Naming Scheme for DE Scales

It recently occurred to me that DE scales can be indexed by a combination of their number of large steps (or small, but I picked large) and their number of total notes. So forget the nonsensical "7 dwarf" scheme of many years ago, here's one that makes sense. The names tell you exactly what the scales are. Less efficient to type than the xL+ys format, but easier to say and more "official" sounding.

Pentatonics:
1L+4s: Monomacro Pentatonic
2L+3s: Dimacro Pentatonic
3L+2s: Trimacro Pentatonic
4L+1s: Tetramacro Pentatonic

Hexatonics:
1L+5s: Monomacro Hexatonic
2L+4s: Dimacro Hexatonic
3L+3s: Trimacro Hexatonic
4L+2s: Tetramacro Hexatonic
5L+1s: Pentamacro Hexatonic

Heptatonics:
1L+6s: Monomacro Heptatonic
2L+5s: Dimacro Heptatonic
3L+4s: Trimacro Heptatonic
4L+3s: Tetramacro Heptatonic
5L+2s: Pentamacro Heptatonic
6L+1s: Hexamacro Heptatonic

Octatonics:
1L+7s: Monomacro Octatonic
2L+6s: Dimacro Octatonic
3L+5s: Trimacro Octatonic
4L+4s: Tetramacro Octatonic
5L+3s: Pentamacro Octatonic
6L+2s: Hexamacro Octatonic
7L+1s: Heptamacro Octatonic

Enneatonics:
1L+8s: Monomacro Enneatonic
2L+7s: Dimacro Enneatonic
3L+6s: Trimacro Enneatonic
4L+5s: Tetramacro Enneatonic
5L+4s: Pentamacro Enneatonic
6L+3s: Hexamacro Enneatonic
7L+2s: Heptamacro Enneatonic
8L+1s: Octamacro Enneatonic

Decatonics:
1L+9s: Monomacro Decatonic
2L+8s: Dimacro Decatonic
3L+7s: Trimacro Decatonic
4L+6s: Tetramacro Decatonic
5L+5s: Pentamacro Decatonic
6L+4s: Hexamacro Decatonic
7L+3s: Heptamacro Decatonic
8L+2s: Octamacro Decatonic
9L+1s: Enneamacro Decatonic

-Igs

P.S. I've been told by Paul that the term "MOS" excludes all scales that have non-octave periods/repeat intervals; I haven't seen that exclusion honored here in discussions, but I think I'm going to stick with the term "DE". I think the term "distributionally even" is slightly more intuitive, anyway.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

The names tell you exactly what the scales are.

Actually, they don't, and I would suggest at least adding "proper" or "improper" somehow.

Less efficient to type than the xL+ys format, but easier to say and more "official" sounding.
>
> Pentatonics:
> 1L+4s: Monomacro Pentatonic

Monomacro Proper Pentatonic
Monomacro Improper Pentatonic?

> P.S. I've been told by Paul that the term "MOS" excludes all scales that have non-octave periods/repeat intervals...

Paul is a stickler but sometimes he gos too far.

> I haven't seen that exclusion honored here in discussions, but I think I'm going to stick with the term "DE". I think the term "distributionally even" is slightly more intuitive, anyway.

It's what's used in academia, if that's a good thing, and there has been a significant shift in that direction here.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
>
> The names tell you exactly what the scales are.
>
> Actually, they don't, and I would suggest at least adding "proper" or "improper" somehow.

Well, propriety is another level of taxonomy all together. I'm not trying to encode *everything*. The pentatonic/hexatonic/heptatonic/etc. part is sort of the "kingdom" (animalia, plantae, etc.), the monomacro/dimacro/etc. part is the "phylum". You can always break it down further, into proper/improper, and then into particular degrees of propriety/impropriety. However, I don't feel that just specifying whether a scale is proper or improper is helpful enough to be worth the extra word. Assuming reasonable bounds of properiety/impropriety, a melody written in a proper scale of a given configuration will work the same when played in an improper version. Compare the 26-EDO and 27-EDO 5L+2s scales, for instance. If I was going to include anything about propriety in the name, it would probably specify the exact ratio of L:s, maybe along the lines of "three-to-two pentamacro heptatonic", but that would uniquely identify the scale and would not serve as a category.

-Igs

On Sat, Jan 29, 2011 at 4:34 PM, cityoftheasleep
<igliashon@...> wrote:
>
> It recently occurred to me that DE scales can be indexed by a combination of their number of large steps (or small, but I picked large) and their number of total notes. So forget the nonsensical "7 dwarf" scheme of many years ago, here's one that makes sense. The names tell you exactly what the scales are. Less efficient to type than the xL+ys format, but easier to say and more "official" sounding.

So many syllables! But good work. I still want to work out the "least
badness temperament" for each MOS though.

> P.S. I've been told by Paul that the term "MOS" excludes all scales that have non-octave periods/repeat intervals; I haven't seen that exclusion honored here in discussions, but I think I'm going to stick with the term "DE". I think the term "distributionally even" is slightly more intuitive, anyway.

He told that to me too, and I think at one point that was a huge
discussion that took place on here and Kraig was involved or
something, and there was Erv Wilson's definition, and stuff, I forget.
Either way I think MOS should probably refer to all of them. Is DE
just another way to say MOS?

-Mike

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

Assuming reasonable bounds of properiety/impropriety, a melody written in a proper scale of a given configuration will work the same when played in an improper version. Compare the 26-EDO and 27-EDO 5L+2s scales, for instance.

You can translate the melody, yes. There are lots of other things you could translate the melody to also. The same? It may not be apples and oranges, but at least oranges and tangerines.

On Sat, Jan 29, 2011 at 8:35 PM, genewardsmith
<genewardsmith@...> wrote:
>
> Assuming reasonable bounds of properiety/impropriety, a melody written in a proper scale of a given configuration will work the same when played in an improper version. Compare the 26-EDO and 27-EDO 5L+2s scales, for instance.
>
> You can translate the melody, yes. There are lots of other things you could translate the melody to also. The same? It may not be apples and oranges, but at least oranges and tangerines.

Gene, have you ever considered what I was proposing above? To find an
algorithm that systematically assigns regular temperaments to MOS's? I
understand that you're the guy who figured out that PB's and MOS's
were the same in the first place, so maybe you'd have a better idea of
how to do that than I do.

So for example, in an idealized heirarchy that this black box
algorithm would spit out, 5+2 is meantone, superpyth, pythagorean,
schismatic, etc. But then, to subdivide it further, 7+5 is meantone,
but 5+7 is superpyth, pythagorean and schismatic, and then you can
subdivide 5+7 further, etc.

7+5 can itself be subdivided into the temperaments with generators
between 26-tet and 19-tet, and those between 19-tet and 12-tet, which
are all assigned to meantone and various higher-limit extensions,
but...

If the generators are south of 19-tet, that means some small interval
will be reversed (I think the magic comma, right?). So if you're
willing to further categorize the regular temperament space into not
only what intervals are tempered out as the temperament "family," but
also associate a corresponding "reversed" interval as the precise
"shade" of the temperament family, it then becomes possible to further
subdivide "meantone" into other "shades" of meantone, and hence makes
it easier to associate with MOS's.

Thoughts?

-Mike

On Sat, Jan 29, 2011 at 8:47 PM, Mike Battaglia <battaglia01@...> wrote:
> So if you're willing to further categorize the regular temperament space into not
> only what intervals are tempered out as the temperament "family," but
> also associate a corresponding "reversed" interval as the precise
> "shade" of the temperament family, it then becomes possible to further
> subdivide "meantone" into other "shades" of meantone, and hence makes
> it easier to associate with MOS's.

Actually, I wonder, what would happen if you classified temperaments,
in general, by intervals that they reverse, rather than temper out?
Would the regular temperaments form a subset of this space, whenever a
combination of intervals tends to be reversed such that an interval
actually gets tempered out?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Jan 29, 2011 at 8:35 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > Assuming reasonable bounds of properiety/impropriety, a melody written in a proper scale of a given configuration will work the same when played in an improper version. Compare the 26-EDO and 27-EDO 5L+2s scales, for instance.
> >
> > You can translate the melody, yes. There are lots of other things you could translate the melody to also. The same? It may not be apples and oranges, but at least oranges and tangerines.
>
> Gene, have you ever considered what I was proposing above? To find an
> algorithm that systematically assigns regular temperaments to MOS's? I
> understand that you're the guy who figured out that PB's and MOS's
> were the same in the first place, so maybe you'd have a better idea of
> how to do that than I do.
>
> So for example, in an idealized heirarchy that this black box
> algorithm would spit out, 5+2 is meantone, superpyth, pythagorean,
> schismatic, etc. But then, to subdivide it further, 7+5 is meantone,
> but 5+7 is superpyth, pythagorean and schismatic, and then you can
> subdivide 5+7 further, etc.
>
> 7+5 can itself be subdivided into the temperaments with generators
> between 26-tet and 19-tet, and those between 19-tet and 12-tet, which
> are all assigned to meantone and various higher-limit extensions,
> but...
>
> If the generators are south of 19-tet, that means some small interval
> will be reversed (I think the magic comma, right?). So if you're
> willing to further categorize the regular temperament space into not
> only what intervals are tempered out as the temperament "family," but
> also associate a corresponding "reversed" interval as the precise
> "shade" of the temperament family, it then becomes possible to further
> subdivide "meantone" into other "shades" of meantone, and hence makes
> it easier to associate with MOS's.
>
> Thoughts?
>
> -Mike
>

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

> Gene, have you ever considered what I was proposing above? To find an
> algorithm that systematically assigns regular temperaments to MOS's?

I've done that from time to time. To code it up formally, you'd need a more refined classification than what Igs proposed, which is another way of stating my objection to it. Putting the MOS inside an edo makes it easy.

I
> understand that you're the guy who figured out that PB's and MOS's
> were the same in the first place, so maybe you'd have a better idea of
> how to do that than I do.

That was Paul Erlich's famous Hypothesis if I understand you correctly.

> So for example, in an idealized heirarchy that this black box
> algorithm would spit out, 5+2 is meantone, superpyth, pythagorean,
> schismatic, etc. But then, to subdivide it further, 7+5 is meantone,
> but 5+7 is superpyth, pythagorean and schismatic, and then you can
> subdivide 5+7 further, etc.

In what form should these be spat out?

On Sat, Jan 29, 2011 at 8:57 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Gene, have you ever considered what I was proposing above? To find an
> > algorithm that systematically assigns regular temperaments to MOS's?
>
> I've done that from time to time. To code it up formally, you'd need a more refined classification than what Igs proposed, which is another way of stating my objection to it. Putting the MOS inside an edo makes it easy.

What do you mean putting the MOS inside an edo makes it easy?

> That was Paul Erlich's famous Hypothesis if I understand you correctly.

Right.

> > So for example, in an idealized heirarchy that this black box
> > algorithm would spit out, 5+2 is meantone, superpyth, pythagorean,
> > schismatic, etc. But then, to subdivide it further, 7+5 is meantone,
> > but 5+7 is superpyth, pythagorean and schismatic, and then you can
> > subdivide 5+7 further, etc.
>
> In what form should these be spat out?

I'm not sure. What do you mean what form? Also, what is your take on
the reversed interval thing? It seems useful to me to distinguish
between meantones where the magic comma is reversed vs those that
aren't.

-Mike

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

> Actually, I wonder, what would happen if you classified temperaments,
> in general, by intervals that they reverse, rather than temper out?

OK, septimal meantone reverses (using POTE tuning or anything reasonable) 1029/1024, 19683/19600, 32805/32768, 2401/2400, 250047/250000, etc. What does that tell you?

On Sat, Jan 29, 2011 at 9:07 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Actually, I wonder, what would happen if you classified temperaments,
> > in general, by intervals that they reverse, rather than temper out?
>
> OK, septimal meantone reverses (using POTE tuning or anything reasonable) 1029/1024, 19683/19600, 32805/32768, 2401/2400, 250047/250000, etc. What does that tell you?

The full series should tell you what generator you're using. If you
figure out how many generators it takes to hit each of the intervals
that are separated by a reversed comma, and you know that the comma is
reversed, then you can keep placing further bounds on the generator
until you have an exact match.

Specifying only some intervals being reversed gives you a range of
generators, and hence there's a kind of correspondence between what
intervals are reversed and what MOS's the generator yields (which is
obvious). Since when we talk about MOS's, we're already just talking
about generator ranges, so why not just expand this out into the
above?

Can we do this with 5-limit meantone, rather than 7-limit for now? I
still have just a rough handle on the concept and this is sort of
confusing. What does the 5-limit meantone approximation yield?

-Mike

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

> What do you mean putting the MOS inside an edo makes it easy?

Picking a particular rational MOS generator. Here's what I sometimes do: say I have a generator such as 18/31. I pick a prime limit, say 7.
The patent val for 31 is <31 49 72 87|. Dividing by 18 and reducing mod 31 symmetrically gives <31 49 72 87|/18 mod 31 = <0 1 4 10|. Now put that together with <31 49 72 87| and saturate the result. For saturation, see:

http://xenharmonic.wikispaces.com/Saturation

The algorithm described there gives [<31 49 72 87|, <-19 -30 -44 -53|], the normal val list for that is [<1 0 -4 -13|, <0 1 4 10|], and the wedgie from either is <<1 4 10 4 13 12||. So we get septimal meantone from 18/31, which is not a huge surprise.

> I'm not sure. What do you mean what form? Also, what is your take on
> the reversed interval thing? It seems useful to me to distinguish
> between meantones where the magic comma is reversed vs those that
> aren't.

Which is to say between where the fifth is sharper than 11/19 and where it's flatter. Why does bringing in 3125/3072 improve matters?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > Actually, I wonder, what would happen if you classified temperaments,
> > in general, by intervals that they reverse, rather than temper out?
>
> OK, septimal meantone reverses (using POTE tuning or anything reasonable) 1029/1024, 19683/19600, 32805/32768, 2401/2400, 250047/250000, etc. What does that tell you?

"Anything reasonable" is wrong; it's damned sensitive to changes in tuning. Adding 1029/1024 or 2401/2400 gives 31et, 19683/19600 19et, 32805/32768 or 250047/250000 12et.

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

> Can we do this with 5-limit meantone, rather than 7-limit for now? I
> still have just a rough handle on the concept and this is sort of
> confusing. What does the 5-limit meantone approximation yield?

Adding 15625/15552 or 3125/3072 to 81/80 gives 19et; adding 128/125 or 648/625 gives 12et; adding 393216/390625 gives 31et; adding 78125/73728 gives 26et; adding 50331548/48828125 gives 43et.

Mike wrote:

> Either way I think MOS should probably refer to all of them. Is DE
> just another way to say MOS?

There are many terms for this type of scale with arcane
differences between them. Common usage will never support
more than one of them. The one we should use should be the
one that gives precedence to Wilson, since he seems to be
the first to notice the full extent of the phenomenon.
That term is "MOS".

-Carl

On Sat, Jan 29, 2011 at 9:39 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What do you mean putting the MOS inside an edo makes it easy?
>
> Picking a particular rational MOS generator. Here's what I sometimes do: say I have a generator such as 18/31. I pick a prime limit, say 7.
> The patent val for 31 is <31 49 72 87|. Dividing by 18 and reducing mod 31 symmetrically gives <31 49 72 87|/18 mod 31 = <0 1 4 10|. Now put that together with <31 49 72 87| and saturate the result. For saturation, see:
>
> http://xenharmonic.wikispaces.com/Saturation
>
> The algorithm described there gives [<31 49 72 87|, <-19 -30 -44 -53|], the normal val list for that is [<1 0 -4 -13|, <0 1 4 10|], and the wedgie from either is <<1 4 10 4 13 12||. So we get septimal meantone from 18/31, which is not a huge surprise.

OK, I understand what's going on. So you find the minimal subgroup of
which 18/31 is a generator and you get septimal meantone then, because
5-limit meantone would yield 75/64, which means a larger lattice than
7/6?

> > I'm not sure. What do you mean what form? Also, what is your take on
> > the reversed interval thing? It seems useful to me to distinguish
> > between meantones where the magic comma is reversed vs those that
> > aren't.
>
> Which is to say between where the fifth is sharper than 11/19 and where it's flatter. Why does bringing in 3125/3072 improve matters?

Because that's also the cutoff between the 12+7 MOS and the 7+12 MOS,
right? So if the goal is to assign temperaments to MOS's, this seems
like a good way to start exploring doing that. Or, I guess, this is
actually the opposite: figuring out what MOS emerges for a temperament
in which an associated interval is not reversed.

But I am interested in seeing what kind of group structure would
emerge if you started categorizing regular temperaments by their
ranges, and their "ranges" cam be determined by not only what
intervals a val reduces to 0, but which ones end up being positive and
which ones end up being negative.

-Mike

On Sat, Jan 29, 2011 at 9:51 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > Actually, I wonder, what would happen if you classified temperaments,
> > > in general, by intervals that they reverse, rather than temper out?
> >
> > OK, septimal meantone reverses (using POTE tuning or anything reasonable) 1029/1024, 19683/19600, 32805/32768, 2401/2400, 250047/250000, etc. What does that tell you?
>
> "Anything reasonable" is wrong; it's damned sensitive to changes in tuning. Adding 1029/1024 or 2401/2400 gives 31et, 19683/19600 19et, 32805/32768 or 250047/250000 12et.

Exactly, that's what I'm getting at. Each reversed interval puts a
greater bound on the generator. We can specify a "fuzzy" generator by
talking about what intervals are 0. We can successively place tighter
bounds on the generator range by placing additional restrictions on
what intervals end up being positive and which end up being negative
(e.g., reversed). This is the whole point of MOS, so it makes sense to
tie them in together.

If you used POTE to generate that series of reversed intervals, it
should be possible to somehow work backwards from it and get the
generator and period that you chose to use, I think.

-Mike

On Sat, Jan 29, 2011 at 9:59 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Can we do this with 5-limit meantone, rather than 7-limit for now? I
> > still have just a rough handle on the concept and this is sort of
> > confusing. What does the 5-limit meantone approximation yield?
>
> Adding 15625/15552 or 3125/3072 to 81/80 gives 19et; adding 128/125 or 648/625 gives 12et; adding 393216/390625 gives 31et; adding 78125/73728 gives 26et; adding 50331548/48828125 gives 43et.

I can see you're already miles ahead of me here. I'm suggesting that
there are benefits to not only plugging in what numbers turn into 0,
but seeing which ones are positive and negative.

Perhaps the way to go is to look at the simplest intervals that can be
reversed and still yield useful information. So, for example, if 81/80
vanishes:

If 16/15 is
reversed: you end up with a 3+2 MOS
tempered out: you end up with 5-equal
not reversed: you end up with a 2+3 MOS

If 16/15 is not reversed, and 25/24 is
reversed: you end up with a 2+5 MOS.
tempered out: you end up with 7-tet.
not reversed: you end up with a 5+2 MOS.

If 16/15 and 25/24 are not reversed, but 128/125 is
reversed: you end up with a 5+7 MOS
tempered out: you end up with 12-equal
not reversed: you end up with a 7+5 MOS

If 16/15, 25/24, and 128/125 are not reversed, but 3125/3072 is
reversed: you end up with a 7+12 MOS
tempered out: you end up with 19-equal
not reversed: you end up with a 12+7 MOS

Do you see what I'm getting at now? Is there something special about
the sequence 16/15, 25/24, 128/125, and 3125/3072 in the context of
meantone?

-Mike

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

> OK, I understand what's going on. So you find the minimal subgroup of
> which 18/31 is a generator and you get septimal meantone then, because
> 5-limit meantone would yield 75/64, which means a larger lattice than
> 7/6?

You lost me. We can do exactly the same thing in the 5-limit and get 5-limit meantone. Do it in the 11-limit and we get meanpop, one of the two 11-limit versions of meantone. Do it in the 13-limit and we get [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|], tempering out 81/80, 105/104,144/143, 196/195; the 13-limit extension of meanpop for which 31, 50 or 81 make good tunings.

> But I am interested in seeing what kind of group structure would
> emerge if you started categorizing regular temperaments by their
> ranges, and their "ranges" cam be determined by not only what
> intervals a val reduces to 0, but which ones end up being positive and
> which ones end up being negative.

That might be useful in that it is still valid for higher rank temperaments, whereas just using the boundry value of MOS generators isn't. But I don't see that it has anything to do with group structure.

On Sat, Jan 29, 2011 at 11:22 PM, genewardsmith
<genewardsmith@...t> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > OK, I understand what's going on. So you find the minimal subgroup of
> > which 18/31 is a generator and you get septimal meantone then, because
> > 5-limit meantone would yield 75/64, which means a larger lattice than
> > 7/6?
>
> You lost me. We can do exactly the same thing in the 5-limit and get 5-limit meantone. Do it in the 11-limit and we get meanpop, one of the two 11-limit versions of meantone. Do it in the 13-limit and we get [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|], tempering out 81/80, 105/104,144/143, 196/195; the 13-limit extension of meanpop for which 31, 50 or 81 make good tunings.

I see what's going on. I meant that in the 7-limit, you end up getting
that 225/224 vanishes because if not, then 75/64 would end up being
its own thing, whereas if 225/224 vanishes, it would be equated with
7/6. And everything is much simpler if 225/224 vanishes, and hence the
lattice is smaller. Or am I on the wrong track?

> > But I am interested in seeing what kind of group structure would
> > emerge if you started categorizing regular temperaments by their
> > ranges, and their "ranges" cam be determined by not only what
> > intervals a val reduces to 0, but which ones end up being positive and
> > which ones end up being negative.
>
> That might be useful in that it is still valid for higher rank temperaments, whereas just using the boundry value of MOS generators isn't. But I don't see that it has anything to do with group structure.

Is there no way to form an organizational structure around what I'm
suggesting? I thought that some kind of extension of the free abelian
group that you're using for regular mapping would be good, but perhaps
it's more like a fuzzy group or something?

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Mike wrote:
>
> > Either way I think MOS should probably refer to all of them. Is DE
> > just another way to say MOS?
>
> There are many terms for this type of scale with arcane
> differences between them. Common usage will never support
> more than one of them. The one we should use should be the
> one that gives precedence to Wilson, since he seems to be
> the first to notice the full extent of the phenomenon.
> That term is "MOS".

But didn't MOS exclude scales with non-octave periods in Erv's usage? That's what Paul told me, so he uses DE (or Well-Formed, too) because it describes the same thing but without the "octave-repeating only" restriction. Is he mistaken?

-Igs

On 30 January 2011 11:11, cityoftheasleep <igliashon@...> wrote:

> But didn't MOS exclude scales with non-octave periods in Erv's usage?  That's what Paul told me, so he uses DE (or Well-Formed, too) because it describes the same thing but without the "octave-repeating only" restriction.  Is he mistaken?

Erv's usage isn't always clear and, from what I've heard, he isn't
much interested in clarifying it. The message that's come through to
us is that the period doesn't have to be an octave, and he's looked at
cases where it isn't. But he does think of the period and generator
as both being consonances. So he probably wouldn't consider a DE
scale that divides the octave into equal parts for convenience or
temperament mapping as being MOS. But all we know is that he doesn't
seem to have considered them at all.

The definition of "distributional evenness" was, I believe, always
framed precisely enough that we can be sure it does include rank 2
tunings with a period that divides the octave. As such, it was the
first definition precisely given with that meaning. I don't think
"well formed" is any clearer than "MOS" in this respect.

Graham

On 30 January 2011 05:47, Mike Battaglia <battaglia01@...> wrote:

> So for example, in an idealized heirarchy that this black box
> algorithm would spit out, 5+2 is meantone, superpyth, pythagorean,
> schismatic, etc. But then, to subdivide it further, 7+5 is meantone,
> but 5+7 is superpyth, pythagorean and schismatic, and then you can
> subdivide 5+7 further, etc.

That sounds like the scale tree. Associating an MOS with a
temperament is what my website does. The rule is to take the best
possible mapping (the one with the lowest error) of each equal
temperament. Because of the geometric properties of TE error and
Cangwu badness, this is very likely to give you the best applicable
for some definition of badness.

Going further, and finding the best applicable rank 2 temperament
according to a particular definition of badness will mean looking at
more equal temperaments.

In Canwu badness space, you can define an angle between two equal
temperaments. The closer to 90 degrees, the better the badness of the
rank 2 temperament corresponds to that of the two equal temperaments.
I wonder if this same angle has anything to do with propriety . . .

> If the generators are south of 19-tet, that means some small interval
> will be reversed (I think the magic comma, right?). So if you're
> willing to further categorize the regular temperament space into not
> only what intervals are tempered out as the temperament "family," but
> also associate a corresponding "reversed" interval as the precise
> "shade" of the temperament family, it then becomes possible to further
> subdivide "meantone" into other "shades" of meantone, and hence makes
> it easier to associate with MOS's.

That's been done. Try the Wikipedia page on Syntonic Temperament.

At the point where an interval reverses, it's a unison vector. So you
can easily find the boundaries.

Graham

Igs wrote:

> > There are many terms for this type of scale with arcane
> > differences between them. Common usage will never support
> > more than one of them. The one we should use should be the
> > one that gives precedence to Wilson, since he seems to be
> > the first to notice the full extent of the phenomenon.
> > That term is "MOS".
>
> But didn't MOS exclude scales with non-octave periods in Erv's
> usage?

That's not clear but even if it were, MOS would still be the
right term for the reason mentioned.

> That's what Paul told me, so he uses DE (or Well-Formed, too)
> because it describes the same thing but without the "octave-
> repeating only" restriction. Is he mistaken?

Probably. He asked Kraig, who gave different answers at
different times. Paul's going on the most recent answer.
Wilson doesn't seem to care enough to say one way or the other.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I still want to work out the "least
> badness temperament" for each MOS though.

What does that mean, though? How do you define "each MOS"? At any level of the scale tree, there will be any number of temperaments between any two nodes. And "badness" is relative to your target harmonies, too. Most MOS scales are horrendous as typical 5-limit or 7-limit temperaments, no matter where along the generator range you are. What I've found is that there are multiple subgroup sweet spots along the generator range for a given MOS scale, which target different primes (and/or odds). So I really don't think it's sensible to look for a "least badness temperament" for any single MOS pattern, unless it's a really big scale (like 21 notes or more). What I'd really love is a program where you can plug in a generator and a period and it will give you the mapping of all the ratios below a certain Tenney Height.

-Igs

On Sun, Jan 30, 2011 at 4:15 AM, Graham Breed <gbreed@...> wrote:
>
> On 30 January 2011 05:47, Mike Battaglia <battaglia01@...> wrote:
>
> > So for example, in an idealized heirarchy that this black box
> > algorithm would spit out, 5+2 is meantone, superpyth, pythagorean,
> > schismatic, etc. But then, to subdivide it further, 7+5 is meantone,
> > but 5+7 is superpyth, pythagorean and schismatic, and then you can
> > subdivide 5+7 further, etc.
>
> That sounds like the scale tree.

Right, and the flip between 5+7 and 7+5 can be represented in regular
temperament parlance by saying that we're working with a meantone
where 128/125 is either reversed or not reversed... yes?

> Associating an MOS with a temperament is what my website does. The rule is to take the best
> possible mapping (the one with the lowest error) of each equal
> temperament. Because of the geometric properties of TE error and
> Cangwu badness, this is very likely to give you the best applicable
> for some definition of badness.

You mean that when we run a rank-2 temperament search, and it spits
out something like "7&12," that it's performing this kind of analysis
in an abstract kind of way?

> Going further, and finding the best applicable rank 2 temperament
> according to a particular definition of badness will mean looking at
> more equal temperaments.

I'm a little confused - is there a way to figure out what the best
temperament for 1L6s is? I would hope it's 11-limit porcupine, but
does some facet of your temperament finder do this?

> In Canwu badness space, you can define an angle between two equal
> temperaments. The closer to 90 degrees, the better the badness of the
> rank 2 temperament corresponds to that of the two equal temperaments.
> I wonder if this same angle has anything to do with propriety . . .

I'll have to go back through Cangwu badness again, that was before I
really started to understand regular mapping. The tuning-math page
where you first introduced it was pretty complex.

It sounds like Cangwu badness is the converse of a similar idea I had
a while ago: that of "compatibility" between two rank-2 temperaments.
So porcupine and meantone are pretty incompatible, because the badness
of the resulting temperament when you combine them is a lot worse than
the temperaments themselves. Maybe this makes more sense to do with
error, rather than badness.

Diminished and augmented, on the other hand, are really compatible,
and meantone and magic are pretty compatible, but mavila and porcupine
aren't really compatible, etc. Meantone and father are horrid
together, but blackwood and porcupine do pretty well, as do srutal and
diminished.

Once you do this, you can figure out a temperament's "affinity" for
other rank-2 temperaments by figuring out its badness-weighted average
compatibility with other temperaments. This might be easily
generalized so that you can combine rank-2 and rank-3 temperaments and
such.

> That's been done. Try the Wikipedia page on Syntonic Temperament.
>
> At the point where an interval reverses, it's a unison vector. So you
> can easily find the boundaries.

Right, I've seen the syntonic temperament page before. I guess what
I'm trying to get at is that you can manually specify generator ranges
for some tuning, like we can call the meantones between 26-tet and
19-tet "infra-meantones" as Margo suggested, for example - but you can
also specify the same exact thing by specifying what intervals have to
end up positive or negative in sign, cents-wise.

There's a homomorphism here, and I think that exploring the second way
more fully might lead to some interesting new forms of analysis. For
example:

Let's call the name of a temperament where the sign of a certain
additional comma is specified to be a "shade" of that temperament. A
badness measure could be derived for each shade of a certain
temperament, because if badness is a function of error and complexity,
the error will change as the generator gets bounded into different
ranges. Once that's done, you can figure out what the lowest-badness
temperament is for any MOS.

So perhaps the magic-reversed shade of meantone ends up winning out
for 7+12, but mavila without a certain shade being specified beats out
the 25/24-reversed shade of meantone for 2+5.

Sounds like you've already worked through a lot of this?

-Mike

On Sun, Jan 30, 2011 at 4:49 AM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > I still want to work out the "least
> > badness temperament" for each MOS though.
>
> What does that mean, though? How do you define "each MOS"? At any level of the scale tree, there will be any number of temperaments between any two nodes.

What do you mean any number of temperaments between any two nodes? Do
you mean any number of sub-MOS's between any two nodes?

If so, I was saying that I'd like to somehow work out what the optimal
tuning is for 5+2 across the entire 5+2 spectrum, by doing something
like figuring out the average error for each temperament across the
spectrum and multiplying it by complexity. Or maybe doing the minimax
error for each temperament across the spectrum and multiplying it by
complexity. Or maybe doing the minimum error for each temperament
across the spectrum and multiplying it by complexity, which is, I
guess, normal badness. Or by doing something new with the reversed
intervals approach that I mentioned, and seeing if that yields
anything useful. There should be some way to do it.

> And "badness" is relative to your target harmonies, too.

I guess to do a full search of this, you'd have to also map things
against different subgroup temperaments as well. Maybe 5+4 would be
found to map best to the 676/675 2.3.13/10 subgroup temperament.

> Most MOS scales are horrendous as typical 5-limit or 7-limit temperaments, no matter where along the generator range you are. What I've found is that there are multiple subgroup sweet spots along the generator range for a given MOS scale, which target different primes (and/or odds). So I really don't think it's sensible to look for a "least badness temperament" for any single MOS pattern, unless it's a really big scale (like 21 notes or more).

Maybe a list of the few least badness ones would be better, as well as
the magic generators that make them work. Gene and Carl were talking
about some kind of "zeta error" concept on tuning-math that would
specify a tuning's error without needing to specify a prime-limit, by
doing some magic with the zeta function. I don't understand it, but
maybe that would be more up your alley.

> What I'd really love is a program where you can plug in a generator and a period and it will give you the mapping of all the ratios below a certain Tenney Height.

I think this is basically the same thing that the saturation thing
that Gene just talked about earlier in this thread.

-Mike

On 30 January 2011 13:57, Mike Battaglia <battaglia01@...> wrote:
> On Sun, Jan 30, 2011 at 4:15 AM, Graham Breed <gbreed@...> wrote:

>> That sounds like the scale tree.
>
> Right, and the flip between 5+7 and 7+5 can be represented in regular
> temperament parlance by saying that we're working with a meantone
> where 128/125 is either reversed or not reversed... yes?

If that's how it works out, and you want do define "+" to be non-commutative.

> You mean that when we run a rank-2 temperament search, and it spits
> out something like "7&12," that it's performing this kind of analysis
> in an abstract kind of way?

If you start here:

http://x31eq.com/temper/net.html

then it finds a temperament according to the best mappings of the
equal temperaments you supplied. If you do a broader search, it finds
equal temperaments in such a way that the best R consistent ETs for a
rank-R temperament get shown. Being short vectors in a lattice,
they're also the nearest to orthogonal. Because they depend on your
choice of badness parameter, you don't always get the same numbers
coming out.

>> Going further, and finding the best applicable rank 2 temperament
>> according to a particular definition of badness will mean looking at
>> more equal temperaments.
>
> I'm a little confused - is there a way to figure out what the best
> temperament for 1L6s is? I would hope it's 11-limit porcupine, but
> does some facet of your temperament finder do this?

1L6s maps to 1&7, right? You can plug that in, and you don't get a
sensible result, because the best mapping for 1-equal is a fairly
arbitrary thing. You can plug in 7&8 and get Porcupine (in the
11-limit, you get Opossum). 6&7 is unknown, and therefore must not be
Porcupine.
> It sounds like Cangwu badness is the converse of a similar idea I had
> a while ago: that of "compatibility" between two rank-2 temperaments.
> So porcupine and meantone are pretty incompatible, because the badness
> of the resulting temperament when you combine them is a lot worse than
> the temperaments themselves. Maybe this makes more sense to do with
> error, rather than badness.

That sounds like "straightness". It's a concept that was floated
qualitatively on tuning-math. Cangwu badness, being a genuine inner
product space, makes it exact and ties it to an angle.

> Diminished and augmented, on the other hand, are really compatible,
> and meantone and magic are pretty compatible, but mavila and porcupine
> aren't really compatible, etc. Meantone and father are horrid
> together, but blackwood and porcupine do pretty well, as do srutal and
> diminished.

That would amount to the angle, or maybe distance, between two planes.
You can follow it up on tuning-math.

> Once you do this, you can figure out a temperament's "affinity" for
> other rank-2 temperaments by figuring out its badness-weighted average
> compatibility with other temperaments. This might be easily
> generalized so that you can combine rank-2 and rank-3 temperaments and
> such.

Angles are, indeed, defined between different ranks.

> Let's call the name of a temperament where the sign of a certain
> additional comma is specified to be a "shade" of that temperament. A
> badness measure could be derived for each shade of a certain
> temperament, because if badness is a function of error and complexity,
> the error will change as the generator gets bounded into different
> ranges. Once that's done, you can figure out what the lowest-badness
> temperament is for any MOS.

You know that error goes up as an interval becomes negative, because
its error exceeds its size.

> Sounds like you've already worked through a lot of this?

We've done all kinds of things before. It's been a long time.

Graham

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

> I'll have to go back through Cangwu badness again, that was before I
> really started to understand regular mapping. The tuning-math page
> where you first introduced it was pretty complex.

Yes, and I'm reluctant to write a Xenwiki article on it until I know what things like "Cangwu badness space" are. It would be nice to have an article number for the most recent pretty complex posting; I was busy with UnTwelve at the time.

On Sun, Jan 30, 2011 at 6:00 AM, Graham Breed <gbreed@...> wrote:
>
> > Right, and the flip between 5+7 and 7+5 can be represented in regular
> > temperament parlance by saying that we're working with a meantone
> > where 128/125 is either reversed or not reversed... yes?
>
> If that's how it works out, and you want do define "+" to be non-commutative.

I guess I mean 5L7s vs 7L5s. But yes, that's how it works out. And the
difference between 7+12 and 12+7 has to do with whether the magic
comma is reversed or not, respectively, etc.

> If you start here:
>
> then it finds a temperament according to the best mappings of the
> equal temperaments you supplied. If you do a broader search, it finds
> equal temperaments in such a way that the best R consistent ETs for a
> rank-R temperament get shown. Being short vectors in a lattice,
> they're also the nearest to orthogonal. Because they depend on your
> choice of badness parameter, you don't always get the same numbers
> coming out.

So how can I use this to assign temperaments to MOS's...? Let's say I
want to find the best 5-limit temperament for 2L5s, which I would
assume is mavila. I put 2, 5 in the equal temperament search and I get
father temperament...? Am I using it wrong?

Actually, after reading below:

> 1L6s maps to 1&7, right?

I didn't realize that. Why do you say that? I would have naively
guessed 1&6...? I think I'm missing something here.

> You can plug that in, and you don't get a
> sensible result, because the best mapping for 1-equal is a fairly
> arbitrary thing. You can plug in 7&8 and get Porcupine (in the
> 11-limit, you get Opossum). 6&7 is unknown, and therefore must not be
> Porcupine.

I see you're doing something with mediants here, but how did you jump
from 1L6s to 1&7? Are you going by Large steps & Total steps?

> > It sounds like Cangwu badness is the converse of a similar idea I had
> > a while ago: that of "compatibility" between two rank-2 temperaments.
> > So porcupine and meantone are pretty incompatible, because the badness
> > of the resulting temperament when you combine them is a lot worse than
> > the temperaments themselves. Maybe this makes more sense to do with
> > error, rather than badness.
>
> That sounds like "straightness". It's a concept that was floated
> qualitatively on tuning-math. Cangwu badness, being a genuine inner
> product space, makes it exact and ties it to an angle.

I have to read Cangwu badness then. Always nice when random ideas I've
had have already been fleshed out in their entirety :)

> > Let's call the name of a temperament where the sign of a certain
> > additional comma is specified to be a "shade" of that temperament. A
> > badness measure could be derived for each shade of a certain
> > temperament, because if badness is a function of error and complexity,
> > the error will change as the generator gets bounded into different
> > ranges. Once that's done, you can figure out what the lowest-badness
> > temperament is for any MOS.
>
> You know that error goes up as an interval becomes negative, because
> its error exceeds its size.

I guess you'd have to be measuring the general 5-limit error for the
generators that correspond to whatever shade we're talking about. So
if we're working with the [26-tet, 19-tet] shade of meantone, which is
really flat, you'd have to figure out some error metric for that whole
generator range and a just 4:5:6. Maybe the way to go at that point is
to integrate over the whole range and divide by the width and get the
mean error, but I'm sure there's probably a more elegant solution.

-Mike

On 30 January 2011 15:26, Mike Battaglia <battaglia01@...> wrote:
> On Sun, Jan 30, 2011 at 6:00 AM, Graham Breed <gbreed@...> wrote:

> So how can I use this to assign temperaments to MOS's...? Let's say I
> want to find the best 5-limit temperament for 2L5s, which I would
> assume is mavila. I put 2, 5 in the equal temperament search and I get
> father temperament...? Am I using it wrong?

Why do you think it shouldn't be Father? 2&7 gives Mavila.

> Actually, after reading below:
>
>> 1L6s maps to 1&7, right?
>
> I didn't realize that. Why do you say that? I would have naively
> guessed 1&6...? I think I'm missing something here.

As you're defining L and s asymmetrically, there's some information
lost if you switch to 1&6. But you can move down a step on the scale
tree. 1L6s has to be 1&7, and 6L1s is 6&7. You always keep the
number on the "L" constant. The large intervals get divided to give
more smaller intervals.

> I have to read Cangwu badness then. Always nice when random ideas I've
> had have already been fleshed out in their entirety :)

http://x31eq.com/badness.pdf

It's a common or garden inner product space. If you don't know what
that is, read a standard text on linear algebra.

Graham

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

> So how can I use this to assign temperaments to MOS's...? Let's say I
> want to find the best 5-limit temperament for 2L5s, which I would
> assume is mavila. I put 2, 5 in the equal temperament search and I get
> father temperament...? Am I using it wrong?

If L=s, then you get 2+5=7 steps. If L=2s, you get 4+5=9 steps. If L=1.5 s, you get 3*2+2*5=16 steps. Putting 7 and 9, 7 and 16 or 9 and 16 into Graham's machine always yields mavila, so you've found mavila.

> Actually, after reading below:
>
> > 1L6s maps to 1&7, right?

Same thing: 1+6=7, 2*1+6=8, 3*1+2*6=15. Putting 7 and 8, 7 and 15, or 8 and 15 into Graham's machine always yields porcupine.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> It's a common or garden inner product space. If you don't know what
> that is, read a standard text on linear algebra.

You keep saying that, but since Cangwu badness is defined over R(x) it seems it would need to be an inner product on R(x).

On Sun, Jan 30, 2011 at 6:40 AM, Graham Breed <gbreed@...> wrote:
>
> As you're defining L and s asymmetrically, there's some information
> lost if you switch to 1&6. But you can move down a step on the scale
> tree. 1L6s has to be 1&7, and 6L1s is 6&7. You always keep the
> number on the "L" constant. The large intervals get divided to give
> more smaller intervals.

