source file: mills2.txt Date: Mon, 18 Dec 1995 07:28:56 -0800 From: "John H. Chalmers" From: mclaren Subject: Chalmers' Constant and a family of constants giving rise to yet another non-just non-equal-tempered scale --- As John Chalmers has pointed out, the iterated absolute log function produces a set of unpredictable numbers unless it starts with a certain number. In that case, the series enters a fixed point. If the function is called the mclaren series, clearly this constant should be named Chalmers' Constant. It lies between 0.399012979 and 0.399012978. This constant is different for each logarithmic base. Thus a family of constants is implied. For base e, the constant lies between 0.567143289 and 0.567143291. For other bases, the constant is different. This implies in turn an infinite number of non-just non-equal-tempered scales. One could, for example, generate one such scale from the constants for the logarithmic bases of the primes: C[1] = fixed point for iterated log base 3 C[2] = fixed point for iterated log base 5 C[3] = fixed point for iterated log base 7 C[4] = fixed point for iterated log base11 C[5] = fixed point for iterated log base 13 C[6] = fixed point for iterated log base 17 C[7] = fixed point for iterated log base 19 C[8] = fixed point for iterated log base 23 and so on. Another scale could be generated by taking C[1] = fixed point for iterated log base pi C[2] = fixed point for iterated log base e C[3] = fixed point for iterated log base F C[4] = fixed point for iterated log base L C[5] = fixed point for iterated log base C C[6] = fixed point for iterated log base G and so on, where F is Feigenbaum's constant, L is the first Liouville number, C is Chapernowne's number, G is the Euler gamma constant, and so on. As musical intervals all these values appear to be particularly intractable from a just intonation point of view; large rational fractions are needed to approximate virtually all of them. Clearly these musical intervals do not fit into the just scheme of things. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 18 Dec 1995 16:31 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id HAA03909; Mon, 18 Dec 1995 07:31:02 -0800 Date: Mon, 18 Dec 1995 07:31:02 -0800 Message-Id: <9512180729.aa02219@cyber.cyber.net> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu