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Old 08-02-2007, 06:51 PM
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I don't regard ET as harmony, and therefore, very strictly speaking, not pure music. I know people write good music with / for it, but it seems like one of the first perversions to come along. It's a system designed by bureaucrats, not mathematicians. In It can be mathematically described by ascending or descending by the 12th root of 2, but is not a harmonic relation, so it is mostly wrong harmonically except when you get to the octave. All of matter would be destabilized if one tried to impose ET on atoms and molecules. One way of looking at radioactivity is that the atom is trying to lose internal dissonance. It keeps degrading until finally all the parts are "in tune". The whole atomic structure stays together because of harmony.

Just harmony is about the only thing I can hear. It's the whole reason I started getting into fret-less instruments. A guitar is impossible to tune because b cannot simultaneously be the 3rd of g and the 5th of e. I wanted to hear some thirds that sounded like perfect constructive wave interference instead of noise. Perfectly constructive wave interference is how I define harmony. But can you really have that with a perfect third between a fundamental and a fifth? The 3rd has to be related perfectly to both the tonic and the fifth. How many notes can you put into a chord before something interferes destructively? Poly-chords are going to be very problematic. So this is not really an easy project, you see. As notes move through a melody, others must adjust. It's easy with just two notes, but 3 or more get complicated.

I changed teachers once because I just could not dig training myself to hear and play as if there were frets. The teacher was trying to get me to hear and play ET. My practice is to tune the notes to whatever the other instruments are at with slight adjustment of position. I can't hear ET. I can listen for perfect constructive interference. It is more subconscious than conscious and I know in practice most notes go by faster than I can hear them just right. But it gets better with practice. As far as the 5ths, they are very close whether one is in equal temperament or in just harmony. 4ths are also not too far out. The rest is pretty much a mess.

Here's a cool chart that ****yzes the differences between ET and pure harmonic music in a couple of different ways. The ratio column is the one I will be using mostly to do the ****ysis. There are a couple of things that can be gleaned right away if you look at G and A on that chart. Before we get into that it is important to recognize that the scale (to me, there is really only one scale per fundamental pitch and then different modes of the one scale) is comprised of simple note relations that follow the series 1:2, 2:3, 3:4, 4:5, 5:6, 6:7, 7:8, 8:9. That is the basis of the scale. It seems 8:9 is the whole step if I remember right. 15:16 is the half-step. Where it tends to be problematic I think, is when one plays chords with stopped notes as the fundamental when another instrument plays the fundamental as an open string. That's the direction to work in to get to why it might sound more like constructive interference with all the instruments tuned in 5ths. But beyond that we need to know what pitches the strings of all the instruments in the ensemble are tuned to. If I get that and get my head right, I can derive some relations that might show something or not. We should divide the work load. Here's what we do. Take the violin, viola, cello, and DB. Make a chart with all the strings pitches tuned in 5ths, and then derive the partials of the different strings in terms of Hz. Then we can use the Hz. to compare the harmonic relations across the strings of the orchestra as different chords are played. What should emerge is a set of preferred positions that play the notes of chords right on a just harmonic position as opposed to one that is off. If we make one chart with the DB in 5ths and another with it in 4ths, that should give us some ideas about what goes on in terms of the constructive interferences we hear, whether these are pure or off the ratios a bit.
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