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Posted by on Sep 24, 2013 in Consonance, Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes | 11 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 15, 2013 in Just Intonation, Septimal Harmony, The Lattice, The Notes | 0 comments

More Blue Tritones

I want to show you some lattice movies of how I’ve used blue tritones in my own music.

Real Girl has several examples. The clearest is a guitar lick in the chorus:

That 7b5 is tasty over the bVI chord. For an instant, it makes a “barbershop seventh,” the 7th harmonic of the root.

Here is a vocal example from the same song:

The melody visits the blue tritone on the way up, and again on the way down. I especially like it on the word “like,” the blues flavor of the septimal note comes through loud and clear without it being strictly blues at all. For me, this fusion of septimal notes to the European collection is the great contribution American music has made to the world. I wrote an early article on this, with some examples, here.

These bits of melody that visit the 7b5 are very similar to the ones that incorporate the 7b3. The septimal flatted third is the melody note of major blues tonality. It functions as the seventh harmonic of the IV chord, just as the 7b5 is the 7th harmonic of the bVI chord. Here’s an example from Flying Dream:

Hear the similarity? Try going back and forth between this video and the guitar lick in the first video.

One of the beauties of the lattice is that the patterns repeat everywhere. If you move a pattern to a different part of the lattice, the new notes will have the same relationship to each other, but the musical context will change and it will convey a different feeling. This is a splendid compositional tool, and helps me greatly in understanding harmony.

Next: Chords on the Lattice

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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 Sep 10, 2013 in Background, Equal Temperament, Just Intonation, Recordings, Septimal Harmony, The Lattice | 2 comments

Premature Nostalgia: Making Friends With Equal Temperament

I just recorded a new song, and it’s a perfect example of how equal temperament and just intonation can get along together.

Here’s the cut:

 

Reading this blog might give you the impression that I’m “against” equal temperament and “for” just intonation, or untempered music.

True, discovering untempered music has been like sailing to a new world. It’s delicious to have 20 or more notes to work with instead of 12, each with its own individual personality.

Equal temperament, however, is a fabulous invention. The lattice of fifths and thirds does not quite repeat. If you start with any note and go in any direction, you will soon encounter almost the same note again, but it will be off by a comma, a small interval, from the original note.

There are no two notes tuned exactly alike on the entire infinite lattice.

Equal temperament flattens out the lattice just a hair so it does repeat. Now there are only twelve notes to work with, and they imply the untempered ones in the ear. This innovation makes lots of things possible in music. Beethoven and Mozart could not exist without it.

It’s sometimes said that equal temperament and just intonation are incompatible with each other, because the notes will be out of tune. I say they can get along fine, you just have to show ’em who’s boss.

I submit for your consideration: Ray Charles.

Ray Charles’ piano is an equal tempered instrument. Ray Charles’ voice is most certainly not. He is singing the exact resonant notes, those blue notes, all tuned just like a gospel choir, which is what he grew up loving. Ray is boss. His voice establishes the tonality of the song. The backup singers, the horns and the standup bass all agree, this song is in the harmonic pocket, and it resonates.

That leaves the piano slightly out of tune, but who cares?

Notes that are slightly out of tune don’t necessarily sound bad — that’s the basis of the “chorus effect.” No two singers in a choir are exactly in tune with each other, and the resulting complexity is a huge part of the sound of the choir.

So if the tonality is established in the ear, maybe the equal tempered notes, which are only a bit off after all, will just enrich the sound a bit.

Listen to how “Hit the Road, Jack” starts off. First the piano intro. ET. Then the horns kick in, and they start to establish the soul of the tune. Then come the backup singers, that gospel choir. When Ray’s voice finally joins them, the pocket is waiting for him, and he proceeds to own it. The piano is now a background instrument.

I think that’s the secret. Put untempered instruments up front, and ET instruments more in the background. This asserts the untempered tonality in the ear.

Playing acoustic guitar and singing is a great playground for this. The acoustic guitar is, in its bones, an equally tempered instrument. Fretted instruments drove the adoption of ET in Europe, even before keyboards did. The voice is the archetypal untempered instrument. It can do anything.

If the guitar is boss, the song will be in equal temperament. If the voice is boss, you can establish any tonality you want (blues, Gypsy, whatever), and the guitar will tag along. You can retune it in the ear, just like Ray retunes his piano.

