A question and a suggestion

I hate to be obtuse, but with all the rank-two 11-limit temperament
discussion going on, I have to ask: What does the two-EDO notation mean?

If you give me the two EDOs, I can go to this page:
http://xenharmonic.wikispaces.com/Creating%20Scala%20scl%20files%20for%20rank%20two%20temperaments
...and that will tell me how to use Graham's temperament finder to generate
scales. But I don't really see how those scales correlate to 11-limit JI.

For instance, when I plug in 22&86, I get a list of temperaments, the first
of which is 22&86ce:
http://x31eq.com/cgi-bin/rt.cgi?ets=22+86&limit=11

When I look at the 22-note scale generated from it:
http://x31eq.com/cgi-bin/scala.cgi?ets=22_86&limit=11&key=9_4_15_1_2_4_5_7_7&tuning=po

...however, I don't see the 11/9 neutral third I would expect from a
typical 11-limit temperament -- the closest matches are 333 cents and 377
cents. Is that because 3/1 is very sharp (it appears that 3/2 maps to 711
cents), and the 333 cents is the flattened 11/9? Or is something else going
on?

So that's the question: What does the two-EDO notation mean?

The suggestion is that the first link in this email, "Creating Scala scl
files [etc]," be placed on the wiki home page. It's one of the more useful
pages on the site, and I found it by accident.

Thanks,
Jake

Each EDO is shorthand for a val; if no letter is specified, it means the
"patent" val for that EDO. So if you take the two vals and concatenate them
into a matrix, you get a mapping matrix for that temperament. The familiar
"reduced matrix" which everyone likes to read can be obtained by putting
this matrix into Hermite normal form.

Sent from my iPhone

On Jan 1, 2012, at 3:37 PM, Jake Freivald <jdfreivald@...> wrote:

I hate to be obtuse, but with all the rank-two 11-limit temperament
discussion going on, I have to ask: What does the two-EDO notation mean?

If you give me the two EDOs, I can go to this page:
http://xenharmonic.wikispaces.com/Creating%20Scala%20scl%20files%20for%20rank%20two%20temperaments
...and that will tell me how to use Graham's temperament finder to generate
scales. But I don't really see how those scales correlate to 11-limit JI.

For instance, when I plug in 22&86, I get a list of temperaments, the first
of which is 22&86ce:
http://x31eq.com/cgi-bin/rt.cgi?ets=22+86&limit=11

When I look at the 22-note scale generated from it:
http://x31eq.com/cgi-bin/scala.cgi?ets=22_86&limit=11&key=9_4_15_1_2_4_5_7_7&tuning=po

...however, I don't see the 11/9 neutral third I would expect from a
typical 11-limit temperament -- the closest matches are 333 cents and 377
cents. Is that because 3/1 is very sharp (it appears that 3/2 maps to 711
cents), and the 333 cents is the flattened 11/9? Or is something else going
on?

So that's the question: What does the two-EDO notation mean?

The suggestion is that the first link in this email, "Creating Scala scl
files [etc]," be placed on the wiki home page. It's one of the more useful
pages on the site, and I found it by accident.

Thanks,
Jake

The notation means the two equal temperaments (no mere EDOs) are possible tunings of the temperament class. There are other rules to make it unique.
22&86ce is fairly complicated. The approximate 11:9 doesn't come up in the 22 note MOS.

Graham

------Original message------
From: Jake Freivald <jdfreivald@...>
To: <tuning@yahoogroups.com>
Date: Sunday, January 1, 2012 3:37:14 PM GMT-0500
Subject: [tuning] A question and a suggestion

I hate to be obtuse, but with all the rank-two 11-limit temperament
discussion going on, I have to ask: What does the two-EDO notation mean?

If you give me the two EDOs, I can go to this page:
http://xenharmonic.wikispaces.com/Creating%20Scala%20scl%20files%20for%20rank%20two%20temperaments
...and that will tell me how to use Graham's temperament finder to generate
scales. But I don't really see how those scales correlate to 11-limit JI.

