What does "3-distributional even" mean?

The subject line really says it all, but I'll give background in case
it helps clarify why I'm asking.

I was goofing off again. I used Scala to create an 8-note MOS with 3/2
period and 7/6 generator. I used the Extend command to get 16 notes
(9/4 period). I rotated the scale such that the 14th note was 1207
cents (236196/117649, to be precise), and truncated the scale.

I could consider this a scale with a stretched octave, or I could
temper out 118098/117649, which is the difference between 2/1 and
236196/117649. (For the latter case, 36 EDO's excellent 3 and 7 make
it a very good tuning for a pure-octave tempered version of this
scale.) It seemed worth playing with, anyway.

Now, I didn't think I had any reason to expect that this scale would
be evenly distributed, even though I had originally generated an MOS,
because I had doubled it and cut off two of the notes off of one end.
(I realize now that I'm probably wrong, but I just have intuition, not
proof.) But I ran "Show Data" in Scala anyway to see what I could see.

To my surprise, Scala told me, "Scale is 3-distributional even". I
don't know what that means, or how it differs from "Scale is
distributional even" and "Scale is maximally even for L / S <= 2"
(both of which I'm pretty sure I understand), which is what it shows
me for the original 8-note (or 16-note) MOS.

The epilogue is that I don't really need all 14 notes. I'm playing
with a 10-note subset, which is only proper and not distributional
even at all. <shrug> But I'm still curious about "3-distributional
even".

Thanks,
Jake

I'd like to try the 14 noye version if you have a scala file of it

Chris
*

-----Original Message-----
From: Jake Freivald <jdfreivald@gmail.com>
Sender: tuning@yahoogroups.com
Date: Wed, 16 May 2012 15:57:10
To: <tuning@yahoogroups.com>
Reply-To: tuning@yahoogroups.com
Subject: [tuning] What does "3-distributional even" mean?

The subject line really says it all, but I'll give background in case
it helps clarify why I'm asking.

I was goofing off again. I used Scala to create an 8-note MOS with 3/2
period and 7/6 generator. I used the Extend command to get 16 notes
(9/4 period). I rotated the scale such that the 14th note was 1207
cents (236196/117649, to be precise), and truncated the scale.

I could consider this a scale with a stretched octave, or I could
temper out 118098/117649, which is the difference between 2/1 and
236196/117649. (For the latter case, 36 EDO's excellent 3 and 7 make
it a very good tuning for a pure-octave tempered version of this
scale.) It seemed worth playing with, anyway.

Now, I didn't think I had any reason to expect that this scale would
be evenly distributed, even though I had originally generated an MOS,
because I had doubled it and cut off two of the notes off of one end.
(I realize now that I'm probably wrong, but I just have intuition, not
proof.) But I ran "Show Data" in Scala anyway to see what I could see.

To my surprise, Scala told me, "Scale is 3-distributional even". I
don't know what that means, or how it differs from "Scale is
distributional even" and "Scale is maximally even for L / S <= 2"
(both of which I'm pretty sure I understand), which is what it shows
me for the original 8-note (or 16-note) MOS.

The epilogue is that I don't really need all 14 notes. I'm playing
with a 10-note subset, which is only proper and not distributional
even at all. <shrug> But I'm still curious about "3-distributional
even".

Thanks,
Jake

I believe it means that every specific interval class comes in three sizes.

-Mike

On Wed, May 16, 2012 at 3:57 PM, Jake Freivald <jdfreivald@...> wrote:
>
> The subject line really says it all, but I'll give background in case
> it helps clarify why I'm asking.
>
> I was goofing off again. I used Scala to create an 8-note MOS with 3/2
> period and 7/6 generator. I used the Extend command to get 16 notes
> (9/4 period). I rotated the scale such that the 14th note was 1207
> cents (236196/117649, to be precise), and truncated the scale.
>
> I could consider this a scale with a stretched octave, or I could
> temper out 118098/117649, which is the difference between 2/1 and
> 236196/117649. (For the latter case, 36 EDO's excellent 3 and 7 make
> it a very good tuning for a pure-octave tempered version of this
> scale.) It seemed worth playing with, anyway.
>
> Now, I didn't think I had any reason to expect that this scale would
> be evenly distributed, even though I had originally generated an MOS,
> because I had doubled it and cut off two of the notes off of one end.
> (I realize now that I'm probably wrong, but I just have intuition, not
> proof.) But I ran "Show Data" in Scala anyway to see what I could see.
>
> To my surprise, Scala told me, "Scale is 3-distributional even". I
> don't know what that means, or how it differs from "Scale is
> distributional even" and "Scale is maximally even for L / S <= 2"
> (both of which I'm pretty sure I understand), which is what it shows
> me for the original 8-note (or 16-note) MOS.
>
> The epilogue is that I don't really need all 14 notes. I'm playing
> with a 10-note subset, which is only proper and not distributional
> even at all. <shrug> But I'm still curious about "3-distributional
> even".
>
> Thanks,
> Jake

Mike, Scala tells me that there are only two one-step interval sizes, which
would seem to preclude having three classes for each interval. There are
two seconds (67 and 167 cents) and three thirds (267, 333, and 433 cents).
Maybe that has something to do with it?

Chris, here's the scale, untempered:

! C:\Program Files (x86)\Scala22\7_6-on-3_2-untempered.scl
!

14
!
17496/16807
54/49
7/6
2916/2401
9/7
157464/117649
486/343
3/2
26244/16807
81/49
7/4
4374/2401
27/14
236196/117649

....and here's the tempered scale (TOP, allowing 2 to be tempered as well):

! C:\Program Files (x86)\Scala22\7_6-on-3_2-tempered.scl
!

