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Undertones are so low that for the most part they're inaudible. You don't have to go down many octaves before they become say quarter notes or eighth notes. A bit more and we're talking about one bar and four bars. Consider that two notes can be many octaves away from each other, and if they're perfectly in tune they will still harmonize. This is also true for undertones. The 12 bars of the blues can be thought of as one octave that is several octaves below audible sound. So the first bar and the sixth bar are a tri-tone apart. Notes played on the 1st, 5th, 9th bars will make an augmented triad. So this reminds us that where we play in the form causes harmonies to be created many octaves below our range of hearing. Even though our ears don't pick up these consonances and dissonances, our intuition may and we certainly perceive them as sounding good. The bigger point that I'm getting at is that form follows the same laws of consonance and dissonance as audible notes do. This is the reason that harmonious proportion is important to achieve when structuring your solos. There is a tri-tone relationship between the first bar of your solo and the mid-point. Between the first bar and the point two-thirds into your solo there is a perfect fifth relationship. If you play every other bar you create a whole-tone scale. You can set up effects that are like chords in a certain way by becoming aware of these nodes/harmonics. In a blues it is just a bit easier to deal with these ratios because the form of the blues perfectly reflects out equal-temperament system.
2/17/06
Undertones as rhythm and form
Undertones are so low that for the most part they're inaudible. You don't have to go down many octaves before they become say quarter notes or eighth notes. A bit more and we're talking about one bar and four bars. Consider that two notes can be many octaves away from each other, and if they're perfectly in tune they will still harmonize. This is also true for undertones. The 12 bars of the blues can be thought of as one octave that is several octaves below audible sound. So the first bar and the sixth bar are a tri-tone apart. Notes played on the 1st, 5th, 9th bars will make an augmented triad. So this reminds us that where we play in the form causes harmonies to be created many octaves below our range of hearing. Even though our ears don't pick up these consonances and dissonances, our intuition may and we certainly perceive them as sounding good. The bigger point that I'm getting at is that form follows the same laws of consonance and dissonance as audible notes do. This is the reason that harmonious proportion is important to achieve when structuring your solos. There is a tri-tone relationship between the first bar of your solo and the mid-point. Between the first bar and the point two-thirds into your solo there is a perfect fifth relationship. If you play every other bar you create a whole-tone scale. You can set up effects that are like chords in a certain way by becoming aware of these nodes/harmonics. In a blues it is just a bit easier to deal with these ratios because the form of the blues perfectly reflects out equal-temperament system.
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Intervallic ratios also generate rhythms. For example, if you take a sounding perfect fifth and drop it by about eleven octaves, it comes out as a symmetrical rhythm that is the resultant of the interferance of the two terms. Since it's 3:2 in the overtone series, it would be like the result of playing a half note and a dotted half note at the same time. So it would literally sound like a half note followed by two quarter notes and then a final half note. The first half note would be accented because that's where the terms coincide. HTe same thing can be done with triads or seventh chords, and more interesting results will occur. It's an interesting field of study.
What this all relates to is a book that Steve Coleman turned me on to called "Harmonic Experience" by a guy named Walter Mathieu (sic). In this book, he relates various intervals with various rhythms. For example, the first interval he deals with is the perfect unison. Mathieu's excercises start with singing unison with a drone (using Saregam, the hindi syllables.) The related rhythmic excercise is to tap two pencils in unison rhythm on a table or something. He reccomends that you spend, say ten minutes a day on each thing. You realize that what seems like the most trivial excercise in the world is in fact quite challenging. Your voice naturally strays from the drone unless you concentrate. Likewise, in pencil tapping, your two taps begin to stray. This ratio is 1:1.
He then moves on to 2:1 with the intervallic manifestation being the octave and the rhythmic excercise being to tap in, say eighth notes in the left hand and quarters in the right hand.
Now you would move on to what HUCBALD is talking about which is the 3:2 ratio. This is essentially the perfect 12th, although Mathieu always does "octave displacement" in which the intervals are collapsed to within an octave of the tonic. He relates this to a 3 against two rhythm (3:2 ratio produces a fifth) which can be fairly easily produced after some practice. Mathieu's theory of harmony is built on these ideas. He slowly builds up a structure of harmonies and related rhythms. Trane came out to California to study with Mathieu and he actually lives down in Sebastopol, I think and sometimes teaches at Sonoma State. He recommends using a guitar to see the ratios of the strings. Very interesting approach to music.
Monk,
I was just about to post a link to Harmonic Experience. It is by far the clearest and most practical book on the acoustic foundations of harmony, tempered and natural tuning that I've ever seen. I'm working through it right now, it is much a practical workbook as a didactic text book.
Mathieu give many practical singing exercises to attune the student to the subtleties of tuning. He's right in the pure Phythagorean AND Hindu tradition. Pure genius, he makes everything so clear.
Monk, I think you meant 'octave reduction' rather than 'octave displacement'. Right?
Dave-
Yes, I blew it and wrote "Octave displacement" instead of "Octave reduction." Please forgive me all. And, please call me Bill. I had to start a new username/blog because I had forgotten what I was and finally just decided to try and make them the same. Could I e-mail you about some blog-related issues as I would like to start a blog that's just reviews. I want to have it similarly designed so that instead of categories, people will just see "reviews" and then I can put the media type in the left paragraph. Sorry that this is unrelated to the subject at hand.