OK, I see.

> > I have to read Cangwu badness then. Always nice when random ideas I've
> > had have already been fleshed out in their entirety :)
>
> http://x31eq.com/badness.pdf
>
> It's a common or garden inner product space. If you don't know what
> that is, read a standard text on linear algebra.

I guess I should really start here to get what you're doing in order?

http://x31eq.com/primerr.pdf

-Mike

On Sun, Jan 30, 2011 at 7:24 AM, genewardsmith
<genewardsmith@...> wrote:
>
> If L=s, then you get 2+5=7 steps. If L=2s, you get 4+5=9 steps. If L=1.5 s, you get 3*2+2*5=16 steps. Putting 7 and 9, 7 and 16 or 9 and 16 into Graham's machine always yields mavila, so you've found mavila.
>
> > Actually, after reading below:
> >
> > > 1L6s maps to 1&7, right?
>
> Same thing: 1+6=7, 2*1+6=8, 3*1+2*6=15. Putting 7 and 8, 7 and 15, or 8 and 15 into Graham's machine always yields porcupine.

OK, I see what's going on now. Thanks. So then, in light of this,
what's the point of giving the MOS scales special names at all? Why
not just something more like this?

2L5s mavila diatonic, or mavila heptatonic
5L2s meantone diatonic, or meantone heptatonic
1L6s porcupine diatonic, or porcupine heptatonic

etc? It would probably be useful to rigorously define what words like
"diatonic" and "chromatic" mean first, though, as well as talk about
scales that fall in between there like 10-note scales, which tend to
have both diatonic and chromatic properties.

Would you say that it's perhaps overly ambitious for the moment, but
at some point we might see a zeta function-based version of the above
that can come up with the best mappings for each MOS without respect
to prime limit?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Jan 30, 2011 at 7:24 AM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > If L=s, then you get 2+5=7 steps. If L=2s, you get 4+5=9 steps. If L=1.5 s, you get 3*2+2*5=16 steps. Putting 7 and 9, 7 and 16 or 9 and 16 into Graham's machine always yields mavila, so you've found mavila.
> >
> > > Actually, after reading below:
> > >
> > > > 1L6s maps to 1&7, right?
> >
> > Same thing: 1+6=7, 2*1+6=8, 3*1+2*6=15. Putting 7 and 8, 7 and 15, or 8 and 15 into Graham's machine always yields porcupine.
>
> OK, I see what's going on now. Thanks. So then, in light of this,
> what's the point of giving the MOS scales special names at all? Why
> not just something more like this?
>
> 2L5s mavila diatonic, or mavila heptatonic
> 5L2s meantone diatonic, or meantone heptatonic
> 1L6s porcupine diatonic, or porcupine heptatonic

Because as I was trying to tell Igs, you at least have to distinguish proper from improper. I looked at the endpoints and the midpoint of the proper range, fed it into Graham's black box, and got the desired result. However, for 2L5s if L=3s you get 11, L=4s 13, L=infinity s 2. 9 and 11 gives God knows what, 9 and 13, Orson, 11 and 13, something else again, and so forth, pretty much a mess.

> Would you say that it's perhaps overly ambitious for the moment, but
> at some point we might see a zeta function-based version of the above
> that can come up with the best mappings for each MOS without respect
> to prime limit?

I don't see how the zeta function could help.

On Sun, Jan 30, 2011 at 7:59 AM, genewardsmith
<genewardsmith@...> wrote:
>
> > OK, I see what's going on now. Thanks. So then, in light of this,
> > what's the point of giving the MOS scales special names at all? Why
> > not just something more like this?
> >
> > 2L5s mavila diatonic, or mavila heptatonic
> > 5L2s meantone diatonic, or meantone heptatonic
> > 1L6s porcupine diatonic, or porcupine heptatonic
>
> Because as I was trying to tell Igs, you at least have to distinguish proper from improper. I looked at the endpoints and the midpoint of the proper range, fed it into Graham's black box, and got the desired result. However, for 2L5s if L=3s you get 11, L=4s 13, L=infinity s 2. 9 and 11 gives God knows what, 9 and 13, Orson, 11 and 13, something else again, and so forth, pretty much a mess.

I guess if you wanted to give MOS's regular temperament names, the way
I envisioned it was something like - for 5L2s, you could figure out
which mappings predominate for the whole 5L2s spectrum, both improper
and proper and give the canonical version the one that has the most
"coverage," where coverage refers to the width of the domain in which
it is the clear victor divided by the min error for that temperament
in that domain or something like that. So for 2L5s you get bug and
mavila as contenders, depending on if it's proper or improper, right?
And then mavila wins out because it covers more of the spectrum with
less error or something like that.

But if you want to differentiate between proper and improper, that's
also the same thing as differentiating between 2L5s into 5L4s and
4L5s, right? So maybe it makes more sense to start lower on the scale
tree, come up with different temperaments for things like 16&13, and
then go one entry higher and at each level higher, give the name to
that MOS the name of the lower-badness temperament of the two entries
below. At least I think that makes sense.

> > Would you say that it's perhaps overly ambitious for the moment, but
> > at some point we might see a zeta function-based version of the above
> > that can come up with the best mappings for each MOS without respect
> > to prime limit?
>
> I don't see how the zeta function could help.

I thought you guys were talking about how it could remove the need to
specify prime limit when doing stuff like this?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> What do you mean any number of temperaments between any two nodes? Do
> you mean any number of sub-MOS's between any two nodes?

Yes. And also various limits that may be considered (3, 5, 7, 11, 13, subgroups, etc.).

> If so, I was saying that I'd like to somehow work out what the optimal
> tuning is for 5+2 across the entire 5+2 spectrum, by doing something
> like figuring out the average error for each temperament across the
> spectrum and multiplying it by complexity.

But optimal for what? How can you meaningfully compare, say, Meantone to Superpyth? Different temperaments do different things. Are you planning on comparing 5-limit temperaments to 7-limit ones? Are you planning on looking at 13-limit temperament or stopping lower? Or going higher, maybe 17-limit? Looking at all the potentially-useful subgroups? And what will the results really tell you, anyway?

Also, if you wanted to do this with respect to MOS patterns, you'd have to evaluate how a temperament performs *at the given number of notes in the MOS*. So for instance, if you were in 3L+4s land, two temperaments you might compare are Dicot and Magic. Magic might be a better temperament in the 5-limit than Dicot, but 7 notes of Magic are going to be worse than 7 notes of Dicot, because in Dicot[7] you at least have bunch of 3/2's to use, whereas in Magic[7] you can barely cobble together enough of the target harmonies to make music. If you just weight by error and complexity, you might not catch that, you've also got to weight for "amount of target harmonies achieved", which would be a function (I think?) of "MOS size" minus "complexity" or something.

> > And "badness" is relative to your target harmonies, too.
>
> I guess to do a full search of this, you'd have to also map things
> against different subgroup temperaments as well. Maybe 5+4 would be
> found to map best to the 676/675 2.3.13/10 subgroup temperament.

Maybe--but who knows? It'd be tricky enough just to compare 5, 7, 11, and 13-limit temperaments, if you start throwing subgroups into the mix, how many subgroups are there? Goodness knows, it might turn out that you'll get a huge number of "first place" matches over a given MOS range, each one corresponding to a different subgroup.

> Maybe a list of the few least badness ones would be better, as well as
> the magic generators that make them work. Gene and Carl were talking
> about some kind of "zeta error" concept on tuning-math that would
> specify a tuning's error without needing to specify a prime-limit, by
> doing some magic with the zeta function. I don't understand it, but
> maybe that would be more up your alley.

All I can say is, "good luck". I'm fairly certain that narrowing it down to a "few" is going to be difficult, arbitrary, and of questionable usefulness.

-Igs

On Sun, Jan 30, 2011 at 12:52 PM, cityoftheasleep
<igliashon@...> wrote:
>
> > If so, I was saying that I'd like to somehow work out what the optimal
> > tuning is for 5+2 across the entire 5+2 spectrum, by doing something
> > like figuring out the average error for each temperament across the
> > spectrum and multiplying it by complexity.
>
> But optimal for what?

It's just a way of assigning names to MOS's that I think makes sense.
Whatever naming convention you choose should probably reflect the fact
that something like 1L6s is pretty intrinsically related to
"porcupine," or so I think.

> How can you meaningfully compare, say, Meantone to Superpyth?

That's what the point of things like badness are.

> Different temperaments do different things. Are you planning on comparing 5-limit temperaments to 7-limit ones? Are you planning on looking at 13-limit temperament or stopping lower? Or going higher, maybe 17-limit? Looking at all the potentially-useful subgroups? And what will the results really tell you, anyway?

There are badness metrics that can compare higher rank temperaments to
lower rank ones. What will the results tell me? I'm proposing a naming
scheme for MOS's that can tie the whole thing into the temperament
names we already know.

> Also, if you wanted to do this with respect to MOS patterns, you'd have to evaluate how a temperament performs *at the given number of notes in the MOS*. So for instance, if you were in 3L+4s land, two temperaments you might compare are Dicot and Magic. Magic might be a better temperament in the 5-limit than Dicot, but 7 notes of Magic are going to be worse than 7 notes of Dicot, because in Dicot[7] you at least have bunch of 3/2's to use, whereas in Magic[7] you can barely cobble together enough of the target harmonies to make music. If you just weight by error and complexity, you might not catch that, you've also got to weight for "amount of target harmonies achieved", which would be a function (I think?) of "MOS size" minus "complexity" or something.

That's a good point, I guess there's Graham complexity for stuff like this.

> > I guess to do a full search of this, you'd have to also map things
> > against different subgroup temperaments as well. Maybe 5+4 would be
> > found to map best to the 676/675 2.3.13/10 subgroup temperament.
>
> Maybe--but who knows? It'd be tricky enough just to compare 5, 7, 11, and 13-limit temperaments, if you start throwing subgroups into the mix, how many subgroups are there? Goodness knows, it might turn out that you'll get a huge number of "first place" matches over a given MOS range, each one corresponding to a different subgroup.

I'm not sure that this is going to be a problem, now that I'm looking
into the Cangwu badness stuff.

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> You can translate the melody, yes. There are lots of other things you could translate the
> melody to also. The same? It may not be apples and oranges, but at least oranges and
> tangerines.
>
Well, of course. I mean, you can translate a melody between any two scales of the same number of notes and the universe will not implode, but having a "pattern of 2nds" in common preserves something significant about the melody that is lost otherwise. Also, the limit for "what values of L:s represents excessive impropriety or excessive equality" varies from scale to scale, so there's no rigorous way to define it. For a scale like 5L+2s, you can get L:s up to 4:1 or even 5:1--maybe even 6:1 on a good day--without losing the commonality with 2:1 or 3:2, but for a scale like 1L+7s, 4:1 doesn't sound ANYTHING like 3:2. And at any rate, the purpose for categorizing DE scales into groups with the same pattern of seconds is really to illustrate common melodic structures between EDOs irrespective of harmony. It's not meant to supplant the names of temperaments or anything, and it's really only useful assuming one is looking at EDOs below a certain size. I figure, anyone who's going to use an EDO above 41 is going to be more interested in meeting specific harmonic goals and less interested in specific melodic patterns, and such a person will be more concerned with the regular temperament paradigm than with the DE paradigm.

What I do like about the DE paradigm better than the regular temperament paradigm is that because harmony is incidental in the former, we can translate a piece of music to any tuning that preserves the pattern of 2nds without having to juggle commas and mappings and error boundaries.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> It's just a way of assigning names to MOS's that I think makes sense.
> Whatever naming convention you choose should probably reflect the fact
> that something like 1L6s is pretty intrinsically related to
> "porcupine," or so I think.

But in 1L+6s, we also find 23, 31, 38, and 39-EDO, of which none are Porcupine temperaments. What you are proposing is to take the taxonomy of the regular temperament paradigm, which groups scales according to shared harmonic properties, and apply it to the DE paradigm, which groups scales according to shared patterns of steps. This cannot possibly result in anything but confusion. The only way it could work is if you are naming MOS's of 11 notes or more, because that's about where scales that share melodic properties begin to share a significant proportion of harmonic properties. But at that size level, what's even the point of using DE properties to group scales? You're basically just talking about temperaments anyway.

> > How can you meaningfully compare, say, Meantone to Superpyth?
>
> That's what the point of things like badness are.

Comparing badness is only valid for temperaments with the same harmonic goals. If you compared Meantone and Superpyth in the 5-limit, Meantone will kick Superpyth's arse. But this obscures that fact that Superpyth is basically Meantone with 7-limit 6:7:9 and 1/(6:7:9) triads instead of 5-limit 4:5:6 and 1/(4:5:6) triads. So which one is "better"? It's an absurd question.

> There are badness metrics that can compare higher rank temperaments to
> lower rank ones. What will the results tell me? I'm proposing a naming
> scheme for MOS's that can tie the whole thing into the temperament
> names we already know.

Which, again, will only sow confusion. Under this proposal, if Magic turns out to be the best temperament in the 3L+4s range, then we'll have to start calling the 3L+4s scales in 10, 13, 16, 17, etc. "Magic heptatonics" which will completely obscure the fact that none of these scales has anything to do with Magic temperament and do not produce harmonies in the way you would expect Magic to do.

-Igs

On Sun, Jan 30, 2011 at 1:33 PM, cityoftheasleep
<igliashon@...> wrote:
>
> > It's just a way of assigning names to MOS's that I think makes sense.
> > Whatever naming convention you choose should probably reflect the fact
> > that something like 1L6s is pretty intrinsically related to
> > "porcupine," or so I think.
>
> But in 1L+6s, we also find 23, 31, 38, and 39-EDO, of which none are Porcupine temperaments. What you are proposing is to take the taxonomy of the regular temperament paradigm, which groups scales according to shared harmonic properties, and apply it to the DE paradigm, which groups scales according to shared patterns of steps. This cannot possibly result in anything but confusion.

Uh, I'm not sure why you say that, especially in light of the entire
discussion that just ensued. There are things that this could possibly
result in other than confusion. There's blissful happiness, for
starters, as well as regularity and order. And then there's mental
clarity, and ease of pedagogical explanation, and so on. There are
wonderful things that it could possibly result in, and since MOS's and
regular temperaments are already related, then there's not nearly as
much confusion as you seem to think it is. That even more so now that
it's just been showed that you can already use Graham's regular
temperament finder to assign temperaments to MOS's.

> The only way it could work is if you are naming MOS's of 11 notes or more, because that's about where scales that share melodic properties begin to share a significant proportion of harmonic properties. But at that size level, what's even the point of using DE properties to group scales? You're basically just talking about temperaments anyway.

That sounds like it has more to do with complexity, which is part of badness.

> > > How can you meaningfully compare, say, Meantone to Superpyth?
> >
> > That's what the point of things like badness are.
>
> Comparing badness is only valid for temperaments with the same harmonic goals. If you compared Meantone and Superpyth in the 5-limit, Meantone will kick Superpyth's arse. But this obscures that fact that Superpyth is basically Meantone with 7-limit 6:7:9 and 1/(6:7:9) triads instead of 5-limit 4:5:6 and 1/(4:5:6) triads. So which one is "better"? It's an absurd question.

Like I said, there are badness metrics that compare temperaments of
different ranks. You can compare bug temperament to superpyth and
figure out that superpyth is better for what it wants to do than bug
is at what it wants to do.

> > There are badness metrics that can compare higher rank temperaments to
> > lower rank ones. What will the results tell me? I'm proposing a naming
> > scheme for MOS's that can tie the whole thing into the temperament
> > names we already know.
>
> Which, again, will only sow confusion. Under this proposal, if Magic turns out to be the best temperament in the 3L+4s range, then we'll have to start calling the 3L+4s scales in 10, 13, 16, 17, etc. "Magic heptatonics" which will completely obscure the fact that none of these scales has anything to do with Magic temperament and do not produce harmonies in the way you would expect Magic to do.

OK, so that's a good point. Maybe it does make more sense if you
distinguish between the proper and improper versions, which seems to
yield more sensible results.

-Mike

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

> I guess if you wanted to give MOS's regular temperament names, the way
> I envisioned it was something like - for 5L2s, you could figure out
> which mappings predominate for the whole 5L2s spectrum, both improper
> and proper and give the canonical version the one that has the most
> "coverage," where coverage refers to the width of the domain in which
> it is the clear victor divided by the min error for that temperament
> in that domain or something like that.

I see you don't have your victory condition worked out, and another consideration is that more accurate temperaments take up less room on the scale of generators. For 5L2s, using my suggestion that you stick to proper and improper ranges, 5L2s proper is meantone. 5L2s improper gives dominant if you go to the 7-limit and look at 12 and 17. Pajara arises out of 12 and 2 if you believe using 2 is legitimate, but also from 12 and 22. 17 and 22 gives quasisuper, 17 and 2 gives something with a 1/17 octave period, 2 and 22 gives a contorted version of pajara.

So for 2L5s you get bug and
> mavila as contenders, depending on if it's proper or improper, right?
> And then mavila wins out because it covers more of the spectrum with
> less error or something like that.

I mostly didn't get bug, but saying mavila would win depends on your victory conditions.

So maybe it makes more sense to start lower on the scale
> tree, come up with different temperaments for things like 16&13, and
> then go one entry higher and at each level higher, give the name to
> that MOS the name of the lower-badness temperament of the two entries
> below. At least I think that makes sense.

What a headache.

> > I don't see how the zeta function could help.
> > I thought you guys were talking about how it could remove the need to
> specify prime limit when doing stuff like this?

You don't need to specify a prime limit, or a val, for edos.

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

> It's just a way of assigning names to MOS's that I think makes sense.
> Whatever naming convention you choose should probably reflect the fact
> that something like 1L6s is pretty intrinsically related to
> "porcupine," or so I think.

Is it? Go into the improper range and 9 and 10 together gives negri, which can also be used on a 7-note scale.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> That even more so now that
> it's just been showed that you can already use Graham's regular
> temperament finder to assign temperaments to MOS's.

No, you can't. Not for improper scales. And you get different results depending on your level of specificity. 5, 7, 12 gives one thing but 5, 7, 17 quite another.

> > Which, again, will only sow confusion. Under this proposal, if Magic turns out to be the
> >best temperament in the 3L+4s range, then we'll have to start calling the 3L+4s scales in > > 10, 13, 16, 17, etc. "Magic heptatonics" which will completely obscure the fact that none > > of these scales has anything to do with Magic temperament and do not produce
> > harmonies in the way you would expect Magic to do.
>
> OK, so that's a good point. Maybe it does make more sense if you
> distinguish between the proper and improper versions, which seems to
> yield more sensible results.

More sensible, sure, but not fully sensible yet. In the example I gave of Porcupine, that 23, 31, 38, and 39 are 1L+6s-compatible but NOT Porcupine temperaments, they are all *still proper*. You also have to consider that propriety means different things depending on the period. For example, in 5L+5s in 30-EDO, you can have L:s be 5:1, but the scale is still technically proper--you'll never have intervals of one class be higher than intervals in the class above or lower than intervals in the class below. I do use the terms "proper" and "improper" to distinguish scales which are instantiated twice in a single EDO, though--such as 25-EDO and its two versions of 5L+5s. And by the way, I've had a bad habit of calling all 5L+5s scales "Blackwood", which I'm going to stop doing, because Blackwood as a composer did not use that scale in other 5n-EDOs, and the temperament named after him refers specifically to a 5-limit temperament close to 15-EDO.

At any rate, trying to link DE scales smaller than 11 or 12 notes to temperaments is insane. What about pentatonics, Mike? Are you going to call all 2L+3s pentatonics "Meantone", despite the fact that this MOS range encompasses about a million different temperaments, including Mavila and Superpyth and Flattone etc. etc.?

I don't want to be a jerk about this, but I think you're barking mad for pursuing this. Affixing a temperament name to a scale that fails to achieve the harmonic goals of the temperament it is named for does an injustice both to the scale and the temperament. Ask Paul if 13-EDO supports Orwell if you don't believe me.

-Igs

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

> OK, so that's a good point. Maybe it does make more sense if you
> distinguish between the proper and improper versions, which seems to
> yield more sensible results.

Somewhat more sensible, but is it sensible enough? For 5L2s proper, we pretty clearly want to call it meantone. For 5L2s improper, we were pushed up to the 7-limit and given a choice between dominant and pajara. Pajara looks better, except for the teeny tiny little fact that it doesn't have any 7 note MOS, of shape 5L2s or any other shape. So we are either left with dominant, or we go up to L=5s, get 27, and then 22 and 27 gives us superpyth. We should discard results like pajara when doing this, or simply not feed in numbers which aren't relatively prime.

On Sun, Jan 30, 2011 at 2:04 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I guess if you wanted to give MOS's regular temperament names, the way
> > I envisioned it was something like - for 5L2s, you could figure out
> > which mappings predominate for the whole 5L2s spectrum, both improper
> > and proper and give the canonical version the one that has the most
> > "coverage," where coverage refers to the width of the domain in which
> > it is the clear victor divided by the min error for that temperament
> > in that domain or something like that.
>
> I see you don't have your victory condition worked out, and another consideration is that more accurate temperaments take up less room on the scale of generators.

Do you mean more complex temperaments, or more accurate? Meantone is
pretty accurate, but has a pretty wide range. Schismatic, on the other
hand, is more accurate, but more complex, and has a less range.
Porcupine is more complex than meantone, but less accurate, but has
less of a valid range of generators than meantone.

I don't have my victory condition worked out... I'm making this up as
I go along. My group theory fu jitsu isn't as strong as yours,
unfortunately. One observation that I think here is important is that
the difference between the proper and improper version of an MOS is
the same as specifying whether a certain interval, or set of
intervals, is/are reversed or not. I'm trying to apply that to all of
this here.

Here's a sample victory condition, although it doesn't make use of the
above observation: if we're analyzing 5L2s, try all generators from
the 5-equal fifth to the 7-equal fifth. Or, if you think it makes more
sense, split it up from the 7-equal fifth to the 12-equal fifth for
the proper version, and then separately deal with the 12-equal to the
5-equal generator for the improper version, and deal with the two
separately.

As you try each generator, figure out which mapping gives you the
least error. You will then end up with data about which mappings have
the least error over different generator ranges, and you can use this
error for some kind of generator-specific badness computation.

Once you have that data, it seems like there should be some kind of
sensible way to pick out the best one and say that that's the most
generally applicable temperament for that MOS.

> So maybe it makes more sense to start lower on the scale
> > tree, come up with different temperaments for things like 16&13, and
> > then go one entry higher and at each level higher, give the name to
> > that MOS the name of the lower-badness temperament of the two entries
> > below. At least I think that makes sense.
>
> What a headache.

Haha, well, it makes sense to me :)

> > > I don't see how the zeta function could help.
> > > I thought you guys were talking about how it could remove the need to
> > specify prime limit when doing stuff like this?
>
> You don't need to specify a prime limit, or a val, for edos.

In Graham's temperament finder you do, so I thought that was a part of this...

-Mike

On Sun, Jan 30, 2011 at 2:29 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > OK, so that's a good point. Maybe it does make more sense if you
> > distinguish between the proper and improper versions, which seems to
> > yield more sensible results.
>
> Somewhat more sensible, but is it sensible enough? For 5L2s proper, we pretty clearly want to call it meantone. For 5L2s improper, we were pushed up to the 7-limit and given a choice between dominant and pajara. Pajara looks better, except for the teeny tiny little fact that it doesn't have any 7 note MOS, of shape 5L2s or any other shape. So we are either left with dominant, or we go up to L=5s, get 27, and then 22 and 27 gives us superpyth. We should discard results like pajara when doing this, or simply not feed in numbers which aren't relatively prime.

If you end up discarding temperaments that don't actually fit the MOS
in question, or just don't feed in contorted numbers to begin with, do
the results end up being sensible? What happens if you do it for the
other MOS's?

-Mike

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

> Here's a sample victory condition, although it doesn't make use of the
> above observation: if we're analyzing 5L2s, try all generators from
> the 5-equal fifth to the 7-equal fifth. Or, if you think it makes more
> sense, split it up from the 7-equal fifth to the 12-equal fifth for
> the proper version, and then separately deal with the 12-equal to the
> 5-equal generator for the improper version, and deal with the two
> separately.

You can't very well look at all of them, because there are an infinite number. You could look at the mediant. For 5L2s, that would be the mediant of 4/7 and 3/5, which is 7/12. Meantone and dominant shoot it out, and meantone wins, I suppose. If you split proper and improper, for proper you get the mediant of 4/7 and 7/12, which is 11/19; meantone. For the improper range, 7/12 and 3/5 gives 10/17. If I work the algorithm I discussed before, it's pretty screwed up unless you treat it as a no-fives system, whereupon it becomes the no-fives temperament tempering out 64/63, which actually seems a reasonable choice.

On Sun, Jan 30, 2011 at 2:22 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > That even more so now that
> > it's just been showed that you can already use Graham's regular
> > temperament finder to assign temperaments to MOS's.
>
> No, you can't. Not for improper scales. And you get different results depending on your level of specificity. 5, 7, 12 gives one thing but 5, 7, 17 quite another.

??? Isn't that what Gene just showed how to do here?
/tuning/topicId_95922.html#95964

If L is 3 and s is 1, then 5*3 + 2*1 = 17. 12&17 temperament yields
dominant, and 17&22 yields something called "quasisuper" in the
7-limit, which I don't really understand how it's different than
superpyth by looking at the mapping.

> > OK, so that's a good point. Maybe it does make more sense if you
> > distinguish between the proper and improper versions, which seems to
> > yield more sensible results.
>
> More sensible, sure, but not fully sensible yet. In the example I gave of Porcupine, that 23, 31, 38, and 39 are 1L+6s-compatible but NOT Porcupine temperaments, they are all *still proper*. You also have to consider that propriety means different things depending on the period. For example, in 5L+5s in 30-EDO, you can have L:s be 5:1, but the scale is still technically proper--you'll never have intervals of one class be higher than intervals in the class above or lower than intervals in the class below. I do use the terms "proper" and "improper" to distinguish scales which are instantiated twice in a single EDO, though--such as 25-EDO and its two versions of 5L+5s.

OK, but if the point of this is to figure out which temperament is
generally most applicable to a certain MOS, which is also a way I'm
proposing to name them. I can't figure out if you think that it'll be
a confusing naming system or if you think there's no utility in
figuring out which temperaments fit which MOS's. Either way, I don't
see why it's unreasonable or a disservice to this scale to still call
the 1L6s MOS "Porcupine," even if there are other valid temperaments
on the fringe that fit it. Again, if you don't like it, you could
still just come up with a list of temperaments rather than just
picking one.

> And by the way, I've had a bad habit of calling all 5L+5s scales "Blackwood", which I'm going to stop doing, because Blackwood as a composer did not use that scale in other 5n-EDOs, and the temperament named after him refers specifically to a 5-limit temperament close to 15-EDO.

OK, but I wouldn't mind giving the 5L5s MOS the name "Blackwood" in
his honor anyway, and since the lowest badness temperament that
supports 5L5s is probably Blackwood.

> At any rate, trying to link DE scales smaller than 11 or 12 notes to temperaments is insane. What about pentatonics, Mike? Are you going to call all 2L+3s pentatonics "Meantone", despite the fact that this MOS range encompasses about a million different temperaments, including Mavila and Superpyth and Flattone etc. etc.?

We're talking about differentiating between proper and improper
scales, and for the improper 2L3s pentatonic, you'd get mavila, and
for the proper one, you'd get meantone. Or, if you'd like, and as I've
said before, you could come up with a list of temperaments that apply
over the range of the MOS that you want. You've said there's no
usefulness in doing that, but then you keep bringing this point up,
which is exactly where the usefulness lies.

> I don't want to be a jerk about this, but I think you're barking mad for pursuing this. Affixing a temperament name to a scale that fails to achieve the harmonic goals of the temperament it is named for does an injustice both to the scale and the temperament. Ask Paul if 13-EDO supports Orwell if you don't believe me.

That's the point of factoring complexity into it.

I've noticed that you have some kind of strong emotional reaction
against this idea and I have no idea why. You throw some different
thing at me every time I bring it up, and then I later on realize it
wasn't a problem anyway. The last time it was that you could
arbitrarily apply any mapping to any MOS, so there's no possible way
to work it out. Which. if you want to really view things that way,
means we can't talk about meantone as an MOS at 7 and 12 notes
anymore. This time it's what, exactly?

-Mike

On Sun, Jan 30, 2011 at 4:10 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Here's a sample victory condition, although it doesn't make use of the
> > above observation: if we're analyzing 5L2s, try all generators from
> > the 5-equal fifth to the 7-equal fifth. Or, if you think it makes more
> > sense, split it up from the 7-equal fifth to the 12-equal fifth for
> > the proper version, and then separately deal with the 12-equal to the
> > 5-equal generator for the improper version, and deal with the two
> > separately.

> You can't very well look at all of them, because there are an infinite number.

I was assuming there exists some clever way to turn the summation into
an integral and work things out that way. No? For starters, you could
always just start with like 32 different mediants or something, all
situated at EDOs, and then run them all through your saturation
algorithm and see what it spits out.

> If you split proper and improper, for proper you get the mediant of 4/7 and 7/12, which is 11/19; meantone. For the improper range, 7/12 and 3/5 gives 10/17. If I work the algorithm I discussed before, it's pretty screwed up unless you treat it as a no-fives system, whereupon it becomes the no-fives temperament tempering out 64/63, which actually seems a reasonable choice.

That's why I was asking about the zeta function, since it would be
nice to not have to deal with prime limits and stuff like this in
general. I thought that the no-fives temperament eliminating 64/63 was
superpyth, but I guess superpyth also equates the 3-limit schismatic
major third and 6/5, or something like that?

Maybe this search needs to be run in such a way that all possible
subgroups are also taken into account, although I'm not sure how to
efficiently compute any of that.

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> You can't very well look at all of them, because there are an infinite number. You could look at the mediant.

If you go through the various 7-note MOS you get some high error temperaments of the kind Igs could love. Or you could stick to the 5-limit, but that removes all the fun of considering things like tempering out 25/24, 49/45 and 35/33, naming it, and then using that name to name something else.

Look. Mike. I've been going round in circles with Paul and Carl off-list over what is essentially the same subject matter here, and the reason I'm getting so frustrated is because you're sort of one step behind the discussions I've been having. I've been brought overwhelmingly to one conclusion: the basis for grouping scales according to temperament is fundamentally different from the basis for grouping scales according to DE/MOS patterns, and it makes no sense to try to shoehorn the two together.

The point of temperament is to achieve specific harmonic goals--defined in terms of JI--with a lower dimensionality than pure JI, for the sake of improving harmonic efficiency and decreasing complexity while maintaing *as much accuracy as possible*. You know this. I know this. But it is rarely spelled out. Fundamental to the nature of temperament is the set of harmonic goals; when you get to the point where a tuning no longer achieves those goals with a reasonable level of accuracy, *that* is where the boundary of the temperament lies.

The point of the distributionally-even/moment-of-symmetry paradigm is to group together differently-tuned scales which share a common pattern of 2nds. All DE scales are bounded by equal scales, and can approach either of these equal scales with arbitrary closeness if we allow arbitrarily-large EDOs. There are thus constraints on how a generator can be tuned and still produce a scale that remains in one scale family, and these constraints are occasionally responsible for the scales in some DE scale families to have a lot in common harmonically, more or less. The 5L+2s scale is a good example of this, since there's barely 40 cents between the lower and upper bounds on the generator, even without distinguishing proper and improper. Other DE scales are not like this. The EDOs that bound a DE family can be handily defined as the EDO corresponding to the number of Large steps and the EDO corresponding to the number of total steps. The fewer the Large steps and/or total steps, the greater the range in generator tunings. 2L+5s goes between 2-EDO and 7-EDO, or between 600 and 686 cents, just about double the range of 5L+2s. 1L+6s goes between 1-EDO and 7-EDO, with a range of 0 to 171 cents! The 1L+4s pentatonic goes from 0 to 240 cents! For scales like this, even dividing into proper and improper does little to restrain the massive harmonic variability.

It's one thing (and perfectly fine to do, quite useful even) to index the optimal generators for however many temperaments you like according to the DE scales they produce. If that's all you're trying to do, that's great. I gave you my blessing in that endeavor last time.

If, on the other hand, you are insisting on finding the "best" temperament in each range of DE scale generators and then naming the whole DE scale family after that temperament, my objections are three-fold. The first is that you are doing a disservice to the temperament paradigm by lumping a bunch of different temperaments together under one name, just because they share the same DE scale. There are a bazillion temperaments with a generator between 4\7 and 7\12, does it really make sense to call them all "Meantone" just because Meantone has the lowest badness? Does anyone on this list ONLY use tunings which are the absolute best at what they do? I don't think so.

The second objection is that what you are doing doesn't make any practical sense. If what you are interested in using musically is the temperament paradigm--i.e. if you have a specific set of harmonic goals that you want to achieve--then you have no reason to use the DE paradigm. You can say everything you need to say about the scales you want to use in the language of the temperament paradigm. If, on the other hand, you are more concerned with melodic structures, and less concerned with what harmonies are specifically being achieved, then you can say all you need to say about the scales you use with the DE paradigm. If you are interested in both, you can (as you have expressed a desire to do) cross-reference according to generator tunings. That, as I said above, would be neat to be able to do.

The third objection is if you're going to give a collective name to a group of things, it would be sensible to name them according to a property they *all* have in common. I mean, say you have two barrels. In one barrel, you have a huge amount of apples, a few oranges, a few bananas, a few kiwis, and a few melons. In another barrel, you have a huge amount of beets, a few carrots, a few heads of lettuce, a few bunches of kale. Now, you could call the first barrel "the apple barrel" and the second barrel "the beet barrel", but wouldn't it make more sense and be more accurate to call the first barrel "the fruit barrel" and the second barrel "the vegetable barrel"? DE scale families all have a pattern of 2nds in common; they do not all have temperaments in common. It's less reasonable to name them after temperaments, even if most of the generators in the range have a temperament in common, than it is to name them after their pattern of seconds.

I just don't understand how you can fail to see that naming DE scale families after the "best" temperaments found in their tuning range will do nothing but blur important distinctions between temperaments. This is undeniable, incontrovertible, pure absolute fact! I've been warned by Paul over and over again not to conflate two temperaments just because they have the same DE scale. I used to insist that it made no sense NOT to call 13-EDO an Orwell temperament, because it has the same 9-note DE scale found in 31-EDO and 22-EDO and generally-speaking you can translate music written in that scale from 31-EDO to 13-EDO and you won't lose much. But I was wrong, because (again) temperaments are bounded by error and 13-EDO is out of bounds for the consonances Orwell is based on. Paul's suggestion was that if you want to name a range of tunings after a temperament, you should define the boundaries of the range according to a damage threshold, which I think is a better idea. You might try that approach.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> I used to insist that it made no sense NOT to call 13-EDO an Orwell temperament, because it has the same 9-note DE scale found in 31-EDO and 22-EDO and generally-speaking you can translate music written in that scale from 31-EDO to 13-EDO and you won't lose much.

You're a scary guy, Igs. A scary, scary guy.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
>
> > I used to insist that it made no sense NOT to call 13-EDO an Orwell temperament, because it has the same 9-note DE scale found in 31-EDO and 22-EDO and generally-speaking you can translate music written in that scale from 31-EDO to 13-EDO and you won't lose much.
>
> You're a scary guy, Igs. A scary, scary guy.
>

Well, in my defense, the 9-note MOS of Orwell doesn't supply much in the way of full 7-limit otonal tetrads. Anyone using Orwell as a complete 7-limit temperament is probably going to use the 13-note MOS at least. Most of what I find the 9-note MOS useful for is playing creepy dissonant stuff, and in that regard I stand by the claim that you could use the 9-note MOS in 13-EDO just as effectively. Obviously if you're using Orwell in its full 7-limit glory, 13-EDO is not gonna work. I'm not THAT crazy. But the point is there's a structural similarity based on the common DE scale and a common size-ordering of interval classes that relates 13-EDO to 22- and 31-EDO, a structural similarity which is not there in 14-EDO or 19-EDO or 24-EDO (ad infinitum). This similarity is what makes what Mike is trying to do so tempting, but I've since learned it's wrong and got nothing to do with temperaments and 13-EDO is NOT Orwell etc. etc.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Most of what I find the 9-note MOS useful for is playing creepy dissonant stuff, and in that regard I stand by the claim that you could use the 9-note MOS in 13-EDO just as effectively.

The very first Orwell piece ever written used the 9-note MOS for two-part harmony, and was quite consonant.

Igs: that was a very thorough explanation. Thanks for going into so
much detail, because there's a lot of connections there I hadn't made.
I still have a few comments:

Your objections to not name the entire MOS family after whichever one
comes out on top were well-thought through and very valid as far as
coming up with a sensible naming system is involved. However, I'm more
interested in trying to find connections between concepts, so that I
can develop musically useful metrics that don't already exist, and to
come to a broader understanding of what's going on. Whether it makes
sense to call MOS's by the name of their best associated is more a
matter of logistics and communication rather than of conceptual
integrity. I would be personally willing to let that small evil slide,
because I think the results would be useful in classifying MOS's, but
you have good arguments about why such a system might not be ideal.
That's fine and the issue of naming is, I guess, not anything that I
have really strong opinions about.

Perhaps most simply I am just trying to advance a conceptual point,
which is that people usually view it such that while the DE scales and
the regular temperaments are two completely different things, stemming
from two completely different paradigms, and that can maybe be loosely
associated, that the associations can be made much tighter if certain
additional mathematical characteristics are taken into consideration.

The first is that when you talk about an MOS to begin with, you're
still doing some kind of quasi-regular mapping, because you're
specifying what the octave is. If this weren't the case, then there'd
be no difference between 5L5s and 4L4s, because you could look at the
5L5s one as just a 4L4s in which the octave is mapped to 960 cents,
which you could always do if you wanted to anyway. So if a rank 2
temperament has two generators, then what an MOS scale is is just a
rank-2 temperament in which only a map for 2/1 is defined in terms of
periods and the other is left open.

The second thing is all aspects of this other generator aren't left
completely open, because by specifying a pattern of large and small
steps, you're effectively placing bounds on the size of the generator.
More specifically, what you're doing is specifying a ratio between So
if you're saying that we have a 5L2s scale, for example, what you're
really saying is that 2 is mapped to 1 period, and the generator
ranges from 3\5 to 5\7 of the size of that period, in cents. If we end
trying to apply the MOS map for 5L2s to a generating interval outside
of this range, we end up with something that's not 5L2s, and hence an
important interval somewhere is getting reversed.

For example, if you're in meantone, and you apply the meantone map to
a generator greater than 720 cents, then 16/15 ends up getting
reversed and you don't get a 5L2s scale anymore. Or if you're less
than 686 cents, 25/24 ends up getting reversed, and you don't end up
getting a 5L2s scale anymore. If you for some reason were weird enough
to apply the porcupine map to this generator, then as the generator
goes beyond 720 cents, it's (10/9)^-6 * (2/1)^3 = 531441/250000 that
gets reversed, which since 250/243 is a unison vector means that 9/2
gets reversed. Huge intervals get reversed when you apply weird
mappings to generators and vice versa.

Furthermore, this works both ways: as you specify a mapping and a set
of intervals that must either be reversed or not reversed, you start
placing MOS-style bounds on the width of the generator. As you end up
specifying an infinite series of intervals that must be reversed, you
successively limit the range of the generator until you approach a
single one. This is analogous to going down the scale tree in
different directions until you reach "the bottom," where you reach a
single generator, and depending on which way you go at each turn, you
get different generators, both defined in cents.

So the two aren't entirely different approaches; rather, they're
almost the same thing. If you give regular mappings bounds by
specifying reversed intervals, you end up with a structure homomorphic
to the MOS's. Specifically, the more reversed intervals you specify,
the tighter your bounds get, and the further down the scale tree you
go.

That is what I am pointing out, no more, no less. I think that that's
a useful realization and that it might lead to a lot of interesting
things later, one of those things being a sensible way to name MOS's.
So your objections, in light of all this, are as follows:

1) This structure is homomorphic, but not isomorphic. While such a
series for a certain mapping does uniquely correspond to an MOS, the
same does not apply in reverse. A certain MOS does not imply a unique,
single regular mapping with a set of reversed intervals, but rather
could correspond to any mapping at all, even stupid ones. And I say
yes, but if you sort the resultant mappings by the error in the range
of the MOS and some normalized measure of complexity, you end up with
a list of how well each temperament corresponds to each MOS.

2) While you can come up with a list, to just pick the largest entry
from each list and say that's "the temperament" for a certain MOS
ignores the fact that different temperaments predominate in different
ranges within a single MOS. This is also true, and as you go further
down the scale tree, this becomes less of a problem, and as you
specify more intervals being reversed, it becomes more clear which
temperament provides the most sensible map for those intervals.

