Erv's letter

I too saw the Ian Stewart Sci Am column and mentioned it to Erv
a couple of days ago when he called me about Sonja's MicroFest in
El Paso. People on both the Gamelan and Tuning list have asked
various questions about this aspect of Erv's work. I can't as yet
say too much, but I'll try to fill in some background.

Erv sent me a letter last summer about the Padovan sequence and
indonesian tunings, but alas I filed it somewhere that I can't locate.
I assumed it was a preliminary personal communication and figured
that more info would be forthcoming. I'm sorry to admit that I didn't
take the time to digest it then and really can't tell you much about it.
Erv is in Mexico breeding plants until Nov 7, when he will be presenting
at Dr. Mommy's MicroFest in El Paso. I expect him back in LA about
the 11th of November and I will ask him for another copy of his letter
and permission to post it to the Gamelan and Tuning Lists.

Pascal's triangle is a triangular array of integers arranged
so that each entry is the sum of the two above (and to either side).
Historically, it appears to be of Chinese origin. By summing along
different diagonals, various numerical constants such as the
Golden Section (I.618034...) may be found. It is not surprising
that the Padovan series constant 1.324718..(aka Plastic Number) would
be among them. (The GS and the PN are the limits of the quotients of
successive terms of their respective series.)

While it's difficult to construct PT in ascii, I'll try.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

Each succeeding row starts and ends with 1 and consists of the sums
of each pair of adjacent numbers in in the row above. Each row
also consists of the Binomial coefficients and the Combinations of
N (the row index) items taken M at a time ( 0 <= M <=N).

The Golden Ratio is also derivable by annexing squares whose sides
are Fibonacci numbers. Quarter circles erected on the diagonals of
these squares approximate a logarthmic spiral. The ratios of the
sides of successive squares approaches the GS and F(n+1)= F(n)
+ F(n-1) The F sequence is 1 1 2 3 5 8 13 21 ....

An analogous construction of equilateral triangles also traces out
a logarithmic spiral and the sides of the triangles form the Padovan
Sequence, 1 1 1 2 2 3 4 5 7 9 16 21 ... P(n+1)= P(n-1)+P(n-2).

For more information, see Ian Stewart, Mathematical Recreations, Tales
of a Neglected Number, Scientific American June 1966 pp102-103.

The Golden Section or Ratio is approximately 1.6180339 or 833 cents.
The Padovan or Plastic Number is about 1.324718 or 487 cents. I
don't recall right now if Erv used cycles of this rather flat fourth
to generate scale or derived the intervals of slendro and pelog like
scales from Pascal's triangle or Padovan spiral. Erv's communications,
alas, are sometimes cryptically concise.




--John

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