Topic: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

2 scales

Thread (22 messages)

From: Mike Battaglia (2011-02-03)
Subject: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

In some ways this is the most efficient temperament I have ever seen,
more so even than meantone.

I have always thought that 4:7:9:11 was a great major sonority to use
for a xenharmonic tonal system. I stumbled on this while looking for
the super-father temperament I mentioned in the last thread, first as
a 2.7.9.11 subgroup temperament, and then as a 2.3.7.11.13
temperament. The generator is a sharp major second.

Firstly, you have the 2.7.9.11.13 temperament eliminating 64/63,
99/98, and 648/637. 17-equal is pretty much ideal for this, 28-equal
as well - when we change the 9 to a 3 we will see that 17-equal is the
clear winner. So there seem to then be two ways to ditch the 9 and tie
this into 2.3.7.11.13, which end up being conflated at 17-equal - I'm
not sure exactly what's going on, but you seem to get a temperament
that's inconsistent in the 9-limit if you use anything but 17-equal
(which is no problem for a 2.7.9.11.13 subgroup temperament, but more
problematic for a 2.3.7.11.13 subgroup temperament). Perhaps it makes
sense to analyze this as a 2.3.7.9'.11.13 temperament, with 9'
denoting the "prime" 9, and then figure out what the two commas are
relative to 9' and 3 that could be tempered, which end up conflated in
17-equal.

Anyway, time for the magic to happen. There are 6, 11, and 17 note
MOS's, the 17-equal 11-note MOS is not proper, mapping 3/2 to two
different interval classes, but this seems to be important
nonetheless.

The 6-note MOS is similar in an abstract sense to the meantone
pentatonic scale in that there's nice 11-limit harmony everywhere you
go. The 11-note MOS is some kind of miracle of mathematics, because
you have full

4:7:9 tetrads on 9 roots of it
4:7:9:11 tetrads on a whopping 7 roots of this scale
4:7:9:11:13 pentads on 4 notes of this scale
And there are endless extensions of this if you throw 3 into it as
well, especially with 17-equal where 3 appears in two classes.

This paradigm really makes 11-equal sing, but I think you need to
explore the 11-note MOS, which despite being the size that it is works
fantastically as a "diatonic" MOS. Try playing with some 4:7:9:11
tetrads and transposing them up and down the scale, you end up getting
some great linked-11 limit triads.

This scale sounds like something Gene would do wonders with. I have no
idea what to call it. There are such a wide range of synesthetic
impressions here that to pick one and name the entire tuning after it
seems like a disservice. It's a very stimulating temperament. My
precise first impression was that I understood what it felt like to be
a well-oiled machine, like all of the kinks in my psyche had been
worked out. So machine temperament, maybe?

Here is a scala file of it:

! C:\Program Files\Scala22\scl\11-17.scl
!
11 out of 17-tet
 11
!
 70.58824
 211.76471
 282.35294
 423.52941
 494.11765
 635.29412
 705.88235
 847.05882
 917.64706
 1058.82353
 2/1


-Mike

From: Michael (2011-02-03)
Subject: Re: [tuning] Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

MikeB>"In some ways this is the most efficient temperament I have ever seen, 
more so even than meantone."

17TET has always been one of my favorites and this scale sounds pretty 
promising.  It seems to have a bunch of 13/9's in it...IMVHO one of the better 
intervals between 3/2 and 7/5 and just about the only interval I like that never 
pops up in my Dimension scale system.
     I'm going to try to compose something with it soon.

From: Mike Battaglia (2011-02-03)
Subject: Re: [tuning] Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

Starting to see what's going on here... This is the 11&17 temperament,
and 11&17 * 2 = 22*34 temperament. Pajara strikes again. This is a
no-5's, 13-limit version of Pajara, kind of. But, on the other hand,
since no 5's are involved, you don't get the period halving that
Pajara gave you. So machine is pajara with twice the generator and
twice the period. I'm not sure that it really fits in the
pajara/diaschismic "family," since there's no diaschisma to temper out
to begin with, and the resulting tonal structure is completely
entirely different.

