Topic: 37 EDO
1 scales
| File | Description | Notes | Period (¢) |
|---|---|---|---|
| 37-EDO_generator11_8 | 11 | 1200.0 |
Thread (5 messages)
From: Jake Freivald (2012-08-22)
Subject: 37 EDO
37 EDO is funny.
Prime 11 is perfect (less than 0.1 cent off), while primes 5, 7, and 13 are
excellent (just a few cents error). Prime 3 isn't great (12 cents off), but
it's usable enough. (I don't mind the sharp 3/2, but I don't like the
counterpart flat 4/3.) All of the following intervals can be found in 37
EDO with less than 9 cents error:
14/13
11/10
7/6
13/11
6/5
11/9
5/4
14/11
13/10
11/8
7/5
10/7
16/11
8/5
7/4
13/10 has less than a cent error. 10/7 and 7/5 have less than two cents. If
you like 2.3.5.7.11.13, this seems like an EDO you can get behind.
Good stuff, right? Problem is, I can't get a decent mapping out of it. The
13-limit patent val is
< 37 59 86 104 128 137 |
and that works fine for 13/11 (<3 cents sharp) and 14/11 (<5 cents sharp),
but the 11/9 gets mapped to 324 cents (23 cents flat). There's a better
11/9 in the pitch set, though: 357 cents, or 10 cents sharp. The natural
thing to do is bring the mapping for 3/1 down to 58 steps, but when I do,
11/9 gets mapped to 389 cents -- which is really that nearly-perfect 5/4. I
could use 60/49 and 49/40 (both about 351 cents) instead of 11/9, but that
seems to add complexity and eliminate the eleven-ness of the darn thing.
So don't ask me what that 357-cent neutral third is, but it's not an 11/9.
:)
Because 13/11 * 14/11 = 3/2, 37 EDO tempers out 364/363. According to my
spreadsheet, 37 EDO (using the patent val) also tempers out 2401/2400 (the
breedsma, so the 49/40 and 60/49 neutral thirds are equated -- if you want
to map the 357-cent neutral third to those intervals instead of 11/9).
Those seem useful. Here are other commas that get tempered out (limiting
myself to the 13 prime limit):
5-limit
| 17 1 -8 > 11.45 393216/390625 Würschmidt's comma
| -16 -6 11 > 37.72 Sycamore comma
| 1 -5 3 > 49.17 250/243 Maximal diesis
7-limit
| -5 -1 -2 4 > 0.72 2401/2400 Breedsma
| 11 1 -3 -2 > 5.36 6144/6125 porwell comma
| 6 0 -5 2 > 6.08 3136/3125 middle second comma
| 0 -2 5 -3 > 21.18 3125/3087 major BP diesis
| -5 -3 3 1 > 21.90 875/864 keema
| 6 -2 0 -1 > 27.26 64/63 septimal comma, Archytas' comma
| 1 -3 -2 3 > 27.99 686/675 senga
11-limit
| 5 -1 3 0 -3 > 3.03 4000/3993 undecimal schisma, Wizardharry
| -7 -1 1 1 1 > 4.50 385/384 undecimal kleisma, Keenanisma
| 16 0 0 -2 -3 > 8.39 65536/65219 orgonisma
| 4 0 -2 -1 1 > 9.86 176/175 valinorsma
| -3 -1 -1 0 2 > 14.37 121/120 undecimal seconds comma
| 2 -2 2 0 -1 > 17.40 100/99 Ptolemy's comma
13-limit
| 3 0 2 0 1 -3 > 2.36 2200/2197 2.36 Parizek comma, petrma
| 2 -1 0 1 -2 1 > 4.76 364/363 4.76 gentle comma
| 2 -1 -1 2 0 -1 > 8.86 196/195 8.86 mynucuma
| -1 -2 -1 1 0 1 > 19.13 91/90 19.13 superleap
I don't really know what a lot of those mean, but they're there if you want
them.
