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Posted by on Aug 26, 2013 in Consonance, Just Intonation, Recordings, The Lattice | 0 comments

100 Girlfriends, Part 2

My new song video, Real Girl, contains many examples of consonance and dissonance, tension and resolution. In my last post, I extracted a phrase from the song and slowed it way down to illustrate how the bass and melody dance, creating and resolving tension in several different ways. Here is the last half of that analysis.

When we last left our heroes, they were on the 4 and b6, quite consonant relative to each other, but still unresolved because the ear remembers where the tonic is. Here is that clip:

Now the melody moves back to the 7. This interval, against the 4, is the dreaded tritone, the devil’s interval, and it’s dissonant indeed.

Then the bass moves up to the 1, lessening the dissonance, and the melody soon joins it, and all is consonant.

But there is still a sense of incompleteness, even though both the bass and melody are smack on the tonic, the most consonant interval of all. What’s up?

The answer is that the ear remembers that the root is still the 4, and we aren’t quite home yet. Getting there requires a cadence, or final resolution. Notice that in this next clip the bass note never moves, but the harmonies and the melody signal that the root has now moved to the 1 and we are home. The bass note has magically changed character.

Here is the complete sequence, annotated.

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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.

 

 

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Posted by on Mar 11, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 1 comment

Tonal Gravity

I believe that the great driving force in tonal music, that creates the drama and story of the music itself (independently of any lyrics), is the longing for home.

Home is the tonic. If a song is in the key of A, all the A’s in their various octaves will sound like home.

Although there are many exceptions, most music begins on the tonic, to show the ear what key the piece is in, and ends on the tonic, to bring the listener home again. In between, the music wanders, out and back again, creating tension and resolution.

One of the beauties of the lattice is that it shows a clear graphical display of this tension.

It’s as though the tonic creates a sort of gravitational field around itself. It acts a lot like real gravity. The chords and notes move in this gravitational field, like planets and moons around a sun. The gravitational field follows a few basic rules:

  1. Movement away from the center creates tension; movement toward the center gives a sense of resolution.
  2. Notes that are overtonal from the center, generated by multiplying, located to the right and up, will feel more resolved. Notes that are reciprocal, generated by dividing, to the left and down, will feel unresolved.
  3. The closer you are to the center in your journey, the stronger the sensations of tension and resolution are. The field is stronger closer in, just like real gravity.
  4. The closer together two notes are, the more consonant, or harmonious, they will be when sounded together. The farther apart they are, the more dissonant they will be, the more they will clash.

Roots generate local gravitational fields. I think of them as Jupiter to the tonic’s Sun. When the root is on the 5, for example, it shifts the gravity field to the east on the lattice, and the 2 and 7 become harmonious, consonant notes, rather than dissonant ones. The tonic still has great influence, so the entire chord feels unresolved — a 5 chord pulls very strongly toward the 1 chord, a property that is heavily relied upon in Western music. As long as the 5 is the root, though, the 2 and 7 will be consonant harmonies, because they are close to the 5 on the lattice.

Here is a movie to show how that works. The music starts with a tonic chord. Then, one at a time, the 2 and 7 are introduced. These notes are dissonant, and create a sense of tension against the tonic.

Then the root moves to the 5, and the character of the 2 and 7 changes. Now they form a major chord based on the 5, a harmonious configuration. They have become moons of Jupiter. Hear how the dissonance goes away? But there is still plenty of tension, as now there are three notes venturing away from the center, pulling the ear back toward home.

Then the root moves back to the 1, and the 2 and 7 collapse back in toward the center. There is a sense of arrival.

This movie illustrates another observation: consonance / dissonance and tension / resolution are not the same thing. They both relate to distance on the lattice, but they do not necessarily track together. When the root moves to the 5, the dissonance goes away, but there is a new tension, a drive to resolve toward the center. The ear remembers where home is, and longs for it.

These principles can be consciously used to create desired effects when writing and arranging. Resolution and consonance give the music beauty, and tension and dissonance give it teeth.

 

 

 

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Posted by on Nov 20, 2012 in The Notes | 0 comments

A Reciprocal Note: The Fourth

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.

One resolution is right next door: the major third. It’s only a half step lower, and it is a point of stable equilibrium.

Aaaaahhhh.

There’s another way to resolve the tension, and that is to move the playing field so that the fourth is in a stable spot. This is done by changing the root. It’s a chord change. Check this out:

The mountain has come to Muhammad.

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.

Next: Mixed Messages

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Posted by on Nov 14, 2012 in The Lattice | 0 comments

Harmonic Space

Now to relate all this to the lattice in the video.

Listening to music is like going on a journey. Most tonal music starts by establishing a center, or basic note, and a basic harmonic framework for the song, such as a major or minor mode. A few melody notes, and a beginning chord, and you have some idea of the space in which the journey will be occurring. Strauss’ Also Sprach Zarathustra (of 2001 fame) is a great example. The famous opening section, called “Sunrise,” gives an extremely clear sense of home. You know exactly where you are, sonically.

By the way, it’s fun to hum this while using an electric toothbrush.

The piece goes on to travel away from this home, and back again, many times. The journey takes place in a space of some sort, an auditory environment.

But what might this space look like? One way to visualize music is staff notation:

It’s beautiful, and if I know how to read it, it will tell me what the music sounds like. It doesn’t do such a good job of showing me why music sounds the way it does. Neither staff notation, nor the 12-tone scale, gives me a particularly clear idea of how music works. Why would this be restful and sonorous:

major

While this, though beautiful in a different way, has tremendous tension? Sounds like the villain (or the cat) is about to pop out and scare you.

aug

Okay, okay, here’s the resolution:major

Aaahh.

If I know a lot about music theory, I can interpret the notation and come up with explanations. The second example is an augmented chord, and yes it sounds like that. But why, Mom, wh-wh-why?

Next: The Lattice

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