Harmonic Division and Sims' Scale

This diagram represents the extent of Sims’ 24 note scale that is generated by pure harmonic division. You can easily see that it is really just a matter of showing the new tones that show up in each new octave up the overtone series. Take the section that has 9:8,11:8,13:8,15:8 and look at the doublings of the previously attained notes as they fit in the spaces. 5x2=10, (3x2)x2=12, 7x2=14. This shows that, in this octave, all of the harmonics are present. 10,12, and 14 are old, 9,11,13, and 15 are new. This process continues for every new octave up the overtone series. 9 now doubles(that is, rises an octave) to 18 to fill the gap of the two new tones 17 and 19. So how is this harmonic division? Let’s look at 5:4 and 11:8. First we’ll put them in the same octave by doubling the 5:4 so it becomes 10:8. The 4 and 8s represent the fundamental, while the 5(10) and 11 represent the harmonic number. So we have 10 and 11 interacting with each other. If we look at the summation tone(as described by Sims) we get 10+11=21, which you will notice is the tone in the next octave between the 5 and 11. Another way to see it is: (10x2)=20 and (11x2)=22. The 21 which fills the gap is the harmonic mean of 20 and 22. By harmonic mean, we mean that numerically each side is equivalent (1) but in terms of space, they are different(the bottom is larger). Such as the interval of the fifth(3) is numerically right between 2 and 4, but we know that a fifth is certainly larger than a fourth. This pattern of division that characterizes the progression of the overtone series is very similar to the division pattern of the golden ratio, expressed in a continued fraction as: 1 plus 1 over 1 plus 1 over 1 plus 1....
How does harmonic division relate to resultant tones? Let’s say now we want to use 5:4 and 9:8 to give us 7:4. First the resultant tone method. 5+9=14 summation and 9-5=4 difference tone. The difference tone fills its name in more than one way. In the harmonic division method, (2x5=10, 2x9=18) the difference tone is the difference from each of these extremities to the harmonic mean or summation tone of 14. This approach, it seems to me, works for all the examples I could think of, including tones that are not in clear harmonic relation to one another. For example: 445 Hz and 764 Hz produce resultants of 319Hz and 1209Hz. (445x2=890, 764x2=1528). 890Hz and 1528Hz are both 319Hz (the difference) away from 1209Hz, the summation and harmonic mean.
To fill out the scale that Sims has used since 1970, we take the upper tetrachord divisions and superimpose them over the lower pentachord. This is clear even at glancing at his full scale (see below) because many of the notes that appear early on(25:24, 13:12, 7:6, 29:24) are ratios based on the 3rd harmonic. By doing this, he takes full advantage of the interval space generated by the upper tetrachord, the space that supplies the perfect evolutionary step for intervallic structure. In this way, he sidesteps the obvious problem of the Fibonacci sequence (being sent too quickly to the harmonic stratosphere) while still preserving and amplifying the characteristic growth structure of the Fibonacci and Golden Section. One of the aspects of the Fibonacci/Golden Section that I find very important and pervasive is the centrifugal force created by the smaller and smaller spiraling of the Fibonacci around the irrational Ratio. I feel that this principle is very much in effect in musical harmony, and that resultant/harmonic mean tones are a potent way to guide the listener’s perceptions into the vortex. The difference being, in a working musical system such as Sims’, there are many focal points that create many vortexes. For a visual on the diagram, look at the zigzag of 5:4, 9:8, 19:16, 37:32. This is the level that I feel the Fibonacci is active.


Sims’ 24 note scale (in ratios):
1:1 – 33:32 – (25:24) – 17:16 – (13:12) – 35:32 – 9:8 – 37:32 – (7:6) – 19:16 – (29:24) – 39:32 – 5:4 – 21:16 – 11:8 – 23:16 – 3:2 – 25:16 – 13:8 – 27:16 – 7:4 – 29:16 – 15:8 – 31:16 – 2:1

Note: The ratios in ( ) are the ones produced from the upper tetrachord of the octave, but superimposed over the bottom to stabilize upper tones with a fifth below and to flesh out the scale with more simple relationships than could be attained purely through harmonic division. You can tell that they all relate to the 3rd harmonic because all of the denominators are doublings of 3. Because all of the ratios ultimately relate back to the fundamental, what this means is ‘make an interval above the fundamental that is the same size as the interval of the 24 harmonic to the 25’.

