Phidiama series, suite

Hi Cameron, and all,

As you know already, 17/13 is certainly a good approximation of (Phi^2)/2 since 13 and 34 belong to the F series, just like many others in the ever-converging list towards (sqrt of 5 + 3) / 4 :

4/3
21/16
17/13
55/42
89/68
72/55 (tiny 3025/3024 skisma under 55/42)...etc.

or in the P series,
9/7
29/22
47/36
38/29
123/94
199/152
161/123... etc.

or in the Golden meantone series,
31/12
25/19
81/62
131/100
106/81 ... etc.

One interesting acoustic property I found with 1.309, or more precisely 2.618, is this -
I will illustrate it in a simpler way by using this Fibonacci Phi^2 F series :
3 8 21 55 144 ... that I multiply by 3 :
9 24 63 165 432...
showing the differencial coherence :
24 - 21 = 3
63 - 55 = 8
165 - 144 = 21, etc.
= slightly beating septimal tones 8/7- like (1/1, 441/440, 1155/1152) but perfectly coherent in their difference tones with the original 3 8 21 55 144... series.

This allows "Phidiama" or (sqrt of 5 + 3) / 2 = 2.618034 to be, independently of Phi, a fractal ratio on its own solution of :
x^2 = 3x - 1

And the scale combining these two "Phidiama" and "3 Phidiama" back to the same octave,
with three commas 48 54-55 63-64 72 165-168 (that I use in the piece "Appel" of my CD) is one of my favorites in the world :

Cylf
1/1
9/8
55/48
21/16
4/3
3/2
55/32
7/4
2/1

- - - - - - - -
Jacques

Posted by: "Cameron Bobro" misterbobro@... misterbobro
Fri May 15, 2009 3:00 am (PDT)

(was "an entertainment tetrachord")

Anyone who has been working with phi in tuning, or following the recent threads,
is going to be familiar with the interval of approx. 466 cents, which appears
many time in different phi tunings: phi mod2, golden sections of golden
sections, and so on.
It is maybe surprisingly close to, and in most situations indistinguishable
from, a Just interval relatively low in the harmonic series, 17/13.

Jacques>

"And the scale combining these two "Phidiama" and "3 Phidiama" back to the same octave,
Cylf 1/19/855/4821/164/33/255/327/42/1"

Pretty cool
   I see how you are taking the F series:
3  8  21  55  144
and the series multiplied by 3
9  24  63  165  432..

...and then taking ratios such as 55/24 and pushing them back into the octave using by multiplying by (1/2)^x to get results like 55/48 that fit within the octave.

   This is quite an amusing scale because it seems both very PHI symmetrical and very JI symmetrical.  The natural occurrence of fractions like 21/16, 9/8, 3/2...all seem to tie this scale to the x/16 harmonic series with a few well-calculated slight deviations that seem to add a lot of extra tonal color.