So since 11-equal supports this, 22 does as well. It's probably better
to think of this as "only" (lol) an 11-limit temperament rather than a
13-limit one in 22. This makes 22-equal a really interesting case of
this temperament: mess with this scale in 22-equal and you get
11-equal but now 3's and 5's are involved as well. The 11-note MOS
changes here and becomes 11-equal, and all of the interesting
"diatonic" transpositional shifts are gone and you now just get
4:7:9:11 everywhere. However, you now get 5's and 3's as well for you
to use, so you can sort of treat 22-equal as an awesomely tonal
11-limit system in which 11-tet is the base 11-limit, no 3's, no 5's
scale. This is analogous to how people use 34-equal as a 5-limit (or
13-limit) system in which 17-tet is the base no-5's scale, or how
people use blackwood as a 5-limit scale in which 5-tet is the base
no-5's scale. This isn't what I had in mind originally, but it's also
an interesting thing to consider. This changes the mapping for 3, but
that might not be a bad thing.

34-equal is probably even better, because you now get a full 13-limit
temperament, and you still get to take advantage of the 11 note MOS.
So perhaps it makes sense for us to define a machine "family," which
is defined over the 2.7.9.11 subgroup, and then give it a 13-limit
child and a 9->3 extension mapping for 17-equal, and then a 5-limit
mapping for 34-equal. And then we should define a different
9->extension mapping for 22-equal.

Lots of neat stuff going on here.

-Mike

From: genewardsmith (2011-02-03)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

--- In [email protected], Mike Battaglia <battaglia01@...> wrote:

> Firstly, you have the 2.7.9.11.13 temperament eliminating 64/63,
> 99/98, and 648/637.

A better way to put it might be eliminating 64/63, 99/98 and 144/143. I don't see why you have the 9: I can't see anything wrong with making it a 3, and certainly 17 handles that well. Of course 28 doesn't, but why does that matter?

If you want you can toss 847/845 into the mix, and for instance use <73 116 170 206 253 271| to tune it; however 5 is very complex under this system; you'd need something like the 39-note MOS to make full use of it. As for no-fives, 12 and 22 note MOS could be used as well as 17.

> Anyway, time for the magic to happen. There are 6, 11, and 17 note
> MOS's, the 17-equal 11-note MOS is not proper, mapping 3/2 to two
> different interval classes, but this seems to be important
> nonetheless.

I don't get that at all. I get 5, 7, 12, 17 for MOS, but no six, and no problem with the fifths. What are you using for a generator? I get that it should be about 43\73, which is 706.849 cents. You might want to compare what you are doing to things Margo has talked about.

From: genewardsmith (2011-02-03)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

--- In [email protected], Mike Battaglia <battaglia01@...> wrote:

> Here is a scala file of it:

OK, this is the 11-note MOS with tuning 3\17, and that involves entirely different commas than was stated. One thing it has going for it is 78/77, but I'm not seeing the efficiency here.

From: Mike Battaglia (2011-02-03)
Subject: Re: [tuning] Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

On Thu, Feb 3, 2011 at 12:29 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > Firstly, you have the 2.7.9.11.13 temperament eliminating 64/63,
> > 99/98, and 648/637.
>
> A better way to put it might be eliminating 64/63, 99/98 and 144/143. I don't see why you have the 9: I can't see anything wrong with making it a 3, and certainly 17 handles that well. Of course 28 doesn't, but why does that matter?

I stumbled on it by accident, but if you make it a 3, you don't get
the awesome 11 note MOS because I think the generator works out
differently. It's a 2.3.7.9'.11.13 temperament, in the sense that
there is a 9' gets mapped differently than 3, and then there's more
than one way to tie 3 and 9' together, is what I think is going on -
and the two ways are conflated in 17-equal.

> If you want you can toss 847/845 into the mix, and for instance use <73 116 170 206 253 271| to tune it; however 5 is very complex under this system; you'd need something like the 39-note MOS to make full use of it. As for no-fives, 12 and 22 note MOS could be used as well as 17.

Well to hell with 5 anyway! Enough with 5. These are very interesting,
although the fact that the 8/7 generator becomes a 3/2 yields like an
entirely different scale structure.

