Post From McLaren

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)


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