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Posted by on Sep 24, 2013 in Consonance, Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes | 7 comments

Three Flavors of Seventh Chord

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:

P1130668

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.

———————————

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.

Next: Straight Line Chords: Aug, Dim, Sus2, Sus4

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Posted by on Sep 13, 2013 in Septimal Harmony, The Notes | 0 comments

The Blue Tritone

I have a favorite note. Don’t tell the others. It’s the septimal flat five, or septimal tritone. I call it 7b5 on the lattice.

There are many reasons why I love this note. One is that Jimi played it, and he’s my favorite musician of them all. Another is that this note is rarely discussed in music theory (try googling it and you will find a few references), which allows me to sort of plant a flag in it. But the biggest reason is that the 7b5 opens up a whole world of melodic and harmonic possibility, and unlocks the minor blues.

The ratio of the septimal flat five is 7/5. It’s a tritone, a note smack in the middle of the octave, between the 4 and the 5. Tritones are famously dissonant. There are three of them in the inner lattice — the 7b5, the #4+, with a ratio of 45/32, and the b5-, whose ratio is 64/45. The 7/5 blue tritone is the most consonant one, by which I mean it has the smallest numbers in its ratio.

Most traditional blues are built on major chords, the I, IV and V, with septimal, or blue, notes in the melody. The 7b3 is especially important — there are entire songs that hang out forever on this note. These blues are major in character — everything happens above the central spine of the lattice.

The 7b5 is different. It lives in the minor part of the lattice, below the central spine, which allows for a whole different set of chordal harmonies. The 7b5 is a blue note that works with songs in minor keys.

Here are a couple of striking examples. First, I invite you to listen to a bit of Dizzy Miss Lizzy, by The Beatles. This is a major blues, played with I, IV and V chords.

In that insistent riff, George Harrison is playing with four notes: the major third (3), the septimal minor third (7b3), the 2 and the 1. He bends the 2 and makes a 7b3, or a 3, or both.

George is exploring a delicious melodic zone that includes four major/blues melody notes in a tight group: the 2-, 2, 7b3, and 3, all in the span of two piano keys. As the I-IV-V progression rocks back and forth from left to right, between dominant and subdominant territory, the melody subtly shifts with it.

Listen again to the intro of the song. The riff repeats, but it’s not always tuned the same. The first two repeats are over a I chord. The riff is sharp, major-third-ish. On the third repeat, the chord changes to a IV, and I hear the tuning fall down into the pocket of the 7b3. It feels to me as though the IV chord allows George to lock into the 7b3, because that note is its seventh harmonic, a beautiful, consonant note. At that point the song goes blue.

Throughout the song, George goes back and forth between that major feeling (the 3) and that blue feeling (the 7b3), over all three chords. Ear candy.

Now listen to Jimi Hendrix exploring the same kind of space, but around the septimal flatted fifth (7b5). This is a minor blues. The chords are i, bVI and bVII.

There is an insistent riff in Voodoo Child (Slight Return) as well, and it’s a lot like the one in Dizzy Miss Lizzy. The pattern is the same, only moved down and to the right on the lattice.

George Harrison is bending the 2, to get the 7b3 and the 3 notes. Jimi Hendrix is bending the 4, to get the 7b5 and 5. It’s another compact, tasty melody zone. Hendrix explores it incredibly well on this song. He cooks up about a half dozen yummy tritone dishes in the space between 0:30 and 0:60.

If I go back and forth between the two songs, the distinction becomes clear. Dizzy Miss Lizzy is major, and the riff centers around the 7b3. Voodoo Child is minor, and the riffs center around the 7b5. Please do click back and forth between the videos.

Want to hear Eric Clapton and Steve Winwood explore the same territory? Here’s a ridiculously good version of Voodoo Chile (the long one from Electric Ladyland) from 2010.

Next: More Blue Tritones

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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 Dec 11, 2012 in Just Intonation, Septimal Harmony | 0 comments

Prime Numbers and the Big Bang

Every prime number generates a new musical universe.

Prime numbers are numbers, greater than one, that can only be evenly divided by themselves and 1. All other numbers are composite — that is, they can be made by multiplying two or more primes.

1) Multiplying by 1 does nothing. 1 is a singularity, the universe before the big bang, the anvil upon which the music is forged.

2) Two starts off the explosion. Multiplying by two creates a universe of octaves, an endless, sterile line of equally spaced mile markers on the road to harmony.

Reminds me of the first chapter of Genesis, where everything is formless until the Creator starts differentiating stuff, day from night, water from sky, land from water, animals from plants, and people from animals. Start multiplying by two and before you know it you have a universe!

I’m also reminded of the current theory as to how our own universe came to be. Here’s a nice summary I found on a physics message board. It’s by Joel Novicio, an undergraduate physics student at the time.

