A case of 37-EDO

Hi all.

How is it possible that I've never heard perhaps a word about 37-EDO? Is it
because there are esentially no stable fifths in this tuning? Well, but
there is almost everything else that you could make music with. The 5-limit,
7-limit, 11-limit and even 13-limit intervals can be approximated quite
"convincingly" here with a maximum error of about 4 cents -- that's not too
much yet, is it? So my favorite chords of 2:5:8:11:14 and 4:7:10:13:16 can
be imitated here much better than in some smaller EDOs.

Petr

> How is it possible that I've never heard perhaps a word about
> 37-EDO?

I'm quite fond of it. The fifths are quite usable. Maybe
Gene can tell us what, if anything, is unique about it
among tunings with 714-cent fifths.

-C.

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:
>
> Hi all.
>
> How is it possible that I've never heard perhaps a word about 37-EDO? Is it
> because there are esentially no stable fifths in this tuning? Well, but
> there is almost everything else that you could make music with.

Andrew Heathwaite agrees: http://www.nonoctave.com/forum/messages/445.html?n=4

But what were you going to do with words? Can't eat 'em...

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > How is it possible that I've never heard perhaps a word about
> > 37-EDO?
>
> I'm quite fond of it. The fifths are quite usable. Maybe
> Gene can tell us what, if anything, is unique about it
> among tunings with 714-cent fifths.

If you mean for me to compare 37 to 42, 37 does much better in the
11 and 13 department. 37 is also a porcupine system, tempering out
250/243 and 64/63. 42 supports what I was calling "tripletone" and
Paul dubbed "augene", despite the fact that I've only written a few
bars of music in it, which tempers out 128/125, 64/63 and 126/125. You
could say in the 7-limit 37 is a surrogate 22 and 42 is a surrogate 27.

Petr Pa��zek wrote:
> Hi all.
> > How is it possible that I've never heard perhaps a word about 37-EDO? Is it
> because there are esentially no stable fifths in this tuning? Well, but
> there is almost everything else that you could make music with. The 5-limit,
> 7-limit, 11-limit and even 13-limit intervals can be approximated quite
> "convincingly" here with a maximum error of about 4 cents -- that's not too
> much yet, is it? So my favorite chords of 2:5:8:11:14 and 4:7:10:13:16 can
> be imitated here much better than in some smaller EDOs.

I mention it on my porcupine temperament page, and I've played around with a 15-note porcupine subset of it, but not having a keyboard with a convenient way to use a full 37 notes per octave, I haven't tried any larger subsets of 37-ET.

Here's what I say about it there: "37-ET is one of the more interesting porcupine scales. Its nearly perfect approximations of the 11th and 13th harmonics contrast with an 11.6-cent sharp fifth. 37-ET also has good approximations of the 17th and 19th harmonics."

http://www.io.com/~hmiller/music/temp-porcupine.html

You can also reduce the size of the octave slightly, which helps with the accuracy of some intervals (at the expense of others). Here's one possibility, a 7-limit TOP tuning:

2/1: 1196.364741 cents (37 steps)
3/1: 1907.716750 cents (59 steps)
5/1: 2780.739669 cents (86 steps)
7/1: 3362.754948 cents (104 steps)

(TOP is a particular kind of optimization that's been discussed on the tuning-math list, which weights the errors according to the complexity of the intervals.)

--- In tuning@yahoogroups.com, "J.Smith" <jsmith9624@...> wrote:

> Well, my nomination for an all-around, practical octave division for
> tempering various tonalities would be 89-EDO. Nice fifths, good 5-
> through 13-limit pitches. What do you say, Gene?

It's come up as a myna (27&31) temperament tuning, since it's
"poptimal" for that in the 7-limit, and does fine in the 11-limit.
That seems to be the main thing it has going for it in the linear
temperaments department; as an all-around utility temperament, it has
a problem in that it doesn't do a good meantone.