> > Anyway, time for the magic to happen. There are 6, 11, and 17 note
> > MOS's, the 17-equal 11-note MOS is not proper, mapping 3/2 to two
> > different interval classes, but this seems to be important
> > nonetheless.
>
> I don't get that at all. I get 5, 7, 12, 17 for MOS, but no six, and no problem with the fifths. What are you using for a generator? I get that it should be about 43\73, which is 706.849 cents. You might want to compare what you are doing to things Margo has talked about.

I think it makes sense to start out with the subgroup temperament to
see what's going on, I might be using nonstandard terminology here.

1) Try starting with 5/28 - you'll see that the resultant scale has
MOS's at 6, 11, and 17.
2) You will then see, in the 11-note MOS, that both mappings for 3/2
that 28-offers occur in this scale over different interval classes;
one of the 7 step intervals is 729 cents, and one of the 6 step
intervals is 686 cents.
3) The magic tuning that makes them both equal seems to be 17-equal,
where they're both 706, but subtend two different interval classes.
This also means that in 17-equal you get a chain of 6 3/2's, but
sometimes the 3/2's are 6 steps and sometimes 7 steps.
4) I have no idea how to classify this, but this particular 11-note
scale is really magical, so if I'm using a nonstandard generator, then
I wish there were some way to formalize nonstandard generators like
this :P
5) If you double 17-equal you get 34-equal, which will give you an
amazing planar version of this temperament in which you can throw 5 in
whenever you want. I'm starting to find planar temperaments more and
more attractive these days.
6) If, instead of messing around with this in 17-equal, you mess with
it in 11-equal, which obviously also supports the 11-note MOS - you
still get 4:7:9:11 all over the place, but the way to tie 3/2 in with
it completely disappears. If you double this you get 22-equal, in
which you also have a 3d version of this temperament in which 5 can
appear whenever you want, but also in which 3 can appear whenever you
want. I'm not sure if it really counts as 3d if 3 is just the square
root of 9, but it never appears in the base 11-note scale.



-Mike

From: Mike Battaglia (2011-02-03)
Subject: Re: [tuning] Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

On Thu, Feb 3, 2011 at 12:54 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In [email protected], Mike Battaglia <battaglia01@...> wrote:
>
> > Here is a scala file of it:
>
> OK, this is the 11-note MOS with tuning 3\17, and that involves entirely different commas than was stated.

?? It tempers out 64/63, 99/98, and 648/637, right?

> One thing it has going for it is 78/77, but I'm not seeing the efficiency here.

Are there other similarly sized MOS's with this much 11-limit harmony
packed in that I'm missing out on?

-Mike

From: genewardsmith (2011-02-03)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

--- In [email protected], Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Feb 3, 2011 at 12:54 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In [email protected], Mike Battaglia <battaglia01@> wrote:
> >
> > > Here is a scala file of it:
> >
> > OK, this is the 11-note MOS with tuning 3\17, and that involves entirely different commas than was stated.
> 
> ?? It tempers out 64/63, 99/98, and 648/637, right?

I get that it doesn't temper out any of those commas. The temperament that does has a generator of a sharp fifth, and is what I described. 3\17 gives you a no-fives linear temperament tempering out 78/77, 512/507 and 352/343, and is supported by the patent vals for 6, 11, 17 and 28. Since 28 has such a good 5 it would make sense to use it if you wanted 5s, and you could still use the 11 or 17 note MOS. This full 13-limit temperament, the 17&28 temperament, tempers out 45/44, 78/77, 352/343 and 1344/1331. You can also use 45edo, with the val <45 71 104 127 156 167|. The 5 is now way flat, but the 3 is so much better it's probably a good idea.

> Are there other similarly sized MOS's with this much 11-limit harmony
> packed in that I'm missing out on?

I've not paid much attention to high error temperaments, but what about 11-limit dominant, as a for instance? You could use 29edo for it. Tempers out 36/35, 56/55 and 64/63. Also 11-limit porcupine isn't too bad for low complexity, and it's more accurate.