Microtonality and Jazz

There are many places to find evidence of microtonality in jazz. A good place to start is a transcription of Robert Johnson's microtonal singing at Tonalsoft, Joe Monzo's site. Louie Armstrong is another player that naturally made use of a broader, more expressive and precise pallette of tones. It makes since that improvising musicians, working from their ear and experience, put to use what would be the logical extension of the Western tonal system. Alot of electric blues players like Jimi Hendrix and Buddy Guy obviously get alot microtonal stuff going on. Have you ever known a good blues player that didn't bend all over the place? It is funny that when current musical theory cannot explain what 'folk' musicians play, it becomes a real wishy-washy, 'unexplainable something' that is, nonetheless, demanded. Bill Frisell, on a track on 'Blues Dream', plays very distictly the two blue thirds(being a 1/6 tone under and a 1/6 tone above the ET minor third).
Then you have saxophonists like Coltrane and Ornette Coleman, who played very intsinctively, intonation wise. This is all very much related to the artistic interpretation of a melody or solo, meaning the players are mostly playing the microtonal intervals intuitively in performance. Another aspect to consider is the actual harmonic progression of jazz within 12-tone ET, and the ways the sytem points beyond itself.
First let's look at blues harmony. In blues, what we think of as a dominant 7th chord is tonicized. So the tonic chord of the tune has a major third and flat seventh - a chord which would demand resolution in Western harmony. But in this case, we are talking about a naturally extended 7th chord derived from the overtone series where the 7th harmonic is in fact a flat seventh, 30 cents lower than the ET interval. So already the harmony of blues demands an interval that cannot be supplied by the piano. This is where the lower of the two blue thirds comes from. The interval 7:6 from the fundamental. (In C it would be the interval from G to 1/6 low Bb, moved down to the lower pentachord:C to 1/6 low Eb). This is called the septimal(7) minor third and represents the first step beyond what the piano can give us.
But many piano players have also made significant contributions to the move toward microtonality. Take for example, Bill Evans. He introduced a way of playing cluster voicings that could treat an entire diatonic scale as consonant harmonic material. This has two things working for it. 1)It simply continues the tradition of extending what is thought of as consonant within the diatonic scale by creating structures that adequately support in our new ears an extended amount of 'resting tones'. 2)Based on the phenomenon of resultant tones(as Sims calls it; I see it also as taking a harmonic mean) when the interval of a minor second is played, it implies a third note in between the two played and an octave up. So by playing cluster voicings, especially with minor seconds, Evans created a microtonal atmosphere 'above' the notes he actually played on the piano. Monk was also known for this same technique.
This is all just to show how surrounded we are by this development and how natural the transition will be. We cannot, however, continue to incessantly imply these advanced harmonies without actually modifying our system. That is what is such a pain in the ass about alot of jazz players; they feel like they are being courageous and on the forefront of harmonic development and bitch about the low level of their audience's ears, when they are really just spewing alot of very unnatural, contrived crap. There becomes a very large disconnect between what we hear and what our cognitive process knows we should be hearing, and as these drift further and further away from each other, the experience of listening becomes more and more an intellectual one(piecing together the puzzle) rather than a wholistic, resonant, energetic, spiritual, bodily one.

I've been working on writing three part harmony with the 18-24 tone scale of Sims that I record with my fretless one part at a time. I also will be getting my 36 tone guitar in just a few days. I believe a full fledged microtonal project with keyboardist Matt Weires will be getting under way.

Portland Jazz Jams Tv

The Chris Mosley Trio recently filmed a live performance and interview with Darren Littlejohn, the founder of PortlandJazzJams.Com.

You can watch the whole thing right here!

Red and Black

The Trio played a great show on the 2nd at Red and Black Cafe, the host of a really great creative music series put on by Mary Sue Tolbin and Paxselin. They have really created a great atmosphere for the musicians and listeners alike. It's a pleasure to feed a roomful of hungry ears and look up to see a roomful of faces looking back at you instead of huddling closer together to continue the conversation. Thank you Mary Sue!

On another note, I just went to the Reel Music Fest showing of a film called the Legacy of Coltrane. It was really just the full reproduction of the Jazz Casual performance and a couple other filmed performances from the Quartet(and one Quintet with Dolphy) interspersed with interviews with 'fellow musicians and friends of John Coltrane' which turned out to be a few(forgive me) rather inane comments from Elvin Jones. It didn't get into his real 'late' period or even interesting discussion the entire time. It was good, though, just to spend some time watching the Quartet at work. I was very much struck by Elvin's playing in particular. What I had in the past experienced more as 'Elvin sure plays alot' turned into a realization of the deep groove going on, especially in his left hand. His snare chruns out some extremely propulsive beats. As for Dolphy, I can't say I have ever liked his playing with the Trane band. To my ears, his playing developed more out of Bebop that is just really different harmonically, whereas Trane was on another level of harmony beyond the stressed out squirelly lines of out-bop. And let's be honest, Dolphy's flute playing is pretty bad. That said, I do enjoy discs like Out To Lunch and the Illinois Concert.