Higher-limit mavila extension spring cleaning

If you manage to read through all of this you have my eternal respect;
there's basically no way to shrink any of this down. But I must do
what needs to be done...!

Mavila has a terrible naming crisis going on - I've heard it sometimes
called "mavila" and sometimes "pelogic," and then it has a ton of
7-limit extensions with different and often conflicting names. I've
seen at least two temperaments called "pelogic," and one called
"hexadecimal/pelogic," and then I see people calling it "Armodue" and
"Hornbostel" as well on the XA. It's a terrible mess as it stands, so
in order to sort this out, I propose the following naming convention,
which more or less splits the mavila umbrella into extensions
supported by 9-equal, 16-equal, and 23(d)-equal. My goal is that when
people talk about "armodue" and "hornbostel" and "pelogic" and
"mavila," they can actually be talking about specific rank-2
temperaments that are subtly consistent with their usage, rather than
having the mishmosh of inconsistency that we do now.

1) "Mavila" is the base 5-limit 135/128 temperament. Anything that
calls the basic, 5-limit, rank-2 135/128 temperament "pelogic" should
just be changed to avoid confusion. This is basically a moot point
these days, as everyone in the community calls it "mavila" now, so
this mainly has to do with identifying and fixing cruft.
2) "Armodue" is the 36/35 7-limit extension, of which 16-equal is a
good tuning. Since many on here don't know what "Armodue" is, I'll go
into greater detail below. This one's currently called either
"Hexadecimal" or "Hexadecimal/Pelogic" or "Pelogic" depending on where
you look.
3) "Pelogic" is the 21/20 extension, which 9-equal supports. This one
is currently just called 7-limit "mavila", but it's worthwhile to
switch it up for reasons explained below.
4) "Hornbostel" is the as of yet unnamed 126/125 extension, because of
its relevance to 23-equal, specifically the 23d val which is optimal
in that it has the lowest TOP-RMS error. I'm sure we're going to start
arguing over patent vals vs TOP-RMS optimal vals here.

These are the 4 main 7-limit extensions of mavila and, in certain
ways, they're more or less consistent with the terminology that's
being used in the community. The 64/63 and 28/27 extensions, for some
reason, keep popping up on the temperament finder, but I have no idea
what to name them or why they're relevant.

(2) The most important of these is Armodue, which gets the 36/35
extension and goes in the direction of 16-equal. There is no reason to
make this name change and the only reason this wasn't called "armodue"
from the start is that we didn't know what Armodue was until recently.
The Armodue theory is a theory of 16-equal based out of Italy that
seems to have been around since at least 2002 (see here:
http://web.archive.org/web/20010712205920/http://www.armodue.com/).
It's one of the few solid, practical, concrete theories available for
a specific tuning other than 12, and is very effective in breaking
16-equal down into a simple theory that people can digest and
understand and make practical music with.

We haven't heard much about it on this list since the site's entirely
in Italian, but the theory's gained somewhat of a following and there
are some adherents on the XA Facebook group, who generally call mavila
itself "armodue." Despite that Armodue is supposedly a theory of
16-equal specifically, it's obvious that the main focus with it is
really as a chromatic scale for mavila, as every aspect of it is more
or less based on mavila[9] (use of mavila[9] as a "superdiatonic"
scale, nominal naming convention of 1-9, 9+7 keyboard layout,
mavila-based staff notation, etc).

Armodue's take on mavila is distinguished from others by focusing on a
specific 7-limit mavila extension that emphasizes that the interval
between "1" and "8" is an approximation to 7/4, which means 36/35
vanishes. This temperament seems to be called "Hexadecimal/Pelogic" on
the xenwiki and elsewhere just "Pelogic" as it stands now. However, as
the Armodue name has historic precedence over all of these, and as the
"armodue" name is already in common use, it makes the most sense to
give this the title of armodue temperament, of which 33/32 and 65/64
serve for excellent extensions compatible with 16-equal. I doubt there
will be any objections to this, as it's clear Armodue was here first,
and this extension now has an inconsistent mishmosh of names, so I'm
going to make a change and call it done.

(3) I'm calling the 21/20 extension, which is supported by 9-equal,
"pelogic." The word "pelogic" has a pretty strange history in that it
can't seem to find a mapping that hasn't been discovered before under
a different name, but in a lot of ways the 21/20 extension is the most
sensible.

The 135/128 temperament, which was originally called "pelogic," and
the 36/35 extension, which is in some places called "pelogic" but
others "hexadecimal" or "hexadecimal/pelogic," both have their names
usurped by earlier discoveries (mavila and armodue respectively).
However, there should still be an official "pelogic" temperament, not
just a hangover from an earlier names, and not one with a haphazard
naming convention too. When people talk about "pelog" or "pelogic" in
contrast to the usual standpoint, especially when they're coming more
from an ethnomusicological standpoint, they're usually talking about
something flatter than 16 or 23-equal, more like 9-equal. And since
9-equal is often cited as being fairly close to how actual pelog music
is tuned, our "pelogic" temperament should be supported by 9-equal.