14
!
64.04955
165.98792
267.92629
331.97585
433.91422
497.96377
599.90214
701.84051
765.89006
867.82843
969.76681
1033.81636
1135.75473
1199.80428

I don't remember if you have a 36-EDO axe, but if you do you could also use
this mode:
Degrees: 2 5 8 10 13 15 18 21 23 26 29 31 34 36

Have fun!

Regards,
Jake

On Wed, May 16, 2012 at 4:32 PM, Jake Freivald <jdfreivald@...> wrote:
>
> Mike, Scala tells me that there are only two one-step interval sizes, which would seem to preclude having three classes for each interval. There are two seconds (67 and 167 cents) and three thirds (267, 333, and 433 cents). Maybe that has something to do with it?

Bah, you're right, that was a typo on my part. The definition of n-DE
we've been using is that each interval comes in "at most" n sizes,
with the usual 2-DE (usually just called "DE") being Paul's substitute
word for MOS.

I can't check the scale now, but I would hope that Scala is using n-DE
to mean max-variety-n, e.g. each interval comes in "at most" n sizes.
Whether or not Manuel has an additional desideratum he's placed on the
"DE" term, I'm not sure...

-Mike

> The definition of n-DE we've been using is that each interval comes in
> "at most" n sizes, with the usual 2-DE (usually just called "DE") being
> Paul's substitute word for MOS.
>
> I can't check the scale now, but I would hope that Scala is using n-DE
> to mean max-variety-n, e.g. each interval comes in "at most" n sizes.
> Whether or not Manuel has an additional desideratum he's placed on the
> "DE" term, I'm not sure...

Mike, that makes sense.

And I realized that I misspoke: When I said there were three thirds, I
was thinking about what *I* consider thirds (major, neutral, or
minor), not what are actually three-step intervals in the scale.
However, looking now, I see that there are at most three n-step
interval sizes for each value of n.

So I was actually wrong, but if you believed what I said, you would
have thought I was right, and as it turns out, the fact that I was
wrong didn't matter. Lucky me. :)

Thanks,
Jake

Nope, only the GR-20 guitar synth for 36 edo.

Thank you for these!

Chris

On Wed, May 16, 2012 at 4:32 PM, Jake Freivald <jdfreivald@...> wrote:

> **
>
>
> Mike, Scala tells me that there are only two one-step interval sizes,
> which would seem to preclude having three classes for each interval. There
> are two seconds (67 and 167 cents) and three thirds (267, 333, and 433
> cents). Maybe that has something to do with it?
>
> Chris, here's the scale, untempered:
>
> ! C:\Program Files (x86)\Scala22\7_6-on-3_2-untempered.scl
> !
>
> 14
> !
> 17496/16807
> 54/49
> 7/6
> 2916/2401
> 9/7
> 157464/117649
> 486/343
> 3/2
> 26244/16807
> 81/49
> 7/4
> 4374/2401
> 27/14
> 236196/117649
>
> ....and here's the tempered scale (TOP, allowing 2 to be tempered as well):
>
> ! C:\Program Files (x86)\Scala22\7_6-on-3_2-tempered.scl
> !
>
> 14
> !
> 64.04955
> 165.98792
> 267.92629
> 331.97585
> 433.91422
> 497.96377
> 599.90214
> 701.84051
> 765.89006
> 867.82843
> 969.76681
> 1033.81636
> 1135.75473
> 1199.80428
>
> I don't remember if you have a 36-EDO axe, but if you do you could also
> use this mode:
> Degrees: 2 5 8 10 13 15 18 21 23 26 29 31 34 36
>
> Have fun!
>
> Regards,
> Jake
>
>

Here is what I did with the untempered version. - I left all of the bad
notes in because they seem to do nice balancing act with the lovely
consonant chord I found at the very end.

http://micro.soonlabel.com/just/jakes_7_6_On-3_2_untempered.mp3

On Wed, May 16, 2012 at 4:32 PM, Jake Freivald <jdfreivald@...> wrote:

> **
>
>
> Mike, Scala tells me that there are only two one-step interval sizes,
> which would seem to preclude having three classes for each interval. There
> are two seconds (67 and 167 cents) and three thirds (267, 333, and 433
> cents). Maybe that has something to do with it?
>
> Chris, here's the scale, untempered:
>
> ! C:\Program Files (x86)\Scala22\7_6-on-3_2-untempered.scl
> !
>
> 14
> !
> 17496/16807
> 54/49
> 7/6
> 2916/2401
> 9/7
> 157464/117649
> 486/343
> 3/2
> 26244/16807
> 81/49
> 7/4
> 4374/2401
> 27/14
> 236196/117649
>
> ....and here's the tempered scale (TOP, allowing 2 to be tempered as well):
>
> ! C:\Program Files (x86)\Scala22\7_6-on-3_2-tempered.scl
> !
>
> 14
> !
> 64.04955
> 165.98792
> 267.92629
> 331.97585
> 433.91422
> 497.96377
> 599.90214
> 701.84051
> 765.89006
> 867.82843
> 969.76681
> 1033.81636
> 1135.75473
> 1199.80428
>
> I don't remember if you have a 36-EDO axe, but if you do you could also
> use this mode:
> Degrees: 2 5 8 10 13 15 18 21 23 26 29 31 34 36
>
> Have fun!
>
> Regards,
> Jake
>
>