From: Mike Battaglia (2011-02-03)
Subject: Re: [tuning] Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

On Thu, Feb 3, 2011 at 1:41 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > ?? It tempers out 64/63, 99/98, and 648/637, right?
>
> I get that it doesn't temper out any of those commas. The temperament that does has a generator of a sharp fifth, and is what I described. 3\17 gives you a no-fives linear temperament tempering out 78/77, 512/507 and 352/343, and is supported by the patent vals for 6, 11, 17 and 28.

OK, I see what's going on. When this was a no-3, 9's subgroup
temperament, 64/63 was tempered out. To move it down to 3, it makes
less sense to temper out 64/63. So it's not 99/98 that vanishes
anymore. Sorry about that.

I think maybe to sell you on this scale, you should check out the
interpretation of it being just a 2.7.9.11 subgroup scale. That alone
yields enough awesome harmonies in the 11-note MOS, whether in 17-tet
or something else, to make this worthwhile IMO. Tuned properly, this
scale is very accurate and if nothing else, it has given me a new look
on 11-tet as a tonal system. If you look at it this way, there is an
easy 13-limit extension here that 17 supports really well, and
tempering 64/63 gives you more 3's, which are just a nice extra bonus
really.

> Since 28 has such a good 5 it would make sense to use it if you wanted 5s, and you could still use the 11 or 17 note MOS. This full 13-limit temperament, the 17&28 temperament, tempers out 45/44, 78/77, 352/343 and 1344/1331. You can also use 45edo, with the val <45 71 104 127 156 167|. The 5 is now way flat, but the 3 is so much better it's probably a good idea.

I like 45. I think I'm also a fan of throwing 5 in there by doing
something like 11&22 and 17&34 as well. The former is interesting
because the 11-note MOS just turns into 11-EDO, which has decently
accurate 4:7:9:11's, and changes the nature of things slightly. I
think the latter is far more powerful because the 11-note MOS now has
really good 3's and there are now 5's with a complexity of 1.

> > Are there other similarly sized MOS's with this much 11-limit harmony
> > packed in that I'm missing out on?
>
> I've not paid much attention to high error temperaments, but what about 11-limit dominant, as a for instance? You could use 29edo for it. Tempers out 36/35, 56/55 and 64/63. Also 11-limit porcupine isn't too bad for low complexity, and it's more accurate.

High error temperaments...? We're definitely on two different pages
here, I must have made a typo or something. The 4:7:9, 4:7:9:11, and
4:7:9:11:13 are the base chords here, and they're ridiculously
accurate in this temperament and ubiquitously placed with only 11
notes... I really must have screwed something up here. Did you play
around with the 11 out of 17 scale I posted? What's high error there?

-Mike

From: Chris Vaisvil (2011-02-03)
Subject: Re: [tuning] Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

grabbed it Mike - thanks!!

On Thu, Feb 3, 2011 at 11:32 AM, Mike Battaglia <battaglia01@...>wrote:

>
>
> In some ways this is the most efficient temperament I have ever seen,
> more so even than meantone.
>
>
> This scale sounds like something Gene would do wonders with. I have no
> idea what to call it. There are such a wide range of synesthetic
> impressions here that to pick one and name the entire tuning after it
> seems like a disservice. It's a very stimulating temperament. My
> precise first impression was that I understood what it felt like to be
> a well-oiled machine, like all of the kinks in my psyche had been
> worked out. So machine temperament, maybe?
>
> Here is a scala file of it:
>
> ! C:\Program Files\Scala22\scl\11-17.scl
> !
> 11 out of 17-tet
> 11
> !
> 70.58824
> 211.76471
> 282.35294
> 423.52941
> 494.11765
> 635.29412
> 705.88235
> 847.05882
> 917.64706
> 1058.82353
> 2/1
>
> -Mike
>  
>

From: genewardsmith (2011-02-03)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

--- In [email protected], Mike Battaglia <battaglia01@...> wrote:

> Did you play
> around with the 11 out of 17 scale I posted? What's high error there?

An easier question is what isn't high error. The fifth, of course, is good, so the major whole tone isn't too awfully sharp. The 11/7 is six cents flat, not bad, and the 11/9 neutral third pretty good at 5.5 cents flat. The 11/6 is 9.5 cents sharp. The 9/7 is 11.6 cents flat. The 11/8 is 13 cents sharp, ouch. The 7/6 is 15.5 cents sharp, double ouch. The 7/5 is 18 cents flat, scream!

So, you have three neutral triads which aren't bad at all.

From: genewardsmith (2011-02-03)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

--- In [email protected], "genewardsmith" <genewardsmith@...> wrote:


The 11/7 is six cents flat, not bad, and the 11/9 neutral third pretty good at 5.5 cents flat. 

Sorry, 11/9 is sharp. Scala tangles me up sometimes still, even with long experience, because it does things backwards: sharp is negative, flat is positive. Don't ask em to explain it.

From: Michael (2011-02-03)
Subject: Re: [tuning] Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

Gene>"The 9/7 is 11.6 cents flat. The 11/8 is  13 cents sharp, ouch. The 7/6 is 
15.5 cents sharp, double ouch. The 7/5  is 18 cents flat, scream!"

   IMVHO, these are the only ones far enough to truly be "high error".  A real 
unique "dead spot" here appears to be around 560 cents or so.  Sure, it has that 
nasty sharp 7/6-ish interval...but so does 12TET.
  Kill the dead spot and, IMVHO, you'd have something competitive with 12TET.  
Gene...do you think you can manage a least squares optimization for Mike B's 
scale?  I wonder if that could kill the 560 cent-ish dead spot...or at least 
bring the error back down to around 10 cents or under (typical higher-error 
range for 12TET)...

From: genewardsmith (2011-02-03)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

--- In [email protected], Michael <djtrancendance@...> wrote:
>
> Gene>"The 9/7 is 11.6 cents flat. The 11/8 is  13 cents sharp, ouch. The 7/6 is 
> 15.5 cents sharp, double ouch. The 7/5  is 18 cents flat, scream!"
> 
>    IMVHO, these are the only ones far enough to truly be "high error".  A real 
> unique "dead spot" here appears to be around 560 cents or so.  Sure, it has that 
> nasty sharp 7/6-ish interval...but so does 12TET.
>   Kill the dead spot and, IMVHO, you'd have something competitive with 12TET.  

You yhink? In the first place, it isn't an et. In the second place, it's got three pretty good 1-11/9-3/2 neutral triads, three pretty good 1-14/11-3/2 unidecimal major triads, three more or less in tune 1-13/11-3/2 triads, but is that enough to make it a winner?

> Gene...do you think you can manage a least squares optimization for Mike B's 
> scale? 

Not a bad idea, but first I'd need to know which of the intervals he wants to optimize.

From: Mike Battaglia (2011-02-03)
Subject: Re: [tuning] Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

On Thu, Feb 3, 2011 at 3:13 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In [email protected], Mike Battaglia <battaglia01@...> wrote:
>
> > Did you play
> > around with the 11 out of 17 scale I posted? What's high error there?
>
> An easier question is what isn't high error. The fifth, of course, is good, so the major whole tone isn't too awfully sharp. The 11/7 is six cents flat, not bad, and the 11/9 neutral third pretty good at 5.5 cents flat. The 11/6 is 9.5 cents sharp. The 9/7 is 11.6 cents flat. The 11/8 is 13 cents sharp, ouch. The 7/6 is 15.5 cents sharp, double ouch. The 7/5 is 18 cents flat, scream!

There's no 5, so who cares about 5?

Do you consider 22-tet in general to be high in error, or 17-equal as
a no-5's temperament to be high in error?

-Mike

From: Mike Battaglia (2011-02-03)
Subject: Re: [tuning] Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

On Thu, Feb 3, 2011 at 4:09 PM, genewardsmith
<genewardsmith@...> wrote:
>
> You yhink? In the first place, it isn't an et. In the second place, it's got three pretty good 1-11/9-3/2 neutral triads, three pretty good 1-14/11-3/2 unidecimal major triads, three more or less in tune 1-13/11-3/2 triads, but is that enough to make it a winner?

I hate neutral triads. But like I said, my goal here was initially to
aim at 4:7:9 and 4:7:9:11. 4:7:9:11:13 came as an afterthought once I
saw how nicely it worked out in 22-equal, and adding 3/2 to the mix
never crossed my mind at all until I saw that it worked out nicely in
17-equal.

