NEGATIVE HARMONY. If we have negative harmony do we also have negative counterpoint? Negative scales and negative chords? Negative intervals? Negative everything and anything!
Positive harmony :: overtone series
Maybe before we even begin to deal with negative harmony we should talk about POSITIVE harmony? Does such a thing exist? Positive harmony?
We might say Arnold Schoenberg’s Theory of Harmony is about positive harmony—even though Schoenberg definitely didn’t use that term. But Schoenberg does speak about the overtone series.
For our purposes and to state the obvious, the overtone series must be over something! Yes? In fact, we know it’s over something.
That something is the fundamental, a base frequency, and that base, in turn, is followed by or it’s the firmament for the overtones, the 1st, 2nd, and 3rd overtones and so on.
Each overtone is an integer multiple of the fundamental. But, curiously, we get the first overtone as a multiple of two! All that means and an example is best, is the pitch we know as A440 could be a fundamental.
The 440 is understood to be a quantity that represents hertz, a term that denotes vibrations per second. A880 which we get by multiplying 440 * 2 is the first overtone.
440 * 3 = 1320. Or we could say 1320 is a number that really means 1320 hertz. 1320 hertz is an octave and a fifth above the fundamental.
While it’s the second overtone, we got to it by multiplying a fundamental by three. All that means is the integer we use as a multiple to calculate a specific overtone is equal to the number of the overtone plus one.
In other words, to get the third overtone two octaves above the fundamental we multiple the fundamental by four. We could save ourselves from the offset of one with different terminology.
But terminology in the end is professional jargon and all we need now for discussion now is the idea of the overtone series. And we don’t, for discussion now, need more than that.
Returning to overtones, they rise in theory forever although in practice the higher they go the less we hear them. There’s a point not far from the first several overtones where we can only measure them—with the appropriate equipment, that is..
If you’d like to hear (part of) the overtone series on a piano: Silently depress middle C. Now bang on the C an octave below.
A short clipped, staccato attack is best. You’ll hear the first overtone. The C on which we banged was the fundamental.
Next, hold down the G over middle C. Play that same C an octave below middle C Just as you did to get the first overtone.
Now you’ll hear the second overtone. For more information and a complete list of the first 32 overtones go here.
With the process just described we can experiment. One experiment, for example, is hold down the overtone you wish to hear.
Plunk out the appropriate fundamental. Depending upon your piano, your dexterity at silently depressing keys, which overtones you choose, and the register in which you do all preceding you may find it’s possible to play melodies.
The melodies, as such, would be comprised of sequences of overtones rather than series of notes. As an improvising pianist who plays solo concerts it’s a technique I use frequently.
Use it in particular, the technique just described, on a well-tuned concert-grade instrument. The sounds we can get from the piano are beyond lovely.
And they’re eerie and maybe even unique as regards timbres we can get from the piano. That’s because there really isn’t much piano repertoire in which composers use this technique—although, yes, of course, it’s out there to be found if we look. But, and very simply, it’s not a common technique and as such it produces a sound that’s not often heard.
The undertone series?
That there’s an undertone series may not be the most intuitive idea in the world. But then neither is the idea of an imaginery number where we begin with some quantity represented by i that when squared yields -1.
How exactly do we square something and end up with -1? The answer, at least intuitively, is we don’t. But take the leap of faith that allows further investigation.
With that leap and further investigation THEN we get to imaginary numbers. And, surprise! They turn out to be useful!
Does that mean if there’s an overtone series there must be an undertone series? Well, if some things, like overtones, are over something else, like a fundamental, then some other things lie beneath?
Does that reasoning mean given an overtone series there is an undertone series? Actually, shouldn’t we expect overtones to have undertones as their complement and vice versa?
And the same with positive harmony based on the overtone series. Shouldn’t we then have negative harmony based on an undertone series?
Finally, if Schoenberg posits positive harmony based on an overtone series then SOMETHING else yet again must be underneath it all. And just like that, as with the cycle of fifths, we’re back were we began.
In that respect—the circular reasoning of back were we began, the existence of overtones and undertones and positive and negative harmony, none of that is unlike or far from the crab canon in J.S. Bach’s Musical Offering.
That’s because while they are things we can describe they may not seem intuitive much less real. Let’s put that into context.
How did J.S. Bach know the theme from The Musical Offering could be rendered within a crab canon? For sure that it could work out like that is far from obvious? But that’s why we know Bach is Bach?
So what is this thing called negative harmony? Does this thing called negative harmony come from the undertone series?
Arnold Schoenberg and klangfarbenmelodie
Putting negative harmony aside, we know, Arnold Schoenberg’s harmony book is about POSITIVE harmony. That’s a reasonable description—positive harmony—given that the harmony Schoenberg describes comes from or out of overtones expressed as positive integers over a fundamental.
Off on a tangent, The Theory of Harmony isn’t light reading—except perhaps for those who want the full harmonic journey of all the way out to the there of atonality with no guaranteed round trip! It’s a journey where Schoenberg’s conclusions travel beyond harmony.