Of course, pelog music doesn't have much to do with actual 7-limit
ratios, so our goal with this is basically to find a mapping that
flattens the fifth more than just normal mavila, and call it "pelogic"
rather than "pelog" as a result to signify that this is mimicry of a
"natural" tuning within the regular temperament paradigm. And for this
goal, an obvious solution presents itself: 21/20-tempering really does
a number on the fifths, even more effectively than 36/35-tempering.
This is actually the lowest-badness 7-limit extension of mavila, so it
makes double sense to assign the "pelogic" name, which is historically
important on this list, to an important extension like that. It's
compatible with 9-equal, which is a triple plus. And, finally, if
you're the kind of guy who likes to imagine that cultures gravitate
towards low-badness tunings, then you can use that argument too...
kind of.

(4) Hornbostel is the original 23-equal guy. While his concept of
"blown fifths" has since been technically debunked, his work is still
inspiring in his approach to ethnomusicology, and he should get some
credit for predicting mavila in some form. There are a few people on
XA who keep referring to 23-equal as "Hornbostel" and to mavila[7] in
general as "Hornbostel" or the "Hornbostel" scale. I don't think his
work with 23-equal ever went beyond the 3-limit, so this is more of a
tribute than a description of an actual temperament he predicted. And
whatever temperament gets the "Hornbostel" name should be more or less
consistent with the way that the 7-limit is used in 23-equal.

23-equal has really good 7/6's, though the rest of the 7-limit isn't
that great. In general, 23-equal is ambiguous in the 7-limit and
presents 2 vals: 23d, where 49/48 vanishes, and 23p where 36/35
vanishes. The first is the lowest error val and also gives you the
good 7/6's, and anecdotally and to my ears 4:5:6:7 sounds better with
the first val. The final nail in the coffin is that the second val
already supports armodue temperament, which defeats the purpose of us
looking for a unique 23-equal 7-limit extension for mavila in the
first place. So 23d it is.

We can't eliminate 49/48 and call that "Hornbostel," because the
generator splits in half and this is already called "superpelog." A
solution presents itself in tempering out 128/125, which is a good
enough extension to appear on the first page of the temperament
finder. So there's your Hornbostel.

I've made changes to the xenwiki now and I assume that Graham's
temperament finder will automatically pick it up once it scrapes the
changes. If anyone has any additions or can think of a better way to
do things than the above feel free to weigh in.

-Mike

On Sat, Aug 13, 2011 at 7:28 AM, Mike Battaglia <battaglia01@...> wrote:
> The final nail in the coffin is that the second val
> already supports armodue temperament, which defeats the purpose of us
> looking for a unique 23-equal 7-limit extension for mavila in the
> first place. So 23d it is.

I just realized that 126/125 is tempered by 16-equal as well, which
doesn't make Hornbostel temperament a uniquely 23-equal thing. But the
main idea I was getting at with this is that, using armodue notation,
1-2## should be 7/6. This is true in 16-equal because 6/5 and 7/6 are
the same thing. But that's fine, because my underlying goal was just
to come up with more fundamental linear temperaments for these
differing concepts. The main thing about mavila in 23-equal is that in
mavila[7], a sharpened "large" step is 7/6, which is what I called
"Hornbostel temperament."

So you can say that
- 9-equal is a mavila temperament, a pelogic temperament, and an
Armodue temperament, and is not a Hornbostel temperament.
- 16-equal is a mavila temperament, an Armodue temperament, and a
Hornbostel temperament, and is not a pelogic temperament.
- 23-equal is a mavila temperament, and can be either a Hornbostel or
an Armodue temperament depending on which mapping you use, and is not
a pelogic temperament.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> We can't eliminate 49/48 and call that "Hornbostel," because the
> generator splits in half and this is already called "superpelog." A
> solution presents itself in tempering out 128/125, which is a good

Typo for 126/125, I assume?

Keenan

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> The Armodue theory is a theory of 16-equal based out of Italy

Not entirely. It also makes use of Valentine[16]:

http://xenharmonic.wikispaces.com/Armodue+theory#Semi-equalized Armodue

Plus just intonation is supposed to be a part of it somehow.

> I've made changes to the xenwiki now and I assume that Graham's
> temperament finder will automatically pick it up once it scrapes the
> changes.

You realize you need to change the Proposed Names page and the Optimal Patent Vals page as well as the Mavila Family page?

On Sat, Aug 13, 2011 at 11:23 AM, Keenan Pepper <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > We can't eliminate 49/48 and call that "Hornbostel," because the
> > generator splits in half and this is already called "superpelog." A
> > solution presents itself in tempering out 128/125, which is a good
>
> Typo for 126/125, I assume?

