source file: mills2.txt Date: Sat, 28 Oct 1995 00:45:59 -0700 From: "John H. Chalmers" From: mclaren Subject: The Glechniszahlen-Reihe Monster --- Along with the Sierpinski gasket and the Wierstrass curve, the Gleichniszahlen-Reihe Monster is one of the more interesting number-theoretic constructs. Consider the sequence: 1 1 1 2 1 1 2 1 1 1 1 1 2 2 1 .. Is there a pattern here? The answer's embarassingly simple: row 2 is "one 1," referring back to the previous row. Row 3 is "2 ones," referring to row 2. And so on. This "likeness sequence" (a loose translation of the German name) grows quickly. Row 27, for example, contains 2017 entries. Clearly the Gleichniszahlen-Reihe Monster can be generalized to any two relatively prime numbers. The monster then takes the form: p q 1 p 1 q 1 1 1 p 1 1 1 q 3 1 1 p 3 1 1 q 1 3 2 1 1 p 1 3 2 1 1 q .. What does this have to do with tuning? Well, the Gleichniszahlen-Reihe Monster offers an attractive method of generating modes from a tuning. It also provides a scheme for traversing ratio space to generate scales (if the units of the sequence are considered as coordinates in ratio space). Lastly, the Gleichniszahlen-Reihe Monster could be used as a melodic engine for algorithmically generating note-sequences in any tuning (if the units of the sequence are considered as indices of an array containing xenharmonic pitches). For more info, see Hilgemeir, M., "Die Gleichniszahlen- Reihe," in Bild der Wissenschaft, vol. 12, 1986, pp. 194-195. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 28 Oct 1995 18:53 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id CAA24365; Sat, 28 Oct 1995 02:54:13 -0700 Date: Sat, 28 Oct 1995 02:54:13 -0700 Message-Id: <9510280952.aa18004@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