And there are plenty of 4:7:9:11 chords all over the place. I say winner!

> > Gene...do you think you can manage a least squares optimization for Mike B's
> > scale?
>
> Not a bad idea, but first I'd need to know which of the intervals he wants to optimize.

I would prefer it first for 4:7:9:11 chords, 4:7:9
chords if the first is impossible, and 4:7:9:11:13 just as an
afterthought. How would you do this - optimize via POTE?

-Mike

From: cityoftheasleep (2011-02-03)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

This is pretty much how I use 11-EDO, the 5L+1s scale.  Though I don't often go for the full tetrads, and tend to voice the chords as 7:8:9 or 8:9:11.  I use this scale on the track "She is My Lilac-Hued Obsession" on _Map_Of_An_Internal_Landscape_.  Only problem with it is the generator is simultaneously a 9/8 and an 8/7, so 7:8:9=1/(7:8:9), meaning you can't use those triads as a basis for a major/minor tonality.  But that doesn't really bother me, because I love how 11-EDO sounds melodically.  I'm working on an 11-EDO trip-hop song right now with an auto-tuned voice that's going to blow your f***ing mind.  It's making me miss my 22-EDO guitar, since I could play 11-EDO on it.

I keep trying to tell people, even the bad EDOs are great for something...

-Igs

--- In [email protected], Mike Battaglia <battaglia01@...> wrote:
>
> In some ways this is the most efficient temperament I have ever seen,
> more so even than meantone.
> 
> I have always thought that 4:7:9:11 was a great major sonority to use
> for a xenharmonic tonal system. I stumbled on this while looking for
> the super-father temperament I mentioned in the last thread, first as
> a 2.7.9.11 subgroup temperament, and then as a 2.3.7.11.13
> temperament. The generator is a sharp major second.
> 
> Firstly, you have the 2.7.9.11.13 temperament eliminating 64/63,
> 99/98, and 648/637. 17-equal is pretty much ideal for this, 28-equal
> as well - when we change the 9 to a 3 we will see that 17-equal is the
> clear winner. So there seem to then be two ways to ditch the 9 and tie
> this into 2.3.7.11.13, which end up being conflated at 17-equal - I'm
> not sure exactly what's going on, but you seem to get a temperament
> that's inconsistent in the 9-limit if you use anything but 17-equal
> (which is no problem for a 2.7.9.11.13 subgroup temperament, but more
> problematic for a 2.3.7.11.13 subgroup temperament). Perhaps it makes
> sense to analyze this as a 2.3.7.9'.11.13 temperament, with 9'
> denoting the "prime" 9, and then figure out what the two commas are
> relative to 9' and 3 that could be tempered, which end up conflated in
> 17-equal.
> 
> Anyway, time for the magic to happen. There are 6, 11, and 17 note
> MOS's, the 17-equal 11-note MOS is not proper, mapping 3/2 to two
> different interval classes, but this seems to be important
> nonetheless.
> 
> The 6-note MOS is similar in an abstract sense to the meantone
> pentatonic scale in that there's nice 11-limit harmony everywhere you
> go. The 11-note MOS is some kind of miracle of mathematics, because
> you have full
> 
> 4:7:9 tetrads on 9 roots of it
> 4:7:9:11 tetrads on a whopping 7 roots of this scale
> 4:7:9:11:13 pentads on 4 notes of this scale
> And there are endless extensions of this if you throw 3 into it as
> well, especially with 17-equal where 3 appears in two classes.
> 
> This paradigm really makes 11-equal sing, but I think you need to
> explore the 11-note MOS, which despite being the size that it is works
> fantastically as a "diatonic" MOS. Try playing with some 4:7:9:11
> tetrads and transposing them up and down the scale, you end up getting
> some great linked-11 limit triads.
> 
> This scale sounds like something Gene would do wonders with. I have no
> idea what to call it. There are such a wide range of synesthetic
> impressions here that to pick one and name the entire tuning after it
> seems like a disservice. It's a very stimulating temperament. My
> precise first impression was that I understood what it felt like to be
> a well-oiled machine, like all of the kinks in my psyche had been
> worked out. So machine temperament, maybe?
> 
> Here is a scala file of it:
> 
> ! C:\Program Files\Scala22\scl\11-17.scl
> !
> 11 out of 17-tet
>  11
> !
>  70.58824
>  211.76471
>  282.35294
>  423.52941
>  494.11765
>  635.29412
>  705.88235
>  847.05882
>  917.64706
>  1058.82353
>  2/1
> 
> 
> -Mike
>