They go on and out to klangfarbenmelodie. Sound-colour-melody as it’s often translated.
And what does klangfarbenmelodie sound like? Here’s how Anton Webern, ONE of Schoenberg’s three most known students, applied klangfarbenmelodie technique to Bach.
By the way, Schoenberg’s other two well-known students? They would be John Cage and Alban Berg, an interesting pair. Yes?
To get to sound-colour melody, Schoenberg theorises about timbre rather than pitch as a basic building block for melody. Or rather, he asks, he wishes, he dreams of timbre as the basic building block.
But Schoenberg definitely doesn’t say how he or anyone can fulfil his dream. But why should he? His book is about harmony, not klangfarbenmelodie.
In that respect, Webern’ very real orchestration of the ricercar from J.S. Bach’s The Musical Offering differs entirely from the imaginary conception to which Schoenberg only mentions. Webern’s orchestration differs entirely from Schoenberg’s idea because it, the orchestration exists!
It’s real! It’s a thing!
Negative harmony, Ernst Levy, and the path forward
Ernst Levy’s book, The Theory of Harmony, was published posthumously after his death in 1981. It may have gone out of print not long after. But once out of print, it became an expensive item if found.
However, Levy’s Theory of Harmony is now available (reprinted) on Amazon at a very reasonable price. If you’d like to read music theory as it isn’t and won’t usually be presented then this is your book.
However, beware! Schoenberg’s harmony book isn’t casual reading. Ernst Levy’s harmony book easily qualifies as difficult reading.
But, without further focus on casual or difficult as words that qualify some specfici piece of literature, that phrase I just used: as it isn’t and won’t be presented … Things got interesting not long ago becuase Jacob Collier, one of the most talented musicians in the UK and really anywhere on this planet Earth, mentioned Levy’s book in a Youtube video.
Jason Collier explains in the video what negative harmony is and from where he learned about it. His explanation, in turn, leads directly to Steve Coleman, arguably, simply, or certainly, one of the most influential jazz musicians of this or any time.
All of a sudden, 800 or so copies of the book sold. Instantly. 800+- is a huge number considering prior sales, who knows?—all prior sales?—probably could be counted on one single hand or with two? Do those 800+- copies constitute a movement—perhaps an undertone renaissance?
Jason Collier and Steve Coleman
It’s easy enough to issue superlatives and I’ve already attributed a bunch to Jacob Collier and Steve Coleman. For that reason, it’s probably best not to assume anything.
Visit Steve Coleman’s web site. Look around. Ask and investigate. Why do so many musicians consider him to so be extremely influential?
So it was written
I wrote a blog post a few years ago about Ernst Levy and negative harmony. The better side of the post was Steve Coleman left a comment about Levy and his ideas. He described learning and teaching with them. My post also links to Steve Coleman’s essays on Levy and negative harmony. Or just google Steve Coleman and the essays are there on his web site.
The fantastically-talented Jason Collier refers to Ernst Levy’s (sadly-overlooked?) Theory of Harmony in a Youtube video. Jason Collier mentions Steve Coleman as perhaps the first to put Levy’s theories into practice.
But, of course, Steve Coleman’s adaptations of Ernst Levy’s theories very likely aren’t a first instance of a usecase. Should be presume or assume that Ernst Levy very likely used his own theories, at least to some degree, to write the music that he did?
Notice, I’ve qualified the way Ernst Levy may have used his theories with the phrase at least to some degree. All with experience composing and improvising know when one is really doing those things—really doing as in totally immersed in the process—that’s when theory flies out the door and inner hearing and imagination come to the fore.
But wait, theres more
Where are the examples?
One last question: What to do with Levy’s ideas once you’ve found them? We might say while the jazz community is the sphere now in which Levy’s ideas are operative, that’s only temporary.
I believe it’s only temporary because Levy’s ideas relate perfectly well to the whole of the Western art music tradition and tonality in particular. Of course, that’s the tradition the practicality in which Levy was a composer.
So, the answer in general for Levy’s ideas, or, rather, what to do with them, is they provide possibility for improvisers and composers. Those same ideas provide the possibility of exquisite tools for music theorists and musicologists.
And for any one individual? Get Ernst Levy’s A Theory of Harmony. Read it! Sit at the piano with it.
Play his examples or make up ideas as his examples might generate.
Explore. See how examples that come from Ernst Levy’s theories relate to music you know. In fact, go ahead. Use his theories to make music.
In other words: Consider. Listen. Absorb feedback, your own or as comes from others. Repeat the loop.
One more reference
Check out Harmonic Experience by W.A. Mathieu. Blurbs on the back of that book come from Terry Riley, John Coltrane, and several other interesting musicians.
Or, given the diversity of experience and style among those who wrote the blurbs maybe we should say many musicians of interest. Here’s a post I wrote some time ago about the blurbs and the book.
Also, and here’s where things became even more interesting: W.A. Mathieu writes about reciprocal tones. What you’ll find is:
Reciprocal tones and the undertone series are one and the same thing.