To see if I could exploit some of the nearly-just intervals in 37 EDO, I've
done something like a billion attempts at using generators to create MOS
scales. Unfortunately, few of them have reasonable size (say, 13 or fewer
tones) and still take advantage of all -- or even many! -- of the great
tones in the scale, and most of them are very improper.
At any rate, thinking I should try to do *something* to try out 37, I
settled on an 11-note MOS that uses 11/8 as a generator. Here it is:
! C:\Program Files (x86)\Scala22\37-EDO_generator11_8.scl
!
11
!
162.16215
259.45945
356.75675
454.05405
551.35135
713.51350
810.81080
908.10810
1005.40540
1102.70270
2/1
It's strictly proper, and almost every mode of this scale has an 11/8,
naturally, since that's the generator. Only this mode has a 3/2. There's a
mix of 11/9-ish neutral thirds, 5/4 major thirds, and 13/11 minor thirds in
the other modes. There are no 14/11 major thirds. Interestingly, many modes
have a 745-cent interval seems to go pretty well with the 551-cent interval
-- I suppose it's like putting a 10/9 or 9/8 over the 11/8. The scale
surprised me in a few ways that I haven't been able to properly exploit and
play around with.
Here's a very brief ditty I did using this scale. It happened to work out
to be about 37 seconds long, which seemed appropriate for 37 EDO, so I made
it fit. I didn't exploit the other modes, so the melody mostly uses the
sharp neutral third, sharp fifth, 11/8, and 8/5.
http://soundcloud.com/jdfreivald/37-seconds
Regards,
Jake
From: cityoftheasleep (2012-08-25) Subject: Re: 37 EDO Why not map it as 2.5.7.9.11.13? It kinda makes more sense that way if you think of it as a subset of 74-ED2, where you'd use the meantone mapping and put 3 on an interval that's halfway between the two 3's of 37-ED2 (just a bit sharp of 697 cents). Or you could do it Mike Battaglia style and map it as 2.3.5.7.9'.11.13, where the 9' isn't 3^2 but its own "prime". -Igs --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote: > > 37 EDO is funny. > > Prime 11 is perfect (less than 0.1 cent off), while primes 5, 7, and 13 are > excellent (just a few cents error). Prime 3 isn't great (12 cents off), but > it's usable enough. (I don't mind the sharp 3/2, but I don't like the > counterpart flat 4/3.) All of the following intervals can be found in 37 > EDO with less than 9 cents error: > > 14/13 > 11/10 > 7/6 > 13/11 > 6/5 > 11/9 > 5/4 > 14/11 > 13/10 > 11/8 > 7/5 > 10/7 > 16/11 > 8/5 > 7/4 > > 13/10 has less than a cent error. 10/7 and 7/5 have less than two cents. If > you like 2.3.5.7.11.13, this seems like an EDO you can get behind. > > Good stuff, right? Problem is, I can't get a decent mapping out of it. The > 13-limit patent val is > < 37 59 86 104 128 137 | > and that works fine for 13/11 (<3 cents sharp) and 14/11 (<5 cents sharp), > but the 11/9 gets mapped to 324 cents (23 cents flat). There's a better > 11/9 in the pitch set, though: 357 cents, or 10 cents sharp. The natural > thing to do is bring the mapping for 3/1 down to 58 steps, but when I do, > 11/9 gets mapped to 389 cents -- which is really that nearly-perfect 5/4. I > could use 60/49 and 49/40 (both about 351 cents) instead of 11/9, but that > seems to add complexity and eliminate the eleven-ness of the darn thing. > > So don't ask me what that 357-cent neutral third is, but it's not an 11/9. > :) > > Because 13/11 * 14/11 = 3/2, 37 EDO tempers out 364/363. According to my > spreadsheet, 37 EDO (using the patent val) also tempers out 2401/2400 (the > breedsma, so the 49/40 and 60/49 