source file: mills2.txt Date: Fri, 21 Jun 1996 11:55:30 -0700 Subject: Post From McLaren From: John Chalmers From: mclaren Subject: nj net scales revisited -- In Tuning Digest 707, Paul Hahn quoted my previous post in which I remarked > For instance, one could choose to define a tuning > in which the just twelfth is to the octave as pi is > to two. This is John Harrison's pi tuning, -- mclaren and Paul added "Erh? I thought in pi tuning the octave was to the major third as pi is to one. The two are not equivalent, at least according to my calculations." -- Paul Hahn Paul is correct so far as he goes. I had in mind (and should have mentioned that I had in mind) Charles Lucy's modification of Harrison's pi tuning. In Lucy's more general modification, the ratio and difference of two intervals (a large interval and a small interval) are used, with octave corrections, to obtain an unlimited number of subsequent pitches. Paul, however was incorrect when he posted "And how is this scale self-similar? The Golden Tuning is because successive trips around the spiral of fifths keep subdividing intervals by Golden Sections, but pi tuning does no such thing." -- Paul Hahn [pi/2]^N and [4/pi]^N with N running from one to infinity produces a set of self-similar intervals insofar as the ratio of one interval to the subsequent or previous one is always pi/2 or 4/pi, depending on whether you go up by pi/2 or down by pi/2. > Or one could choose to define a tuning in which the > interval of 2 octaves + 7/6 is to the 1/1 as > 4.6692106 (Feigenbaum's constant). > Or one could choose to define a tuning in which the > octave + 11/8 is to the 1/1 as 2.718281828:1 > And so on. -- mclaren Paul Hahn goes on to say: "It's these kind of proposals of Brian's that baffle me sometimes. I cannot imagine any fashion in which Feigenbaum's or Euler's constants could possibly be aurally significant in any way." -- Paul Hahn The inability to imagine X is generally not a valid criticism of X. Rather it is an implied critique of the person making the comment. Arthur C. Clarke's famous bon mot ought to be modified as follows: "Whenever a respected music theorist says that something is aesthetically possible, he is quite probably right. But whenever a respected music theorist says that something is aesthetically impossible, he is almost certainly wrong." --mclaren ( One might also recall Clarke's words to the effect that any sufficiently advanced technology is indistinguishable from magic. I might suggest that "Any sufficiently advanced music is indistinguishable from noise." JC) Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 21 Jun 1996 20:58 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id LAA10963; Fri, 21 Jun 1996 11:58:47 -0700 Date: Fri, 21 Jun 1996 11:58:47 -0700 Message-Id: <25960621185252/0005695065PK3EM@MCIMAIL.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu