Modalities

Cameron has been opining about rank two temperaments as "modalities" of rank one temperaments. I'm still not sure what exactly is up his nose, but it got me to wondering if rank three temperaments are modalites of rank two temperaments.

A rank two temperament such as meantone or miracle can serve to structure a rank one temperament such as 31 or 72 equal. This can happen in various ways. For one thing, if the rank two temperament is generated by octave and another interval ("generator") then the interval classes of the rank one temperament can be sorted into nearer and father in terms of generator steps, or "complexity". So, in meantone, fifths and thirds are relatively simple, and 7/4 more complex, and so if we are using it in 31et we may expect an emphasis on triads, even though 31et has just as much 7 in it as it does 3 and 5. We can also use the temperament to define MOS scales which go naturally with its use, and to define two-dimensional "generalized" keyboards which work with the temperament in a natural way, using less complex intervals as steps.

For rank three temperaments, scales are not so simple, involving at least three different sizes of scale steps as with 5-limit JI, and the three dimensional keyboards one can construct conceptually don't seem very practical. Sorting interval classes into near and far can be done, but the question of just how it should be done raiss a lot of tuning-math style issues. As an example of what "modality" might mean even so, let's consider septimal meantone.

81/80, 126/125 and 225/224 are all tempered out by septimal meantone. If we temper out 225/224, then 7 ~ 2^(-5) 3^2 5^2, and so a square-shaped Euler genus is a natural scale to temper, with the digaonal pointing right along the direction of the sevens and so allowing a lot of septimal harmony to be worked in. If we take the
genus which consists of the divisors of 3^5 5^5, we obtain 36 notes per octave, which collapses to 31 notes if we temper out 81/80 also.
We may then feel motivated to tweak it further to 31 equal, but we still have a two-dimensional picture with the Euler genus which perhaps counts as a modality.

If we turn now to the 126/125 modality, we find that
7 ~ (5/3)^3 (3/2), and so now constructing a scale such as
(5/3)^i (3/2)^j, i from 0 to 3N-1, j from 0 to N-1 makes sense. These produce a lot of 15625/15552 kleismas, and regarding 126/125 planar ("starling") temperament as a "modality" of keemun/kleismic may make more sense. It can also be seen as a "modality" of myna, sensi or diaschismic. If you add a little more 3 to the mix, for instance
by (5/3)^i (3/2)^j with i from 0 to 5, j from 0 to 3, it makes more sense in terms of meantone, with its emphasis on the fifth. That scale has 24 tones, which reduce to 19 on tempering out 81/80.

Since 81/80 is 5-limit, as a modality of septimal meantone it has the circle of fifths along one axis,and the 7 added separately, for instance by means of 7/5, where six six-note scales stacked by 7/5 gets us up to 36 notes again.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Cameron has been opining about rank two temperaments as >"modalities" of rank one temperaments. I'm still not sure what >exactly is up his nose, but it got me to wondering if rank three >temperaments are modalites of rank two temperaments.

Actually, I specifically said ALL ranks but anyway, rank two as a modality of rank one is the example I had in mind.
>
> A rank two temperament such as meantone or miracle can serve to >structure a rank one temperament such as 31 or 72 equal. This can >happen in various ways. For one thing, if the rank two temperament >is generated by octave and another interval ("generator") then the >interval classes of the rank one temperament can be sorted into >nearer and father in terms of generator steps, or "complexity". So, >in meantone, fifths and thirds are relatively simple, and 7/4 more >complex, and so if we are using it in 31et we may expect an emphasis >on triads, even though 31et has just as much 7 in it as it does 3 >and 5. We can also use the temperament to define MOS scales which go >naturally with its use, and to define two-dimensional "generalized" >keyboards which work with the temperament in a natural way, using >less complex intervals as steps.

