Overtonal notes, generated by multiplying, are restful, stable — they have positive polarity, pulling toward the center. Reciprocal notes, generated by division, are restless, unstable — they push. I call this negative polarity. Mixed-polarity notes have both, and I’ve chosen to simply add their overtonal and reciprocal components together to get the total polarity.
Here again is the graph of the 13 most central notes of the lattice.
The stable notes are gravity wells, and the unstable ones are peaks. Melodies and harmonies dance in this gravity field. Higher points represent tension, lower ones resolution, and the lower they are, the more resolved and stable. The tonic major triad, most stable of all, occupies the lowest spots — 1, 3 and 5.
The polarity map of the major scale looks like this:
The notes are all overtonal except the 4, which is strongly reciprocal, and the 6, which is mixed and slightly unstable.
Here’s a split screen video showing the major scale, against a tonic drone, on both the lattice and the octave. This is an example of how the lattice serves as a Rosetta Stone, a translator between harmonic and melodic space.
Can you hear the push/pull quality of the notes? Each note has its own feeling against the steady 1.
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 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!”
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.
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.
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.
Every single note in this infinite matrix is tuned to a different pitch. You can go out to Mars and beyond, and you will never see the same note twice.
In pure vocal music, this is not as hard as it seems. The voices will tune up to each other, and it’s natural to sing the pure intervals. So if the piece makes its way to the far north, by small steps, the voices may be singing something like a #3, #7 and ##5, and it will be a nice pure major chord.
Want to hear this in action? Here’s a video of a piece by Guillaume de Machaut. I find this music exhilarating. Check out the astonishing note at 3:00. It’s not jazz, not classical, not blues, it’s adventurous harmony on the lattice of fifths and thirds.
In the 1300’s, before temperament started taking over in Europe, there was a flourishing of untempered music, both secular and spiritual, called Ars Nova. Machaut was one of the greatest composers of that era. In Harmonic Experience, Mathieu shows a map of a Machaut piece that wanders amazingly far on the lattice, staying in tune all the way.
The only instruments that can really play like this have infinitely variable pitch. Voice is top dog, although the fretless stringed instruments can do it too (standup bass, violin, etc).
For instruments of fixed pitch, such as pianos, organs, guitars, lutes and accordions, the tuning of the lattice in just intonation is an absolute nightmare. How do you accommodate all those pitches? The keyboard to the right gives it the old college try. (Photo is from a gallery of such keyboards at H-pi Instruments.) Yipe! ltbb_035-44k
Fixed pitch instruments work just fine if you stay in a small part of the lattice, and stay in one key. But after Ars Nova, European composers and listeners got more and more interested in wandering the map, and in changing keys, or modulation. So they started to look for compromise tunings, in which one note could represent several nearby ones, close enough in pitch that the ear would tend to interpret them as the pure note.
For example, there are two major seconds on the inner lattice. In just intonation, the 2 (ratio 9/8) is tuned to 204 cents. The other major second, 2- (ratio 10/9) is tuned to 182 cents. If the major second on your instrument is tuned to, say, 193 cents, it will be right in the middle and you can use it to play both notes, slightly but perhaps acceptably out of tune.
There are many possible ways to “temper” the scale, and each one compromises different notes. Over the next few hundred years after Ars Nova, tunings evolved through a bunch of meantone tunings, which detuned fifths and left thirds quite pure, through well temperament, which spreads out the detuning enough that it becomes possible to play in all keys. During this lattice study I discovered, to my surprise, that Bach’s Well-Tempered Clavier was not written for equal temperament. In ET, all keys sound exactly the same, but if Bach is played in the original tuning, each key sounds slightly different. Such key coloration was an integral part of the music, and composers took it into account.
Twelve-tone ET completely flattens out the lattice, so that each block of twelve tones (the different colors in the top picture) is tuned exactly the same. It’s sort of like a map projection, in which the the geography is slightly distorted so that the curved surface of the earth can be represented on a flat page. In ET, the fifths are very much in tune (only off by 2 cents), and both major and minor thirds are considerably compromised (off by 14 and 16 cents respectively, quite audible). The minor seventh (Bb in the key of C) is the farthest off of the ET notes, 18 cents. Click here to hear the JI and ET minor sevenths compared.
This is how the central portion of the lattice looks in equal temperament (in the key of C):
Whew! Familiar territory. There’s the tonic major chord, C-E-G, and the relative minor, A-C-E, and so on, and it’s easy to see how they relate to each other.
When you start expanding the ET lattice, it’s a simple repeat. Starting with the 10/9 major second:
No pesky commas, it’s just another D. Note that a new chord has appeared, the minor chord on the second degree of the scale — D-F-A, called the ii chord and very common in jazz. Here’s the whole lattice, converted to ET.
Now the blocks repeat exactly. Think of the lattice as a horizontal surface, extending to the north, south, east and west, and imagine the pitch of the notes as the vertical dimension. The untempered lattice has a tilt to it — up to the east and down to the west, by 22 cents per block, and down to the north, up to the south, by 41 cents per block. The equal tempered version is flat. You can wander at will, and play everything with just 12 pitches.
I oversimplified the ET names in order to show the repetition. For example, in the yellow block just north of the center, Ab really should be G#. In ET, these are exactly the same in pitch, but calling the yellow one G# helps in understanding where it is on the lattice and how it might be used in a composition. The following lattice shows a more informative way to name the ET notes. The pitches of the notes in the blocks are still exactly the same — 100, 200, 300 cents and so on. A C## is just the same as a D, in equal temperament. The same in pitch, but not in function.
This lattice explains why classical music has such oddities as double flats, double sharps, and weird notes such as E#. Why not just say F? E# and F are tuned the same, but they are in different places on the lattice, and if you see an E#, you know you’re in the northern zone.
Beethoven, who helped usher in the Romantic period, used equal temperament to roam the lattice like a wild tiger. Some of his music goes so far out on the map that quadruple flats appear. Click here for some crazy Beethoven stuff — the text is pretty dense but just look at the music notation!
The notes are all overtonal except the 4, which is strongly reciprocal, and the 6, which is mixed and slightly unstable.
The picture to the right shows the 
