17 (plus2) notes extension of Soria sequences

Dear Margo and all,

In my message #88078 of 25 April 2010, I gave some other version of the Soria tuning delivered with Ethno2, "soria12.scl", developped from a higher convergent Airos series.
I am giving here extended versions to 17 tones of both, also as an illustration of how different recurrent series of one same sequence can have very different musical colors.
Following Margo Schulter's inspired comments on how a choice of different pitches for Eb and Bb would be pertinent in the Persian music context, I also extended both series in order to integrate double versions of those two notes, which came out to introduce two commas for both Eb and Bb of around 15 c. in the first one, and 7 c. in the second.
(the fractal solution of Soria would introduce between 17 fifths or fourths modulo octaves a smaller comma of exactly 5,335236671 c. (Airos^17 / 2^7) in favor of the fourths direction).
For those who wish to give correct Persian names and symbols to these close-to-17 edo scales,
they should of course refer to the very instructive article posted by Margo yesterday 10th of May 2010 on this list :
Persian koron/sori notation and 17-note systems

For Ethno2 users, and if the user presets update has not arrived (I didn't check, and I encourage all of you to request it from MOTU), you would need to use the 24 tones mapping, this means repeating five of the notes, or adding five new ones (such as further extensions of the series, or also 13/8 transpositions, as suggested by the Airos algorithm).
One can also reduce the scale to 17 notes by making a choice between the double notes.

In order to help the comprehension of these extensions from 12 to (17+2), I did not changed the 12 original tones positions and I defined the new tones by whole numbers in reference to the same whole numbers used for C = 1/1 and its octaves.
It means that I used roundings, but with a high precision, that would not affect the -c or eq-b performances of the series.
For those who would be willing to use further extensions of the series in the same way, this is the method I used :

First of all, know that I already tried all possible useful extensions in the Airos direction (cycle of fourfths of around 494.4315 c.) ;
Extensions then should be done in the Soria direction (cycle of fifths of around 705.5685 c.), in which the series converges, but through whole numbers that need to be divided by 13^n, each time with increasing n.
The Soria "superpyth" recurrent algorithm is : 13H(i+3) = 16H(i+2) + 8H(i),
or in resumed form 13x^3 = 16x^2 + 8.
I will take an example with the first two additional tones of Soria17-b :
The last terms of the series in the 12 notes version are
1276 : 1918 : 2883
The next one will be (16*2883 + 8*1276)/13 = 56336/13 ~4333.5385, which I rounded to 8667/2
For the following ones I do not use the precedent roundings even if quite good, but the N / 13^n versions :
The next one is then (16*56336/13 + 8* 1918)/13 = 1100848/169 ~6513.8935 which I rounded to 52111/8, and so on :
39165/4
29435/2
44245/2
33253
99967/2

My personal impression is that with this new commas option the first series (a) is acceptable, has lots of character but also some irregularities, such as the semifourths in end of cycle for the low versions of Eb and Bb, which can also be seen as tasteful variations.
The second series (b) has much less deviations, a more smooth and uniform musicality, and would be the one I would advise perhaps to start with in order to explore the more general possibilities of this 17+n system.

! soria17_plus2a.scl
!
12 from a 17-notes cycle, equal-beating extended fifths (705.5685 c.) sequence
19
!
236211/227840
481/445
4011/3560
2067/1780
8339/7120
278213/227840
1133/890
9503/7120
19643/14240
128/89
533/356
22191/14240
723/445
754/445
1233/712
24889/14240
3267/1780
1703/890
2/1
! Airos recurrent sequence x^3 = 13 - 8x, Dudon 2008
! Eq-b of fourths, and septimal minor thirds with harmonic 7ths :
! 8(4 - 3x) = 3x^3 - 7 = (32x^3 - 56x)/13
! Airos fourths of 494.4315 c.
! -c with the 13th harmonic : (13 * 890) - 9503 = 2067, etc.

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! soria17_plus2b.scl
!
12 from a 17-notes cycle, equal-beating extended fifths (705.5685 c.) sequence
19
!
29435/28352
959/886
15995/14176
33253/28352
8345/7088
8667/7088
2259/1772
37727/28352
39165/28352
638/443
5319/3544
44245/28352
2883/1772
1503/886
12443/7088
99967/56704
52111/28352
6791/3544
2/1
! Airos recurrent sequence x^3 = 13 - 8x, Dudon 2008
! Eq-b of fourths, and septimal minor thirds with harmonic 7ths :
! 8(4 - 3x) = 3x^3 - 7 = (32x^3 - 56x)/13
! Airos fourths of 494.4315 c. (approx. 1276/959)
! -c with the 13th harmonic : (13 * 2883) - (32 * 959) = 6791, etc.

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Jacques