Hi,

Just to bring a different persepective here:

More interesting to me than alternate tunings, are simply the types of unpredictable variances when gets when real instruments go "out of tune". To me, music that is accidentally "out of tune" sounds quite rich and interesting.

I used PD recently to generate long tones in eight instruments on a very simple set of four pitch classes, in any one of three octaves. However, a simple Markov chain allows the pitch to drift either up or down by some randomly chosen amount (something between 0 and a quarter tone I believe). The sound turned out to be far richer than I imagined. The best thing is, that the same algorithm can also generate the events for my score of real instruments, though I think I must draw the score by hand to get the right look. So I will probably have PD generate text files. I might even generate images for the score with GEM, but I don't know how to make curves that have the randomness of the ones I draw, so in this case I'd probably trace the images and humanize the look of the lines.

John Cage "Ryanji" pieces were something of an inspiration for this - where he draws pitch curves using the edges of rocks.

~David

On 8/10/06, Chuckk Hubbard badmuthahubbard@gmail.com wrote:

I have to disagree with this. You could play music that sounds just as natural using a series of 6 or 8 perfect fifths. *Any* number of notes separated by fifths sounds natural.

12-tone equal temperament is useful because 2^(7/12) is close to 3/2. In 19-tet, 2^(5/19) is close to 6/5 (minor third), 2^(6/19) is close to 5/4 (major third), and 2^(11/19) is close to 3/2. That makes these exponential divisions of 2 useful for *fooling* the ear, but in all cases they are approximations, and 19-tet's substitute for 3/2 is farther from it than 12-tet's.

I say if you want 3/2, use 3/2.

I don't think 5-tone equal temperament is a substitute for the pentatonic scale. The pentatonic scale is made up of decidedly uneven intervals. Javanese gamelan tuning is notoriously non-standardized, adjusted individually for different ensembles. They usually don't even use octaves. http://www2.hmc.edu/~alves/laras.html

Also, it is not true that all intervals are based on fifths. Anceint Chinese theorists suggested this, but their anceint Chinese secret was that the Ch'in used intervals not derived this way, such as 8/7. Ptolemy and Didymos both suggested tuning thirds based on the 5th harmonic, 5/4, rather than Pythagoras' 81/64, which is more out of tune than 12-tone equal temperament. Islamic theorists used intervals like 18/17, 81/68, and 27/22, supposedly because it was simply easier to tell people where to put the frets that way. This part is more against the Jeans quote than the article, though.

BTW, he mentions the idea of 7 plus or minus 2. George Miller's essay on this idea is awesome: http://www.well.com/~smalin/miller.html One thing he mentions, that I think dispels the idea that the 7-tone scale is inviolable, is that folks can recall longer and longer series if they form a vocabulary of smaller parts, i.e., become more familiar with the material. How else could we differentiate 26 letters, or remember 10-digit phone numbers? Or recognize hundreds of people on sight? Cross-categorizing- Identifying a fat, bald man in a blue shirt is far easier than recognizing someone based on any one of those criteria. A melody is far easier to recognize than a single note.

-Chuckk

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On 8/10/06, Charles Henry czhenry@gmail.com wrote:

...

I don't think 5-tone equal temperament is a substitute for the pentatonic scale. The pentatonic scale is made up of decidedly uneven intervals.

Yes, it's true. They're two different things. I disagree with the John S. Allen article that there's a connection between the pentatonic scale, diatonic scale and fibonacci sequences. One is a subset of the other. We're not get better approximations of just intonation as we move from one to the other. I'm saying that we get better and better approximations, in a monotone decreasing series, as we move through the equal tempered scales using the fibonacci sequence. I'd trust in the just intonation system, if it weren't so hard to modulate keys.

You'd get still better approximations with 1 million divisions of an octave. Is there any relation between integer ratios and logarithmic divisions based on the Fibonacci sequence? Seems like that's really the only question. I guess I'd be hesitant to say no, with it phrased that way. I like just intonation because there are so many more possibilities to modulate; which lend themselves better to some instruments than others, of course. My JI Sequencer was an attempt to make this more intuitive, for non-real-time composition. (I intend to clean it up and post it soon, hopefully today) Toby Twining's 13-limit vocal works are the best JI I've heard: http://www.justintonation.net/soundfiles.html#twining The top set of examples. I think the Kyrie is the best, but the Sequence I, which is actually a Dies Irae, is pretty haunting (not so much in this exerpt).

The small-numbered ratios of frequencies create consonant intervals, because of the harmonic overlap (when using a harmonic series). These are the local minima of dissonance as a function of interval. I think that what we hear in these more outlandish tuning ratios are only approximations of the small-numbered ratios (don't ask, how small is small-numbered? cause I really don't know...It's pretty hard to tell).

What do you think about the 4/3? At times this interval was forbidden in counterpoint as a dissonance, and obviously the 4th harmonic of the higher tone and the 5th of the lower clash, and even more so the 5th of the higher with the 7th of the lower. Yet it is a small-number ratio, and most consider it consonant, if only as a suspension. Then again, chords built entirely of perfect 4ths are common in jazz, and not at all unsettling (I would call this a Law of Pragnanz situation). Harry Partch insisted the ear could adapt to arbitrarily high prime ratios. He only went as high as 11, and that tends to be pushing it for most people. 7 is essential though, IMO. I use 11-derived intervals, usually only at the higher end of a chord. At any rate, insisting that all scales are built on perfect fifths is a regression to Pythagoras.

For Islamic theorists, supposedly al-Farabi described positioning the frets so that, for instance, one fret goes at 8/9 of the string, the next at 8/9 of the remaining length, and the next one halfway between them. Halfway between 72/81 and 64/81 is 68/81, or 81/68 of the frequency. Definitely not small, but easy to place, and maybe having some of the interesting quality you're talking about with a distinctly Arabian sound...

There's something strange about those mistuned harmonics that gives an instrument its richness. I know that Plomp studied this back in the 60s "Beats of mistuned consonances" in JASA. When we detune two pure tones away from a small-numbered ratio (like 3/2), there are beats, that do not correspond to difference tones, cubic differences, or other distortion products. It's not a physical phenomenon, not part of the cochlea, but something to do with the brain and how it groups the harmonics together (could possibly be related to stochastic ghost resonance, mentioned by Chialvo in Chaos, which has to do with the way that harmonic ratios of impulses sync up along single neurons)

Is this related to how they say we hear certain difference tones even when the real tones are isolated in either ear? Or maybe just coincidence. A lot of people say that just intonation "lacks the energy" of 12-tet, and JI enthusiasts counter that 12-tet lacks the restfulness of JI. Anecdotally, Western theorists pushed for more and more modulation (intellectual stimulation?), while some Eastern music still is happy to rest on one chord for a long time, or to keep one tonic drone through a whole piece. Ben Johnston suggested tuning temperament was behind the drug addiction and restlessness of so many jazz and rock musicians, lol.

What is it with people and their morbid fascination with fibonacci sequences?

For all I know you only said that for my benefit, lol. I'm not a number expert, but I actually find the plain old prime series more mysterious. -Chuckk