A modified neutral third sequence

Dear Jacques and all,

Please let me share an idea that occurred to me for a
neutral third sequence which I suspect may have already
been proposed here more than once.

Your Mohajira sequence "issued from 13" is:

4 4 6 7 9 11 13 16
18 22 26 32 39 48 59 72 88

Considering this, a variant occurred to me, however
coherent or otherwise, starting with 9:11, the simplest
ratio for a neutral third:

2 3 3 4 5 6 7 9 11 13 16
9 11 14 17 21 26 32 39 48 59 72 88

From 26 on, my series simply quotes yours. The main
difference is that mine includes 14:17:21. As George
Secor pointed out to me around late 2001 or early 2002
when I mentioned my liking for these ratios, 14:17:21
is the division of the fifth into two neutral thirds
requiring the lowest maximum odd number, by comparison
to 18:22:27 or 26:32:39.

An interesting touch is that in your series, the one
ratio between adjacent terms not forming a neutral
third is 22:26 or 11:13; and my modified version,
we have 11:14.

In regular temperaments around 704 cents, 13:11, 14:11,
and 14:17:21 are all very close to just; but I'm not
sure if that has any connection with these two series.
With the diminished fourth around 367-368 cents, this
interval nicely represents either 21:17 or 26:21 in the
alternative version of this series.

Best,

Margo Schulter
mschulter@...

Margo wrote :

> Dear Jacques and all,
>
> Please let me share an idea that occurred to me for a
> neutral third sequence which I suspect may have already
> been proposed here more than once.
>
> Your Mohajira sequence "issued from 13" is:
>
> 4 4 6 7 9 11 13 16
> 18 22 26 32 39 48 59 72 88
>
> Considering this, a variant occurred to me, however
> coherent or otherwise, starting with 9:11, the simplest
> ratio for a neutral third:
>
> 2 3 3 4 5 6 7 9 11 13 16
> 9 11 14 17 21 26 32 39 48 59 72 88

I understand you can be interested in this sequence ! But is it a sequence ?
It breaks the x^5 - x^4 = 1/2 algorithm and it is not a Mohajira sequence, at least of that type.
Would it be another sequence ?
17 : 21 : 26 : 32 belongs typically to a Buzurg sequence, but it leaves it here for a Mohajira.
A good option would be "Buzurg 8", based on the differential coherence of :
x^8 - x^7 = 1
or if you prefer, where each new term is x^8 = x^7 + 1
It would converge normally towards 361.2794748 c., which is too high for your purpose, but by integrating a segment of a scale with a lower generator (mohajira), it would be an interesting alternative !
Since 8 terms are needed to start a series, the choices are open.
The basic Buzurg 8 series goes :
(6, 7) 9 11 14 17 21 26 32 39 48 59 73 90 111 137 169 208 256 315 388 478 589 726 895 1103
and is identical to your proposition up to 59 !
Both have 3 octave repetitions : 9 : 11 > 72 : 88, or 26 : 32 > 208 : 256
Another option would be to start from 17, where the intervals are more regular :
(6, 7, 9, 11, 13) 17 21 26 32 39 48 59 72 89 110 136 168 207 255 314 386 475 585 721 889
octave repetitions : 17 : 21 > 136 : 168
All these series are cycling close to 10 tones /octave, not 17.
For 17 there would be other sequences and one of them is Soria !
- - - - - - - -
Jacques

> From 26 on, my series simply quotes yours. The main
> difference is that mine includes 14:17:21. As George
> Secor pointed out to me around late 2001 or early 2002
> when I mentioned my liking for these ratios, 14:17:21
> is the division of the fifth into two neutral thirds
> requiring the lowest maximum odd number, by comparison
> to 18:22:27 or 26:32:39.
>
> An interesting touch is that in your series, the one
> ratio between adjacent terms not forming a neutral
> third is 22:26 or 11:13; and my modified version,
> we have 11:14.
>
> In regular temperaments around 704 cents, 13:11, 14:11,
> and 14:17:21 are all very close to just; but I'm not
> sure if that has any connection with these two series.
> With the diminished fourth around 367-368 cents, this
> interval nicely represents either 21:17 or 26:21 in the
> alternative version of this series.
>
> Best,
>
> Margo Schulter