From: genewardsmith (2011-02-04)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

--- In [email protected], Mike Battaglia <battaglia01@...> wrote:

> Do you consider 22-tet in general to be high in error, or 17-equal as
> a no-5's temperament to be high in error?

Borderline.

From: genewardsmith (2011-02-04)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

--- In [email protected], Mike Battaglia <battaglia01@...> wrote:

> I would prefer it first for 4:7:9:11 chords, 4:7:9
> chords if the first is impossible, and 4:7:9:11:13 just as an
> afterthought. How would you do this - optimize via POTE?

I'd compute a lesfip scale:
http://xenharmonic.wikispaces.com/Lesfip+scales

From: dasdastri (2011-02-04)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

for anyone who might be interested,an old piece in that very scale in 11 

http://homemademusic.com/daniel_stearns/info_238.php


--- In [email protected], "cityoftheasleep" <igliashon@...> wrote:
>
> This is pretty much how I use 11-EDO, the 5L+1s scale.  Though I don't often go for the full tetrads, and tend to voice the chords as 7:8:9 or 8:9:11.  I use this scale on the track "She is My Lilac-Hued Obsession" on _Map_Of_An_Internal_Landscape_.  Only problem with it is the generator is simultaneously a 9/8 and an 8/7, so 7:8:9=1/(7:8:9), meaning you can't use those triads as a basis for a major/minor tonality.  But that doesn't really bother me, because I love how 11-EDO sounds melodically.  I'm working on an 11-EDO trip-hop song right now with an auto-tuned voice that's going to blow your f***ing mind.  It's making me miss my 22-EDO guitar, since I could play 11-EDO on it.
> 
> I keep trying to tell people, even the bad EDOs are great for something...
> 
> -Igs
> 
> --- In [email protected], Mike Battaglia <battaglia01@> wrote:
> >
> > In some ways this is the most efficient temperament I have ever seen,
> > more so even than meantone.
> > 
> > I have always thought that 4:7:9:11 was a great major sonority to use
> > for a xenharmonic tonal system. I stumbled on this while looking for
> > the super-father temperament I mentioned in the last thread, first as
> > a 2.7.9.11 subgroup temperament, and then as a 2.3.7.11.13
> > temperament. The generator is a sharp major second.
> > 
> > Firstly, you have the 2.7.9.11.13 temperament eliminating 64/63,
> > 99/98, and 648/637. 17-equal is pretty much ideal for this, 28-equal
> > as well - when we change the 9 to a 3 we will see that 17-equal is the
> > clear winner. So there seem to then be two ways to ditch the 9 and tie
> > this into 2.3.7.11.13, which end up being conflated at 17-equal - I'm
> > not sure exactly what's going on, but you seem to get a temperament
> > that's inconsistent in the 9-limit if you use anything but 17-equal
> > (which is no problem for a 2.7.9.11.13 subgroup temperament, but more
> > problematic for a 2.3.7.11.13 subgroup temperament). Perhaps it makes
> > sense to analyze this as a 2.3.7.9'.11.13 temperament, with 9'
> > denoting the "prime" 9, and then figure out what the two commas are
> > relative to 9' and 3 that could be tempered, which end up conflated in
> > 17-equal.
> > 
> > Anyway, time for the magic to happen. There are 6, 11, and 17 note
> > MOS's, the 17-equal 11-note MOS is not proper, mapping 3/2 to two
> > different interval classes, but this seems to be important
> > nonetheless.
> > 
> > The 6-note MOS is similar in an abstract sense to the meantone
> > pentatonic scale in that there's nice 11-limit harmony everywhere you
> > go. The 11-note MOS is some kind of miracle of mathematics, because
> > you have full
> > 
> > 4:7:9 tetrads on 9 roots of it
> > 4:7:9:11 tetrads on a whopping 7 roots of this scale
> > 4:7:9:11:13 pentads on 4 notes of this scale
> > And there are endless extensions of this if you throw 3 into it as
> > well, especially with 17-equal where 3 appears in two classes.
> > 
> > This paradigm really makes 11-equal sing, but I think you need to
> > explore the 11-note MOS, which despite being the size that it is works
> > fantastically as a "diatonic" MOS. Try playing with some 4:7:9:11
> > tetrads and transposing them up and down the scale, you end up getting
> > some great linked-11 limit triads.
> > 
> > This scale sounds like something Gene would do wonders with. I have no
> > idea what to call it. There are such a wide range of synesthetic
> > impressions here that to pick one and name the entire tuning after it
> > seems like a disservice. It's a very stimulating temperament. My
> > precise first impression was that I understood what it felt like to be
> > a well-oiled machine, like all of the kinks in my psyche had been
> > worked out. So machine temperament, maybe?
> > 
> > Here is a scala file of it:
> > 
> > ! C:\Program Files\Scala22\scl\11-17.scl
> > !
> > 11 out of 17-tet
> >  11
> > !
> >  70.58824
> >  211.76471
> >  282.35294
> >  423.52941
> >  494.11765
> >  635.29412
> >  705.88235
> >  847.05882
> >  917.64706
> >  1058.82353
> >  2/1
> > 
> > 
> > -Mike
> >
>

