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Putting Some Numbers on Tonal Gravity

Posted by on Oct 22, 2013 in Consonance, Just Intonation, The Lattice, Tonal Gravity | 0 comments

I believe the sensation of tonal gravity is the most important driver of tension and resolution in tonal music, music that has a central key note.

The tonic is like a sun, creating a gravitational field around it. The lattice is a beautiful map of this gravitational field, in harmonic space.

Tonal gravity acts like real gravity, in that it’s strongest close to the center. The farther out you get, the weaker the effect.

There is a difference, though — unlike physical gravity, which only attracts, tonal gravity has two polarities — it can attract or repel. Briefly, if an interval is generated by multiplication, it will pull toward the tonic, as though to say, “You’re going the right way, you’re just not there yet.” If the interval is generated by division, the sensation is more of a push — it still points to the tonic, but now it’s saying “It’s over that way, go!”

Overtonal notes are stable, reciprocals are unstable. Reciprocal intervals create tension, overtonal ones create resolution.

The article on Polarity goes into detail, with examples.

The lattice can be divided into four quadrants, each with a characteristic tonal gravity. The northeast quadrant is entirely overtonal. This is the world of Major/Dominant: overtonal thirds, overtonal fifths.

The southwest quadrant is entirely reciprocal. Every interval is generated by division.

The northwest and southeast are zones of mixed polarity, overtonal on one axis and reciprocal on the other one. Here’s a map of the quadrants and the compass points:

Green is overtonal, stable, tonal gravity pulls.

Red is reciprocal, unstable, tonal gravity pushes.

Blue is mixed, gravity pulls on one axis and pushes on the other.

For a couple of years now, I’ve been chewing on how to represent the tonal gravity of the lattice mathematically. To describe what I experience, the equations should behave as follows:

1. Overtonal intervals have one polarity, and reciprocals have the opposite polarity,
2. Gravity gets weaker the farther one gets from the center, and
3. Multiplying or dividing by 2 does not affect tonal gravity. This is to account for the octave phenomenon — going up or down an octave does not change a note’s position on the lattice.

Here’s my latest approach. I’m not presenting this as some kind of truth — but it nicely matches my own perceptions, and it leads to some interesting graphs. Any input you may have is welcome — feel free to comment, or email me from the Contact page. Here goes:

I will call the direction and magnitude of the tonal gravity field P, for Polarity.

Intervals are expressed as a ratio of two numbers, numerator and denominator, N/D. For example, a perfect fifth is 3/1, or N=3, D=1.

1. For purely overtonal notes, of the form N/1: P = 2/N.
2. For purely reciprocal notes, of the form 1/D: P = -2/D.
3. For compound notes, with both overtonal and reciprocal components, add the overtonal and reciprocal gravities together: P = 2/N – 2/D.
4. The ratio of the tonic, the 1, is 2/1.

I’d better explain that last one, because it would seem at first glance that the ratio of the tonic would be 1/1.

But what is the actual tonic? It has no specific pitch. It is not a ratio. It is an abstraction, the anvil upon which all notes are forged, the sound of one hand clapping. If a song is in the key of A, all of the A’s from subsonic to ultrasonic are actually octaves of the tonic, created by multiplying by two. It is impossible to say that any one of these A’s is “the” tonic — the tonic is “A-ness,” that thing which connects the numbers 110, 220, 440, 880, to infinity in both directions. I submit that the “1” of the lattice, which is a real pitch (or set of pitches, an octave apart, just like all the other notes) is in fact the octave, and its ratio is 2/1.

Here’s another drawing of the inner lattice. Instead of the note names, I’ve filled in the ratios, and the value of P.

The green notes all have positive polarity, getting weaker as they get farther out. The red ones have negative polarity, also fading with distance. The blue ones have different polarities. Sometimes the overtonal part dominates, sometimes the reciprocal.

The b3, just southeast of the tonic, is a mixed-polarity note. Its ratio is 3/5, combining an overtonal fifth, P = .67, with a reciprocal third, P = -.40. If I just add the two gravities together, I get a positive net polarity of .27.

This makes sense. The minor third is considered to be a stable interval, though not as stable as the major third.

Both the major and minor triads consist entirely of stable intervals with positive P, which helps explain their special place in music.

So: now that I have some values for P, I can graph the tonal gravity of these 13 inner notes against the octave, in order of pitch.

Positive polarity is at the bottom, so that the feel is the same as real gravity. Unstable notes are up on mountain peaks, and when they resolve to stable ones they slide down into the gravity wells of the stable notes.

