"Minimum denominator" 13-tone per 2/1 octave JI

If this doesn't prove PHI is tied to JI I'll be damned... :-D

Also, if this scale doesn't qualify as "a lot of information fit into a simple logic", I'd be very interested to know what's better at that goal (or if someone already has made a "really weird" JI scale that looks like this...b/c I haven't found one in Scala's library). :-)
---------------------------------------------------------------
Here's the new scale (and the interval gaps between notes)

scale intervals between notes (from SCALA)
17/16 17/16
9/8 18/17
19/16 19/18
5/4 20/19
4/3 16/15
7/5 21/20
28/19 20/19
14/9 19/18
21/13 27/26 (golden octave APX 1.615)
--------------------------------------
Note that 21/13 * 5/4 = about 2.02 thus making it about 13-tones per 2/1 octave. I also tested it as a "chord" (IE all notes played at once) and couldn't find anything out of place (therefore posing little need to make an "adaptive" version of the scale).

Note that the process took about 1 week of testing 2 hours per day all experimenting with different ratio combinations to get the denominator number down.

Mind my posing an outright challenge about the PHI-meantone generated tuning IE PHI^y/2^x...

I think it's blatantly obvious that this proves PHI-meantone is not just irrational-number mumbo-jumbo and CAN indeed be tied back to JI mathematics.
If you look at the JI-style ratios above vs. PHI mean-tone they are all only off by a just few cents...

******************************
The only thing I see on the surface that could use some improvement is reducing the 27/26 to something more "low-limit" IE more like 21/20 by changing 21/13 and perhaps 14/9 to something more efficient that way. But everything I've tried can't seem to both estimate PHI well at the octave and reduce to a lower denominator fraction.

-Michael

Say, you should re-read my posts on this topic. I specifically mentioned 21/13 as a phi approximation that links to JI- and you've left out the slickest bit there, which I posted before, which is the fact that it ties a 7 otonality to a 13 utonality via a superparticular interval (21/13 * 13/12 = 7/4).

--- In tuning@yahoogroups.com, "djtrancendance" <djtrancendance@...> wrote:
>
> If this doesn't prove PHI is tied to JI I'll be damned... :-D
>
> Also, if this scale doesn't qualify as "a lot of information fit into a simple logic", I'd be very interested to know what's better at that goal (or if someone already has made a "really weird" JI scale that looks like this...b/c I haven't found one in Scala's library). :-)
> ---------------------------------------------------------------
> Here's the new scale (and the interval gaps between notes)
>
> scale intervals between notes (from SCALA)
> 17/16 17/16
> 9/8 18/17
> 19/16 19/18
> 5/4 20/19
> 4/3 16/15
> 7/5 21/20
> 28/19 20/19
> 14/9 19/18
> 21/13 27/26 (golden octave APX 1.615)
> --------------------------------------
> Note that 21/13 * 5/4 = about 2.02 thus making it about 13-tones per 2/1 octave. I also tested it as a "chord" (IE all notes played at once) and couldn't find anything out of place (therefore posing little need to make an "adaptive" version of the scale).
>
> Note that the process took about 1 week of testing 2 hours per day all experimenting with different ratio combinations to get the denominator number down.
>
> Mind my posing an outright challenge about the PHI-meantone generated tuning IE PHI^y/2^x...
>
> I think it's blatantly obvious that this proves PHI-meantone is not just irrational-number mumbo-jumbo and CAN indeed be tied back to JI mathematics.
> If you look at the JI-style ratios above vs. PHI mean-tone they are all only off by a just few cents...
>
> ******************************
> The only thing I see on the surface that could use some improvement is reducing the 27/26 to something more "low-limit" IE more like 21/20 by changing 21/13 and perhaps 14/9 to something more efficient that way. But everything I've tried can't seem to both estimate PHI well at the octave and reduce to a lower denominator fraction.
>
> -Michael
>

--I specifically mentioned 21/13 as a phi approximation that links to JI

(21/13)...which I used as seems to work pretty well... ;-)
    Sorry my memory did not serve me well enough to mention that you'd mentioned it before...  I also came up with a few others IE 34/21 = 1.6190476 and 13/8 = 1.625 but I found 21/13 matched against 14/9 (the last note before the octave) the best.

  I'll definitely check out your old posts...it's still surprising me how close the PHI tuning can be linked to u-tonal/o-tonal relationships.
*****************************************************
---(21/13 * 13/12 = 7/4).
    But how would/could I fit 13/12 into the scale?  Between 17/16 and 19/16 it doesn't seem to produce very basic intervals...  Note the idea is not to make the scale work best relative to just one root (IE C)...but to all possible roots. :-)

-Michael

--- On Fri, 3/27/09, Cameron Bobro <misterbobro@...> wrote:

From: Cameron Bobro <misterbobro@...>
Subject: [tuning] Re: "Minimum denominator" 13-tone per 2/1 octave JI
To: tuning@yahoogroups.com
Date: Friday, March 27, 2009, 6:14 AM

Say, you should re-read my posts on this topic. I specifically mentioned 21/13 as a phi approximation that links to JI- and you've left out the slickest bit there, which I posted before, which is the fact that it ties a 7 otonality to a 13 utonality via a superparticular interval (21/13 * 13/12 = 7/4).

--- In tuning@yahoogroups. com, "djtrancendance" <djtrancendance@ ...> wrote:

>

> If this doesn't prove PHI is tied to JI I'll be damned... :-D

>

> Also, if this scale doesn't qualify as "a lot of information fit into a simple logic", I'd be very interested to know what's better at that goal (or if someone already has made a "really weird" JI scale that looks like this...b/c I haven't found one in Scala's library). :-)

> ------------ --------- --------- --------- --------- --------- -

> Here's the new scale (and the interval gaps between notes)

>

> scale intervals between notes (from SCALA)

> 17/16 17/16

> 9/8 18/17

> 19/16 19/18

> 5/4 20/19

> 4/3 16/15

> 7/5 21/20

> 28/19 20/19

> 14/9 19/18

> 21/13 27/26 (golden octave APX 1.615)

> ------------ --------- --------- --------

> Note that 21/13 * 5/4 = about 2.02 thus making it about 13-tones per 2/1 octave. I also tested it as a "chord" (IE all notes played at once) and couldn't find anything out of place (therefore posing little need to make an "adaptive" version of the scale).

>

> Note that the process took about 1 week of testing 2 hours per day all experimenting with different ratio combinations to get the denominator number down.

>

> Mind my posing an outright challenge about the PHI-meantone generated tuning IE PHI^y/2^x...

>

> I think it's blatantly obvious that this proves PHI-meantone is not just irrational-number mumbo-jumbo and CAN indeed be tied back to JI mathematics.

> If you look at the JI-style ratios above vs. PHI mean-tone they are all only off by a just few cents...

>

> ************ ********* *********

> The only thing I see on the surface that could use some improvement is reducing the 27/26 to something more "low-limit" IE more like 21/20 by changing 21/13 and perhaps 14/9 to something more efficient that way. But everything I've tried can't seem to both estimate PHI well at the octave and reduce to a lower denominator fraction.

>

> -Michael

>