Ragismatic temperament and two sequences leading to the ragisma

Hi again.

I've been reading some webpages about 7-limit 3D temperaments. Interestingly enough, this topic was among my great favorites a couple of years ago. To my surprise,

I've found out that most of the temperaments on the webpages used generators approximating integer factors. In contrast, at the time I was involved in these things "to a larger extent", I often ended up with completely different generators as I was trying to find such generators which you only require a couple of them to get a full 4:5:6:7 chord. If, in contrast, I try to make the sequence of "steps" very long, I can clearly show that there are other intervals which could well serve as possible generators.

For example, let's take the 4375/4374 temperament which I was calling "ragismatic" just to later find out that someone else started calling it "ragismic" (still not sure if my term wasn't actually older, though). If we say that "X=3/1" and "Y=5/1", then we have to rise by 7 Xs and fall by 4 Ys (which is quite a lot of tones to go through) just to approximate a 7/2. Because of this, I can see a possibility of repeating interval patterns if one of the generators was 9/5. Therefore, I think there could be a way to achieve similar results with less tones if one generator was 9/5 in this temperament.

To make an illustration of some possible interval sequences leading to the ragisma, I've made one of 16 steps and another of 17 steps. Here they are listed.

! ragipu16.scl

!

16-step ragisma pump (1/3, 10/7, 7/2)

16

!

1/3

7/6

7/18

5/9

5/27

35/54

35/162

245/324

175/162

175/486

1225/972

1225/2916

875/1458

875/4374

6125/8748

4375/4374

! ragipu17.scl

!

17-step ragisma pump (7/6, 5/1, 2/7)

17

!

7/6

1/3

5/3

35/18

5/9

35/54

5/27

25/27

175/162

25/81

125/81

875/486

125/243

875/1458

125/729

875/4374

4375/4374

Petr

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:

> For example, let's take the 4375/4374 temperament which I was calling
> "ragismatic" just to later find out that someone else started calling it
> "ragismic" (still not sure if my term wasn't actually older, though). If we
> say that "X=3/1" and "Y=5/1", then we have to rise by 7 Xs and fall by 4 Ys
> (which is quite a lot of tones to go through) just to approximate a 7/2.
> Because of this, I can see a possibility of repeating interval patterns if
> one of the generators was 9/5. Therefore, I think there could be a way to
> achieve similar results with less tones if one generator was 9/5 in this
> temperament.

Yep, as pointed out on the ragismic (ragismatic?) page, 10/9 is low in complexity in it. In fact, looking at the spectrum of the temperament

http://xenharmonic.wikispaces.com/Spectrum+of+a+temperament

we get 10/9, 6/5, 4/3, 5/4, 9/8, 7/6, 8/7, 9/7, 7/5. Hence, looking for comma pumps with a lot of 10/9 and 6/5 is a place to start, especially as 10/9, 6/5 and 2 generate the 5-limit. Since 7/6 is the lowest complexity 7-limit interval, we head there (or to 12/7). 7/6 is equivalent to 729/625, and 12/7 to 1250/729. Solving an equation with two variables and two unknowns points me to (10/9)^2 (5/3)^2/2 = 1250/729, so one comma pump would be 5/3-10/9-7/12-5/3-10/9-1/2.

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

> we get 10/9, 6/5, 4/3, 5/4, 9/8, 7/6, 8/7, 9/7, 7/5. Hence, looking for comma pumps with a lot of 10/9 and 6/5 is a place to start, especially as 10/9, 6/5 and 2 generate the 5-limit. Since 7/6 is the lowest complexity 7-limit interval, we head there (or to 12/7). 7/6 is equivalent to 729/625, and 12/7 to 1250/729. Solving an equation with two variables and two unknowns points me to (10/9)^2 (5/3)^2/2 = 1250/729, so one comma pump would be 5/3-10/9-7/12-5/3-10/9-1/2.
>

I think we might call, modulo inversion, permutation and octave equivalence, the comma pump we get in this way the "minimal comma pump" for the comma in question. Other examples would be 81/80: 4/3-5/6-4/3-2/3; 2401/2400: 7/5-7/8-7/5-7/8-4/3; 385/384: 5/4-11/16-7/8-4/3; 676/675: 15/13-15/13-3/4.