neutral thirds are equated -- if you want > to map the 357-cent neutral third to those intervals instead of 11/9). > Those seem useful. Here are other commas that get tempered out (limiting > myself to the 13 prime limit): > > 5-limit > | 17 1 -8 > 11.45 393216/390625 Würschmidt's comma > | -16 -6 11 > 37.72 Sycamore comma > | 1 -5 3 > 49.17 250/243 Maximal diesis > > 7-limit > | -5 -1 -2 4 > 0.72 2401/2400 Breedsma > | 11 1 -3 -2 > 5.36 6144/6125 porwell comma > | 6 0 -5 2 > 6.08 3136/3125 middle second comma > | 0 -2 5 -3 > 21.18 3125/3087 major BP diesis > | -5 -3 3 1 > 21.90 875/864 keema > | 6 -2 0 -1 > 27.26 64/63 septimal comma, Archytas' comma > | 1 -3 -2 3 > 27.99 686/675 senga > > 11-limit > | 5 -1 3 0 -3 > 3.03 4000/3993 undecimal schisma, Wizardharry > | -7 -1 1 1 1 > 4.50 385/384 undecimal kleisma, Keenanisma > | 16 0 0 -2 -3 > 8.39 65536/65219 orgonisma > | 4 0 -2 -1 1 > 9.86 176/175 valinorsma > | -3 -1 -1 0 2 > 14.37 121/120 undecimal seconds comma > | 2 -2 2 0 -1 > 17.40 100/99 Ptolemy's comma > > 13-limit > | 3 0 2 0 1 -3 > 2.36 2200/2197 2.36 Parizek comma, petrma > | 2 -1 0 1 -2 1 > 4.76 364/363 4.76 gentle comma > | 2 -1 -1 2 0 -1 > 8.86 196/195 8.86 mynucuma > | -1 -2 -1 1 0 1 > 19.13 91/90 19.13 superleap > > I don't really know what a lot of those mean, but they're there if you want > them. > > To see if I could exploit some of the nearly-just intervals in 37 EDO, I've > done something like a billion attempts at using generators to create MOS > scales. Unfortunately, few of them have reasonable size (say, 13 or fewer > tones) and still take advantage of all -- or even many! -- of the great > tones in the scale, and most of them are very improper. > > At any rate, thinking I should try to do *something* to try out 37, I > settled on an 11-note MOS that uses 11/8 as a generator. Here it is: > > ! C:\Program Files (x86)\Scala22\37-EDO_generator11_8.scl > ! > > 11 > ! > 162.16215 > 259.45945 > 356.75675 > 454.05405 > 551.35135 > 713.51350 > 810.81080 > 908.10810 > 1005.40540 > 1102.70270 > 2/1 > > It's strictly proper, and almost every mode of this scale has an 11/8, > naturally, since that's the generator. Only this mode has a 3/2. There's a > mix of 11/9-ish neutral thirds, 5/4 major thirds, and 13/11 minor thirds in > the other modes. There are no 14/11 major thirds. Interestingly, many modes > have a 745-cent interval seems to go pretty well with the 551-cent interval > -- I suppose it's like putting a 10/9 or 9/8 over the 11/8. The scale > surprised me in a few ways that I haven't been able to properly exploit and > play around with. > > Here's a very brief ditty I did using this scale. It happened to work out > to be about 37 seconds long, which seemed appropriate for 37 EDO, so I made > it fit. I didn't exploit the other modes, so the melody mostly uses the > sharp neutral third, sharp fifth, 11/8, and 8/5. > > http://soundcloud.com/jdfreivald/37-seconds > > Regards, > Jake >
From: monz (2012-08-26) Subject: Re: 37 EDO Hi Jake and Igs, I posted here about 37-edo not too long ago. Have you seen my Encyclopedia page about it? http://tonalsoft.com/enc/number/37-edo/37edo.aspx ... i should mention that for some reason the graph that is supposed to end the section "Some 41-limit JI ratios mapped to 37-edo" is not appearing ... i don't know why -- the image file is uploaded to my website and the html code on the page seems to be correct. -monz --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote: > > Why not map it as 2.5.7.9.11.13? It kinda makes more sense that way if you think