EXACTLY. That's is exactly what is implied in "functions as a modality". That is why I asked for subsets of several temperaments using the same EDO. Order of appearance, hierarchy, not just "the stuff", are standard implications of "modality", in all fields AFAIK.
>
> For rank three temperaments, scales are not so simple, involving at >least three different sizes of scale steps as with 5-limit JI, and >the three dimensional keyboards one can construct conceptually don't >seem very practical. Sorting interval classes into near and far can >be done, but the question of just how it should be done raiss a lot >of tuning-math style issues. As an example of what "modality" might >mean even so, let's consider septimal meantone.
>
> 81/80, 126/125 and 225/224 are all tempered out by septimal >meantone. If we temper out 225/224, then 7 ~ 2^(-5) 3^2 5^2, and so >a square-shaped Euler genus is a natural scale to temper, with the >digaonal pointing right along the direction of the sevens and so >allowing a lot of septimal harmony to be worked in. If we take the
> genus which consists of the divisors of 3^5 5^5, we obtain 36 notes >per octave, which collapses to 31 notes if we temper out 81/80 also.
> We may then feel motivated to tweak it further to 31 equal, but we >still have a two-dimensional picture with the Euler genus which >perhaps counts as a modality.
>
> If we turn now to the 126/125 modality, we find that
> 7 ~ (5/3)^3 (3/2), and so now constructing a scale such as
> (5/3)^i (3/2)^j, i from 0 to 3N-1, j from 0 to N-1 makes sense. These produce a lot of 15625/15552 kleismas, and regarding 126/125 planar ("starling") temperament as a "modality" of keemun/kleismic may make more sense. It can also be seen as a "modality" of myna, sensi or diaschismic. If you add a little more 3 to the mix, for instance
> by (5/3)^i (3/2)^j with i from 0 to 5, j from 0 to 3, it makes more sense in terms of meantone, with its emphasis on the fifth. That scale has 24 tones, which reduce to 19 on tempering out 81/80.
>
> Since 81/80 is 5-limit, as a modality of septimal meantone it has the circle of fifths along one axis,and the 7 added separately, for instance by means of 7/5, where six six-note scales stacked by 7/5 gets us up to 36 notes again.
>

Looks like a good fundamental concept to me. I was quite taken aback that it wasn't immediately clear- the term is used, stretched from its older meanings in relation to the traditional modes, in other muscial organizations as well, such as "set theory", and this very afternoon in the library I found that it is used in "noise" and musique concrete kinds of organizations as well.

On 21 April 2010 04:10, cameron <misterbobro@...> wrote:

> Actually, I specifically said ALL ranks but anyway, rank two as
> a modality of rank one is the example I had in mind.

What you said was (slightly cleaned up):

"Someone said there's a big difference between 31 and 1/4 comma? . . .
One is a big pile of intervals on a grid, the other is a modality.
Yes, there is a very big difference."

Doesn't that mean one is rank 1 and the other is rank 2, but only one
is a modality? That's what I've been trying to understand for the
last 4 days.

> EXACTLY. That's is exactly what is implied in "functions as a modality".
> That is why I asked for subsets of several temperaments using the
> same EDO. Order of appearance, hierarchy, not just "the stuff",
> are standard implications of "modality", in all fields AFAIK.

So, back to your original remark, how is quarter comma meantone better
than 31-equal? Can't you just treat 31-equal as a meantone?

http://en.wikipedia.org/wiki/Modality_%28theology%29

"Modality in Protestant and Catholic Christian theology, is the
structure and organization of the local or universal church. In
Catholic theology, the modality is the universal Catholic church. In
Protestant theology, the modality is variously described as either the
universal church (that is, all believers) or the local church."

Graham

I think he's trying to say that in general, rank x+1 temperaments are
"modalities" of rank x temperaments that "contain" them. The point he
was making is that by itself, 31-tet is conceptually almost
meaningless since you don't see how any of the notes relate to one
another. It's only when you start to view it as a type of meantone (or
whatever your favorite temperament is that 31-tet contains) that it
starts to get any type of structure to it.

A comparable analogy would be using 12-tet for atonal, serialist music
vs using it for tonal, "meantone"-based music. He's saying that if you
just threw 31 notes into the hands of a novice, they wouldn't have any
clue what to do with it and would just start screwing around with
random intervals, find most of them dissonant, and give up. It's when
you start to use regular mapping and see what rank-2 temperaments are
"modalities" of 31-tet that you can then proceed to make music with
it. And by extension, meantone itself is useless until you start to
think of it as a way of using 5-limit JI. So 5-limit JI would be a
"modality" of meantone, which would also be a "modality" of 31-tet.

At least that's what I think he's saying.

-Mike

On Wed, Apr 21, 2010 at 2:29 AM, Graham Breed <gbreed@...> wrote:
>
>
>
> On 21 April 2010 04:10, cameron <misterbobro@...> wrote:
>
> > Actually, I specifically said ALL ranks but anyway, rank two as
> > a modality of rank one is the example I had in mind.
>
> What you said was (slightly cleaned up):
>
> "Someone said there's a big difference between 31 and 1/4 comma? . . .
> One is a big pile of intervals on a grid, the other is a modality.
> Yes, there is a very big difference."
>
> Doesn't that mean one is rank 1 and the other is rank 2, but only one
> is a modality? That's what I've been trying to understand for the
> last 4 days.
>
> > EXACTLY. That's is exactly what is implied in "functions as a modality".
> > That is why I asked for subsets of several temperaments using the
> > same EDO. Order of appearance, hierarchy, not just "the stuff",
> > are standard implications of "modality", in all fields AFAIK.
>
> So, back to your original remark, how is quarter comma meantone better
> than 31-equal? Can't you just treat 31-equal as a meantone?
>
> http://en.wikipedia.org/wiki/Modality_%28theology%29
>
> "Modality in Protestant and Catholic Christian theology, is the
> structure and organization of the local or universal church. In
> Catholic theology, the modality is the universal Catholic church. In
> Protestant theology, the modality is variously described as either the
> universal church (that is, all believers) or the local church."
>
> Graham
>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 21 April 2010 04:10, cameron <misterbobro@...> wrote:
>
> > Actually, I specifically said ALL ranks but anyway, rank two as
> > a modality of rank one is the example I had in mind.
>
> What you said was (slightly cleaned up):
>
> "Someone said there's a big difference between 31 and 1/4 comma? . . .
> One is a big pile of intervals on a grid, the other is a modality.
> Yes, there is a very big difference."
>
> Doesn't that mean one is rank 1 and the other is rank 2, but only >one
> is a modality? That's what I've been trying to understand for the
> last 4 days.

Yes, without a modality any set of intervals actually is just a pile of intervals. The same pile can be very different things depending on the modality. For example, 12-tET addressed by the modality of 12-tone serialism (in which case it's more logically called 12-edo, with numbers instead of note names, the latter practice being standard) and 12-tET addressed as a meantone tuning.

> So, back to your original remark, how is quarter comma meantone >better
> than 31-equal? Can't you just treat 31-equal as a meantone?

When you treat 31-equal as meantone it becomes meantone. Just like when take so-and-so many steps of 53 as a generator and such-n-such as period, you have Orwell. Other generator? Hanson. Or schismatic.

Why can't you just treat 53-equal as Orwell? Well of course you can. And would you fail to distinguish between this and 53-equal? Of course not.

-Cameron Bobro

> For example, 12-tET addressed by the modality of 12-tone serialism

I was wrong.

-Mike

Yeah, Mike!

I thought, wrongly it seems, that it would be immediately obvious simply by me saying, one's a big pile, one's a modality.