Here are some tricks for making friends with acoustic guitar (or any tempered instrument):

1) Sing solidly in tune, with the tonality coming from you, and not from the guitar. Don’t follow the guitar, lead it. The song is the melody, it is your voice, and you are accompanying that voice with guitar notes.

I like to think of the guitar as playing the grid lines on the map. The guitar notes are perfectly equally spaced, and are excellent reference points. The guitar tells me where I am. We completely agree on one note, the tonic. I use the tonic on the guitar as my true home base.

My voice is playing the actual territory.

2) Sing louder than the guitar.

This isn’t all that easy. The guitar is projecting outward, so it sounds louder to the audience than it does to me. The voice is right there in my head, so it sounds quieter to the audience than it does to me. If I sound balanced to myself, the audience will hear way more guitar than vocal. I hear this all the time at open mics.

I’ve found that in an acoustic setting, I have to sing twice as loud as my guitar (from my own point of view) for it to sound balanced out in front of me.

It gets easier with more JI instruments. In “Premature Nostalgia,” the fretless bass and backing vocals are all in strict just intonation. The guitar is truly a backing instrument, and the tonality of the song feels secure.

3) There is a third, more subtle thing you can do to bring the guitar closer to just intonation. The most clearly out-of-tune note on acoustic guitar is the major third. It’s already 14 cents sharp even when perfectly tuned, and the slightest unintentional string bend will take it into some really grating territory. Choose chord voicings that de-emphasize major thirds, and your guitar will sound a lot sweeter. I wrote an article illustrating this effect, here.

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Posted by on Aug 7, 2013 in Consonance, Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes | 0 comments

More Mirror Twins

Mirror twins are pairs of intervals, exactly opposite each other on the lattice.

The two intervals are reciprocals of each other, which means their ratios are flipped — if one is 5/3, the other is 3/5.

Harmonic distance is the same for each interval — the only difference is polarity. Listening to mirror twin pairs gives a good idea of what polarity sounds like.

The clearest example is the fifth/fourth pair, multiplying and dividing the tonic by 3.

Beautiful, consonant notes, one with overtonal energy, and the other with reciprocal energy.

The next closest pair is the major third / minor sixth. This has a different flavor. Now the tonic is multiplied and divided by 5.

The overtonal third feels stable and restful, though not quite as much so as the fifth. These notes are a bit farther from the center than the 5 and 4. The reciprocal sixth sounds more dissonant than the 4.

The next closest note to the center is the septimal flatted seventh, or harmonic seventh. The ratio of this note is 7/1, and its mirror twin is 1/7. I have not yet consciously used the mirror-seventh, and it’s not on my drawing of the lattice. The note is the septimal major second, at 231 cents, a dissonant interval indeed. The yellow lens shows where I would put it on the lattice.

Oy! That should put to rest the idea that just intonation is all about consonance! The septimal major second is nastier than anything equal temperament has to offer. I like the word “untempered” for this music because it better captures the wild and wooly nature of JI. “Just Intonation” sounds a bit stuffy to me, and the natural intervals of whole number ratios are anything but academic, they are burned in at a very basic level. Equal temperament is brilliant, but it’s actually the headier and less visceral of the two. IMO.

The next pair is a little further out — each note requires two moves on the lattice.

The ratios are 9/1 and 1/9. I still hear the 2 as stable, though it is less consonant than the previous notes. The b7- is suitably dissonant. It cranks up the tension in dominant-seventh-type chords, the workhorse tension-resolution chords of classical music.

I hear the effect of both tension and resolution diminishing somewhat, as tonal gravity gets weaker farther from the tonic.

These last two videos each contain a minor seventh. One is overtonal, the other reciprocal. The septimal flatted seventh, or harmonic seventh, is a stable, resolved note, the signature of barbershop harmony.

Septimal sevenths abound in this music, and they are sweet and consonant and stable.

The b7-, on the other hand, is dissonant and tense. It makes the ear want to change.

In equal temperament, these two notes are played exactly the same. ET weakens and obscures the difference, but it still can come through because of context.

The common “… and many more” tag, sung at the end of Happy Birthday, is a great example. That last note, “more,” is a harmonic seventh, 7/1, the stable, beautiful barbershop note at 969 cents. If you play “and many more” on a piano, the ear will hear the last note as a septimal seventh, only with less impact, because it is very sharp, at 1000 cents.

Next: Why Can We Hear Harmony?

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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 chorus, Be 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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