Topic: What does "3-distributional even" mean?
2 scales
| File | Description | Notes | Period (ยข) | Limit |
|---|---|---|---|---|
| 7_6-on-3_2-tempered | 14 | 1199.8 | ||
| 7_6-on-3_2-untempered | 14 | 1206.6 | 7 |
Thread (8 messages)
From: Jake Freivald (2012-05-16) Subject: 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
From: chrisvaisvil@gmail.com (2012-05-16) Subject: Re: [tuning] What does "3-distributional even" mean? 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
From: Mike Battaglia (2012-05-16) Subject: Re: [tuning] What does "3-distributional even" mean? 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
From: Jake Freivald (2012-05-16) Subject: Re: [tuning] What does "3-distributional even" mean? 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
From: Mike Battaglia (2012-05-16) Subject: Re: [tuning] What does "3-distributional even" mean? 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
From: Jake Freivald (2012-05-16) Subject: Re: [tuning] What does "3-distributional even" mean? > 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
From: Chris Vaisvil (2012-05-19) Subject: Re: [tuning] What does "3-distributional even" mean? 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 > >
From: Chris Vaisvil (2012-05-19) Subject: Re: [tuning] What does "3-distributional even" mean? 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 > >