calebprime
17th September 2006, 12:12 PM
Here's another post headed straight to the bottom, with a bullet! :D
If, as I expect, there is little interest, well, let this thread go the way of the Edmund Fitzgerald.
http://cimss.ssec.wisc.edu/wxwise/fitz.html
Keep in mind that this poster has very limited math skills. In highschool, I was playing guitar in a bar band and math was first thing in the morning. (I do mean "math" with an "a", not an "e".)
Recently I've been trying to find formulas that generate self-similar musical patterns, specifically 12-tone patterns. This has paid off in the sense that I've found some formulas that generate these patterns very successfully.
(This might have been considered sexy in 1975, but now, well, Milton Babbitt is 90 years old and Elliott Carter is maybe 120, and Robert Morris has retired.)
Quadratic residues, binomial coefficients, Stirling numbers, Fibonacci series, Golomb rulers, Sidon Sets, and Collatz-type sequences have sometimes yielded good self-similar 12-tone series. I don't really understand what I'm doing, but I can apply these concepts.
Before the ship is abandoned, maybe someone here can help explain 2 more things.
These are a Costas Arrays and something I found by Kevin Brown called cyclic divisibility.
Costas Arrays, or Welsh Costas:
http://www.hindawi.com/GetArticle.aspx?doi=10.1155/JAM/2006/26385
In this case, I'd be happy if someone could generate or locate a list of all Costas arrays of size 12. I haven't been able to find one. I'm really looking for the specific constructions themselves, not how many there are.
cyclic divisibility:
http://64.233.161.104/search?q=cache:Lp4pMOyQ7ggJ:www.mathpages.com
quote:
"For another example, consider the polynomial f(x) = x^2 - 4x - 9,
whose discriminant is 4*13 = 52. Every sequence of residues mod 13
that satisfies this recurrence must have a period dividing 156. One
such sequence, with a period of 12, is
1 2 4 8 3 6 12 11 9 5 10 7
which gives an integer with the decimal equivalent
13984822237218 = (2)(3)(7)(41)(61)(157)(847997)
and we can verify that this numbers and all its rotations, reversals,
and 13s-complements share a common divisor of (2)(3)(7)(61)(157) with
13^12 - 1.
For just one more example, consider the same polynomial, but this time
evaluate the recurrence modulo 2(13) = 26. One solution sequence,
having a period of 12, is
1 2 17 8 3 6 25 24 9 18 23 20
which gives an integer with the decimal equivalent of
76753544008735881 = (3^6)(7)(19)(31)(37)(149)(4632011)
This number, and all its rotations, reflections, and 13s-complements
share a common factor of (3^4)(7)(19)(31)(37) with 26^12 - 1..."
End quote]
again:
"Every sequence of residues mod 13
that satisfies this recurrence must have a period dividing 156. One
such sequence, with a period of 12, is
1 2 4 8 3 6 12 11 9 5 10 7
my question: what are the other sequences of residues mod 13 with a period of 12?
There are two questions here. Hope I've narrowed it down so that someone with an interest in number theory and the saintliness of Mother Teresa (!) could provide the answers for me. Even a tip about where I could post these questions would be appreciated.
To summarize, I've got 2 questions:
1) a list of all Costas arrays of size 12.
2) f(x) = x^2 - 4x - 9,
Every sequence of residues mod 13 with a period of 12 that satisfies this equation.
I'm really looking for the specific answers, not a general explanation.
thanks.calebprime :boggled:
p.s. I ask people these questions, and mostly, they just back away slowly. Is it my personal hygiene?
If, as I expect, there is little interest, well, let this thread go the way of the Edmund Fitzgerald.
http://cimss.ssec.wisc.edu/wxwise/fitz.html
Keep in mind that this poster has very limited math skills. In highschool, I was playing guitar in a bar band and math was first thing in the morning. (I do mean "math" with an "a", not an "e".)
Recently I've been trying to find formulas that generate self-similar musical patterns, specifically 12-tone patterns. This has paid off in the sense that I've found some formulas that generate these patterns very successfully.
(This might have been considered sexy in 1975, but now, well, Milton Babbitt is 90 years old and Elliott Carter is maybe 120, and Robert Morris has retired.)
Quadratic residues, binomial coefficients, Stirling numbers, Fibonacci series, Golomb rulers, Sidon Sets, and Collatz-type sequences have sometimes yielded good self-similar 12-tone series. I don't really understand what I'm doing, but I can apply these concepts.
Before the ship is abandoned, maybe someone here can help explain 2 more things.
These are a Costas Arrays and something I found by Kevin Brown called cyclic divisibility.
Costas Arrays, or Welsh Costas:
http://www.hindawi.com/GetArticle.aspx?doi=10.1155/JAM/2006/26385
In this case, I'd be happy if someone could generate or locate a list of all Costas arrays of size 12. I haven't been able to find one. I'm really looking for the specific constructions themselves, not how many there are.
cyclic divisibility:
http://64.233.161.104/search?q=cache:Lp4pMOyQ7ggJ:www.mathpages.com
quote:
"For another example, consider the polynomial f(x) = x^2 - 4x - 9,
whose discriminant is 4*13 = 52. Every sequence of residues mod 13
that satisfies this recurrence must have a period dividing 156. One
such sequence, with a period of 12, is
1 2 4 8 3 6 12 11 9 5 10 7
which gives an integer with the decimal equivalent
13984822237218 = (2)(3)(7)(41)(61)(157)(847997)
and we can verify that this numbers and all its rotations, reversals,
and 13s-complements share a common divisor of (2)(3)(7)(61)(157) with
13^12 - 1.
For just one more example, consider the same polynomial, but this time
evaluate the recurrence modulo 2(13) = 26. One solution sequence,
having a period of 12, is
1 2 17 8 3 6 25 24 9 18 23 20
which gives an integer with the decimal equivalent of
76753544008735881 = (3^6)(7)(19)(31)(37)(149)(4632011)
This number, and all its rotations, reflections, and 13s-complements
share a common factor of (3^4)(7)(19)(31)(37) with 26^12 - 1..."
End quote]
again:
"Every sequence of residues mod 13
that satisfies this recurrence must have a period dividing 156. One
such sequence, with a period of 12, is
1 2 4 8 3 6 12 11 9 5 10 7
my question: what are the other sequences of residues mod 13 with a period of 12?
There are two questions here. Hope I've narrowed it down so that someone with an interest in number theory and the saintliness of Mother Teresa (!) could provide the answers for me. Even a tip about where I could post these questions would be appreciated.
To summarize, I've got 2 questions:
1) a list of all Costas arrays of size 12.
2) f(x) = x^2 - 4x - 9,
Every sequence of residues mod 13 with a period of 12 that satisfies this equation.
I'm really looking for the specific answers, not a general explanation.
thanks.calebprime :boggled:
p.s. I ask people these questions, and mostly, they just back away slowly. Is it my personal hygiene?