From: genewardsmith (2011-02-04)
Subject: Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

--- In [email protected], Mike Battaglia <battaglia01@...> wrote:

> I would prefer it first for 4:7:9:11 chords, 4:7:9
> chords if the first is impossible, and 4:7:9:11:13 just as an
> afterthought. How would you do this - optimize via POTE?

OK. Two observations: I optimized only for 11:9:7:4, but 13:11:9:7:4 fits in perfectly. Second, the 28et MOS is so close to the result it seems to me you may as well use it.

! mike.scl
Mike 11:9:7:4 Lesfip scale
! Approximately 23232323233 in 28et
11
!
85.99880
215.04394
299.89634
428.87771
514.62336
643.60472
728.45713
857.50226
943.50107
1071.75053
1200.00000

From: Mike Battaglia (2011-02-04)
Subject: Re: [tuning] Re: Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament

On Thu, Feb 3, 2011 at 11:01 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In [email protected], Mike Battaglia <battaglia01@...> wrote:
>
> > Do you consider 22-tet in general to be high in error, or 17-equal as
> > a no-5's temperament to be high in error?
>
> Borderline.

Alright, then this is at least a "borderline" temperament then :)

> OK. Two observations: I optimized only for 11:9:7:4, but 13:11:9:7:4 fits in perfectly. Second, the 28et MOS is so close to the result it seems to me you may as well use it.
>
> ! mike.scl
> Mike 11:9:7:4 Lesfip scale
> ! Approximately 23232323233 in 28et
> 11
> !
> 85.99880
> 215.04394
> 299.89634
> 428.87771
> 514.62336
> 643.60472
> 728.45713
> 857.50226
> 943.50107
> 1071.75053
> 1200.00000

Alright, thanks. I guess I see why you call it high error now, since
the worst thing here seems to be the 7/4 being sharp, assuming at
least you prioritize the consonance of 7/4 over 9/7 like I do. I still
prefer 17-equal, since you end up with better 3/2's as well, and
that's worth the sharper 11/8 in my view.

The real reason I like this scale so much is that I think it's
undeniably "tonal," and in a uniquely 11-limit sense. Once I work out
my Kontakt woes I'll try and write some stuff with it. I like this
scale so much, in fact, that it left me with some kind of new personal
understanding about the "big picture," visually speaking, somehow left
colors looking brighter all day long. Which was pleasant for me, but
you might not like it as much if you're more into much higher-accuracy
temperaments. Perhaps some hobbits are in order here, or maybe a
corresponding planar temperament.

-Mike