Classes of 5-limit superparticular scales

Three tones:

(6/5)(5/4)(4/3) = 2
(10/9)(6/5)(3/2) = 2
(16/15)(5/4)(3/2) = 2

Four tones:

(10/9)(6/5)^2(5/4) = 2
(16/15)(6/5)(5/4)^2 = 2
(25/24)(6/5)^2(4/3) = 2
(81/80)(10/9)(4/3)^2 = 2

Five tones:

(10/9)^2(9/8)(6/5)^2 = 2
(16/15)^2(9/8)(5/4)^2 = 2
(25/24)(10/9)(6/5)^3 = 2

Six tones:

(25/24)^2(16/15)(6/5)^3 = 2
(81/80)(10/9)^3(6/5)^2 = 2

Seven tones:

(16/15)^2(10/9)^2(9/8)^3

Nine tones:

(25/24)^2(16/15)^4(9/8)^3

Ten tones:

(81/80)^3(16/15)^2(10/9)^5 = 2

Fifteen tones:

(81/80)^3(25/24)^5(16/15)^7 = 2

Interestingly, no twelve tones.

Aside from these, we have 2/1 = 2 in the 2-limit, and
(4/3)(3/2) = 2 and (9/8)^2(4/3) = 2 in the 3-limit.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Three tones:
>
> (6/5)(5/4)(4/3) = 2
> (10/9)(6/5)(3/2) = 2
> (16/15)(5/4)(3/2) = 2
>
/.../
> Six tones:
>
> (25/24)^2(16/15)(6/5)^3 = 2
> (81/80)(10/9)^3(6/5)^2 = 2

Not quite following this post, Gene...

> Seven tones:
>
> (16/15)^2(10/9)^2(9/8)^3

=?

> Nine tones:

Eight tones was your last message? So the =x is
the number of connected permutations for each set
of superparticulars? More with eight tones than
with 1-7 or 9, 10, 12, 15!?

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> > (25/24)^2(16/15)(6/5)^3 = 2
> > (81/80)(10/9)^3(6/5)^2 = 2
>
> Not quite following this post, Gene...

The only 5-limit superparticulars are 2/1,3/2,4/3,5/4,6/5,9/8,10/9,
16/15,25/24 and 81/80. I took these three at a time, and calculated whether or not they lead to a partition of the octave, and if so, how many of each tone were required. So, for intance, 2 25/24's one 16/15 and three 6/5 make up an octave, and by arraging them in various ways give us six-tone scales.

> > Seven tones:
> >
> > (16/15)^2(10/9)^2(9/8)^3

> Eight tones was your last message?

Seven tones--the same steps as above, which it seems is the only seven-tone possibility.

>>> Seven tones:
>>>
>>> (16/15)^2(10/9)^2(9/8)^3
>>
>> Eight tones was your last message?
>
> Seven tones--the same steps as above, which it seems is the only
> seven-tone possibility.

Right, they all =2, means they sum to an octave, duh. So you only
checked for connectednes in the 7-tone case. Got it, thanks.

-Carl