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Posted by caleb First, I’ll answer your questions literally. Second, I’ll provide some more examples. If you have the time and energy for some back-and-forth, later I’ll show some extensions of this stuff and some applications. 1) First, the formula using i ^ n mod p: what does that mean? Is "^" a logarithm sign? If so, do you look up the integer in the log table? Are the letter variables only, or is "i" Interval and "n" Number, etc.? I hope I'm not alone in my lack of knowledge on this. No, you’re not alone! My bad! The standard terminology, I think, is that we’re doing the “indices of power residues”, or “discrete logarithms”. Sorry about i^n mod p coinage: “i” stands for integer and “p” stands for any prime number where i is relatively prime to p. Yes, ^ stands for “to the nth power”. Perhaps you or someone could suggest a better way to say it. There’s a calculator on the Web by Dario Alpern that calculates these. www.alpertron.com.ar/DILOG.HTM I had someone take his calculator and hot-rod it in lots of ways to be specialized for these kinds of series. You can have it if you like, just give me your e-mail address. Mine is calebmrgn@yahoo.com. So: 2^n mod 13 is an example. The series is: 1,2,4,8,3,6,12,11,9,5,10,7,(1) and the indices are: 0,1,4,2,9,5,11,3,8,10,7,6. The indices, translated into pitch-classes, are C,C#,E,D,A,F,B,Eb,Ab,Bb,G,F#. This is the famous Mallalieu series, which is the mother of all self-similar 12-tone series. It’s also a favorite series in cryptography and sonar/radar! But I digress. Ask me to explain indices, if you’re not already familiar with this concept. From my database of series, some more examples: 15 ^ n mod 577. 13 ^ n mod 1373. 28 ^ n mod 431. Etc. Etc. I’ve done maybe a thousand of these. There are, as you know, an infinite number of primes, but only around 40 million 12-tone series if you hold the first note constant (at “C”, for example.) Therefore, using this method, different combinations of numbers can and do create the same 12-tone series. For example, 10 ^ n mod 499, 136 ^ n mod 613, and 170 ^ n mod 283 all produce the following series: Another resource book you can get from an online used book dealer like Alibris is A Table of Indices and Power Residues. The only problem with this last book is that it only gives the “least primitive root”, that is, the smallest integer that is relatively prime to some prime number. Therefore, it omits a great many useful series. Another name for “self-similar” is “multiple-order-function”, or MOF, for short. “Self-similar” is a general term used by some math people and the music theorist Tom Johnson. He has a book called Self-Similar Melodies. MOF is the term used by theorists like Robert Morris, Andrew Mead, and Phillip Batstone. Stylistically, I’m sort of half-way in between the hardcore modernists like Morris and the minimalists like Johnson. So I have a strong interest in series with tonal associations. Sort of the jazzoid Berg school of 12-tone, if you will. Or sort of Third Stream, although the term is no longer fashionable. 2) Next, by self similar, are you comparing hexachords? Are you saying that when you lay out the entire array of certain rows, you'll get something similar in some of the permutations? More like the latter. I’m not so much thinking in terms of cells or hexachords, although these interest me. There’s a good study of cell-content preserved in different partitions called 'Having your cake and eating it too: the property of reflection in twelve-tone rows (or, further extensions on the Mallalieu complex). There’s also a good paper by Tuukka Ilomaki with yet another take on self-similarity that I could provide you, as an attachment. These are a little different than what I’m after. What I don’t like about “Having Your Cake” article is that it only looks at unordered content. So it’s not really about similar melodic contours. Similar melodic contours are precisely what interest me. No, in “i ^ n mod p”, when translated into a 12-tone row , the every-other note permutation starting on the second note, and the every-3rd permutation starting on the 3rd note all strongly resemble the original series, but the resemblance is “fuzzy”—that is—plus or minus a semitone or two. The “fuzziness” varies from series to series. So cell-content is usually not preserved. However, as you know, if you partition 12 notes into 2 hexachords, often they are identical. That’s not really what I’m after. I tend to like broken symmetry, or near symmetry, and tonal “fields”. Here’s an example: 131 ^ n mod 2767 or 157 ^ n mod 181 The indices of these series make the same contour, or 12-tone row! Indices of 131 ^ n mod 2767: Indices of 157 ^ n mod 181: Series in pitch-classes, transposed to C as starting note: Line it up with the “every 2cnd permutation” starting on second note: C E C# G Eb F A Bb D F# B Ab original note strong similarity. Line it up with the “every 3rd permutation” starting on third note: C E C# G Eb F A Bb D F# B Ab original also good similarity. Looking at the first 5 notes: C, E, C#, G, Eb: (I’ve put the important notes in CAPS:) p/11/2: C f# d E g# a C# f a# G b Eb i/7/3: C E d a# a f C# g# b G Eb f# 5m/p/7/2: C f# a# g# E a f C# d b G Eb Last 5 notes are reproduced in more places: p/9/2: A# e c D F# g B d# G# f a c# p/2/11: A# D F# eb a f g B c e Ab c# r/11/1: A# f c# a g# e D F# c d# B g a# f c# a G# also, r/3/8, ri/4/2, ri/7/5, (and also in 4 of the 5m series--that is--the series created by multiplying the original series intervals by 5) In informal tests, a random series will have a mof content of around 3, meaning that the first 5 notes can typically be found in 3 other transformations within a range of 6 notes. But random series almost never have this degree of self-similarity. Yes, I do like the results, very much. Remind me to speak about the positivism of theorists like Morris, Babbitt, and Mead—their scrupulous avoidance of subjective, emotional terminology when talking theory. But they certainly have great passion as composers…in their way. Next post: what’s any of this stuff good for, etc., more examples, more responses, MOFs by other techniques, etc. Thanks for making me a happy guy, Mark Q. Ask away.
on 10/3/2006, 5:58 pm, in reply to "Re: mallalieu & self-sim series"
24.63.115.63
First, thanks for your interest. It gives me great pleasure to discuss this stuff. Don’t be reluctant to ask me the same questions over and over until I make this stuff clear. Also, don’t be afraid to challenge me if you see errors in what I say. I consider myself to be well-versed in music theory and pretty childish when it comes to math.
“Relatively prime to” means: produces a series of (p-1) different numbers before it “loops”. That is, if something is relatively prime to the prime number 19, it will produce a “loop” of 18 different numbers. If prime is 23, the loop will be 22. And so on.
C,D,Ab,Eb,B,A,F,E,G,C#,F#,Bb. There are more primes than series, if you limit yourself to 12 tones.
Raw: 2766 762 11 1524 409 773 1653 2286 22 1171 2742 1535
Rank: 11 3 0 6 2 4 8 9 1 5 10 7
Raw: 180 29 4 58 24 33 75 87 8 53 178 62
Rank: 11 3 0 6 2 4 8 9 1 5 10 7
C, E, C#, G, Eb, F, A, Bb, D, F#, B, Ab
E G F Bb F# Ab C C# Eb A D B every 2cnd
4 3 4 3 3 3 3 3 1 3 3 3 intervals between series
C# F D Ab E Eb Bb B C G A F# every 3rd
1 1 1 1 1 10 1 1 10 1 10 10 intervals between series
Test the series for other self-similar properties:
The series isn’t particularly rich in literal multiple-order-function--if you’re looking for embedded versions of the first 5 notes, although they do occur. Let’s look at the LAST 5 notes: a#,d,f#,b,g#:
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