Yahoo Tuning Groups Ultimate Backup tuning 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

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

🔗cityoftheasleep <igliashon@...>

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
>

🔗monz <joemonz@...>

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
> >
>

🔗Jake Freivald <jdfreivald@...>

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