The Big Bang singularity is a point of zero volume, but very high mass, which makes the density infinite. This singularity contained all of the matter and energy in the Universe. The initial moment of the cyclopean explosion very well remains a mystery — however, astronomers and physicists believe that after the tiniest fraction of a second, the strong nuclear force and the electromagnetic force separated, which probably caused the Universe to begin inflating. The Big Bang itself created space, time, and all of the matter and energy we know today.

OK, maybe I’m getting a little bit woo-woo here, but really I don’t think this is a trivial or accidental connection. The musical universe arises from the numbers. So does the physical one, at its deepest levels. I think that’s why we perceive music as beautiful.

I am stretching it now, but guess what is thought to have happened next after the splitting of the forces? Quarks! Quarks are the building blocks of protons and neutrons, almost all the matter we’re familiar with. And they come in threes.

3) Three makes it interesting. Keep multiplying and dividing by 3 and you can get an equivalent for every key on the keyboard, and many more. The notes never repeat, as you multiply and divide, so this universe is infinite as well.

This is the central spine of the lattice. The crucial notes 4 (perfect fourth, 4/3) and 5 (perfect fifth, 3/2) are multiples of 3. They are the backbone of music, and in my opinion, the fact that these are almost exactly in tune in equal temperament is a big reason why ET has been able to be so successful. If the 4 and the 5 were as far out of tune as the major third is, I don’t think ET could ever have been adopted.

Pythagoras based his musical scale entirely on 3 and 2. His followers expanded this, compounding it many times into what is now called Pythagorean tuning.

The first few notes generated by this tuning are beautiful. The 5 (x3) and 4 (÷3) are perfect consonances. The 2 (3×3) is really sweet. I personally like the Pythagorean sixth (3x3x3, 6+ on my map). But apparently the ear can’t follow compounds of 3 forever. By the time you get to the Pythagorean major third (3x3x3x3) you have a dissonant note. It’s on the central spine of the lattice, just off the border of my map, to the east of the 6+.

Here’s a 5/4 major third, with the tonic, and then in the context of a major chord.

just 3

Now here it is in Pythagorean tuning. It’s even sharper than the equal tempered version. Ouch!

pythagorean 3

The universe of threes is infinite, but still somewhat limited musically.

4) Four doesn’t add anything new, it’s just two octaves, every second mile marker.

5) Five, on the other hand, combines with three to create a vast and wondrous universe, the world of the lattice, and adds many more flavors of consonance, dissonance and beauty. The twelve tones I’ve just described, and virtually all of European classical music, can be found in this universe.

6) Six, like four, adds nothing fundamental. It’s 3×2, and generates only Pythagorean intervals.

7) Aha.

America doesn’t export much any more. Except culture. American music, and the movies, have spread worldwide.

Strange turn of events considering that 100 years ago, America was pretty raw. It imported much of its culture from Europe. But when it imported the music of Africa, and combined it with the music of Europe, blues and jazz and rock and roll were born, and the world’s music is still ringing like a bell. Go Johnny go!

http://www.youtube.com/watch?v=6ofD9t_sULM

In my opinion, the great advance in this music (harmonically, at least), is the incorporation of the prime number seven.

Next: Seven

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Posted by on Nov 9, 2012 in Just Intonation | 0 comments

Between the Keys

I grew up thinking that music was made with a particular set of twelve notes, the ones on the piano keyboard. I had a vague sense that there were other scales in the world, but I thought of them as “more primitive” or perhaps subsets of the 12-tone scale, like that pseudo-Asian music you make if you play around on the black keys of the piano. I certainly didn’t know that those 12 notes, now so unconsciously established that hardly anyone in Western culture even questions them, are a relatively recent invention. In Europe, where they first caught on, they were fought bitterly for a century or so before they became the norm. Even now, much of the world still does not tune to these notes, although they are still spreading.

But I also grew up deeply aware of blues singers, and that notes sung “blue” could not be duplicated on the piano, or on the guitar without bending strings. Something was always different about rock, country and other blues-influenced music. All my favorite music had this quality in common — somehow richer in sound, with more heart, and it wasn’t just feel. And it wasn’t just blues either — almost all vocal harmony had “it” too, regardless of genre.

When I was a teenager, I heard a tiny phrase that hit me like Sirius falling from the sky in the Truman Show. Here it is, fair use excerpt:

The Note

Hear it there, at the end? In the right channel, George Harrison plays something you absolutely cannot play on a piano, yet it is perfectly in tune. There is a wealth of information in that little phrase — it points to a whole world living there, in between the keys. That lick has stuck with me for all these years, a sign in the sky, that there was a lot more to know about music than I had been taught in textbooks.

Next: The Chord of Nature

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