Topic: Thirds positivity

1 scales

Thread (3 messages)

From: Gene Ward Smith (2006-06-12)
Subject: Thirds positivity

Here's an idea I've mentioned before, but now I'm wondering about
using it for rank three scales, which were asked about. Given 2 and a
set of thirds for the p-limit, for example [2,5/4,6/5,7/6,11/9] for
the 11-limit, a comma q is "positive" if either q or 1/q has
nonnegative values for all exponents of the thirds, when these are
used to represent 
p-limit intervals. The point is, a positive interval allows for
temperaments with circles of thirds, like meantone. 

There are lots of 5-limit positive temperaments: schismatic, meantone,
augmented, semithirds, diaschismic, amity, porcupine, sensi, 5-limit
orwell, parakleismic, etc. However, 7 and 11 limit opens up new vistas:
4000/3969, 1029/1024, 5120/5103, 6144/6125, 65625/65536, 2401/2400,
250047/250000 among others in the 7-limit. Lots of good ones in there.
In the 11-limit, a little less to feed on: 385/384 and 6250/6237. 

Anyway, it seems to me there are possibilities here for scale
construction, from for example (5/4)*(6/5)^3*(7/6)^4 = 2401/600 (a
mere eight thirds!), or (5/4)^4(6/5)^4(7/6)^3 = 1029/128, or
(5/4)^6*(6/5)^7*(7/6) = 5103/320.

From: Gene Ward Smith (2006-06-12)
Subject: Re: Thirds positivity

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> Anyway, it seems to me there are possibilities here for scale
> construction, from for example (5/4)*(6/5)^3*(7/6)^4 = 2401/600 (a
> mere eight thirds!), or (5/4)^4(6/5)^4(7/6)^3 = 1029/128, or
> (5/4)^6*(6/5)^7*(7/6) = 5103/320.

Propriety let me down on this one, in the sense that one can certainly
construct scales this way, but you can check all possibilities and not
find any proper ones in general, it appears. None, at least, for
2401/2400 and 5120/5103. I'm listing what you get from 5120/5103 by
using a regular pattern for the circle of thirds, with alternating
minor and major (leading to lots of triads) followed by the single
7/6. It's way improper, but kind of interesting from other points of view.

! circ5120.scl
Circle of seven minor, six major, and one subminor thirds in 531-et
14
!
24.858757
205.649718
230.508475
316.384181
411.299435
522.033898
616.949153
702.824859
727.683616
908.474576
933.333333
1019.209040
1114.124294
1200.000000
! [11, 91, 102, 140, 182, 231, 273, 311, 322, 402, 413, 451, 493, 531]
! [11, 80, 11, 38, 42, 49, 42, 38, 11, 80, 11, 38, 42, 38]

From: yahya_melb (2006-06-12)
Subject: Re: Thirds positivity

Hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" 
<genewardsmith@...> wrote:
>
> Here's an idea I've mentioned before, but now I'm wondering
> about using it for rank three scales, which were asked about.
> Given 2 and a set of thirds for the p-limit, for example
> [2,5/4,6/5,7/6,11/9] for the 11-limit, a comma q is 
> "positive" if either q or 1/q has nonnegative values for all
> exponents of the thirds, when these are used to represent 
> p-limit intervals. The point is, a positive interval allows
> for temperaments with circles of thirds, like meantone. 
> 
> There are lots of 5-limit positive temperaments: schismatic,
> meantone, augmented, semithirds, diaschismic, amity, porcupine,
> sensi, 5-limit orwell, parakleismic, etc. However, 7 and 11
> limit opens up new vistas:
> 4000/3969, 1029/1024, 5120/5103, 6144/6125, 65625/65536,
> 2401/2400, 250047/250000 among others in the 7-limit. Lots of
> good ones in there.
> In the 11-limit, a little less to feed on: 385/384 and 6250/6237. 
> 
> Anyway, it seems to me there are possibilities here for scale
> construction, from for example (5/4)*(6/5)^3*(7/6)^4 = 2401/600
> (a mere eight thirds!), or (5/4)^4(6/5)^4(7/6)^3 = 1029/128, or
> (5/4)^6*(6/5)^7*(7/6) = 5103/320.

Yes, there's certainly some scope here.  The very first of these
needs only the comma 2401/2400 to be tempered out to give scales
with three distinctive flavours of thirds.  How they will pan out
depends, I guess on whether the scales built with them are able
to approximate other desired intervals closely enough.  Worth
looking into, I think.

All of these involve three distinct generating thirds to repeat
at a near-octave period.  What kinds of scales exist with just
two generating thirds?  (Choose your own period.)

Regards,
Yahya