There is that tasty melody zone I mentioned a few posts back. The whole region from 2- to 3 is stable.

The 4 is an isolated peak, and it’s easy to imagine a 4 sliding into the pocket of the 3, or the 5. This is what happens when a Sus4 chord resolves.

The 7 is lightly stable but hanging on by its fingernails — it’s called the leading tone, because it “wants” so badly to resolve to the 1. The tonal gravity of the 7 is usually thought of in terms of melodic pull — here’s a graphic demonstration that it has harmonic pull as well.

There is an unstable region from b6 through b7, with all the mirror twins of that stable melody zone. A melody will sound unstable, unresolved as long as it stays in that region.

This is the gravitational field in which the music moves, a sort of tonal skate park.

Straight Line Chords: Aug, Dim, Sus2, Sus4

Posted by on Oct 14, 2013 in The Lattice, The Notes | 0 comments

Major and minor chords look like triangles on the lattice. They are closed loops, which I think contributes to their stable feeling. Up a major third, up a minor third, and down a fifth (due West on the lattice) brings you right back where you started. The intervals interlock, and they are all consonant, reinforcing the sense of harmonic rest.

Other combinations of notes are more open. Many chords look like straight lines on the lattice.

Augmented chords are stacks of major thirds:

Diminished chords are stacks of minor thirds:

These chords often show up in my animations as fleeting transitional harmonies. In Flying Dream, for an instant, there’s even a stack of three major thirds:

To my ear, this stack of thirds has a distinctive sound, almost like cloth ripping.

A Suspended Second chord is a straight line, like an augmented chord, but on the horizontal axis. It’s a stack of two fifths. At the time the Sus2 and Sus4 chords were named, the 2 and 4 were usually “suspended,” or held over from the previous chord, as a tension to be resolved. Now they are often used as full chords.

The Police used these stacks of fifths a lot. Andy Summers’ guitar part for Every Breath You Take is full of Sus2 chords, alternating with major and minor thirds.

To me, Sus2 feels lightly stable, and wistful. All the intervals are overtonal, and quite consonant, but unlike the major and minor triads, the Sus2 doesn’t come full circle. If it keeps going, it will never return home, but climb on up the endless spiral of fifths.

It’s a great way to end a certain sort of song, finished but with a sense of longing.

The Suspended Fourth consists of a root, a perfect fifth, and a perfect fourth. The Sus4 looks the same on the lattice as the Sus2. The difference is the root.

This chord has a wonderful tension. The root establishes home. The 5 is overtonal, stable, with strong tonal gravity that attracts. The 4 is reciprocal, unstable, with a strong tonal gravity that repels.

Resolving to the 3 is satisfying indeed, as the unstable 4 slides from its unstable peak into the stable gravity well of the major third.

Sus4 chords are all over rock music. Pinball Wizard is a study, here it is by Townshend on acoustic guitar:

Be Love at Sacred Grounds

Posted by on Oct 2, 2013 in Recordings | 0 comments

I love open mics. I have two favorites in San Francisco that I make every time I’m in town — Hotel Utah on Mondays, and Sacred Grounds on Thursdays.

It is in places like Sacred Grounds that the culture begins, where it wells up right from the ground, not manufactured or corporate or in service of any agenda but the beauty of human creativity.

Last week I sang Be Love at Sacred Grounds, and my dear friend Claudia took a video. Here it is. I hope you enjoy it. It really gets the flavor of the place.

Breakup Songs

Posted by on Sep 25, 2013 in Recordings | 0 comments

I keep working my way closer to recording in strict just intonation. Here’s one I did today, of my song Breakup Songs.

The acoustic guitar is in equal temperament, and the bass and vocals are untempered. I love singing harmony when the fretless bass is playing lattice notes. Sometimes I feel like I’m sliding along a groove in the tonal gravity field.

Three Flavors of Seventh Chord

Posted by on Sep 24, 2013 in Consonance, Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes | 9 comments

Chords and other collections of notes have consistent, recognizable shapes on the lattice. A major chord is a triangle sitting on its base, a minor chord is a triangle on its point. Yesterday’s post has videos showing these chords.

In the songs I know and write, the next most common chords after major and minor triads are seventh chords.

By convention, a “seventh chord” means a triad, with a minor seventh added. If the added seventh is a 7, or major seventh, it’s called a “major seventh” chord.

