocal.

This makes the minor third a stable note, although less stable than the major third. Songs can end on a tonic minor chord and they will still sound finished.

Next: Leading the Ear

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Posted by on Jun 10, 2013 in Just Intonation, The Lattice, The Notes, Tonal Gravity |

Home

Tonal music is music that has a particular key center, or home note. Not all music is tonal, but most is, worldwide.

The key note is at the center of the lattice of fifths and thirds. All other notes are generated from this one. I call it the 1. It’s also called the tonic. When we say a song is “in the key of A,” we mean that A is the tonic.

This isn’t any particular A. In the key of A, every one of the ten or so A’s within the range of human hearing is a tonic, or perhaps more accurately some octave of the tonic. The tonic itself is an abstract concept, of “A-ness.”  In concert pitch, A is defined as a vibration of 440 cycles per second (called Hertz, or Hz), and any octave of this, up or down, is also a tonic. Thanks to a remarkable (and handy) quirk of human perception, multiplying or dividing a pitch by 2 does not change its essential character. So 220Hz is also an A, as are 110, 55, 27.5 — and 880, 1760 and so on forever.

The tonic doesn’t even have to be one of the 12 equal-tempered notes — it can be halfway between A and A#, and it will still work just as well. The rest of the notes are simply calculated from that home note. The resulting music will be in tune with itself, and will sound fine, even though it has no relation to concert (A=440) tuning. In learning songs from old recordings, I’ve found that many are in between two official keys. The instruments are tuned to each other, but not to any outside reference. They sound great.

The tonic sounds like home. The great driver of tonal music is the sense of departure from, and return to, home.

Be Love, like many tonal songs, starts right off with the tonic. It makes a statement, with the very first note: “This is where home is.”

Again and again throughout the song, the music departs from home, creating tension, and then returns to it, relieving the tension. The following clip contains two such homecomings, at 0:07 and again right at the end.

Then, finally, the song ends with the tonic. Ahhhh. Journey complete, the lattice has been explored, and after many adventures Sam Gamgee is back in Hobbiton.

Not all songs begin and end on the tonic. If you want the song to sound resolved, finished, end it on the tonic. If you want it to sound unresolved, unfinished, end it on another note. It’s a powerful tool. Listen to the end of Cream’s Sunshine of Your Love.

Have you ever had the experience of the audience clapping at the wrong time, in the middle of a song? It’s embarrassing!

Usually it happens when you pause for dramatic effect, and the audience thinks you are finished. You can send a strong signal that the song is not over by pausing on a chord that is clearly not the tonic. Then, when you do want the audience to clap, give them a big tonic chord and they’ll know what to do.

Next: The Compass Points

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