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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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Posted by on Jun 6, 2013 in Just Intonation, Recordings, The Lattice, Tonal Gravity |

Be Love

After two years of working mostly with existing material, I’m happy to be writing songs again. This one took me over completely for a few days, and then I spent another couple of weeks recording it and animating it on the lattice.

Be Love is a simpler song than Flying Dream, and I think it does a better job of illustrating what the lattice is all about. A couple of notable moments:

  • Several times, all the notes suddenly collapse to the 1, the note in the center. Check out the feeling of arrival, or homecoming in the music when it happens. It’s especially powerful going into the first chorus at 1:12. This is a real-time demo of tonal gravity.
  • During the verse, the song hangs out in the left part of the lattice, and then for the chorus it moves to the right. This is an example of a change in mode — the song stays in the same key (1 is still the center) but two of the scale notes change. The language of pure music doesn’t translate literally to English, or to emotion, but it evokes its own sensations that can support, or contradict, the words and feelings in the lyrics.

Next: Home

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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.

Next: Cadences

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Posted by on Mar 6, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

Rosetta Stone

Almost all Western music, including my own, lives in the world of tonal harmony. This means:

  • There can be, and usually are, multiple notes playing at the same time.
  • There is a key center, or tonic, around which the notes are arranged. The tonic doesn’t always sound — it’s an intangible presence, the home from which you leave on your harmonic journey, and to which you will hopefully return.

The multiple notes can have different functions:

  • Roots are the fundamental notes of chords. A G chord has its root on G. Roots are local centers that move the ear around the lattice as they change.
  • Harmonies flesh out the chord. In a G major chord, the harmony notes are B and D. They stake out more lattice territory and add definition to the chord. Is it a G major, minor, seventh? The harmonies establish this.
  • Melodies dance in the harmonic field set up by the tonic, roots and harmonies. They have more freedom than the others. Melodies travel fast and light, and though they can sing the same notes as the others, they can also travel farther afield, further embellishing the chord, or leading the ear toward the next chord in the progression, or lingering on the last one after it has changed.

All this action is happening in two musical spaces at once.

Piano-keyboard

Melodic space is the world of scales. It’s organized in order of pitch. The piano keyboard is a perfect representation of melodic space.

full lattice all-01

Harmonic space is the world of ratios. Multiply a note by a small whole number ratio, and you have moved a small distance in harmonic space. Multiply by large numbers, and you have moved a large distance. The lattice is a map of harmonic space.

The two worlds are not the same. Often, they are opposites. The perfect fifth is a small move harmonically but it’s a mile in the melody — bass singers have to jump all over the place in pitch. Small melodic moves tend to be big harmonic ones. A chromatic half step, the distance between the 3 and b3, is only 70 cents, less than the distance between neighboring keys on the piano. But on the lattice, it’s a long haul — down a third, down another third, and up a fifth.

Writing and arranging a song is sort of like designing (rather than solving) a crossword puzzle. There are two intersecting, independent universes, Up and Down. To design the puzzle, you work back and forth between the two, massaging them until they don’t conflict, and each one makes sense on its own.

All of the notes live in both harmonic and melodic space. They may have a foot in one more than the other — the roots tend to move small distances on the lattice, the melodies usually move small distances in pitch, and the harmonies tend to bridge the two, moving melodically while staking out the form of the music on the lattice. But every note moves in both spaces, all the time.

Rosetta_stone_(photo)A great advantage of the lattice is that it serves as a sort of Rosetta Stone, a bridge or translator between the two worlds.

The Rosetta Stone was carved in 196 BC and rediscovered in 1799. It immediately became famous because it repeats the same text three times, in three different languages. It was the key that allowed scholars to decipher Egyptian hieroglyphs.

The lattice bridges the two musical spaces by means of the patterns it presents to the eye.

When two or more notes are plotted on the lattice, they will form a particular visual pattern. Any time you see this pattern, no matter where on the lattice it is, the relationship between the notes of the pattern will be exactly the same, in both harmonic and melodic space.

3-01For example, this pattern shows an interval of a major third. The ratio of the frequencies of these two notes is 5/4 (or 5/2, or 5/1 — twos don’t count, they just shift the note by an octave). Any time you see two notes in this formation, no matter where they are, you know they have the following relationship to each other:

  • Harmonic space: When the notes are sounded simultaneously, they will have the characteristic sound of a pure major third.
  • Melodic space: When you move from one note to the other, you are traveling a distance of 386 cents, or about four semitones on the piano.

Getting familiar with these patterns, and learning to recognize them wherever they are, has made it easier for me to think in harmonic and melodic space at the same time, which makes writing and arranging music much easier.

Next: Tonal Gravity

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