A minor seventh is an interval of ten half steps, or two shy of an octave. There are three different minor sevenths in the inner lattice, and each one makes chords with a different sound and function — that is, if you are playing in just intonation, or untempered. In equal temperament, the minor sevenths all sound the same, but there is still profit in knowing that they are different, because they function differently in chord progressions.

The three notes are:

• The b7, at 1018 cents. The ratio is 9/5.
• The b7-, or dominant-type seventh, at 996 cents. The ratio, octave reduced so it lands in the same octave as the tonic, is 16/9.
• The 7b7, at 969 cents. This is 7/4, the harmonic, or barbershop seventh, a consonant note that appears in the actual harmonic series of the tonic.

Here are some movies in just intonation, so you can hear the differences.

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First, the b7, added to a minor chord.

A pretty sound, I like it! In equal temperament, this note is at 1000 cents, 18 cents flat of the b7, a clearly audible difference. Here’s the same movie in ET:

Both the b3 and b7 are decidedly flat. The b3 especially sounds different, a lot more dissonant and “beating.”

I wrote a post a while ago, exploring this minor seventh and how it sounds in an untempered chord progression. It’s here.

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The next minor seventh is enormously important. This is the dominant-type seventh, b7-, 996 cents. It is fortunate that it is so close to the equal tempered note, 1000 cents, because that means its effect is barely diminished in ET — and it is a really important note in classical music.

The reason it’s called a dominant-type seventh is because it most often shows up with the dominant, or V chord. The note two steps south of the 5 is the 4 — and when you add a 4 to a V chord you get this:

The 4 is a powerful note in this context. It has strong tonal gravity, with reverse polarity. I’ve written several posts about this — here’s one  about polarity, here’s one about the use of dominant seventh chords.

Here’s how the chord sounds when it’s built on the 1, in just intonation.

There is strong dissonance when that seventh comes in, and it’s dissonance with a purpose — the chord “wants” badly to resolve somewhere. In this case, it wants to resolve to the 4, the empty space in the middle of the chord. The 1, 3 and 5 are all in the harmonic series of the 4 — that is, they all appear in its “chord of nature,” the overtones that accompany a natural sound. So these notes sort of point to the 4. They point to the 1 even more strongly, though, until that b7- comes into the picture.

When you add the new note, the b7-, something new happens. This note points hard to the 4, and in a different way. It’s as though it says, “home is over there, go!”

Here’s a more detailed discussion.

The entire note collection “wants” to collapse to its center, like a gravitational collapse. The b7- helps to locate that center on the 4.

This effect is often used to move the ear to a IV chord. For example, if you want to start the bridge of a song on the IV, it helps to hit a I7 first. If you’re playing a song in G, and want to go to a C chord, a quick G7 will make the change seem more inevitable. Here’s that move in slo-mo.

The pull of the dominant-seventh-type chord is so strong that it is the sharpest tool in the kit for changing keys, or modulating. Classical composers use it for this constantly.

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The last of the three is a beauty. This is the 7b7, the quintessential note of barbershop harmony, the harmonic seventh, 7/4. The b7- is highly dissonant, the b7 rather neutral, and the 7b7 highly consonant. It sounds (and looks) like this:

This is a resolved chord. In fact, if the consonance and stability of an interval are determined by the smallness of the numbers in its ratio, these are the four most consonant notes of all — 1/1, 3/1, 5/1 and 7/1.

Here is another opportunity to compare just intonation with equal temperament. The harmonic seventh and the dominant seventh sound exactly the same in ET. I believe that a good composer knows, consciously or not, which one is meant.

A good example is the “… and many more” ending so commonly added to Happy Birthday. It is clearly not a dominant type — it’s intended to mean the end of the song, even to put a stronger period on it than the major triad by itself. It’s a quote, or a parody of blues harmony. Play it on the piano and it will be tuned exactly like a I7 chord, but the ear can tell, by context, that there is no move expected, to the IV or anywhere, because it’s heard that little melody a thousand times, and it belongs at the end of a song.

But the signal is so much clearer when the tuning sends the message too! The 7b7 is at 969 cents, a third of a semitone flatter than the piano key.

By the way, I think this is why a common definition of “blue note” is “sung flatter than usual.” I believe the blue notes are the world of multiples of seven, and these just happen to be flatter than the closest notes in the worlds of 3 and 5, the basic lattice.

Here is a video of the 7b7 chord that starts with the harmonic seventh, goes to the equal-tempered seventh, and back to the 7b7.

Quite a difference. ET works because it implies the JI note, and the ear figures out what it’s supposed to be hearing. But the visceral impact is lessened a lot — in this case, IMO, completely.

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