Diamonds are forever

Here are some assorted old posts of mine which might give rise to some
thought.

/tuning-math/message/12208

/tuning-math/message/12327

/tuning-math/message/12228

/tuning-math/message/12374

/tuning-math/message/12376

/tuning-math/message/12379

/tuning-math/message/12387

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> Here are some assorted old posts of mine which might give rise to some
> thought.
>
> /tuning-math/message/12208
>
> /tuning-math/message/12327
>
> /tuning-math/message/12228
>
> /tuning-math/message/12374
>
> /tuning-math/message/12376
>
> /tuning-math/message/12379
>
> /tuning-math/message/12387
>

thanks for that list. for the sake of simplicity and from a musical
perspective, given say 3 primes, what is the best way to form a
complete diamond with the smallest number of superparticular intervals
and hopefully not too many shit intervals (like smaller than 30 cents).

so for 2,3 and 5 u have the 5lim 9 tone dwarf. for 2, 3, 7 u have the
arrangement 7 tone 8/7,49/48,8/7,9/8,8/7,49/48,8/7 that makes a
diamond or u can have a more complete scale with an in complete
diamond like 9/8, 64/64,9/8,49/48,64/63,9/8,64/63,49/48,9/8,64/63,9/8
however thats not that fantastic. some arrangements of a scall will
be a diamond and others of the same scale wont. do u have a method to
determine?

so the properties i guess i have in mind for a good scale are
-being a complete diamond
-all intervals being superparticular
-the least number of unique intervals compared to the number of tones
-not too many small tones

i am not familiar 'epimorphic' can you explain what it is and why is
it important?

can u tell me some scales that fit these properties for different
primes. there cant be too many of them. i have sadly lost the use of
my software (maple) to write small programs to find the results i want
so looking for an alternative.

sorry i am not familiar with all ur jargon but to be honest i want to
get some results as fast as possible with minimal effort!
thanku!

--- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@...> wrote:

> thanks for that list. for the sake of simplicity and from a musical
> perspective, given say 3 primes, what is the best way to form a
> complete diamond with the smallest number of superparticular intervals
> and hopefully not too many shit intervals (like smaller than 30 cents).

If you have n primes, then the minimum number of indendent generators
you need is n. Hence you will need n superparticular ratios, whose
exponents define an nxn unimodular (determinant +-1) matrix. But I
don't think you'll have much luck unless you stick to a p-limit, and
use all primes <= p.

One way to keep the smallest intervals and largest intervals from
diverging to widely is to slice the biggest superparticular into two
smaller ones. At first, you can do this by the square-triangle method,
where superparticulars with triangle numerators are factored into two
with square numerators, and with square numerators into square times
triangle.

So:

(4/3)(3/2) = 2 (2 steps, 3 limit)
(9/8)(4/3)^2 = 2 (3 steps, 3 limit)
(10/9)^2 (9/8) (6/5)^2 = 2 (5 steps, 5 limit)
(16/15)^2 (10/9)^2 (9/8)^3 = 2 (7 steps, 5 limit)
(21/20)^3 (16/15)^2 (15/14)^3 (10/9)^2 = 2 (10 steps, 7 limit)
(25/24)^2 (21/20)^3 (16/15)^4 (15/14)^3 = 2 (12 steps 7 limit)
(36/35)^3 (25/24)^5 (21/20)^3 (16/15)^4 = 2 (15 steps 7 limit)
(36/35)^7 (28/27)^4 (25/24)^5 (21/20)^3 = 2 (19 steps 7 limit)
(49/48)^3 (36/35)^10 (28/27)^4 (25/24)^5 = 2 (22 steps 8 limit)

Things start to get more complicated around here. 25/24 can factor
as (49/48)(50/49), sticking to the 7-limit but breaking the
square-triangle pattern, or we can factor it as (45/44)(55/54) and go
up to the 11-limit. But even before that, we could have factored
10/9 as (15/14)(28/27) and not (16/15)/(25/24). So this method merely
gives you a start on the possibilities. A complete survey of
superparticular representations of the 5 and 7 limit, with a bound on
the largest/smallest ratio, is what you really want; I think somewhere
in the archives you could find it since I've done this before.

Anyway, if you choose enough notes (27) to get the 7-limit diamond
with distinct consistency you should be able to fill things out using
four superparticulars. Splitting 25/24 as (49/48)(50/49) will give you
27 notes, so I'd try {50/49, 49/48, 36/35, 28/27}. The 7-limit diamond
already has a 50/49, two 49/48s, and two 36/35s, so it's a matter of
suitably splitting up the rest. There will, of course, be various
resulting scales, so you need a way of cherry picking, such as chord
counts.

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

> There will, of course, be various
> resulting scales, so you need a way of cherry picking, such as chord
> counts.

Convexity of the resulting scale in the breed plane might be a good
additional condition. If we start from here:

/tuning-math/message/12208

we get a scale which can be split up further to make for
superparticular ratio inervals.

Of course, one question which needs to be asked is why superparticular
rations?