It's a sound idea. That should be clear by now. And, while it wasn't immediately clear here and probably not immediately clear in many places, I guarantee that it is clear elsewhere, where musicians are used to such ideas from specific areas of studies. I know this is so from experience. And right here on pg. 108 of Hugill's Digital Musician what do I see? "Modalities". He extends it here to organizational principles in noise and musique concrete.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I think he's trying to say that in general, rank x+1 temperaments are
> "modalities" of rank x temperaments that "contain" them. The point he
> was making is that by itself, 31-tet is conceptually almost
> meaningless since you don't see how any of the notes relate to one
> another. It's only when you start to view it as a type of meantone (or
> whatever your favorite temperament is that 31-tet contains) that it
> starts to get any type of structure to it.
>
> A comparable analogy would be using 12-tet for atonal, serialist music
> vs using it for tonal, "meantone"-based music. He's saying that if you
> just threw 31 notes into the hands of a novice, they wouldn't have any
> clue what to do with it and would just start screwing around with
> random intervals, find most of them dissonant, and give up. It's when
> you start to use regular mapping and see what rank-2 temperaments are
> "modalities" of 31-tet that you can then proceed to make music with
> it. And by extension, meantone itself is useless until you start to
> think of it as a way of using 5-limit JI. So 5-limit JI would be a
> "modality" of meantone, which would also be a "modality" of 31-tet.
>
> At least that's what I think he's saying.
>
> -Mike
>
>
> On Wed, Apr 21, 2010 at 2:29 AM, Graham Breed <gbreed@...> wrote:
> >
> >
> >
> > On 21 April 2010 04:10, cameron <misterbobro@...> wrote:
> >
> > > Actually, I specifically said ALL ranks but anyway, rank two as
> > > a modality of rank one is the example I had in mind.
> >
> > What you said was (slightly cleaned up):
> >
> > "Someone said there's a big difference between 31 and 1/4 comma? . . .
> > One is a big pile of intervals on a grid, the other is a modality.
> > Yes, there is a very big difference."
> >
> > Doesn't that mean one is rank 1 and the other is rank 2, but only one
> > is a modality? That's what I've been trying to understand for the
> > last 4 days.
> >
> > > EXACTLY. That's is exactly what is implied in "functions as a modality".
> > > That is why I asked for subsets of several temperaments using the
> > > same EDO. Order of appearance, hierarchy, not just "the stuff",
> > > are standard implications of "modality", in all fields AFAIK.
> >
> > So, back to your original remark, how is quarter comma meantone better
> > than 31-equal? Can't you just treat 31-equal as a meantone?
> >
> > http://en.wikipedia.org/wiki/Modality_%28theology%29
> >
> > "Modality in Protestant and Catholic Christian theology, is the
> > structure and organization of the local or universal church. In
> > Catholic theology, the modality is the universal Catholic church. In
> > Protestant theology, the modality is variously described as either the
> > universal church (that is, all believers) or the local church."
> >
> > Graham
> >
>

On 21 April 2010 03:42, genewardsmith <genewardsmith@...> wrote:
> Cameron has been opining about rank two temperaments as
> "modalities" of rank one temperaments. I'm still not sure
> what exactly is up his nose, but it got me to wondering if
> rank three temperaments are modalites of rank two temperaments.

Rank three temperaments obviously relate to rank two temperaments in
the same way that rank two temperaments relate to rank one
temperaments. It's about adding an equal temperament (or val) or
removing a unison vector (or whatever the reader may prefer to call
that).

As ways of thinking, though, I think rank two is the sweet spot. I'm
more likely to think of higher rank scales as lumpy versions of MOSs.
Like 5-limit JI is meantone with comma problems. Or marvel (225:224
planar) is what I get by writing magic temperament and enforcing
correct spelling in tripod notation.

Then again, the tripod scale is planar, so perhaps that proves your
point. If you don't know what it is, see here:

http://x31eq.com/magic/tripod.pdf

or note that these intervals map to it:

5/3
14/9
35/24
4/3
5/4
6/5
16/15
1/1
15/14

Is that using ratios as a modality for a regular temperament? It
shows you can use them with temperaments anyway. One of Kyle Gann's
arguments for JI was that he likes using ratios. That's why I mention
it. They can be easier to work with in a temperament because you have
a choice of which ratio to use for which note, and you can go for the
simpler one.

Graham

Oh- you used the same example with 12-tET I did (we were posting at the same time), cool.

There is something to be careful of: being extremely coherent and concise, I have been careful throughout to say things like "function as a modality..." rather than IS a modality.

Very important distinction. Gene got this, I believe, you see how he went to a lattice and not an EDO in his examples.