3) You finally assert that psychoacoustic tuning error is more
important for drawing error bounds for temperaments rather than which
intervals get reversed. I think that the two are both important and
work together. The harmonic structure of what's going on is important,
and you want to make sure that you don't end up with a temperament
whose error is too high. The melodic structure is also extremely
important for what's going on, and although blackwood and porcupine
have the same 5-limit error in 15-tet, they sound completely different
from one another. They are both important things to keep track of, but
the deal here is that the series of reversed intervals and its subsets
correspond to different generator ranges, which also happen to
correlate perfectly with different MOS's, which are basically ways of
storing melodic patterns, which are very important when it comes to
making music.

Hope that's thorough enough. Maybe folks have already realized all of
this, I dunno.

-Mike

On 31 January 2011 07:17, Mike Battaglia <battaglia01@...> wrote:

> Perhaps most simply I am just trying to advance a conceptual point,
> which is that people usually view it such that while the DE scales and
> the regular temperaments are two completely different things, stemming
> from two completely different paradigms, and that can maybe be loosely
> associated, that the associations can be made much tighter if certain
> additional mathematical characteristics are taken into consideration.

They're hardly different paradigms, and MOS/DE scales are already part
of the regular mapping paradigm. The mapping you choose will
constrain the range of MOS scales that might apply. And the choice of
MOS scale (with a precise tuning) will constrain the mappings that
might apply. The labels may disagree.

It is possible to treat an MOS scale without considering a mapping to
ratios, or other special intervals, at all. That would entail a
different paradigm.

> Furthermore, this works both ways: as you specify a mapping and a set
> of intervals that must either be reversed or not reversed, you start
> placing MOS-style bounds on the width of the generator. As you end up
> specifying an infinite series of intervals that must be reversed, you
> successively limit the range of the generator until you approach a
> single one. This is analogous to going down the scale tree in
> different directions until you reach "the bottom," where you reach a
> single generator, and depending on which way you go at each turn, you
> get different generators, both defined in cents.

The point at which an interval reverses is a ways an equal
temperament. An equal temperament is also the point at which two
mappings apply.

The strong regular mapping credo would be that, once you specify the
mapping, the tuning isn't important because it takes care of itself.
But you can still think of tunings first and use mappings to explain
the harmony.

Graham

On 31 January 2011 02:53, cityoftheasleep <igliashon@...> wrote:

> The point of temperament is to achieve specific harmonic goals--defined in terms of JI--with a lower dimensionality than pure JI, for the sake of improving harmonic efficiency and decreasing complexity while maintaing *as much accuracy as possible*.  You know this.  I know this.  But it is rarely spelled out.  Fundamental to the nature of temperament is the set of harmonic goals; when you get to the point where a tuning no longer achieves those goals with a reasonable level of accuracy, *that* is where the boundary of the temperament lies.

The point of a temperament is to reconcile harmonic and melodic
constraints. A rank 2 temperament class will lead to a specific MOS
family. You may have melodic properties in mind when you look for a
temperament.

Yes, the boundaries of temperament classes are harmonic, and harmonic
and melodic boundaries may disagree. The tunings of different
temperament classes may also overlap. But all boundaries will likely
be drawn at equal temperaments (or EDOs if you take a melodic purist's
position).

Graham

On Mon, Jan 31, 2011 at 6:48 AM, Graham Breed <gbreed@...> wrote:
>
> They're hardly different paradigms, and MOS/DE scales are already part
> of the regular mapping paradigm. The mapping you choose will
> constrain the range of MOS scales that might apply. And the choice of
> MOS scale (with a precise tuning) will constrain the mappings that
> might apply. The labels may disagree.
>
> It is possible to treat an MOS scale without considering a mapping to
> ratios, or other special intervals, at all. That would entail a
> different paradigm.

Not unless you at least specify a map for 2 in terms of periods,
right? Otherwise 4L4s and 5L5s would be the same scale. So it seems to
me that an MOS will always include some kind of quasi-map; you have to
at least say how many periods that 2/1 is. And by then specifying a
pattern of large and small steps, you're also saying what
pseudo-intervals in the mapping are going to be reversed; you're just
not saying exactly what combination of primes you're mapping the
reversed intervals to. So it seems like the MOS concept is partly a
"fuzzy mapping" paradigm, in which the generators don't have exact
values, etc.

> > Furthermore, this works both ways: as you specify a mapping and a set
> > of intervals that must either be reversed or not reversed, you start
> > placing MOS-style bounds on the width of the generator. As you end up
> > specifying an infinite series of intervals that must be reversed, you
> > successively limit the range of the generator until you approach a
> > single one. This is analogous to going down the scale tree in
> > different directions until you reach "the bottom," where you reach a
> > single generator, and depending on which way you go at each turn, you
> > get different generators, both defined in cents.
>
> The point at which an interval reverses is a ways an equal
> temperament. An equal temperament is also the point at which two
> mappings apply.

Wait a second, I just realized something. When you say a temperament
like 7&12 is meantone, is that the same thing that I'm saying here?
Or, more specifically, like you just said, these intervals that end up
reversing do so at the boundaries of some equal temperament, which is
the point where that interval's size is 0 and you end up with a rank-1
temperament. So when you make statements like "meantone is the 5-limit
7&12 temperament," is that the same thing as saying that the ideal
5-limit mapping for the 12L7s mapping is meantone, because all of this
stuff I'm talking about with intervals reversing is really the same
thing as that?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> The first is that when you talk about an MOS to begin with, you're
> still doing some kind of quasi-regular mapping, because you're
> specifying what the octave is.

It's more like you're *assuming* what the octave is. The DE paradigm works just as well if you're looking at scales that fill the 3/1, or the 3/2, or the sqrt 2. To use the naming convention I suggested, no such assumption is required; all you have to assume is that all the scales you're discussing span the same interval. The whole "xL+ys" thing is actually an equation, and when we're talking about EDO DE scales, we're tacitly saying "xL+ys=1200 cents", or maybe in other terms "xL+ys=n-EDO" if we're specifying sizes of L and s in # of scale steps instead of cents. In fact, it's quite possible to use the DE paradigm without using the terms "generator" and "period" at all, and may in fact be simpler to do it that way. That's one reason I'm actually starting to prefer the term "DE" to "MOS".

> The second thing is all aspects of this other generator aren't left
> completely open, because by specifying a pattern of large and small
> steps, you're effectively placing bounds on the size of the generator.

Yes, I mentioned this specifically. Did you miss that part? Can't blame you, it was a bit of a long-winded reply. To repeat what I said: while this is true, the tightness of the bounds depends not just on the size of the scale but also on how many large steps are in the scale. The EDO bounds of an MOS scale are defined as the EDOs equal to the number of Large steps and equal to the number of total steps. So 5L+2s is between 5-EDO and 7-EDO, but 1L+6s is between 1-EDO and 7-EDO. I think the problem is that you've only been looking at 5L+2s, which has a comparatively narrow range of generators.

Since the size of the scale is also a variable that determines the bounds of the generators, it will also be true that pentatonics are a bit more difficult to categorize. As I said previously (and you may have missed), for a pentatonic of 1L+4s, you're looking at a generator range between 1\1 and 1\5, or about 240 cents! So the point is, the number of temperaments that will fall within a DE family will vary greatly depending on the size of the scale and the number of large steps it has. So on the one hand, you'll have an easy time drawing a correspondence between 9L+1s and (I think) Negri temperament, and a very difficult time drawing a correspondence between 1L+4s and any one temperament.

*I suggest you look at all the 1L DE scales before proceeding.*

> So the two aren't entirely different approaches; rather, they're
> almost the same thing. If you give regular mappings bounds by
> specifying reversed intervals, you end up with a structure homomorphic
> to the MOS's. Specifically, the more reversed intervals you specify,
> the tighter your bounds get, and the further down the scale tree you
> go.

I'm not sure I understand what you mean when you say an interval gets "reversed".

> 2) While you can come up with a list, to just pick the largest entry
> from each list and say that's "the temperament" for a certain MOS
> ignores the fact that different temperaments predominate in different
> ranges within a single MOS. This is also true, and as you go further
> down the scale tree, this becomes less of a problem, and as you
> specify more intervals being reversed, it becomes more clear which
> temperament provides the most sensible map for those intervals.

Yes, the further toward the canopy of the scale-tree you go, the closer to specific regular temperaments you get, but the higher in EDOs you get, too. At the level at which it becomes possible to make a meaningful connection between a DE family and a temperament, it is no longer possible to generalize that connection back toward the root of the scale tree.

> 3) You finally assert that psychoacoustic tuning error is more
> important for drawing error bounds for temperaments rather than which
> intervals get reversed. I think that the two are both important and
> work together.

If both are important, then both must be taken into account. I think what you should do, if you want to do this properly, is figure out the range of generators for each temperament that keeps the approximated harmonies below a certain damage threshold, presumably defined as "the point in which the approximations are no longer psychoacoustically identifiable as the intervals they are meant to approximate". Then correlate those ranges with ranges of DE families. At that point, I will gladly accept a naming convention that names DE families after temperaments, but I kind of doubt anyone is really up to the task of calculating error boundaries *for all the good 13-limit (and lower) temperaments, as well as subgroup temperaments*.

Really, there's no point in any more argument. I think if you actually pursue this, the ways in which it is problematic will become apparent pretty quick. First, you've got to set a badness threshold to constrain the infinity of temperaments, then you've got to decide what prime limit (and what subgroups) you are willing to consider tempering, then you've got to calculate the bounds based on psychoacoustic damage, then you've got to index by generator range and compare the badness of all the temperaments (and subgroups) that have overlapping generator ranges and single out those that have the lowest badness (which may be misleading depending on the variety of subgroup temperaments you look at). The results from this will undoubtedly be useful, but DANG that sounds like a lot of work! And I do not expect the results to cover anything below heptatonic scales, and most probably not lower than decatonic in many areas of the generator spectrum. So it still won't be a viable alternative to the DE scale-family naming scheme I proposed. But I don't really think this is about names, anyway.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> I think the problem is that you've only been looking at 5L+2s, which has a comparatively narrow range of generators.

One way to look at that problem is this: I've shown how, given a period expressed as 1/N of an octave, and a generator expressed as p/q of an octave, and a prime limit, you can compute a rank two temperament in that prime limit with that period and generator. For very exceptional edos like 31 that will always give you sensible results at least up to the 11 limit. But what if you are interested in the region around 7/11? Since 11 isn't very good, you can't expect very good results and will find yourself reduced to grubbing around for some subgroup. If you go up to a high enough denominator you will be able to find something decent between 5/8 and 2/3, but why should it give its name to the whole interval?

On Mon, Jan 31, 2011 at 1:20 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > The first is that when you talk about an MOS to begin with, you're
> > still doing some kind of quasi-regular mapping, because you're
> > specifying what the octave is.
>
> It's more like you're *assuming* what the octave is.

An MOS, or a DE scale, specifies a pattern of large and whole steps.
That's the same thing, mathematically, as specifying a ratio that the
generator and period are in some proportional ratio to one another (or
in some range of ratios). But to do this, you have to say how many
periods maps to 2, or else there is no difference between 4L4s and
5L5s.

> The DE paradigm works just as well if you're looking at scales that fill the 3/1, or the 3/2, or the sqrt 2.

If you have one period mapping to sqrt(2), that means that your map
for 2 is 0 generators, 2 periods. If you're going to have things fill
the 3/2 instead, you're specifying that 3/2 maps to 0 generators, 1
period. Some kind of mapping is implicit.

> To use the naming convention I suggested, no such assumption is required; all you have to assume is that all the scales you're discussing span the same interval. The whole "xL+ys" thing is actually an equation, and when we're talking about EDO DE scales, we're tacitly saying "xL+ys=1200 cents", or maybe in other terms "xL+ys=n-EDO" if we're specifying sizes of L and s in # of scale steps instead of cents. In fact, it's quite possible to use the DE paradigm without using the terms "generator" and "period" at all, and may in fact be simpler to do it that way. That's one reason I'm actually starting to prefer the term "DE" to "MOS".

Right... Well, first off, if you want to give all DE scales that fit
the same imprint the same name, regardless of whether the period is
3/2 or 2/1 or sqrt(2), then what's the difference between 4L4s and
5L5s and 1L1s? What's the difference between 8L2s and 4L1s? 8L2s is
just 4L1s, but you're mapping the period differently. You still
haven't broken away from mapping.

Here's another question: when people say that Blackwood is the 10&15
temperament, why is that okay? That's basically the same thing anyway,
since "10&15" can be turned into a corresponding MOS on the scale
tree, right?

> The EDO bounds of an MOS scale are defined as the EDOs equal to the number of Large steps and equal to the number of total steps. So 5L+2s is between 5-EDO and 7-EDO, but 1L+6s is between 1-EDO and 7-EDO. I think the problem is that you've only been looking at 5L+2s, which has a comparatively narrow range of generators.

That could be true.

> Since the size of the scale is also a variable that determines the bounds of the generators, it will also be true that pentatonics are a bit more difficult to categorize. As I said previously (and you may have missed), for a pentatonic of 1L+4s, you're looking at a generator range between 1\1 and 1\5, or about 240 cents! So the point is, the number of temperaments that will fall within a DE family will vary greatly depending on the size of the scale and the number of large steps it has. So on the one hand, you'll have an easy time drawing a correspondence between 9L+1s and (I think) Negri temperament, and a very difficult time drawing a correspondence between 1L+4s and any one temperament.
>
> *I suggest you look at all the 1L DE scales before proceeding.*

OK, but to be fair, you should at least admit that 1L5s scales that
have a generator of 20 cents could probably be ignored for whatever
naming convention that you come up with. While your example of the
7-limit porcupine-ish scale above was good (what is the name of that
temperament?), 1\1 and 1\5 has a definite sweet spot. Or 1L9s scales
that have a generator of 20 cents, etc.

Tangentially, I would personally support a hybrid naming system that
gives credit to a temperament whenever it really does dominates a
certain MOS, and comes up with a "neutral" name or a portmanteau of
the winners when no clear winner dominates.

> > So the two aren't entirely different approaches; rather, they're
> > almost the same thing. If you give regular mappings bounds by
> > specifying reversed intervals, you end up with a structure homomorphic
> > to the MOS's. Specifically, the more reversed intervals you specify,
> > the tighter your bounds get, and the further down the scale tree you
> > go.
>
> I'm not sure I understand what you mean when you say an interval gets "reversed".

LOL geez man, that's the whole point of my rant here! :)

An interval is reversed when it gets mapped to a negative size in
cents. So in 19-equal, for example, the Pythagorean comma is reversed,
because if you go up 12 fifths and down whatever many octaves, you end
up at a lower pitch than you started at. This is because meantones
equate the diesis and the Pythagorean comma, and unless they're both
getting tempered out as in 12-tet, one of them is going to be reversed
with respect to the other.

So let's go to 12L7s scales vs 7L12s scales, and let's say you apply
the meantone mapping here. The difference between a 12L7s scale vs a
7L12s scale, expressed in regular temperament parlance, is the
difference between whether a diesis is smaller than a chromatic
semitone or larger than a chromatic semitone. If the diesis is
smaller, you get a 12L7s scale, and if it's larger, you get a 7L12s
scale. If the diesis ends up being larger than a chromatic semitone,
then the Natural Order Of Things has been broken: the difference
between them, which is the magic comma, is now negative. And the
boundary between 12L7s and 7L12s scales is 19-equal, which is a magic
temperament.

The same applies if you look at 5L2s scales vs 2L5s scales, but you
decide to keep the meantone mapping the whole time: when you flip over
to a "meantone" generator that generates a 2L5s scale, that's the same
thing as saying that 25/24 has been reversed. In this case, it makes
more sense to apply the mavila mapping, but this is just for the sake
of theory - likewise, if you apply the mavila mapping and you pick a
generator that makes a 5L2s scale, 25/24 also gets reversed.

If you look at 7L5s scales vs 5L7s scales, and if you apply the
meantone mapping here, the difference between 7L5s vs 5L7s translates
over to regular temperament parlance as the difference between whether
or not 128/125 gets reversed. You could be absurd and apply the
porcupine mapping here, and say that the ~700 cent sized generator is
a 10/9, and if you do that, the interval getting reversed maps to 9/2
instead of 128/125, which is silly.

Your point, then, is that sometimes more than one mapping applies, and
the boundaries between when superpyth makes sense to use vs meantone
have more to do with the temperament's error from 10:12:15 vs 6:7:9
and all that, and less to do with reversed intervals. And I'm saying,
yes, you're right, but the latter is equally as important, because it
dictates the melodic structure of what's going on. The point is simply
that when you specify a DE scale, you're specifying which intervals
get reversed without specifying what they map to, but the two aren't
really separate.

I brought all of this up and Graham then pointed out that this is
isomorphic to when people refer to meantone as "the 12&19
temperament." Why isn't porcupine the 12&19 temperament? At least,
that's what I was understanding his meaning to be. etc.

> Yes, the further toward the canopy of the scale-tree you go, the closer to specific regular temperaments you get, but the higher in EDOs you get, too. At the level at which it becomes possible to make a meaningful connection between a DE family and a temperament, it is no longer possible to generalize that connection back toward the root of the scale tree.

That's fine, but a decent list of all of the top 4 that lie below each
node would be enough for me.

> > 3) You finally assert that psychoacoustic tuning error is more
> > important for drawing error bounds for temperaments rather than which
> > intervals get reversed. I think that the two are both important and
> > work together.
>
> If both are important, then both must be taken into account. I think what you should do, if you want to do this properly, is figure out the range of generators for each temperament that keeps the approximated harmonies below a certain damage threshold, presumably defined as "the point in which the approximations are no longer psychoacoustically identifiable as the intervals they are meant to approximate". Then correlate those ranges with ranges of DE families. At that point, I will gladly accept a naming convention that names DE families after temperaments, but I kind of doubt anyone is really up to the task of calculating error boundaries *for all the good 13-limit (and lower) temperaments, as well as subgroup temperaments*.

I think there will be a few cases in which this approach works really
well. The reason I suggested using the zeta function for this is that
it has been claimed on tuning-math that you could assume from the
Riemann hypothesis a measure of tuning-error which removes the need to
specify prime limit at all. I'm not sure if this handles subgroups or
not. Gene then posted on here that equal temperaments don't need limit
to be specified, and my brain exploded, so I'm not sure where we stand
on that.

> Really, there's no point in any more argument. I think if you actually pursue this, the ways in which it is problematic will become apparent pretty quick. First, you've got to set a badness threshold to constrain the infinity of temperaments

This is something that logflat badness does really well, no? I'm still
learning about it.

> then you've got to decide what prime limit (and what subgroups) you are willing to consider tempering

Unless we can work out the zeta error thing.

> then you've got to calculate the bounds based on psychoacoustic damage

This is redundant, because if we're actually comparing temperaments in
some generator range, then the ones with so much error that they're
unrecognizable don't need to be pruned off beforehand, because they'll
naturally end up at the bottom of the list.

> then you've got to index by generator range and compare the badness of all the temperaments (and subgroups) that have overlapping generator ranges and single out those that have the lowest badness (which may be misleading depending on the variety of subgroup temperaments you look at).

Yeah, this part's going to suck. I think there's probably some simple
way to do it if you work a lot of the math out.

> And I do not expect the results to cover anything below heptatonic scales, and most probably not lower than decatonic in many areas of the generator spectrum. So it still won't be a viable alternative to the DE scale-family naming scheme I proposed. But I don't really think this is about names, anyway.

I like the DE scale-family naming scheme you proposed, but it's just
the same thing as saying xLns in Latin :)

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > It's more like you're *assuming* what the octave is.
>
> An MOS, or a DE scale, specifies a pattern of large and whole steps.
> That's the same thing, mathematically, as specifying a ratio that the
> generator and period are in some proportional ratio to one another (or
> in some range of ratios). But to do this, you have to say how many
> periods maps to 2, or else there is no difference between 4L4s and
> 5L5s.
>

Right, you do have to specify a period (and thus, I suppose, a mapping) before hand and assume that all the scales you are talking about share it, or else 5L5s and 1L1s will be the same thing. But the point I was trying to make is that that specification is arbitrary, i.e. not built into the pattern of steps at all. So if you're talking about 4L+5s spanning the 2/1, that might be something related to Orwell temperament (at least, maybe in the strictly-proper range), but if you're talking about it spanning the 3/1, that's Bohlen-Pierce! Same DE scale, different period, drastically different properties. Since you want to name DE scale families after temperaments, the names will be period-specific, so you'll need sets of names for different periods, too.

> Right... Well, first off, if you want to give all DE scales that fit
> the same imprint the same name, regardless of whether the period is
> 3/2 or 2/1 or sqrt(2), then what's the difference between 4L4s and
> 5L5s and 1L1s? What's the difference between 8L2s and 4L1s? 8L2s is
> just 4L1s, but you're mapping the period differently. You still
> haven't broken away from mapping.

Ah, but that's not true. You only have to assume that all the scales you're talking about span the *same* period for 1L+1s to be different from 4L+4s. You don't need to know what the period is, just that it is the same for all the scales you're looking at. So the mapping doesn't actually need to be specified.

> Here's another question: when people say that Blackwood is the 10&15
> temperament, why is that okay? That's basically the same thing anyway,
> since "10&15" can be turned into a corresponding MOS on the scale
> tree, right?

Well, note the 15. As I've hinted at before, once you get up to >11-note MOS's, the range of generators is pretty narrow for most scales. I personally don't think 10&15 is specific enough for Blackwood, though, in terms of constraining the generators to make good 5-limit harmonies.

> OK, but to be fair, you should at least admit that 1L5s scales that
> have a generator of 20 cents could probably be ignored for whatever
> naming convention that you come up with. While your example of the
> 7-limit porcupine-ish scale above was good (what is the name of that
> temperament?), 1\1 and 1\5 has a definite sweet spot.

1L+5s is between 1\1 and 1\6. But yes, it does have a solid sweet-spot (in the 3-limit, anyway) just a bit sharp of 1\7, around 5\34 I'd say. And another one for 2.5-subgroups at a bit sharp of 2\13, around 5\31. But yeah, generally speaking the sweet spot is between 1\7 and 1\6. But (as I'm sure you're aware from your frequent discussions with me) not everyone cares for the "sweet spots".

> Tangentially, I would personally support a hybrid naming system that
> gives credit to a temperament whenever it really does dominates a
> certain MOS, and comes up with a "neutral" name or a portmanteau of
> the winners when no clear winner dominates.

I could go for that, I suppose. But I still don't see why, if you want to talk about a temperament, you don't just talk about the temperament. Why go through all this trouble in the first place, when temperaments and DE scales are already clearly defined?

> An interval is reversed when it gets mapped to a negative size in
> cents. So in 19-equal, for example, the Pythagorean comma is reversed,
> because if you go up 12 fifths and down whatever many octaves, you end
> up at a lower pitch than you started at. This is because meantones
> equate the diesis and the Pythagorean comma, and unless they're both
> getting tempered out as in 12-tet, one of them is going to be reversed
> with respect to the other.

Oh, interesting! I've been trying to get someone to explain to me what commas have to do with temperament boundaries like, FOREVER, and no one's told me this before.

> So let's go to 12L7s scales vs 7L12s scales, and let's say you apply
> the meantone mapping here. The difference between a 12L7s scale vs a
> 7L12s scale, expressed in regular temperament parlance, is the
> difference between whether a diesis is smaller than a chromatic
> semitone or larger than a chromatic semitone. If the diesis is
> smaller, you get a 12L7s scale, and if it's larger, you get a 7L12s
> scale. If the diesis ends up being larger than a chromatic semitone,
> then the Natural Order Of Things has been broken: the difference
> between them, which is the magic comma, is now negative. And the
> boundary between 12L7s and 7L12s scales is 19-equal, which is a magic
> temperament.

Interesting. So when an interval is reversed between two temperaments, is that when you get antiodromias when you translate music from one to the other (an antiodromia, if I'm using the word correctly, is when majors and minors switch places, like between Mavila and Meantone). I didn't realize this could be described in terms of commas.

So does that mean that you can describe all scales that share a common DE scale in terms of a common comma that isn't reversed? Like, is there a comma that meantone and mavila share that explains why they have the same 5-note MOS? If so, this might be a game-changer.

> I brought all of this up and Graham then pointed out that this is
> isomorphic to when people refer to meantone as "the 12&19
> temperament." Why isn't porcupine the 12&19 temperament? At least,
> that's what I was understanding his meaning to be. etc.

Yes, I know this convention. It works, I think, because at the level of a 19-note MOS the generator range is so narrow that pretty much anything is going to be a meantone.

> I think there will be a few cases in which this approach works really
> well. The reason I suggested using the zeta function for this is that
> it has been claimed on tuning-math that you could assume from the
> Riemann hypothesis a measure of tuning-error which removes the need to
> specify prime limit at all. I'm not sure if this handles subgroups or
> not. Gene then posted on here that equal temperaments don't need limit
> to be specified, and my brain exploded, so I'm not sure where we stand
> on that.

A measure of tuning error that removes the need to specify prime limit? Head xplod.

> I like the DE scale-family naming scheme you proposed, but it's just
> the same thing as saying xLns in Latin :)

Greek, actually. ;->

-Igs

On Mon, Jan 31, 2011 at 10:22 PM, cityoftheasleep
<igliashon@...> wrote:
>
> > An MOS, or a DE scale, specifies a pattern of large and whole steps.
> > That's the same thing, mathematically, as specifying a ratio that the
> > generator and period are in some proportional ratio to one another (or
> > in some range of ratios). But to do this, you have to say how many
> > periods maps to 2, or else there is no difference between 4L4s and
> > 5L5s.
> >
>
> Right, you do have to specify a period (and thus, I suppose, a mapping) before hand and assume that all the scales you are talking about share it, or else 5L5s and 1L1s will be the same thing. But the point I was trying to make is that that specification is arbitrary, i.e. not built into the pattern of steps at all. So if you're talking about 4L+5s spanning the 2/1, that might be something related to Orwell temperament (at least, maybe in the strictly-proper range), but if you're talking about it spanning the 3/1, that's Bohlen-Pierce! Same DE scale, different period, drastically different properties. Since you want to name DE scale families after temperaments, the names will be period-specific, so you'll need sets of names for different periods, too.

Therefore the properties of the DE scale will to depend on the
mapping, so it really isn't a separate paradigm from regular mapping
at all. Unless I'm misunderstanding you here.

> > Right... Well, first off, if you want to give all DE scales that fit
> > the same imprint the same name, regardless of whether the period is
> > 3/2 or 2/1 or sqrt(2), then what's the difference between 4L4s and
> > 5L5s and 1L1s? What's the difference between 8L2s and 4L1s? 8L2s is
> > just 4L1s, but you're mapping the period differently. You still
> > haven't broken away from mapping.
>
> Ah, but that's not true. You only have to assume that all the scales you're talking about span the *same* period for 1L+1s to be different from 4L+4s. You don't need to know what the period is, just that it is the same for all the scales you're looking at. So the mapping doesn't actually need to be specified.

OK, good point.

> Well, note the 15. As I've hinted at before, once you get up to >11-note MOS's, the range of generators is pretty narrow for most scales. I personally don't think 10&15 is specific enough for Blackwood, though, in terms of constraining the generators to make good 5-limit harmonies.

> > An interval is reversed when it gets mapped to a negative size in
> > cents. So in 19-equal, for example, the Pythagorean comma is reversed,
> > because if you go up 12 fifths and down whatever many octaves, you end
> > up at a lower pitch than you started at. This is because meantones
> > equate the diesis and the Pythagorean comma, and unless they're both
> > getting tempered out as in 12-tet, one of them is going to be reversed
> > with respect to the other.
>
> Oh, interesting! I've been trying to get someone to explain to me what commas have to do with temperament boundaries like, FOREVER, and no one's told me this before.

It's just something I realized recently. Apparently everyone has
mapped this all out before, no pun intended. Graham's temperament
finder describes temperaments like meantone in terms like 7&12, and to
do so is apparently somehow equivalent on a deep level to what I'm
doing, so to describe a regular temperament by two equal temperaments
already assigns temperament names to entries on the scale tree, which
is how it's always worked. The difference is, if we're not talking
about there being a prime limit anymore, which mapping for 7 wins out
for 5L2s scales in general? The temperament where 64/63 vanishes, or
the one where 225/224 vanishes? There's no clear victor there, which
is, I believe, the point you're making.

> > So let's go to 12L7s scales vs 7L12s scales, and let's say you apply
> > the meantone mapping here. The difference between a 12L7s scale vs a
> > 7L12s scale, expressed in regular temperament parlance, is the
> > difference between whether a diesis is smaller than a chromatic
> > semitone or larger than a chromatic semitone. If the diesis is
> > smaller, you get a 12L7s scale, and if it's larger, you get a 7L12s
> > scale. If the diesis ends up being larger than a chromatic semitone,
> > then the Natural Order Of Things has been broken: the difference
> > between them, which is the magic comma, is now negative. And the
> > boundary between 12L7s and 7L12s scales is 19-equal, which is a magic
> > temperament.
>
> Interesting. So when an interval is reversed between two temperaments, is that when you get antiodromias when you translate music from one to the other (an antiodromia, if I'm using the word correctly, is when majors and minors switch places, like between Mavila and Meantone). I didn't realize this could be described in terms of commas.
>
> So does that mean that you can describe all scales that share a common DE scale in terms of a common comma that isn't reversed? Like, is there a comma that meantone and mavila share that explains why they have the same 5-note MOS? If so, this might be a game-changer.

Yes, but the results are kind of stupid. Meantone and mavila share a
common mapping for 3/2, so the two commas there that must NOT be
reversed are 3-limit commas, and they're 9/8 and 256/243. All along
2L3s, these commas aren't reversed. When it is reversed, you end up
with something that maps closer to father temperament. The boundary at
which 256/243 reverses is 5-equal, which tempers out 256/243 and is
hence a blackwood temperament, and the boundary at which 9/8 reverses
is 2-equal, which is just another way of saying what you already know.

The common interval that reverses for both meantone and mavila when
the 2L3s scale becomes improper is the 3-limit apotome, 2187/2048.
Perhaps the best way to go, when naming MOS's is involved, is to pick
the smallest temperament "clan" that contains most of the winners,
such that the 7-note MOS would be called pythagorean or something, or
some suitable name to suggest that the generator should be mapped to
3/2, which is surely common among all of the victors in that
temperament. Not sure how this would work for something like 1L6s
though.

One interesting thing is that the 5-limit commas that flip as the 2L3s
scale becomes improper are flipped for meantone and mavila - for
meantone, as the scale becomes improper, 25/24 flips around, making
major thirds minor thirds and vice versa. For mavila, as the scale
becomes improper, 25/24 finally becomes un-flipped around, as 25/24
would have to be reversed above that.

-Mike

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> So does that mean that you can describe all scales that share a common DE scale in terms of a common comma that isn't reversed? Like, is there a comma that meantone and mavila share that explains why they have the same 5-note MOS? If so, this might be a game-changer.

Mavila and meantone intersect on a 7-note MOS. Mavila tempers out 135/128, and meantone 81/80. Feed these two commas in to temperament finding software such as mine or Graham's, and <7 11 16| pops out.

> > I brought all of this up and Graham then pointed out that this is
> > isomorphic to when people refer to meantone as "the 12&19
> > temperament." Why isn't porcupine the 12&19 temperament? At least,
> > that's what I was understanding his meaning to be. etc.
>
> Yes, I know this convention. It works, I think, because at the level of a 19-note MOS the generator range is so narrow that pretty much anything is going to be a meantone.

Neither 12 nor 19 temper out 250/243,so there's no mystery why it is called 15&22 and not 12&19.

On Tue, Feb 1, 2011 at 12:17 AM, genewardsmith
<genewardsmith@...> wrote:
>
> Mavila and meantone intersect on a 7-note MOS. Mavila tempers out 135/128, and meantone 81/80. Feed these two commas in to temperament finding software such as mine or Graham's, and <7 11 16| pops out.

You mean a 5-note MOS, right? Mavila and meantone differ as far as
7-note MOS's are concerned...

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 1, 2011 at 12:17 AM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > Mavila and meantone intersect on a 7-note MOS. Mavila tempers out 135/128, and meantone 81/80. Feed these two commas in to temperament finding software such as mine or Graham's, and <7 11 16| pops out.
>
> You mean a 5-note MOS, right? Mavila and meantone differ as far as
> 7-note MOS's are concerned...

There's no real boundry between them with pentatonic scales; a 7-note MOS is precisely and only where the two come together at a boundry. Stick mavila and meantone together, and out pops 7. Stick porcupine and meantone together, and again out pops 7, but the generators don't correspond and so they don't have a boundry. Stick mavila and porcupine together, and out pops 15. Stick mavila and kwazy together, and out pops a variant and fairly useless val for 80edo.

On Sat, Jan 29, 2011 at 9:51 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > OK, septimal meantone reverses (using POTE tuning or anything reasonable) 1029/1024, 19683/19600, 32805/32768, 2401/2400, 250047/250000, etc. What does that tell you?
>
> "Anything reasonable" is wrong; it's damned sensitive to changes in tuning. Adding 1029/1024 or 2401/2400 gives 31et, 19683/19600 19et, 32805/32768 or 250047/250000 12et.

To elucidate on this, I think that the series of reversed intervals
encodes the exact generator you ended up using. To say that a certain
interval is reversed is to say that the generator is sharp or flat of
this ET or that ET, and as you specify more, you end up tying the
generator into closer and closer boundaries until at infinity you end
up with just one.

-Mike

On Tue, Feb 1, 2011 at 12:43 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Tue, Feb 1, 2011 at 12:17 AM, genewardsmith
> > <genewardsmith@...> wrote:
> > >
> > > Mavila and meantone intersect on a 7-note MOS. Mavila tempers out 135/128, and meantone 81/80. Feed these two commas in to temperament finding software such as mine or Graham's, and <7 11 16| pops out.
> >
> > You mean a 5-note MOS, right? Mavila and meantone differ as far as
> > 7-note MOS's are concerned...
>
> There's no real boundry between them with pentatonic scales; a 7-note MOS is precisely and only where the two come together at a boundry. Stick mavila and meantone together, and out pops 7. Stick porcupine and meantone together, and again out pops 7, but the generators don't correspond and so they don't have a boundry. Stick mavila and porcupine together, and out pops 15. Stick mavila and kwazy together, and out pops a variant and fairly useless val for 80edo.

I see, so by this definition, you're saying the "MOS" they intersect
at is 7-equal?

If you stick mavila and porcupine together, how do you get 15? Do you mean 7?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
>
> Yes, but the results are kind of stupid.

Stupid, like hell! Dagnabbit, this is exactly what I was looking for like, a YEAR ago.

> Meantone and mavila share a
> common mapping for 3/2, so the two commas there that must NOT be
> reversed are 3-limit commas, and they're 9/8 and 256/243. All along
> 2L3s, these commas aren't reversed. When it is reversed, you end up
> with something that maps closer to father temperament.

I suspected as such, but I never knew how to figure it out. Mike, you have basically just told me that you can map commas to nodes on the Scale Tree. People have been telling me for the last year or so that there is no correspondence between temperaments and DE scales but you've just demonstrated that there totally IS and that there is a sensible way to bound temperaments in terms of commas as opposed to error. For "Bob"'s sake, there's a perfect taxonomy of scales staring us in the face and it's been here the whole time.

My question is, how did you work out the commas that get reversed? I'd love to work out all the commas for maybe the first four or five levels of the Scale Tree.

Crikey, Mike, this is brilliant! We can make a full taxonomy of scales this way, using a hierarchy of increasing specificity based on adding commas.

-Igs

On Tue, Feb 1, 2011 at 1:07 AM, cityoftheasleep <igliashon@...> wrote:
>
> Stupid, like hell! Dagnabbit, this is exactly what I was looking for like, a YEAR ago.

But your objections about multiple mappings applying are still valid.
If you're crazy enough to apply the porcupine mapping to that size
generator, for example, the "comma" that gets reversed is 9/2.

The reason it works in this case is that the implicit assumption being
made is that whatever tuning you pick for 5L2s, or 2L3s in your
example, or whatever mapping it is, you're probably going to map the
generator to be a 3/2. This only works because both mavila, meantone,
superpyth, and generally all of the generators for all of the valid
temperaments in that range all share the common characteristic that
there's a 3/2 generator there.

Come to think of it, this also works for dicot and magic as well; the
generator there for both is an approximate 5/4. So no matter whether
or not you use dicot or magic to generate that MOS, as long as it's
generating a 3L2s scale, you're stating that 128/125 must not be
reversed, putting 3-equal as an upper bound, and that 78125/65536 must
not be reversed, putting 7-equal as a lower bound.

Again, unless you're dumb enough to apply the pajara generator to
dicot or something.

> > Meantone and mavila share a
> > common mapping for 3/2, so the two commas there that must NOT be
> > reversed are 3-limit commas, and they're 9/8 and 256/243. All along
> > 2L3s, these commas aren't reversed. When it is reversed, you end up
> > with something that maps closer to father temperament.
>
> I suspected as such, but I never knew how to figure it out. Mike, you have basically just told me that you can map commas to nodes on the Scale Tree.

LOL, we've switched sides, but now I've seen the wisdom in your
objections as well. What happens for a 1L6s scale, where you're
bounded by 1-equal and 6-equal? The generator could range from
anything up to 0 cents to 171 cents - what's the common generator
there that yields the common interval that's reversed?

But at that point I'd argue that there is really very little point in
grouping all 1L6s scales together at all, because 1L6s scales with a
35 cent generator really have very little to do with 1L6s scales with
a 170 cent generator, I doubt there's going to be much melodic
coherence at all there. How often are you actually going to use a 1L6s
scale that has a 35 cent generator? For taxonomical purposes I have no
problem throwing away outliers like that, but the situation might
still get tricky in other cases. Are there times in which more than
one mapping for the generator clearly makes sense across the range of
a given MOS, under the -reasonable- size constraints under which that
MOS will be used, and to pick one would be bad?

> People have been telling me for the last year or so that there is no correspondence between temperaments and DE scales but you've just demonstrated that there totally IS and that there is a sensible way to bound temperaments in terms of commas as opposed to error. For "Bob"'s sake, there's a perfect taxonomy of scales staring us in the face and it's been here the whole time.

It still depends on you applying a reasonable mapping, however.
There's no correspondence between mappings and equal temperaments
either; sometimes more than one mapping can make sense for a certain
tuning, as with 11/8 in 68-tet. The entire point of the regular
mapping paradigm is to pick good mappings, though so I never
understood the conceptual problem here.

I was all gung-ho about just picking the best mapping for a certain
MOS and calling it a night, but you raised the good point that the
2.3.7 64/63 "superpyth" and meantone both have nice little niches for
5L2s. The saving grace here that makes it all work out, at least in
the way that you want, is that they both share a common mapping for 3
and 2.

But is it really the saving grace? To delve further into it, I'd
respond to it that they really -don't- have two nice little separate
niches for 5L2s, because they both end up sounding very similar
anyway. The superpyth supermajor triads clearly resemble mistuned
5-limit major triads, and both the superpyth subminor and meantone
minor triads are both high enough in entropy to sound "minor" anyway,
although in both cases the constituent triads can be clearly
distinguished. And, furthermore, triads like 4:6:7 are still decently
recognizable whether you use meantone and the 6:7's are sharp, or
whether you use superpyth and they aren't, so it seems like the clear
winner here is dominant temperament, which conflates the two (at least
the 2.3.7 subgroup version of superpyth I mentioned) and covers most
of the spectrum. If we apply the dominant map to 5L2s, one simple
interval that gets reversed as the scale flips from proper to improper
is 50/49.

A corresponding analogy probably exists for the 7-limit
pseudo-porcupine you mentioned earlier; conflate that with regular
porcupine, and you have a reasonable mapping that covers most of the
spectrum.

In the 11-limit, this breaks down, as there are now two clear maps for
11/8, and whichever one you pick, 128/121 flips around as the scale
becomes improper. It either goes from reversed to not reversed or vice
versa depending on which map you used.

OK, so there's no clear 11-limit winner, but a clear 7-limit winner
(dominant), a clear 5-limit winner (meantone), and a clear 3-limit
winner (pythagorean). The 5L2s temperament assumes that 9/8 and 20,
assuming that you map 3/2 to the generator. What name would you pick?
Pythagorean, meantone, etc? I say run the zeta error algorithm on all
of them and pick the winner, which will probably be meantone.

> My question is, how did you work out the commas that get reversed? I'd love to work out all the commas for maybe the first four or five levels of the Scale Tree.
>
> Crikey, Mike, this is brilliant! We can make a full taxonomy of scales this way, using a hierarchy of increasing specificity based on adding commas.

It still assumes that you can at least provide a decent map for the
generator. Assuming you can at least do that, you can work out the
intervals that are reversed based on that single generator. If not, or
if an MOS exists where more than one mapping makes sense for the
generator, you have to work out the reversed intervals in terms of
nondescript variables like p and g, which you can then map out later.

-Mike

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

> If you stick mavila and porcupine together, how do you get 15? Do you mean 7?

Sorry, I computed blackwood and porcupine. If you feed 135/128 and 250/243 into the maw of the machine, out pops <7 11 16|.

On Tue, Feb 1, 2011 at 2:14 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > If you stick mavila and porcupine together, how do you get 15? Do you mean 7?
>
> Sorry, I computed blackwood and porcupine. If you feed 135/128 and 250/243 into the maw of the machine, out pops <7 11 16|.

How does the black box actually work? Does it figure out that
blackwood is 5L5s, and porcupine is 1L6s, and they share common-sized
MOS's at 5L10s and 7L8s, and hence 15-equal handles both of them?

-Mike

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

> It's just something I realized recently. Apparently everyone has
> mapped this all out before, no pun intended. Graham's temperament
> finder describes temperaments like meantone in terms like 7&12,
> and to do so is apparently somehow equivalent on a deep level to
> what I'm doing, so to describe a regular temperament by two equal
> temperaments already assigns temperament names to entries on the
> scale tree, which is how it's always worked.

Herman Miller was the first to notice strange patterns on a
2-D plot of ETs, where the [x,y] coords for each ET give the
error in primes 3 and 5.

/tuning/files/HermanMiller/ET-Scales.png

Paul Erlich quickly refined this using a triangular plot

/tuning/files/PaulErlich/treezuma.gif

and it became evident that linear temperaments are lines on
such plots. Since it takes two points to define a line, an LT
can therefore be uniquely identified by a pair of ETs.

Gene quickly pointed out that in the dual of this space, LTs
are points and ETs are lines

http:/groups.yahoo.com/group/tuning-math/files/paul/dualzoomer.gif

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> I suspected as such, but I never knew how to figure it out. Mike, you have basically just told me that you can map commas to nodes on the Scale Tree.

He did? What's the mapping? Which node does 81/80 correspond to, for instance? If you say 81/80 runs from 4/7 to 7/12, then it gets mapped to 11/19, which is the mediant. But how do you know meantone conks out at 7/12? You can say, well, at 4/7 25/24 gets reversed, but does it really? It's also zero at 10et and 17et, why doesn't it reverse there? If it gets reversed, why isn't it negative for 9et or 16et? On the other hand, 3125/3072 does reverse at 11/19, being negative at 4/7, and in between at 15/26. We would need to look at what this reversing business really means to make anything of it.

>
> http:/groups.yahoo.com/group/tuning-math/files/paul/dualzoomer.gif
>
> -Carl
>

s/b

/tuning-math/files/paul/dualzoomer.gif

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> > I suspected as such, but I never knew how to figure it out.
> > Mike, you have basically just told me that you can map commas
> > to nodes on the Scale Tree.
>
> He did? What's the mapping?

5-limit commas do map to points in the space I just linked to

/tuning-math/files/paul/dualzoomer.gif

-Carl

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

And, furthermore, triads like 4:6:7 are still decently
> recognizable whether you use meantone and the 6:7's are sharp, or
> whether you use superpyth and they aren't, so it seems like the clear
> winner here is dominant temperament, which conflates the two (at least
> the 2.3.7 subgroup version of superpyth I mentioned) and covers most
> of the spectrum.

Meantone is soooo much better in the 7-limit than dominant that it clearly kicks ass in proper 5L2s, so going for dominant is only a consequence of lumping all of 5L2s together, a bad idea in the first place if you are bringing regular temperaments into it.

If we apply the dominant map to 5L2s, one simple
> interval that gets reversed as the scale flips from proper to improper
> is 50/49.

And yet, it's zero for both 7/12 and 13/22. It's negative for 10/17, but that has some pretty sorry tunings in comparison.

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

> > Sorry, I computed blackwood and porcupine. If you feed 135/128 and 250/243 into the maw of the machine, out pops <7 11 16|.
>
> How does the black box actually work? Does it figure out that
> blackwood is 5L5s, and porcupine is 1L6s, and they share common-sized
> MOS's at 5L10s and 7L8s, and hence 15-equal handles both of them?

Nah, it just figures out that the only noncontorted 5-limit val of any size which tempers out both 256/243 and 250/243 is <15 24 35|.

On Tue, Feb 1, 2011 at 2:40 AM, genewardsmith
<genewardsmith@...> wrote:
>
> He did? What's the mapping? Which node does 81/80 correspond to, for instance? If you say 81/80 runs from 4/7 to 7/12, then it gets mapped to 11/19, which is the mediant. But how do you know meantone conks out at 7/12? You can say, well, at 4/7 25/24 gets reversed, but does it really? It's also zero at 10et and 17et, why doesn't it reverse there?

If you were to actually apply the meantone map to 10-et or 17-et, it
wouldn't be reversed there. It just gets silly to do so, but it gets
less silly to apply the dicot map there, so it makes more sense to say
that 10et and 17et are dicot temperaments than meantone temperaments.
That is, unless you actually do apply the meantone map to 17-et, which
is how I prefer to use it anyway; I like to leave the supermajor
triads as "major" triads, and the neutral triads as "neutral" triads.
If I had a 17-tet guitar I'd probably be playing supermajor chords
more than neutral ones.

> If it gets reversed, why isn't it negative for 9et or 16et? On the other hand, 3125/3072 does reverse at 11/19, being negative at 4/7, and in between at 15/26.

Maybe I'll just explain what I'm trying to say this way: if you apply
the meantone map to a 700 cent generator, and you then make the
generator flat, as it gets flatter than 7-equal, 25/24 suddenly flips
around to a negative interval. If you were dumb enough to apply the
porcupine map to a 700 cent generator, as you pass 7-equal, I think
it's 9/2 that suddenly becomes the reversed interval.

My point was that at the boundaries where an MOS flips, some interval
gets tempered out, and on one side of that boundary, it will be
reversed, and on the other, it won't. Also that MOS's, in the form of
5L5s and such, also implicitly specify a map for 2 as 5 periods, which
is why 5L5s and 4L4s can be differentiated in the first place. The
claim was made that DE scales and regular mapping are completely
different things, and now I'm starting to realize that they aren't.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > > I suspected as such, but I never knew how to figure it out.
> > > Mike, you have basically just told me that you can map commas
> > > to nodes on the Scale Tree.
> >
> > He did? What's the mapping?
>
> 5-limit commas do map to points in the space I just linked to
>
> /tuning-math/files/paul/dualzoomer.gif

Yeah, I know, but that's points in a projective plane, and we were talking about fractions of an octave.

On Tue, Feb 1, 2011 at 2:58 AM, genewardsmith
<genewardsmith@...> wrote:
>
> > How does the black box actually work? Does it figure out that
> > blackwood is 5L5s, and porcupine is 1L6s, and they share common-sized
> > MOS's at 5L10s and 7L8s, and hence 15-equal handles both of them?
>
> Nah, it just figures out that the only noncontorted 5-limit val of any size which tempers out both 256/243 and 250/243 is <15 24 35|.

OK, so I get this, and have always understood this concept
practically, but I think that there's something deeper going on that
I'm missing.

So in light of what Igs is saying - how does this work? What is the
val that represents porcupine, or is it that there are two vals? It
seems like the entire point of regular mapping is already to mix MOS
scales and temperaments, and that that was the point all along, and it
seems like the whole concept of the "val" is in some way already a
generalization of MOS.

-Mike

On Tue, Feb 1, 2011 at 2:55 AM, genewardsmith
<genewardsmith@...> wrote:
>
> And, furthermore, triads like 4:6:7 are still decently
> > recognizable whether you use meantone and the 6:7's are sharp, or
> > whether you use superpyth and they aren't, so it seems like the clear
> > winner here is dominant temperament, which conflates the two (at least
> > the 2.3.7 subgroup version of superpyth I mentioned) and covers most
> > of the spectrum.
>
> Meantone is soooo much better in the 7-limit than dominant that it clearly kicks ass in proper 5L2s, so going for dominant is only a consequence of lumping all of 5L2s together, a bad idea in the first place if you are bringing regular temperaments into it.

I said dominant over septimal meantone because we don't even get to
see any meantone 7-limit action in 5L2s to begin with, except I guess
with 7/5. For 7L5s, septimal meantone clearly delivers over dominant,
but for 5L2s, the best we get is 7/5 whereas with dominant we get 7/4,
7/2, and 7/1. Perhaps that could be formalized somehow.

BTW, is there no other name for septimal meantone than "septimal
meantone?" :\ What a mouthful.

> If we apply the dominant map to 5L2s, one simple
> > interval that gets reversed as the scale flips from proper to improper
> > is 50/49.
>
> And yet, it's zero for both 7/12 and 13/22. It's negative for 10/17, but that has some pretty sorry tunings in comparison.

It's not zero for 13/22 if you were to actually apply the dominant map
there. If you apply the pajara map, yes.

-Mike

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

> So in light of what Igs is saying - how does this work? What is the
> val that represents porcupine, or is it that there are two vals?

Porcupine is rank two, so there are two vals.

On Tue, Feb 1, 2011 at 3:14 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > So in light of what Igs is saying - how does this work? What is the
> > val that represents porcupine, or is it that there are two vals?
>
> Porcupine is rank two, so there are two vals.

And the first number of each of those vals is always an MOS that that
temperament supports?

-Mike

On Tue, Feb 1, 2011 at 2:40 AM, Carl Lumma <carl@...> wrote:
>
> and it became evident that linear temperaments are lines on
> such plots. Since it takes two points to define a line, an LT
> can therefore be uniquely identified by a pair of ETs.

And hence an MOS can be uniquely identified by a pair of ETs, then
each linear temperament can be uniquely identified by an MOS...?

-Mike

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

> If you were to actually apply the meantone map to 10-et or 17-et, it
> wouldn't be reversed there.

OK, but that means you are starting from a linear temperament and getting to the MOS later on, which I didn't think was the point of the exercise.

If we take the map <1 f 4f 10f-3| and apply it to any 7-limit interval, we get either zero or something we can equate to zero and solve for f; if the 7-limit interval is a comma, we may expect to get a fraction of the octave which actually makes sense for meantone. Where does this get us?

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

> > Porcupine is rank two, so there are two vals.
>
> And the first number of each of those vals is always an MOS that that
> temperament supports?

Vals aren't MOS, they are vals. You can make the vals two equal temperament vals, but they don't need to be. Graham gives it both ways.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 1, 2011 at 2:40 AM, Carl Lumma <carl@...> wrote:
> >
> > and it became evident that linear temperaments are lines on
> > such plots. Since it takes two points to define a line, an LT
> > can therefore be uniquely identified by a pair of ETs.
>
> And hence an MOS can be uniquely identified by a pair of ETs, then
> each linear temperament can be uniquely identified by an MOS...?

A pair of et vals defines a rank two temperament, not a MOS. In the 5-limit, that means they also define a comma.

On Tue, Feb 1, 2011 at 3:26 AM, genewardsmith
<genewardsmith@...> wrote:
>
> > And hence an MOS can be uniquely identified by a pair of ETs, then
> > each linear temperament can be uniquely identified by an MOS...?
>
> A pair of et vals defines a rank two temperament, not a MOS. In the 5-limit, that means they also define a comma.

Maybe I should move this to tuning-math. But OK, so let me try to be clear:

- A pair of ET vals defines a rank two temperament; i.e., meantone is 5&7.
- Two ET's also define an MOS, i.e. 5-ET and 7-ET bound 5L2s.
- Therefore, a pair of ET vals already inherently gives a rank-two
temperament a corresponding MOS.

Right?

-Mike

On Tue, Feb 1, 2011 at 3:21 AM, genewardsmith
<genewardsmith@...> wrote:
>
> OK, but that means you are starting from a linear temperament and getting to the MOS later on, which I didn't think was the point of the exercise.
>
> If we take the map <1 f 4f 10f-3| and apply it to any 7-limit interval, we get either zero or something we can equate to zero and solve for f; if the 7-limit interval is a comma, we may expect to get a fraction of the octave which actually makes sense for meantone. Where does this get us?

I was just pointing out that the relationship between MOS's and
temperaments aren't completely disparate. I'm right now so confused
about how vals work that I don't have any response other than that.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 1, 2011 at 1:07 AM, cityoftheasleep <igliashon@...> wrote:
> >
> > Stupid, like hell! Dagnabbit, this is exactly what I was looking for like, a YEAR ago.
>
> But your objections about multiple mappings applying are still valid.
> If you're crazy enough to apply the porcupine mapping to that size
> generator, for example, the "comma" that gets reversed is 9/2.

Yes, okay, I was awake all friggin' night thinking about this and I realized this is more of a problem that I at first thought (if that's possible). For instance, I'm pretty sure we could define commas for the entire scale tree without leaving the 3-limit if we didn't care how high we wanted to take the complexity of ratios we use to define the commas. We could, for instance, say that the comma that vanishes at each n-ET is just the difference between n-3/2's and some number of octaves, and that comma will be reversed on either side of each ET...but what good will that do us? It clearly makes more sense to look at 3-EDO as vanishing the difference between three 5/4's and the 2/1 (128/125) rather than the difference between three 81/64's and the 2/1 (531441/524288 or something horrendous like that), since 5/4 has a stronger psychoacoustic identity...and now I see what Paul's been on about for so long.

So anyway, when we're looking at the 5L+2s range between 7-EDO and 5-EDO, it seems like we're not looking at 5-limit mappings. We can define a lot in this space by 3-limit commas, i.e. that at 7-EDO we're not worried about 25/24 at this level but actually 2187/2048, which vanishes at 7-EDO and is reversed on either side. This, along with 256/243 (vanishing at 5-EDO) gives us the boundary conditions for 5L+2s, and explains why we can translate music between 19 and 22 and 26 and 27 and 12-EDOs (ad infinitum) based on a pythagorean structure! 3/2 works the same in all of them, so as long as we don't worry about 5/4 at all we're fine.

But then it gets a little more complicated. I know the Pythagorean comma that vanishes at 12-EDO is reversed between Meantone and Schismatic, and I also know that at 12-EDO both the Schisma and the Syntonic Comma vanish (and that the Schisma does not vanish in Meantone, nor the Syntonic Comma in Schismatic) but I have no idea what (if any) 5-limit comma is reversed between Meantone and Schismatic. But maybe that's irrelevant, because the Pythagorean comma ALSO happens to be very close to the difference between a major 3rd and a diminished 4th. I have no friggin' clue, really.

So the problem seems to be one of limits. When do you transition to the next prime limit? When do you use subgroups? It seems like we can't really map commas to the scale tree without some way of figuring this out.

> LOL, we've switched sides, but now I've seen the wisdom in your
> objections as well. What happens for a 1L6s scale, where you're
> bounded by 1-equal and 6-equal? The generator could range from
> anything up to 0 cents to 171 cents - what's the common generator
> there that yields the common interval that's reversed?

Yeah, it seems like the 1L scales, where the generator is bounded on one side by 0 cents, are the most problematic. That's where you're on the outermost branches of the scale tree, and if you follow the outermost line it goes 1\1-1\2-1\3-1\4-1\5-1\6-1\7...1\infinity. So actually, it seems like the sensible thing to do is just ignore any scale that's got 1-EDO or 0-EDO as a bound. This also sets some limit on impropriety, which is the biggest problem for all 1L scales.

> I was all gung-ho about just picking the best mapping for a certain
> MOS and calling it a night, but you raised the good point that the
> 2.3.7 64/63 "superpyth" and meantone both have nice little niches for
> 5L2s. The saving grace here that makes it all work out, at least in
> the way that you want, is that they both share a common mapping for 3
> and 2.

Right. Because the approach I'm taking (or trying to take) is to group all the MOS scales by commas *they have in common*, not by the commas of the best temperament in the range.

-Igs

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

> I was just pointing out that the relationship between MOS's and
> temperaments aren't completely disparate.

Here's something which may be relevant to this quest. I have a list of 36 7-limit commas which I often use as a sort of standard list. If I do what I suggested, namely apply <1 f+1 4f 10f-3| to them and solve for f, I get stuff out of the range from 11/19 to 7/12 only for commas of low complexity or very high complexity: 250/243 gives 4/7, and 78125000/78121827 does also. I have a list of 5-limit commas which includes some large ones and a number of absurdly high complexity. Only for 16/15 and 27/25, which are awfully big to call commas, did it stray off the reservation and give 3/5. Some very complex commas gave 4/7, 15/26, 17/29, 26/45, and 19/33 but mostly even they stuck to the range from 11/19 to 7/12. However, the complex commas were all over the map inside that range: 105/181, 61/105, 60/103, 62/107, 79/136 etc. These are all places where a comma reverses.

On Tue, Feb 1, 2011 at 1:22 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > I was just pointing out that the relationship between MOS's and
> > temperaments aren't completely disparate.
>
> Here's something which may be relevant to this quest. I have a list of 36 7-limit commas which I often use as a sort of standard list. If I do what I suggested, namely apply <1 f+1 4f 10f-3| to them and solve for f, I get stuff out of the range from 11/19 to 7/12 only for commas of low complexity or very high complexity: 250/243 gives 4/7, and 78125000/78121827 does also. I have a list of 5-limit commas which includes some large ones and a number of absurdly high complexity. Only for 16/15 and 27/25, which are awfully big to call commas, did it stray off the reservation and give 3/5. Some very complex commas gave 4/7, 15/26, 17/29, 26/45, and 19/33 but mostly even they stuck to the range from 11/19 to 7/12. However, the complex commas were all over the map inside that range: 105/181, 61/105, 60/103, 62/107, 79/136 etc. These are all places where a comma reverses.

What's your goal here...? To see which EDOs result if you combine
various interesting commas with septimal meantone...?

Here's another question: what do you make of the fact that the phrase
"12-equal" denotes, at the bare minimum, a contorted 2-limit map? What
does "7L5s" denote, anything? Seems like a 2.x subgroup map, where x
is unspecified, or is there some different perspective I'm lacking?

-Mike

On Tue, Feb 1, 2011 at 11:13 AM, cityoftheasleep
<igliashon@...> wrote:
>
> Yes, okay, I was awake all friggin' night thinking about this and I realized this is more of a problem that I at first thought (if that's possible). For instance, I'm pretty sure we could define commas for the entire scale tree without leaving the 3-limit if we didn't care how high we wanted to take the complexity of ratios we use to define the commas.

But that's only because in the cases I mentioned, the generator is
itself in the 3-limit. I don't think a 3-limit comma will suffice for
porcupine - or actually, I guess, in an indirect way, perhaps it
could. That's a difficult question to answer.

> We could, for instance, say that the comma that vanishes at each n-ET is just the difference between n-3/2's and some number of octaves, and that comma will be reversed on either side of each ET...but what good will that do us? It clearly makes more sense to look at 3-EDO as vanishing the difference between three 5/4's and the 2/1 (128/125) rather than the difference between three 81/64's and the 2/1 (531441/524288 or something horrendous like that), since 5/4 has a stronger psychoacoustic identity...and now I see what Paul's been on about for so long.

Yeah, that's the whole point of picking a mapping... Since the
generator for both dicot and magic is ~5/4, there will also be
corresponding reversed commas there too.

Here's something else to consider:

By itself, the rank-1 temperament consisting of a 100 cent generator
actually means nothing. Is it 12-equal? Is it 13-equal with stretched
octaves? Is it 24-equal with the octaves having 1200 cent error? etc.
By saying "12-equal" we're already specifying part of the map, e.g. 2
maps to 12 generators. In fact, no matter how large the generator is,
even if it's 700 cents, if we map 2 to 12 generators, we're in
12-equal as far as the map is concerned. So the phrase "12-equal"
already specifies a map - a contorted 2-limit map, specifically. So
we've already jumped into regular mapping when we even talk about
things like "12-equal." And this is an undeniable fact before we even
start figuring out what's a sensible map for 3, for 5, etc... it's
part of the definition of "12-equal."

OK, so by talking about 12-EDO to begin with, we are already at least
specifying a 2-limit map. So what happens when we talk about an MOS
scales, in which we also map the octave (let's assume we're just
dealing with octave-equivalent MOS's for now, and deal with tritaves
and such later - the math should still work out). So we've mapped the
octave, but we now admit that a second generator is present, so we're
in rank 2. We don't map the generator to anything - let's just call
it "g" for now, but we bound it such that certain intervals, specified
only in terms of "g," must end up being reversed/not reversed.
Depending on what mapping you pick for "g," you end up with different
such intervals. So if you can at least pick a sensible mapping for "g"
for every MOS, you can then end up classifying temperaments that way.

