Re: What's the name of this scale please?

David Z. Kasler <[EMAIL PROTECTED]> wrote:
> Sorry about the confusion I created by trying to do a calculation in my head
> before having a second cup of coffee.
> The question you might be asking is:
> "What does 729 have to do with C C# E F G G# A# C?"

Hello, there, and the curious alternative answer that quickly occurred to
me -- likely an aside to this thread -- is that if this scale is tuned in
Pythagorean intonation (based on a chain of pure 3:2 fifths or 4:3
fourths, respectively around 702 and 498 cents), then a diminished fifth
such as C#-G has a ratio of 1024:729 (~588 cents), and an augmented fourth
or tritone such as G-C# a ratio of 729:512 (~612 cents).

In Pythagorean, using an octave of C-C as the reference, we have these 
ratios, with rounded values in cents and deviations from 12-tone equal 
temperament for those who might use this information to set up the tuning 
on a synthesizer or computer software to control a sound card or the like:


         C       C#       E    F      G     G#      A#     C
ratios: 1/1  2187/2048  81/64 4/3    3/2 6561/4096 59049/ 2/1
                                                   32768   

cents:   0      114      408  498    702    816     1020  1200
12-et+/- C 0    C#+14    E+8  F-2    G+2    G#+16   Bb+20  C 0

There are some interesting features of this scale like the diminished 
fourth G#-C at 8192:6561 (~384.36 cents), only about 1.95 cents narrow of 
a pure 5:4 major third (~386.31 cents). This contrasts with the more 
active regular Pythagorean major thirds such as C-E and E-G# at 81:64 
(~407.82 cents). Thus C#-F-G# is very close to a pure 4:5:6 (the 
Renaissance ideal), while C-E-G at 64:81:96 is a more active tuning of 
this kind of sonority in a typical 13th-14th century style.

Anyway, in Pythagorean intonation, the number 729 helps define the integer 
ratio of a tritone (equal to three 9:8 whole-tones), and also its octave 
complement of the diminished fifth.
      
Most appreciatively,

Margo Schulter
[EMAIL PROTECTED]