Hopefully it is obvious how it would be possible to obviate the "ginormous ET" and "JI vs ET" arguments/misunderstandings by presenting regular temperaments as modalities of various systems.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I think he's trying to say that in general, rank x+1 temperaments are
> "modalities" of rank x temperaments that "contain" them. The point he
> was making is that by itself, 31-tet is conceptually almost
> meaningless since you don't see how any of the notes relate to one
> another. It's only when you start to view it as a type of meantone (or
> whatever your favorite temperament is that 31-tet contains) that it
> starts to get any type of structure to it.
>
> A comparable analogy would be using 12-tet for atonal, serialist music
> vs using it for tonal, "meantone"-based music. He's saying that if you
> just threw 31 notes into the hands of a novice, they wouldn't have any
> clue what to do with it and would just start screwing around with
> random intervals, find most of them dissonant, and give up. It's when
> you start to use regular mapping and see what rank-2 temperaments are
> "modalities" of 31-tet that you can then proceed to make music with
> it. And by extension, meantone itself is useless until you start to
> think of it as a way of using 5-limit JI. So 5-limit JI would be a
> "modality" of meantone, which would also be a "modality" of 31-tet.
>
> At least that's what I think he's saying.
>
> -Mike
>
>
> On Wed, Apr 21, 2010 at 2:29 AM, Graham Breed <gbreed@...> wrote:
> >
> >
> >
> > On 21 April 2010 04:10, cameron <misterbobro@...> wrote:
> >
> > > Actually, I specifically said ALL ranks but anyway, rank two as
> > > a modality of rank one is the example I had in mind.
> >
> > What you said was (slightly cleaned up):
> >
> > "Someone said there's a big difference between 31 and 1/4 comma? . . .
> > One is a big pile of intervals on a grid, the other is a modality.
> > Yes, there is a very big difference."
> >
> > Doesn't that mean one is rank 1 and the other is rank 2, but only one
> > is a modality? That's what I've been trying to understand for the
> > last 4 days.
> >
> > > EXACTLY. That's is exactly what is implied in "functions as a modality".
> > > That is why I asked for subsets of several temperaments using the
> > > same EDO. Order of appearance, hierarchy, not just "the stuff",
> > > are standard implications of "modality", in all fields AFAIK.
> >
> > So, back to your original remark, how is quarter comma meantone better
> > than 31-equal? Can't you just treat 31-equal as a meantone?
> >
> > http://en.wikipedia.org/wiki/Modality_%28theology%29
> >
> > "Modality in Protestant and Catholic Christian theology, is the
> > structure and organization of the local or universal church. In
> > Catholic theology, the modality is the universal Catholic church. In
> > Protestant theology, the modality is variously described as either the
> > universal church (that is, all believers) or the local church."
> >
> > Graham
> >
>

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

>
> Is that using ratios as a modality for a regular temperament? It
> shows you can use them with temperaments anyway. One of Kyle Gann's
> arguments for JI was that he likes using ratios. That's why I mention
> it. They can be easier to work with in a temperament because you have
> a choice of which ratio to use for which note, and you can go for the
> simpler one.
>
>
> Graham
>

Ta-da! You posted this as I was typing my previous post. "Hopefully it should be obvious..." and you typed exactly the point! So we're definitely on the same sheet of paper.

Gotta run, more when I get the chance.

On 21 April 2010 10:35, Mike Battaglia <battaglia01@...> wrote:
> I think he's trying to say that in general, rank x+1 temperaments are
> "modalities" of rank x temperaments that "contain" them.

There we are, it's been confirmed. That's perfectly clear and he
should have said it in the first place. But how is it something we
haven't been talking about on these lists? What happened to all that
stuff about MOSs, regular mappings, and being jumped on every time we
mentioned an ET?

> The point he
> was making is that by itself, 31-tet is conceptually almost
> meaningless since you don't see how any of the notes relate to one
> another. It's only when you start to view it as a type of meantone (or
> whatever your favorite temperament is that 31-tet contains) that it
> starts to get any type of structure to it.

31-tet is a sequence of equally spaced intervals, with full
transposability. It has the structure of homogeneity. It also
implies all the different rank 2 mappings consistent with it, which
makes it much more powerful than meantone. But power can be a
dangerous thing, of course.