For instance, when I plug in 22&86, I get a list of temperaments, the first
of which is 22&86ce:
http://x31eq.com/cgi-bin/rt.cgi?ets=22+86&limit=11

When I look at the 22-note scale generated from it:
http://x31eq.com/cgi-bin/scala.cgi?ets=22_86&limit=11&key=9_4_15_1_2_4_5_7_7&tuning=po

...however, I don't see the 11/9 neutral third I would expect from a
typical 11-limit temperament -- the closest matches are 333 cents and 377
cents. Is that because 3/1 is very sharp (it appears that 3/2 maps to 711
cents), and the 333 cents is the flattened 11/9? Or is something else going
on?

So that's the question: What does the two-EDO notation mean?

The suggestion is that the first link in this email, "Creating Scala scl
files [etc]," be placed on the wiki home page. It's one of the more useful
pages on the site, and I found it by accident.

Thanks,
Jake

On 1/1/2012 3:37 PM, Jake Freivald wrote:
> I hate to be obtuse, but with all the rank-two 11-limit temperament
> discussion going on, I have to ask: What does the two-EDO notation mean?
>
> If you give me the two EDOs, I can go to this page:
> http://xenharmonic.wikispaces.com/Creating%20Scala%20scl%20files%20for%20rank%20two%20temperaments
> ...and that will tell me how to use Graham's temperament finder to generate
> scales. But I don't really see how those scales correlate to 11-limit JI.
>
> For instance, when I plug in 22&86, I get a list of temperaments, the first
> of which is 22&86ce:
> http://x31eq.com/cgi-bin/rt.cgi?ets=22+86&limit=11

You'll want to add a "p" after the EDO to use Graham's temperament finder (if it doesn't have another letter). Just "86" by itself could represent more than one temperament, but "86p" is the nearest prime mapping (a.k.a. "patent val"), the temperament which uses the closest approximation of each of the primes.

http://x31eq.com/cgi-bin/rt.cgi?ets=22p%2686p&limit=11

> So that's the question: What does the two-EDO notation mean?

One way of finding or describing regular temperaments is to use a combination of ETs (two for a rank 2 temperament). You can think of the temperament as a kind of weighted average of the two ETs. If the EDO has a letter after it, it represents a temperament having one prime mapped to the second nearest approximation. If it has more than one letter, like that "86ce", more than one of the primes uses the less accurate mapping (in this case 5 and 11).

So 86p is this:
<86 136 200 241 298]

and 86ce is this:
<86 136 199 241 297]

The "reduced mapping" on the page is just a more convenient mapping of the same temperament, where the first generator is an octave or fraction of an octave, and the second one is reduced to be smaller than the first one. (I use the convention that the second generator is smaller than half the first one, but not everyone follows this convention.)

To relate the temperament to JI, you can either use the generator tunings with the reduced mapping or the step tunings with the equal temperament mapping. I'll use the reduced mapping here.

2 3 5 7 11
[< 2 4 4 7 6 ]
< 0 -9 7 -15 10 ]>

This is a shorthand notation for describing the mapping of the primes 2, 3, 5, 7, and 11. To get the prime 2 (the octave), just take 2 of the first generator (599.838). To get the prime 3, take 4 of the first generator (599.838) and -9 of the second (55.289).

4 * 599.838 - 9 * 55.298 = 1901.67

If you have the mapping of the primes, you can calculate the tempered equivalent of any interval by finding its prime factors.

Sorry I failed to respond to this earlier. Thanks very much for helping
understand what this notation (e.g., 12&17) means, and Herman, thanks even
more for the detailed discussion of doing the multi-dimensional mappings.
(I say "multidimensional" because I presume it works the same way for,
e.g., 12&17&21.) Very clear, very helpful.

Thanks,
Jake