of it as a subset of 74-ED2, where you'd use the meantone mapping and put 3 on an interval that's halfway between the two 3's of 37-ED2 (just a bit sharp of 697 cents). Or you could do it Mike Battaglia style and map it as 2.3.5.7.9'.11.13, where the 9' isn't 3^2 but its own "prime". > > -Igs > > --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@> wrote: > > > > 37 EDO is funny. > > > > Prime 11 is perfect (less than 0.1 cent off), while primes 5, 7, and 13 are > > excellent (just a few cents error). Prime 3 isn't great (12 cents off), but > > it's usable enough. (I don't mind the sharp 3/2, but I don't like the > > counterpart flat 4/3.) All of the following intervals can be found in 37 > > EDO with less than 9 cents error: > > > > 14/13 > > 11/10 > > 7/6 > > 13/11 > > 6/5 > > 11/9 > > 5/4 > > 14/11 > > 13/10 > > 11/8 > > 7/5 > > 10/7 > > 16/11 > > 8/5 > > 7/4 > > > > 13/10 has less than a cent error. 10/7 and 7/5 have less than two cents. If > > you like 2.3.5.7.11.13, this seems like an EDO you can get behind. > > > > Good stuff, right? Problem is, I can't get a decent mapping out of it. The > > 13-limit patent val is > > < 37 59 86 104 128 137 | > > and that works fine for 13/11 (<3 cents sharp) and 14/11 (<5 cents sharp), > > but the 11/9 gets mapped to 324 cents (23 cents flat). There's a better > > 11/9 in the pitch set, though: 357 cents, or 10 cents sharp. The natural > > thing to do is bring the mapping for 3/1 down to 58 steps, but when I do, > > 11/9 gets mapped to 389 cents -- which is really that nearly-perfect 5/4. I > > could use 60/49 and 49/40 (both about 351 cents) instead of 11/9, but that > > seems to add complexity and eliminate the eleven-ness of the darn thing. > > > > So don't ask me what that 357-cent neutral third is, but it's not an 11/9. > > :) > > > > Because 13/11 * 14/11 = 3/2, 37 EDO tempers out 364/363. According to my > > spreadsheet, 37 EDO (using the patent val) also tempers out 2401/2400 (the > > breedsma, so the 49/40 and 60/49 neutral thirds are equated -- if you want > > to map the 357-cent neutral third to those intervals instead of 11/9). > > Those seem useful. Here are other commas that get tempered out (limiting > > myself to the 13 prime limit): > > > > 5-limit > > | 17 1 -8 > 11.45 393216/390625 Würschmidt's comma > > | -16 -6 11 > 37.72 Sycamore comma > > | 1 -5 3 > 49.17 250/243 Maximal diesis > > > > 7-limit > > | -5 -1 -2 4 > 0.72 2401/2400 Breedsma > > | 11 1 -3 -2 > 5.36 6144/6125 porwell comma > > | 6 0 -5 2 > 6.08 3136/3125 middle second comma > > | 0 -2 5 -3 > 21.18 3125/3087 major BP diesis > > | -5 -3 3 1 > 21.90 875/864 keema > > | 6 -2 0 -1 > 27.26 64/63 septimal comma, Archytas' comma > > | 1 -3 -2 3 > 27.99 686/675 senga > > > > 11-limit > > | 5 -1 3 0 -3 > 3.03 4000/3993 undecimal schisma, Wizardharry > > | -7 -1 1 1 1 > 4.50 385/384 undecimal kleisma, Keenanisma > > | 16 0 0 -2 -3 > 8.39 65536/65219 orgonisma > > | 4 0 -2 -1 1 > 9.86 176/175 valinorsma > > | -3 -1 -1 0 2 > 14.37 121/120 undecimal seconds comma > > | 2 -2 2 0 -1 > 17.40 100/99 Ptolemy's comma > > > > 13-limit > > | 3 0 2 0 1 -3 > 2.36 2200/2197 2.36 Parizek comma, petrma > > | 2 -1 0 1 -2 1 > 4.76 364/363 4.76 gentle comma > > | 2 -1 -1 2 0 -1 > 8.86 196/195 8.86 mynucuma > > | -1 -2 -1 1 0 1 > 19.13 91/90 19.13 superleap > > > > I don't really know what a lot of those mean, but they're there if you want > > them. > > > > To see if I could exploit some of the nearly-just intervals in 37 EDO, I've > > done something like a billion attempts at using generators to create