> A comparable analogy would be using 12-tet for atonal, serialist music
> vs using it for tonal, "meantone"-based music. He's saying that if you
> just threw 31 notes into the hands of a novice, they wouldn't have any
> clue what to do with it and would just start screwing around with
> random intervals, find most of them dissonant, and give up. It's when
> you start to use regular mapping and see what rank-2 temperaments are
> "modalities" of 31-tet that you can then proceed to make music with
> it. And by extension, meantone itself is useless until you start to
> think of it as a way of using 5-limit JI. So 5-limit JI would be a
> "modality" of meantone, which would also be a "modality" of 31-tet.

If you go up to the 11-limit, 31-equal is almost entirely consonant,
if not that well approximated. I think the wolf fourth/fifth is the
only pair of intervals large enough to be in the 11-limit that isn't.
But, yes, you would be screwing around randomly, and the result would
be atonal.

Another thing: equal temperament implies a mapping, and the mapping is
another thing that could be a modality. With the mapping, you
wouldn't have random intervals, you'd have a hierarchy of intervals
approximating different odd limits. And still atonal melody.

Graham

> On 21 April 2010 10:35, Mike Battaglia <battaglia01@...> wrote:
> > I think he's trying to say that in general, rank x+1 temperaments are
> > "modalities" of rank x temperaments that "contain" them.
>
> There we are, it's been confirmed. That's perfectly clear and he
> should have said it in the first place. But how is it something we
> haven't been talking about on these lists? What happened to all that
> stuff about MOSs, regular mappings, and being jumped on every time we
> mentioned an ET?

I don't think it is, hahaha... it was just an interesting choice of
word, "modality."

> > The point he
> > was making is that by itself, 31-tet is conceptually almost
> > meaningless since you don't see how any of the notes relate to one
> > another. It's only when you start to view it as a type of meantone (or
> > whatever your favorite temperament is that 31-tet contains) that it
> > starts to get any type of structure to it.
>
> 31-tet is a sequence of equally spaced intervals, with full
> transposability. It has the structure of homogeneity. It also
> implies all the different rank 2 mappings consistent with it, which
> makes it much more powerful than meantone. But power can be a
> dangerous thing, of course.

What is "homogeneity?" And what you are saying is not in disagreement
with what I'm saying (or really what Cameron is saying), which is, I
think, that equal temperaments in general are almost useless until you
apply some form of regular mapping to them. And by extension, rank-2
temperaments are useless until you apply some form of regular mapping
to them, and so on until you eventually reach JI. Except I think I
agree more with what you said a few messages ago, which is that
5-limit JI makes more sense when viewed as a type of meantone with
comma problems. I came to the same conclusion about four days ago, and
notions of any theory for the foundations of music that would make any
sort of sense were basically destroyed as a result.

> Another thing: equal temperament implies a mapping, and the mapping is
> another thing that could be a modality. With the mapping, you
> wouldn't have random intervals, you'd have a hierarchy of intervals
> approximating different odd limits. And still atonal melody.

Do equal temperaments themselves imply mappings? I thought the idea
was that you could figure out how to apply a mapping to it in a way
that makes musical sense, but that you could theoretically apply more
than one mapping to an equal temperament. Maybe I have the wrong
understanding.

-Mike

On 21 April 2010 12:28, Mike Battaglia <battaglia01@...> wrote:

> What is "homogeneity?" And what you are saying is not in disagreement
> with what I'm saying (or really what Cameron is saying), which is, I
> think, that equal temperaments in general are almost useless until you
> apply some form of regular mapping to them. And by extension, rank-2
> temperaments are useless until you apply some form of regular mapping
> to them, and so on until you eventually reach JI. Except I think I
> agree more with what you said a few messages ago, which is that
> 5-limit JI makes more sense when viewed as a type of meantone with
> comma problems. I came to the same conclusion about four days ago, and
> notions of any theory for the foundations of music that would make any
> sort of sense were basically destroyed as a result.

"Homogeneity" means "all the same".

Yes, regular mappings are where it's at. You know I identified the
regular mapping paradigm, which is a different way of thinking about a
load of different things. Maybe a paradigm is a set of modalities.
So if you follow that paradigm (as you seem to) it's natural to start
looking for mappings into whatever scales you happen to be dealing
with. Plenty of composers, though, use equal temperaments as is, and
have no need of our mappings. They even pretend they aren't
approximating any ratios. And if you give them a mapping, they won't
know what to do with it, because it isn't what they were looking for.
That's a sign they're operating under a different paradigm. As are
just intonationists, but coming from the other direction.

Yes, you can view 5-limit JI as a meantone with comma problems. But
you can also view it as a lumpy magic, in which case you can go
further before you hit the commas. It's useful to have different
mappings to apply.

Again, the paradigm shift. Musicians are taught to think in meantone,
whether they know it or not. Tell them they can have two different
sizes of tones, and they won't know what to do with them. They'll
only see problems or complications. Tell them they can use other
mappings and they won't understand. They assume the meantone mapping
has to be applied without thinking about it. Or, after breaking free
of meantone, they end up with atonality because there's no other
structure to apply.

> Do equal temperaments themselves imply mappings? I thought the idea
> was that you could figure out how to apply a mapping to it in a way
> that makes musical sense, but that you could theoretically apply more
> than one mapping to an equal temperament. Maybe I have the wrong
> understanding.

An equal temperament, as some of us define it, implies a mapping from
primes to tempered intervals. An equal division of the octave, at
least in theory, needn't. Those are the definitions. So in 12 note
equal temperament a 3:2 maps to 7 steps. That's different to applying
a rank 2 mapping to an equal temperament, which is like thinking of
two equal temperaments at the same time.

Some EDOs can have more than one mapping. 24-equal, for example, in
the 7-limit. It's related to inconsistency.

Another regular mapping idea is that the tuning follows from the
mapping. Once you temper out enough commas to get an equal
temperament, the scale steps will be roughly equal. And if you apply
a rank 2 mapping to an equal temperament, you'll naturally end up with
an MOS. That's one reason the mappings are so important. Whether the
exact tuning is equally tempered, a rank 2 TOP, or a JI periodicity
block is less important than knowing what mappings apply.

Graham

> "Homogeneity" means "all the same".

Yes, but do you mean it in the sense that the structure is the same in any key?

> Yes, regular mappings are where it's at. You know I identified the
> regular mapping paradigm, which is a different way of thinking about a
> load of different things. Maybe a paradigm is a set of modalities.
> So if you follow that paradigm (as you seem to) it's natural to start
> looking for mappings into whatever scales you happen to be dealing
> with. Plenty of composers, though, use equal temperaments as is, and
> have no need of our mappings. They even pretend they aren't
> approximating any ratios. And if you give them a mapping, they won't
> know what to do with it, because it isn't what they were looking for.
> That's a sign they're operating under a different paradigm. As are
> just intonationists, but coming from the other direction.

Right, as far as paradigms are concerned, yes. As far as
psychoacoustics and the foundations of music go, the "pretending they
aren't approximating any ratios" approach is "wrong." But like I said
earlier, clearly temperament has more of a psychoacoustic function
than just to approximate JI... the existence of "puns" being the main
point there.

If I knew nothing about JI, I wouldn't even know that most of these
"puns" were even "puns" at all. Some of them are obvious, like when a
chord moves up in 12-tet minor thirds and ends up back at the octave.
But I would have never known that Cmaj-Am-Dm-Gmaj had any element of
inharmonicity to it at all.

> Again, the paradigm shift. Musicians are taught to think in meantone,
> whether they know it or not. Tell them they can have two different
> sizes of tones, and they won't know what to do with them. They'll
> only see problems or complications. Tell them they can use other
> mappings and they won't understand. They assume the meantone mapping
> has to be applied without thinking about it. Or, after breaking free
> of meantone, they end up with atonality because there's no other
> structure to apply.

Yes, but at the same time, meantone implies its own unique brand of
"music" which JI doesn't have. It's almost like JI is built off of the
real-life harmonic series, and meantone is built off of this slightly
askew inharmonic series. So I suppose I don't fully understand exactly
how meantone relates to 5-limit JI.

My initial understanding was that the point of temperament was to
basically cut down on the amount of notes that you have to deal with.
So when dealing with schismatic temperament, for example, there's no
need to have separate notes to deal with the pythagorean Fb and the
actual 5/4 over a C, since they're so close that the ear "tempers"
them together anyway.

The fact that the syntonic pun carries a musical meaning that doesn't
seem to exist in JI basically destroyed that hypothesis, and now I'm
back to square 1 as far as understanding how music works.

> > Do equal temperaments themselves imply mappings? I thought the idea
> > was that you could figure out how to apply a mapping to it in a way
> > that makes musical sense, but that you could theoretically apply more
> > than one mapping to an equal temperament. Maybe I have the wrong
> > understanding.
>
> An equal temperament, as some of us define it, implies a mapping from
> primes to tempered intervals. An equal division of the octave, at
> least in theory, needn't. Those are the definitions. So in 12 note
> equal temperament a 3:2 maps to 7 steps. That's different to applying
> a rank 2 mapping to an equal temperament, which is like thinking of
> two equal temperaments at the same time.

How do you figure out how many steps map to xxx interval? Just by
rounding it off to the nearest tempered interval? In 12-equal, how
would you map 11:8?

-Mike

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

> Again, the paradigm shift. Musicians are taught to think in meantone,
> whether they know it or not.

Then theoreticians, trying to explain how they were taught to think, come up with the idea of looking at notes in terms of both 7 and 12 and are deeply distressed when someone whispers the forbidden word, "meantone".

On 21 April 2010 13:13, Mike Battaglia <battaglia01@...> wrote:
>> "Homogeneity" means "all the same".
>
> Yes, but do you mean it in the sense that the structure is the same in any key?

A key's already what we can call a modality you're imposing.
(Naturally, it's a mode in a smart suit.) An equal temperament is a
set of notes related only to each other, like Schoenberg's slogan.
You can pick any notes out to be a tune or chord and then transpose
them up or down to wherever you like. That's the nature of an equal
temperament. And Schoenberg's idea was to write in all the keys at
once. That wouldn't make sense in an open ended meantone.

> How do you figure out how many steps map to xxx interval? Just by
> rounding it off to the nearest tempered interval? In 12-equal, how
> would you map 11:8?

This is something my software has to do because I build mappings out
of equal temperaments. What I do is have a fairly clever routine that
finds all the mappings that could apply to a given division of the
octave within a given cutoff. But, yes, you can also round prime
intervals to the nearest whole number of steps or work out the mapping
that gives the lowest error, however you measure it. I'd probably
call 11:8 a tritone in 12-equal. But it's really something that isn't
in 12-equal, and leads you to quartertones.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 21 April 2010 10:35, Mike Battaglia <battaglia01@...> wrote:
> > I think he's trying to say that in general, rank x+1 temperaments are
> > "modalities" of rank x temperaments that "contain" them.
>
> There we are, it's been confirmed. That's perfectly clear and he
> should have said it in the first place. But how is it something we
> haven't been talking about on these lists? What happened to all that
> stuff about MOSs, regular mappings, and being jumped on every time we
> mentioned an ET?

There I reasons why I didn't "say so in the first place". Right off the bat, avoiding jargon. "Rank 1"? Yeah right. More importantly, a modality of an EDO is NOT what a regular temperament "IS". A regular temperament can function that way: that is not its identity.

A regular temperament can function as a modality of a set of rational intervals as well as of an EDO. You and Gene just demonstrated that. First time ever I've seen a clean concise dismissal of the "regular temperament "vs" JI" debate from you guys. Show me anywhere in the history of this list such a clean butchering of that false dichotomy. You won't find it, because the terminology and anscheuung I just introduced haven't been used. You see, describing the same thing in a different way is not an irrelavant or superfluous act. Doing so is probably the bulk of what constitutes "art", do you really think that any-ol' redneck has less first-hand idea of fucking and fighting than Homer? Nope.