David,
I like this concept of creating nodes in space/time through harmonic principles of a musical form. I think this is very relevant to understanding the depth of structure to a form like the blues. I do feel, though, that there is a certain disconnect between the larger idea and the intervals you assign. Take the tri-tone. This is saying that the form split in half is like an octave split in half. But that is not at all what the diagram you show above indicates. For it, the wave form with two waves instead of one represents an octave above the fundamental(2:1), not a symmetrical split. Then there is the whole-tone scale. If we play on every other bar of a 12 bar form, we are dividing it into six parts. We see this happen on the diagram you show, but this tone would be the doubling of 3, meaning a fifth. If we went all the way through 12, the actual notes that would have been generated would be(in C: C,D,E,F#,G,Bb). I feel it is more difficult to reconcile the 12-tone ET space with harmonic form than you recongnize in the article. But maybe i'm missing some of your thought process. At any rate, this is great stuff to be thinking and talking about.
You're right Chris. The diagram I included had nothing to do with what I was talking about except that it illustrated harmonics. Sometimes you need graphics and just have to settle with what you can find. I'm glad you were paying attention. :-)
If I understand the physics of this correctly, the vibration whose wave peaks in the 1st & 7th measures would be beating at twice the frequency of the "fundamental" produced by the wave peaking in all 1st measures. This of course would be the octave. Continuing the mathematics of this, the 1st & 6ths bars produce a 9th, not a tritone. There wouldn't appear to be a tritone above this "fundamental" until you start considering positions within measures. The frequency between the middle of the 1st bar & the mid-point of all the bars may be a tritone above the "fundamental", depending on how many bars are in the cycle. The frequency of the beginning point & the mid-point won't be a tritone, but an octave of the "fundamental". The point 2 3rds into a "solo" may have any relationship with the "fundamental" as defined above, though my math does agree that its frequency will be a 5th above the vibration whose wavelength is equal to the length of the "solo". (So will the point 1 3rd into the "solo".) Playing every other bar doesn't create a whole-tone scale. When you divide a wavelength into 12 equal divisions, each length measured from the beginning to 1 of the divisions is dividing the wavelength more or less than 1/2. In other words, each measurement from the beginning to 1 of the divisions is multiplying the frequency by more or less than 2. This is not equivalent to multiplying the frequency by 2^(1/12), which is what would be required to create a chromatic scale (& which would then give you the overtone series if you played every other measure). When you play every other bar of a 12-bar cycle, you play the fundamental, the 5th, the 5th, the 8ve, the 5th, & then a sharp b3. The 12-bar blues isn't easier to set up harmonies with. 1/3rd of them don't fit easily into the tempered scale or the overtone series. On the other hand, a cycle whose number of bars is a perfect square (such as 8 or 16) will at least have harmonies which fit into the overtone series.
Thanks for bringing up an interesting idea. Le'me know if you don't get my math (I expect you will), or if you do get it but can show me how I'm off.
hucbald: A 5th doesn't produce a 1/2-note & 2 1/4-notes, but a triplet. (3:2 = 3/2.)
chris mosley: By my calculations, the notes above C generated by the waves whose lengths span from the 1st bar to the other 11 bars are, respectively: C G G C G Eb C A G F Eb & Db. (Also, the Eb, A & Db are sharper than in the tempered scale.)
Brian,
i'm not sure how you arrived at these tones. i would say the progression of the overtone series is: C,C,G,C,E(12 cents below ET),G,Bb(32 cents below),C,D,E,F#(50 cents below). These are the first 11 harmonics. The Db comes in at 17, the Eb at 19 and the A at 27, I believe.
Let's call whatever length of time the 12 measures takes "Time Span". If the frequency of a wave having one cycle for each Time Span is equal to, let's say, that of C4 a.k.a. "Middle C", i.e. 261.63 Hz., then the frequency of a wave having one cycle for each passage of 11/12ths of Time Span will be equal to 12/11ths of 261.63 Hz, which is 285.41 Hz, i.e. D4 less 49 cents. The frequency of a wave having one cycle for each passage of 10/12ths of Time Span will be equal to 12/10ths of 261.63 Hz, which is 313.96 Hz, i.e. Eb4 + 16 cents. The frequency of 9/12ths of Time Span will be 348.84 Hz, which is F4 less 2 cents. 8/12ths of Time Span is 392.45, G4 + 2 cents. 7/12ths is 448.51, A4 + 33 cents. 6/12ths is 523.26, C5. 5/12ths is 627.91, Eb5 + 16 cents. 4/12ths is 784.89, G5 + 2 cents. 3/12ths is 1046.52, C6. 2/12ths is 1569.78, G6 + 2 cents. 1/12th is 3139.56, G7 + 2 cents. To summarize: C, D, Eb, F, G, A, C, Eb, G, C, G, G. (I gave these results in reverse order previously & accidentally put the original "C" on the wrong end. Apparently I also didn't estimate the cents for D correctly & put it closer to Db, but the calculations here should be correct to the cent.)
You're right about the Overtone Series (including Db being the 17th, Eb the 19th & A the 27th), but the process above is different: For each step in the process above we're making a cycle of a wave fit with a Time Span which is decreasing by one 12th of it's original length. In other words we're making its wavelength to be 11/12ths of the original wavelength, then 10/12ths, then 9/12ths, etc. In still other words we're making its frequency equal 12/11ths of the original frequency, then 12/10ths, then 12/9ths etc. The Overtone Series is produced by a different & simpler process: The frequency of each partial is increasing by a constant number of Hz which is always equal to the frequency of the fundamental.
Nice work Brian. That's exactly what I was getting at with this post.
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