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hypothesis - the only harmonic ratio that is necessary for LCC to be derived (or the circle of fifths for that matter) is 3:2, the perfect fifth (russell merely calls it a "fifth")

my view on harmonic ratios is:

first, that it is easily demonstrable that the first non-octave interval produced by the overtone series is what classical theory calls the perfect fifth. guitarists may play the twelfth fret harmonic followed by the seventh fret harmonic to hear this. overtones on wind or brass also...

secondly, twelve-tone equal-temperament alters this ratio slightly (2 cents flat) so that the circle of fifths may produce octaves and unisons of exactly twelve pitch classes.

further, if you play a fifth of a fifth of a fifth, it has been suggested that a ratio of 27:8 (3/2x3/2x3/2) results and that the human ear may perceive this but what i hear is an octave above a major sixth (ideal frequency ratio 5:3) and if you tune a pitch to exactly 27:16 (an octave below a fifth of a fifth of a fifth) all i am going to hear is a slightly out of tune sixth.

you see, i am simply not going to buy into the suggestion that i am going to be able to perceive the difference between something like 27:16 and 28:16 or worse still, 243:128...

what i am trying to say is that 3:2 is a fifth and the circle of slightly adjusted fifths gives us the familiar pitch classes that APPROXIMATE other harmonic ratios and that no more than the simplest ratios need be dealt with:

2:1 the perfect octave or 12.00 half steps

3:2 the perfect fifth or 7.02 half steps

4:3 the perfect fourth or 4.98 half steps

5:4 the major third or 3.86 half steps

6:5 the minor third or 3.16 half steps

major sixths are inverted minor thirds - 6:5 becomes 5:6 or 5:3 (8.84 half-steps)

minor sixths are inverted major thirds - 5:4 becomes 4:5 or 8:5 (8.14 half-steps)***

these are the most commonly accepted ratios of musical intervals. multiple major seconds may be produced but 9:8 is the most widely cited although 10:9 shows up as well as 8:7 and others. the half-step is usually defined as 16:15.

you see, whomever has created, discovered, or otherwise invented equal-temperament has saved us from a whole lot of trouble. i am not suggesting that players of professional groups aren't making microtonal adjustments in order to blend their harmonies to maximum consonance. of course they are; however, what i am suggesting is that for me to perceive 243:128 (a pythagorean major seventh) as being unique from something close like 15:8 (the most commonly accepted ratio for the major seventh) other than being slightly out of tune...

the term "absurd" comes to mind. see also "the science of musical sound" by john r pierce...

LCC is based on the fifth. george russell told me so himself. the ratio 3:2 is all that is necessary to generate not only the circle of fifths but the entire LCC.

another two cents

***ps - all of the math for converting intervals in terms of harmonic ratios to half-steps can be derived from:

R = 2 raised to the power of (H/12) or R = 2^(H/12)

where R is the harmonic ratio and H = number of half-steps in twelve tone equal temperament.

therefore H = 12 log R / log 2

dogbite wrote: "the term "absurd" comes to mind"

I'm told dogs only bite if they feel threatened or sense fear.

What's so threatening about what I'm writing? If it's all wrong, it'll all evaporate like a puddle of dog pee. If it's not, something good can only come out of it.

Let's not be so adversarial already. Remember, I'm not advocating a tuning system, and if you could wait to read my further installments, I'm also not saying that the "real" M6 is the 27th partial. The ladder actually implies that, but I agree with your statement that our ears probably prefer 5:3. Actually part of my whole problem with the ladder.

Thanks for the formula, though. I can get back into cents finally.

actually strachs, i have no problem whatsoever with what you're writing. i was really trying to point out that the ladder (in my view) is simply a series of fifths rather than a series of compounding harmonic ratios. there is a mention of the interval 3072:243 (which i will assume for the moment leads to the pythagorean comma) in the book itself (but not by russell) that i find truly exasperating not because i am irritated with the source but more in a pragmatic sense: what on earth am i supposed to do with these kinds of numbers on a musical instrument?

but no, back to that ladder: the way i view it is that it's a just a series of equal-tempered fifths; therefore, introducing (for example) the 27th partial seems unnecessary to me, that's all. so no strachs, don't get me wrong: i absolutely champion your exploration into harmonic ratios both simple and complex and thus eagerly await any and all conclusions you have drawn from your studies.

i will therefore sit and watch on the sidelines as you walk us through your thoughts. it was not my intention to interrupt you (this is why i posted all this on a separate thread) or otherwise derail your train of thought in this matter. carry on, and enjoy the formula; it's real interesting when applied to that lattice you guys were talking about...

I have a feeling that the Dog, being the expert cipherer, knows the difference between an ET fifth and and a pure fifth.
BTW I played a gig about six weeks ago with a pianist who did much of the arranging on Gaucho and quite a few other sides. He told me about an interesting book, The Just Intonation Primer. Interesting stuff but would drive my OCD piano tuner even more nuts.

joegold wrote:"
They're very smooth sounding and *harmonious*/restful.

The guy that told me about the Just Intonation book describe them as 'sad..they just sit there'. I don't think he meant that in a derogatory way, more like an impression.

"But a ladder of pure 5ths only exists within Pythagorean tuning..."

i was talking about equal tempered fifths.

by the way, of course you already know that a pure perfect fifth is 3:2 (pure number 1.5) and that an equal tempered fifth is 2 raised to the power (7/12) or 2^(7/12) or 1.498307..., an irrational number which by definition is not a ratio between two integers at all...

and yes, of course there is a difference; however, i was not attempting to provide a historical overview of just, meantone, or any other tuning system...

i did get the distinct impression that strachs was leading up to something and that i was distracting him from ultimately doing so. i could go on for days about string lengths, harmonic ratios and node positions but i'm going to sit out for a moment to see what strach has in store inside those future installments of which he was speaking...

chat soon?