But somehow, in some deep mathematical way, something else is going
on, in that every linear temperament can uniquely be specified by two
equal temperaments. So there's some deep stuff going on here that I
don't fully understand. Well, two equal temperaments are also enough
to uniquely specify an MOS, right? The simple fact that we're picking
a mapping for 2 (or 3 if we're dealing with the tritave or whatever)
seems to be enough for the math to work itself out in some deep
mathematical group theory way that Gene and Graham seem to understand
and I don't.

> So anyway, when we're looking at the 5L+2s range between 7-EDO and 5-EDO, it seems like we're not looking at 5-limit mappings. We can define a lot in this space by 3-limit commas, i.e. that at 7-EDO we're not worried about 25/24 at this level but actually 2187/2048, which vanishes at 7-EDO and is reversed on either side. This, along with 256/243 (vanishing at 5-EDO) gives us the boundary conditions for 5L+2s, and explains why we can translate music between 19 and 22 and 26 and 27 and 12-EDOs (ad infinitum) based on a pythagorean structure! 3/2 works the same in all of them, so as long as we don't worry about 5/4 at all we're fine.

This is because intervals with greater complexity tend to diverge more
than those with lower complexity as the size of the generator is
varied. If you're in meantone and you move the generator 5 cents, you
end up moving 5/4 by 20 cents. So it works out that an awful lot of
these scales have a common mapping for the generator which is most
obviously sensible.

> But then it gets a little more complicated. I know the Pythagorean comma that vanishes at 12-EDO is reversed between Meantone and Schismatic, and I also know that at 12-EDO both the Schisma and the Syntonic Comma vanish (and that the Schisma does not vanish in Meantone, nor the Syntonic Comma in Schismatic) but I have no idea what (if any) 5-limit comma is reversed between Meantone and Schismatic.

That would be 128/125, I think, right?

> So the problem seems to be one of limits. When do you transition to the next prime limit? When do you use subgroups? It seems like we can't really map commas to the scale tree without some way of figuring this out.

Magic and fairy tales seem to be involved, since despite this
seemingly obvious realization, two equal temperaments can somehow
uniquely specify a linear temperament, which is clearly impossible,
and yet it isn't.

I guess the fact that equal temperaments can often have more than one
mapping is what reconciles this, although this usually happens more
with higher ETs, not lower ones, so I'm still stumped.

Perhaps one way to go is just to pick a clear mapping when it does
exist, and not worry about any type of "limits." So in porcupine, for
example, 3 is usually mapped to 4 generators, which we can say with a
clear head regardless of what we map one generator to. So if we at
least specify a contorted 3-limit mapping we're moving in the right
direction.

> > LOL, we've switched sides, but now I've seen the wisdom in your
> > objections as well. What happens for a 1L6s scale, where you're
> > bounded by 1-equal and 6-equal? The generator could range from
> > anything up to 0 cents to 171 cents - what's the common generator
> > there that yields the common interval that's reversed?
>
> Yeah, it seems like the 1L scales, where the generator is bounded on one side by 0 cents, are the most problematic. That's where you're on the outermost branches of the scale tree, and if you follow the outermost line it goes 1\1-1\2-1\3-1\4-1\5-1\6-1\7...1\infinity. So actually, it seems like the sensible thing to do is just ignore any scale that's got 1-EDO or 0-EDO as a bound. This also sets some limit on impropriety, which is the biggest problem for all 1L scales.

Well, you have far more experience with this than I do, so outside of
these somewhat pathological cases, are there other MOS's that have
this problem? Does a reasonably common generator exist for other cases
that aren't bound by 1-EDO? Dicot and magic share 5/4, for example.

> Right. Because the approach I'm taking (or trying to take) is to group all the MOS scales by commas *they have in common*, not by the commas of the best temperament in the range.

If we can at least find one common interval, whether the generator or
otherwise, for every decent mapping within an MOS, then we're moving
in the right direction.

-Mike

On Tue, Feb 1, 2011 at 2:58 PM, Mike Battaglia <battaglia01@...> wrote:
>
>> Right. Because the approach I'm taking (or trying to take) is to group all the MOS scales by commas *they have in common*, not by the commas of the best temperament in the range.

After thinking more about this, it seems like the simpler, more
all-encompassing, and overall better goal is to group all of the MOS
scales by the -mappings- they have in common. If the mapping turns out
to be some common comma, ok, but it seems like more often it's going
to be a common generator.

So maybe the question is, what should we name the temperament family
that supports the val [<1 0| <0 1|]? That is, what should we name the
temperament family that supports mapping 2 to a period and 3 to a
generator? Examples of temperaments that fall into this family are
meantone, mavila, superpyth, etc.

Maybe "family" isn't the right word here, more like "temperament
primitive" or something. Perhaps we should call this entity "diatonic"
temperament, and call specifically the 3-limit case "pythagorean"
tuning, which is what it's already called anyway. That way we can just
name 5L2s after the new "diatonic" temperament primitive, and be done
with it.

So that still doesn't handle the case where both 2L5s and 5L2s would
both be a part of this "diatonic" temperament. However, we still have
the issue of intervals being reversed. So with this primitive,
although both are diatonic, the 2L5s case ends up yielding way more
reversed intervals, so we can concretely quantify that and label it
"anti-diatonic." I wonder how well this would extend then to other MOS
cases.

-Mike

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

> What's your goal here...? To see which EDOs result if you combine
> various interesting commas with septimal meantone...?

To see where commas flip. If they aren't too simple or too complex, I surmise in a reasonable range for the temperament. Most very complex commas seem to be in a reasonable range also.

> Here's another question: what do you make of the fact that the phrase
> "12-equal" denotes, at the bare minimum, a contorted 2-limit map?

Not much.

What
> does "7L5s" denote, anything?

Seven large steps, and five small ones?

Seems like a 2.x subgroup map, where x
> is unspecified, or is there some different perspective I'm lacking?

Maybe there's one I'm lacking; I don't see where this is coming from.

On Tue, Feb 1, 2011 at 3:57 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What's your goal here...? To see which EDOs result if you combine
> > various interesting commas with septimal meantone...?
>
> To see where commas flip. If they aren't too simple or too complex, I surmise in a reasonable range for the temperament. Most very complex commas seem to be in a reasonable range also.

OK, I see. That's pretty similar to what I'm suggesting here then; if
really simple commas are flipping for a certain generator and mapping,
it's probably a bad mapping.

> What
> > does "7L5s" denote, anything?
>
> Seven large steps, and five small ones?

And also that the generator has to vary between 3/5 and 4/7 of the
size of the period, and also that 2/1 is mapped to 1 period, right?
Or, if we're representing this as as 2.g subgroup temperament, where g
represents an anonymous pseudo-generator, that pseudo-certain
intervals must be/not be reversed, specified in terms of g.

It doesn't just specify large and small steps, there's some kind of
quasi-mapping involved here. Just suggesting a train of thought. Maybe
it seems simple to you but for me it's leading to some interesting
things.

> Seems like a 2.x subgroup map, where x
> > is unspecified, or is there some different perspective I'm lacking?
>
> Maybe there's one I'm lacking; I don't see where this is coming from.

See the very last message I wrote to Igs for more of an idea on where
I want to go with this.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 1, 2011 at 2:58 PM, Mike Battaglia <battaglia01@...> wrote:

> After thinking more about this, it seems like the simpler, more
> all-encompassing, and overall better goal is to group all of the MOS
> scales by the -mappings- they have in common. If the mapping turns out
> to be some common comma, ok, but it seems like more often it's going
> to be a common generator.

Aren't commas derived from mappings? So if two temperaments share a mapping, don't they share commas as well?

> So maybe the question is, what should we name the temperament family
> that supports the val [<1 0| <0 1|]?

Kingdom phylum class order family genus species...it works for biologists, why not for us? That val could be a temperament kingdom. I don't know how to decide what to call it...I'd like to see more temperament kingdoms first to figure out a consistent way of naming and differentiating them.

-Igs

> To see where commas flip.

The flipping thing has potential. As everybody knows,
Bosanquet classified ETs according to the number of steps,
positive or negative, certain commas come out as. -C.

On Tue, Feb 1, 2011 at 4:08 PM, cityoftheasleep <igliashon@...> wrote:
>
> > After thinking more about this, it seems like the simpler, more
> > all-encompassing, and overall better goal is to group all of the MOS
> > scales by the -mappings- they have in common. If the mapping turns out
> > to be some common comma, ok, but it seems like more often it's going
> > to be a common generator.
>
> Aren't commas derived from mappings? So if two temperaments share a mapping, don't they share commas as well?

Sure, I just meant that it might be more simple conceptually to think
about common mappings, since vals deal with mappings and not commas.
Maybe since we're throwing in the stuff about reversed intervals, it
makes sense to think about both.

> > So maybe the question is, what should we name the temperament family
> > that supports the val [<1 0| <0 1|]?
>
> Kingdom phylum class order family genus species...it works for biologists, why not for us? That val could be a temperament kingdom. I don't know how to decide what to call it...I'd like to see more temperament kingdoms first to figure out a consistent way of naming and differentiating them.

Temperament kingdom, temperament primitive, whatever. But here's a
snag: what's the "common generator" for 3L4s scales: 5/4, or 11/9? Are
proper 3L4s scales mohajira, or dicot? Maybe here's where we should
take a cue from psychoacoustics and say that it's 5/4.

-Mike

On Tue, Feb 1, 2011 at 4:08 PM, Carl Lumma <carl@...> wrote:
>
> > To see where commas flip.
>
> The flipping thing has potential. As everybody knows,
> Bosanquet classified ETs according to the number of steps,
> positive or negative, certain commas come out as. -C.

You might have to supply some of the missing glue here, because the
stuff you posted where the ET's fit in lines, and then how it turns
out that every linear temperament can be uniquely specified by two
ET's, has left me uncertain about the nature of numbers.

-Mike

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

> So maybe the question is, what should we name the temperament family
> that supports the val [<1 0| <0 1|]?

Pythagorean?

On Tue, Feb 1, 2011 at 4:24 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > So maybe the question is, what should we name the temperament family
> > that supports the val [<1 0| <0 1|]?
>
> Pythagorean?

I think that probably makes more sense than my "diatonic" suggestion.
I still like the idea of calling them "primitives." So mavila is born
out of the pythagorean primitive, or something like that.

Perhaps then the "diatonic" name could denote a specific scale size,
e.g. somewhere between 7 and 10 notes, so we end up with the full name
for 5L2s being "Pythagorean diatonic." This is then differentiated
from 2L5s because any Pythagorean primitive temperament that generates
a 2L5s scale is going to have the apotome reversed, by definition.
Since we have simpler commas reversed with 2L5s than with 5L2s, 2L5s
could get the name "Pythagorean anti-diatonic."

At first glance this pattern seems to hold reasonably well, e.g. 5L5s
gets Blackwood diatonic, and since it's symmetrical I guess there is
no anti-diatonic. Then 5L10s gets you Blackwood chromatic, and 10L5s
gives you Blackwood anti-chromatic, etc.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> The flipping thing has potential. As everybody knows,
> Bosanquet classified ETs according to the number of steps,
> positive or negative, certain commas come out as. -C.

It's more or less the same thing as giving a generator which is a fraction of an octave, it seems to me. Say we look at 29/50. Looking at the continued fraction for 2^(29/50), we probably don't want to consider it an approximation to 145/97 in the 97-limit, so we call it an approximation to 3/2. Another way to look at it is that dividing the patent val <50 79 116 140| by 29 and reducing mod 50 gives <0 1 4 10|, and putting that together with <50 79 116 140|, saturating, and reducing to a normal list gives [<1 0 -4 -13|, <0 1 4 10|], which is meantone, which has a generator of a fifth. The Minkowski basis for the commas of <50 79 116 140| is [81/80, 126/125, 16807/16384], and the [81/80, 126/125] part is meantone. So the flipping comma is 16807/16384, which is positive for meantones with fifths sharper than 2^(29/50), and negative when the fifth is flatter.

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

> Perhaps then the "diatonic" name could denote a specific scale size,
> e.g. somewhere between 7 and 10 notes, so we end up with the full name
> for 5L2s being "Pythagorean diatonic."

According to Wikipedia, "diatonic" means any mode of 5L2s. And that's all. That's also how Blackwood used it.

On Tue, Feb 1, 2011 at 8:08 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Perhaps then the "diatonic" name could denote a specific scale size,
> > e.g. somewhere between 7 and 10 notes, so we end up with the full name
> > for 5L2s being "Pythagorean diatonic."
>
> According to Wikipedia, "diatonic" means any mode of 5L2s. And that's all. That's also how Blackwood used it.

Sure, but doesn't this setup make some kind of intuitive sense as
well? Lots of useful tunings end up fitting into a small,
"pentatonic"-ish sized MOS, then a slightly larger "diatonic" one,
then an even larger "chromatic" one. The 676/675 temperament has a
pentatonic MOS, a 9-note "diatonic" MOS which I believe could be
pretty useful for setting up a tonality, and a 14-note chromatic MOS.
Blackwood has a 5-note pentatonic MOS, a 10 note "diatonic" MOS
(blackwood[10] is clearly doing something closer to meantone[7] rather
than meantone[12]) and a 15-note "chromatic" MOS, etc. When we talk
about miracle, I've heard it said about how you have to use more
"chromatic" harmonies, because the 10-note MOS doesn't cover
everything. It makes sense to me.

There are admittedly lots of scales that have both diatonic and
chromatic features, like whitewood[14], but I still think it's a
pretty good setup to try and shoot for. Maybe instead of going for a
certain scale size, you could somehow work it out another way.

-Mike

On Tue, Feb 1, 2011 at 8:17 PM, Mike Battaglia <battaglia01@...> wrote:
> Maybe instead of going for a certain scale size, you could somehow work it out another way.

Here's a way I feel might make sense to distinguish between the two:
average triadic consonance for all triadic permutations in a certain
scale.

"Pentatonic" scales, which probably shouldn't really be called
pentatonic scales under this naming convention, tend to have very high
average triadic consonance. It's pretty hard to find a sour triad in
Meantone[5], for example, and the same applies with Island[5] (which
is the tentative name I gave to the 676/675 2.3.13/10 subgroup
temperament) and Blackwood[5]. Moving to meantone[7] makes it a bit
easier, you have a decent blend of consonance and dissonance there,
and same with island[9] and blackwood[10]; these scales all share that
same property, which I feel is a good fit for the "diatonic" name.
Then meantone[12], island[14], blackwood[15] have more random triads
being consonant than dissonant, and this is chromatic; you have to
pick and choose the colors you want to use from scales like this
carefully.

Makes sense to me, and to hell with Wikipedia!

-Mike

On Tue, Feb 1, 2011 at 8:24 PM, Mike Battaglia <battaglia01@...> wrote:
> Then meantone[12], island[14], blackwood[15] have more random triads
> being consonant than dissonant

Sorry, I meant more triads being dissonant than consonant. And it
doesn't really have to be triads either, you could probably do the
same with tetrads or probably even dyads.

-Mike

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

It's pretty hard to find a sour triad in
> Meantone[5], for example, and the same applies with Island[5] (which
> is the tentative name I gave to the 676/675 2.3.13/10 subgroup
> temperament) and Blackwood[5].

I was meaning to ask about that. I thought the idea was that "Island" was the name for the whole 676/675 complex, "Barbados" was the subgroup temperament above, and "Parizekmic" Petr's temperment. So what is it?

On Tue, Feb 1, 2011 at 8:42 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> It's pretty hard to find a sour triad in
> > Meantone[5], for example, and the same applies with Island[5] (which
> > is the tentative name I gave to the 676/675 2.3.13/10 subgroup
> > temperament) and Blackwood[5].
>
> I was meaning to ask about that. I thought the idea was that "Island" was the name for the whole 676/675 complex, "Barbados" was the subgroup temperament above, and "Parizekmic" Petr's temperment. So what is it?

Wait, I think you're right. So what is "island" then, a "clan?"

-Mike

On Tue, Feb 1, 2011 at 8:42 PM, genewardsmith
<genewardsmith@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> > wrote:
> >
> > It's pretty hard to find a sour triad in
> > > Meantone[5], for example, and the same applies with Island[5] > (which
> > > is the tentative name I gave to the 676/675 2.3.13/10 subgroup
> > > temperament) and Blackwood[5].
> >
> > I was meaning to ask about that. I thought the idea was that > "Island" was the name for the whole 676/675 complex, "Barbados" was > the subgroup temperament above, and "Parizekmic" Petr's temperment. > So what is it?
>
> Wait, I think you're right. So what is "island" then, a "clan?"
>
> -Mike

Hi Gene and Mike,

"Parizekmic" is certainly the most appropriate name for Petr's temperament, including my "Bala-ribbon" -c version.
For the linear others I also proposed "Bali-Bala", but nobody seems to care !
(with the "Eq-Bala" triple equal-beating of 15/13, 20/13, and 4/3 solution) :
/tuning/topicId_95448.html#95812

And also proposed "Bala" and "Balasept" alternatives with a 5.7.13.15.21 mapping for the same temperament :

/tuning/topicId_95448.html#95642
/tuning/topicId_95448.html#95687
/tuning/topicId_95448.html#95701

I could not follow very well the whole discussion, but will be interested to know in what boxes you class those and what's the idea of the classification.
- - - - - - -
Jacques

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

> Wait, I think you're right. So what is "island" then, a "clan?"

Island itself is a rank five temperament, which would have a family if anyone wanted to worry about that, but the whole complex is not like anything I've ever tried to name--maybe it's an archipelago, consisting of regular temperaments of various ranks, subgroup temperaments, and linear recurrence scales.

On Wed, Feb 2, 2011 at 9:09 PM, genewardsmith
<genewardsmith@...> wrote:
>
> Island itself is a rank five temperament, which would have a family if anyone wanted to worry about that, but the whole complex is not like anything I've ever tried to name--maybe it's an archipelago, consisting of regular temperaments of various ranks, subgroup temperaments, and linear recurrence scales.

Maybe the val is [<1 2] <0 -2]>, in the 3-limit? That seems to be the
defining characteristic for all of these.

This corresponds better to what we were calling "primitives" and
"kingdoms" before, and what I said are really more like "paradigms." A
clan is a higher-rank temperament with lower-rank children, and a
family is a lower-rank temperament also with equivalent-rank but lower
limit children, and now we're starting from the middle and working
outward. So in the scale tree, in the 5-limit 4L3s scales have a 6/5
generator, and 3L4s scales have a 5/4 generator. Thus 3L1s scales, the
parent for both of these have a generator that could be either 5/4 or
6/5. The parent structure in this case equates 5/4 and 6/5 as a
generator, and all temperaments that have 5/4 or 6/5 generators are
children of this structure - like dicot, magic, kleismic, etc. It's
not a family, and it's definitely not a clan, so while Igs thought it
good to call it a "Kingdom," I think it makes more sense to think of
it as a "paradigm." For 2L1s scales, the paradigm includes all scales
where 45/32, 4/3, 5/4 are generators.

Maybe a kingdom can be used to denote something like where 5/4 is a
generator, without specifying how 3 is mapped, and a paradigm can be
used to lump different kingdoms together. Anyway, I bring it up
because some offshoot of this would seem to apply here, with island
temperament above. This can be used to categorize how things in the
scale tree work, but it also applies here.

I also wonder if a paradigm could be used to denote a structure
whereby meantone is the parent, and 5-limit JI and pythagorean are
children; another case of starting from the middle and branching
outward. I can't figure out if I'm talking about three different
things or one overarching thing here.

-Mike

At 01:08 AM 2/2/11 -0000, genewardsmith wrote:
>
>
>--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
>> Perhaps then the "diatonic" name could denote a specific scale size,
>> e.g. somewhere between 7 and 10 notes, so we end up with the full name
>> for 5L2s being "Pythagorean diatonic."
>
>According to Wikipedia, "diatonic" means any mode of 5L2s. And that's all.
That's also how Blackwood used it.

Wikipedia's definition requires that two half steps be maximally separated
by whole steps. Neither the melodic minor scale (LsLLLLs) nor the
Neapolitan major scale (sLLLLLs) have that property, so they are not
mentioned at <http://www.rev.net/~aloe/music/diatonic.html>.

How wide is Blackwood's definition?

Peace,
Beco dos Gatinhos

--- In tuning@yahoogroups.com, ALOE@... wrote:

> >According to Wikipedia, "diatonic" means any mode of 5L2s. And that's all.
> That's also how Blackwood used it.
>
> Wikipedia's definition requires that two half steps be maximally separated
> by whole steps.

And that, by definition, is 5L2s, which refers only to MOS and not other scales of interest.