raintree's scale (was: (unknown))

> From: raintree goldbach <raintree_goldbach@hotmail.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, December 16, 2001 2:27 AM
> Subject: [tuning] (unknown)
>
>
>
> I need someone to evaluate the following tuning system;
>
> c 1/1
> c# 256/243
> d 4096/3645
> d# 32/27
> e 512/405
> f 4/3
> f# (64/45 1024/729)
> g 16384/10935 this fifth is almost perfectly circular
> g# 128/81
> a 2048/1215
> a# 16/9
> b 256/135

In prime-factor notation, for 2^x * 3^y * 5^z :

x y z

| 8 -3 -1 | 1108 b 256/135
| 4 -2 0 | 996 a# 16/9
| 11 -5 -1 | 904 a 2048/1215
| 7 -4 0 | 792 g# 128/81
| 14 -7 -1 | 700 g 16384/10935
| 6 -2 -1 | 610 f# 64/45
| 10 -6 0 | 588 f# 1024/729
| 2 -1 0 | 498 f 4/3
| 9 -4 -1 | 406 e 512/405
| 5 -3 0 | 294 d# 32/27
| 12 -6 -1 | 202 d 4096/3645
| 8 -5 0 | 90 c# 256/243
| 0 -0 0 | 0 c 1/1

Assuming "8ve"-invariance, which means ignoring the powers of 2,
the 5-limit lattice is:

() ----- 1024----256----128----32----16----4----1
/\ 729 243 81 27 9 3 1
/ \ f# c# g# d# a# f c
/ \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ /
16384----4096----2048----512----256----64
10935 3645 1215 405 135 45
g d a e b f#

Technically, if "C"=1:1, then "F" is indeed 4:3, but all the all
other ratios, which you labeled in the chain of "4ths" from "A#"
to "G", should be known by the enharmonic equivalents "Bb" to "Abb".
Or alternatively, if you'd like to keep the A#...G chain, then
4:3 should be "E#" and 1:1 "B#".

But in effect, what you've got here is a 5-limit system which you
seem to be trying to equate with a skhismic temperament.

When you write "this fifth is almost perfectly circular", what
you mean is that 16384:10935 is only a skhisma (~2 cents) lower
than the typical Pythagorean "5th" with ratio 3:2, so that it
very closely resembles the "5th" of 700 cents found in 12-EDO,
which is a closed system.

Assuming that under ordinary circumstances a difference of a skhisma
between two pitches is indistinguishable, this scale is essentially
the same as the typical Pythagorean scale 3^-6 Gb...3^6 F# :

For 3^x,

gb---db---ab---eb---bb---f---c---g---d---a---e---b---f#
x = -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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