source file: mills2.txt Date: Mon, 28 Oct 1996 10:19:21 -0800 Subject: Erv's letter From: John Chalmers 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 Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 28 Oct 1996 19:26 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA20403; Mon, 28 Oct 1996 19:26:16 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA20393 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA23293; Mon, 28 Oct 1996 10:26:14 -0800 Date: Mon, 28 Oct 1996 10:26:14 -0800 Message-Id: <3235762173.298371@ycrdi.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu