87-edo, 46-edo

Is anybody here familiar with writing in 87-edo? Or just know something about it's properties?

It looks good in the 15-limit, scoring lowest in all edos to 100 for RMS error (0.812609 according to my calculations, using all odd harmonics from 3 to 15), and having an error range (i.e. diff between max and min errors for the harmonics 1-15) of 1.12 cents. *

Does anyone else have any more to say about it....it doesn't seems to pop up a lot in the archives, but it seems like it should given it's properties. I haven't checked to see if it's consistent in the 15-limit.

94-edo also looks good, but slightly less so, and at additional cost of notes, so it's not very efficient.

It's been mentioned before by Gene, and I can see why now---46-edo really is more promising than it seems that anyone has realized and written for. Up to 53, in the 15-limit, the top three edos are 53,41 and 46:

RMS_error error_range

53: 1.606348 46: 4.511450
41: 2.081022 53: 5.433184
46: 2.098984 41: 6.962770

*By the way, I weighed each harmonic by dividing the error in each by log2(harmonic), in case anyone is comparing with my calculations.

-A.

From: "Aaron K. Johnson" <aaron@akjmusic.com>
To: <tuning@yahoogroups.com>
Sent: Monday, June 04, 2007 6:21 AM
Subject: [tuning] 87-edo, 46-edo

> Does anyone else have any more to say about it....it doesn't seems to > pop up a lot in the archives, but it seems like it should given it's > properties. I haven't checked to see if it's consistent in the 15-limit.

It is. And so is 94-edo, up to 23-limit in fact.

~D.

Hi Aaron,

I took a quick look at some of the properties of
87-edo in the 5-prime-limit:

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> Is anybody here familiar with writing in 87-edo?
> Or just know something about it's properties?

87-edo has a perfect-5th which is about 1.5 cents
larger than pythagorean/JI. But it's 3rds are very
interesting, in that it gives both very good approximations
to the JI major and minor 3rds, and also to the 12-edo
versions.

In the 5-limit bingo-card lattice, 87-edo tempers
out the following commas, among others:

comma ....... 2-3-5-monzo

kleisma ..... [-6 -5, 6>
semithirds .. [38 -2, -15>
escapade .... [32 -7, -9>

("Semithirds" and "escapade" are the family names,
i don't know if these commas have names.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> Is anybody here familiar with writing in 87-edo? Or just know
something
> about it's properties?

I haven't written in it, but I'm slogging away with 118 and it
supports hemithirds like 118 does. Also kleismic, rodan, etc.

> It looks good in the 15-limit, scoring lowest in all edos to 100
for RMS
> error (0.812609 according to my calculations, using all odd
harmonics
> from 3 to 15), and having an error range (i.e. diff between max and
min
> errors for the harmonics 1-15) of 1.12 cents. *

It's 15-limit consistent and is certainly a logical choice there.

> Does anyone else have any more to say about it....it doesn't seems
to
> pop up a lot in the archives, but it seems like it should given
it's
> properties.

As far as I know Barbour was the first to notice it, as (of course) a
5-limit system. I could analyze it a lot more than this if you'd like.

> 94-edo also looks good, but slightly less so, and at additional
cost of
> notes, so it's not very efficient.

It's different; 94 tempers out the schisma, 87 the kleisma. 87 also
tempers out 245/243, 1029/1024, 3136/3125, 5120/5103, 385/384,
441/440, 896/891 etc and from this in turn many properties follow, a
discussion which could go on for some time.

> It's been mentioned before by Gene, and I can see why now---46-edo
> really is more promising than it seems that anyone has realized and
> written for.

Well, in fact, music *has* been written for it. I would urge more to
be done with it. And speaking of 46, the 46&87 temperament is rodan.
It's one of the notable ways to organize 87, with a low complexity on
3 and 7. OE part of the wedgie is <<3 17 -1 -13 -22 ... || and
tempering out 245/243, 1029/1024, 385/384, 441/440

> Up to 53, in the 15-limit, the top three edos are 53,41 and 46:

Here's something to ponder:

http://www.research.att.com/~njas/sequences/A117559