Yeah, whoops. Starling + mavila.

-Mike

On Sat, Aug 13, 2011 at 1:55 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The Armodue theory is a theory of 16-equal based out of Italy
>
> Not entirely. It also makes use of Valentine[16]:
>
> http://xenharmonic.wikispaces.com/Armodue+theory#Semi-equalized Armodue
>
> Plus just intonation is supposed to be a part of it somehow.

Yeah, these are more or less afterthoughts in the armodue theory. This
is only a partial translation of the work. The paper has some sections
about diminished[8] in 16-equal as well as some Messiaen "modes of
limited transposition" stuff, but the basic gist of the whole thing is
mavila in 16-equal. Almost every part of the theory is geared around
that - the note names, the keyboard layout, the staff layout, the
naming of mavila[9] as a "superdiatonic" scale, etc.

> > I've made changes to the xenwiki now and I assume that Graham's
> > temperament finder will automatically pick it up once it scrapes the
> > changes.
>
> You realize you need to change the Proposed Names page and the Optimal Patent Vals page as well as the Mavila Family page?

I've already changed the proposed names page, and there was no mention
of any 7-limit mavila extensions on the optimal patent val page, so I
think it's more or less worked out now.

-Mike

OK, I'm going to make one change to this, after to talking with
Graham. This setup leaves us with two undesirable features:

1) There's no 7-limit temperament with the name "mavila"
2) The "Hornbostel" temperament, which was supposed to lock us into
23, applies to 16 as well

The simple solution is to reinvent what I called "Hornbostel"
temperament above, and call it "mavila" or "septimal mavila" instead.
There are three reasons why this is desirable

1) It's of about the same complexity as septimal meantone
2) Septimal meantone tempers out 81/80 and 126/125, and septimal
mavila tempers out 81/80 and 126/125.
3) The generator changes only a cent compared to 5-limit mavila.

Then, we can give the name "Hornbostel" to the unnamed 875/864
temperament, which means that 36/35 and 25/24 are equated. This is
desirable for the following reasons:
1) It means that the distance between 5/4 and 6/5 is the same as the
distance between 6/5 and 7/6. This is an even more intuitive way to
use 23-equal in the 7-limit than the original way I mentioned, which
made 1-2## equal to 7/6 -- this way makes 1-3bb equal to 7/6 instead.
It's more intuitive in that it puts 7/6 and 6/5 in the same interval
class, save for chromatic alterations.
2) Unlike septimal meantone, this one really does only work in
23-equal, specifically 23d.
3) Hornbostel didn't really do much work with 23-equal outside of the
3-limit, so this is more or less just a tribute to him and so that
when people talk about 23-equal as supporting "Hornbostel" they can be
talking about an actual temperament.
4) The 126/125 one above, is more important and less tied down to
23-equal anyway, and so should be called "mavila" or "septimal mavila"
if you enjoy being pedantic :)

I'm going to make the change but need to run, so this is just a quick note.

-Mike

On Sat, Aug 13, 2011 at 7:28 AM, Mike Battaglia <battaglia01@...> wrote:
> If you manage to read through all of this you have my eternal respect;
> there's basically no way to shrink any of this down. But I must do
> what needs to be done...!

On Sat, Aug 13, 2011 at 5:44 PM, Mike Battaglia <battaglia01@...> wrote:
>
> 1) It's of about the same complexity as septimal meantone
> 2) Septimal meantone tempers out 81/80 and 126/125, and septimal
> mavila tempers out 81/80 and 126/125.

I mean 135/128 and 126/125. Blah, rush rush rush.

-Mike

On Sat, Aug 13, 2011 at 5:44 PM, Mike Battaglia <battaglia01@...> wrote:
>
> 1) It means that the distance between 5/4 and 6/5 is the same as the
> distance between 6/5 and 7/6. This is an even more intuitive way to
> use 23-equal in the 7-limit than the original way I mentioned, which
> made 1-2## equal to 7/6 -- this way makes 1-3bb equal to 7/6 instead.

For posterity's sake, this should say 1-4bb.

-Mike

On Thu, Oct 20, 2011 at 2:15 AM, Mike Battaglia <battaglia01@...> wrote:
> On Sat, Aug 13, 2011 at 5:44 PM, Mike Battaglia <battaglia01@...> wrote:
>>
>> 1) It means that the distance between 5/4 and 6/5 is the same as the
>> distance between 6/5 and 7/6. This is an even more intuitive way to
>> use 23-equal in the 7-limit than the original way I mentioned, which
>> made 1-2## equal to 7/6 -- this way makes 1-3bb equal to 7/6 instead.
>
> For posterity's sake, this should say 1-4bb.

Argh, I screwed it up even worse.

- Hornbostel is the temperament whereby 1-2# is 7/6.
- Septimal Mavila is the temperament whereby 1-4bb is 7/6.

Damn.

-Mike