MOS > > scales. Unfortunately, few of them have reasonable size (say, 13 or fewer > > tones) and still take advantage of all -- or even many! -- of the great > > tones in the scale, and most of them are very improper. > > > > At any rate, thinking I should try to do *something* to try out 37, I > > settled on an 11-note MOS that uses 11/8 as a generator. Here it is: > > > > ! C:\Program Files (x86)\Scala22\37-EDO_generator11_8.scl > > ! > > > > 11 > > ! > > 162.16215 > > 259.45945 > > 356.75675 > > 454.05405 > > 551.35135 > > 713.51350 > > 810.81080 > > 908.10810 > > 1005.40540 > > 1102.70270 > > 2/1 > > > > It's strictly proper, and almost every mode of this scale has an 11/8, > > naturally, since that's the generator. Only this mode has a 3/2. There's a > > mix of 11/9-ish neutral thirds, 5/4 major thirds, and 13/11 minor thirds in > > the other modes. There are no 14/11 major thirds. Interestingly, many modes > > have a 745-cent interval seems to go pretty well with the 551-cent interval > > -- I suppose it's like putting a 10/9 or 9/8 over the 11/8. The scale > > surprised me in a few ways that I haven't been able to properly exploit and > > play around with. > > > > Here's a very brief ditty I did using this scale. It happened to work out > > to be about 37 seconds long, which seemed appropriate for 37 EDO, so I made > > it fit. I didn't exploit the other modes, so the melody mostly uses the > > sharp neutral third, sharp fifth, 11/8, and 8/5. > > > > http://soundcloud.com/jdfreivald/37-seconds > > > > Regards, > > Jake > > >
From: Jake Freivald (2012-08-27) Subject: Re: [tuning] Re: 37 EDO Igs, good to see you here -- I thought you had dropped off. Since I don't mind the very-sharp 3/2, but I still want to use the kinda-sharp 11/9, I think it makes the most sense to do it Battaglia-style (9' is its own prime). I remember seeing people talk about that way of thinking, but had forgotten it. It suits the way I'm considering the scales I've generated in it. Thanks for the thought. Monz, I had forgotten that 37 was the object of your recent discussion. I had just been looking through EDOs in a spreadsheet that calculates intervals and commas and the like (yes, I'm the life of the party). I especially like your diagrams on the encyclopedia! Excellent view into what's going on. One of the things I liked about dabbling in 37 is the perfect 11/8 that can have a major second (a little flat) piled on top of it -- I haven't previously found myself liking 745 cents over the root, but when played on top of a pure 11/8 (551 cents) it sounds pretty good. It doesn't sound like an out-of-tune 3/2, as it sometimes does, but it still seems stretched somehow without being discordant. Anyway, I wish I had better capabilities to use 13+ tone scales, because I think this EDO could be really cool for extended scales. Thanks, Jake
From: monz (2012-08-27) Subject: Re: 37 EDO Thanks to Graham Breed for helping me finally track down the missing hyphen which was preventing that graph from loading. All is well now. -monz --- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote: > > Hi Jake and Igs, > > I posted here about 37-edo not too long ago. > Have you seen my Encyclopedia page about it? > > http://tonalsoft.com/enc/number/37-edo/37edo.aspx > > ... i should mention that for some reason the graph that is > supposed to end the section "Some 41-limit JI ratios > mapped to 37-edo" is not appearing ... i don't know why -- > the image file is uploaded to my website and the html code > on the page seems to be correct. > > -monz