question about Magic

I think this may be a question for Monzo or for Gene:

I was looking at the tonalsoft site about the Magic family of tunings

http://tonalsoft.com/enc/m/magic.aspx

where there are versions of magic for 5-limit, 7-limit and 11-limit
JI. My specific question is about the 11-limit information. If I
understand this stuff correctly (and I may not) the 11-limit is based
on integer powers of 2, 3, 5, 7, and 11, so it is a 5 dimensional
system. In the "mapping" column, there is a 2-dimensional mapping

[<1 0 2 -1 6|, <0 5 1 12 -8|]

It seems that to reduce a 5-d system to a 2-d mapping, one would need
to use 3 commas. Obviously, the magic comma is one, but what are the
other two? If there are two others, how were they chosen? If there are
not two others (that is, if I misunderstand the table) how is the
mapping derived?

Thanks,

Bill Sethares

BTW, shouldn't this go on tuning-math?

Keenan

--- In tuning@yahoogroups.com, "Bill Sethares" <sethares@...> wrote:

> It seems that to reduce a 5-d system to a 2-d mapping, one would need
> to use 3 commas. Obviously, the magic comma is one, but what are the
> other two? If there are two others, how were they chosen? If there are
> not two others (that is, if I misunderstand the table) how is the
> mapping derived?

The mapping isn't unique to the temperament, and how Graham constrained
it to make it unique, presuming he did, is a question you need to ask
him.

My way of answering such questions is to first wedge the two "vals" of
the mapping together to get the bival, or wedgie, for the temperament.
This contains information allowing the triprime commas to be written
down, which can be used to reduce to a comma basis. This basis can then
be Tenney-Minkowski reduced; the result is {100/99, 225/224, 245/243}.

Also of interest, and containing the magic comma, is the Hermite comma
sequence: [3125/3072, 875/864, 4125/4096]. The point of this is more
easily seen in terms of exponents:

[|-10 -1 5 0 0>, |-5 -3 3 1 0>, |-12 1 3 0 1>]

Note now that 875/864 has a 7-exponet of one, and 4125/4096 has a 7-
exponent of zero and an 11-exponent of one. They thus show how
to "bridge" up to higher prime limits from the 5-limit.

Bill Sethares wrote:
> I think this may be a question for Monzo or for Gene:
> > I was looking at the tonalsoft site about the Magic family of tunings > > http://tonalsoft.com/enc/m/magic.aspx
> > where there are versions of magic for 5-limit, 7-limit and 11-limit
> JI. My specific question is about the 11-limit information. If I
> understand this stuff correctly (and I may not) the 11-limit is based
> on integer powers of 2, 3, 5, 7, and 11, so it is a 5 dimensional
> system. In the "mapping" column, there is a 2-dimensional mapping
> > [<1 0 2 -1 6|, <0 5 1 12 -8|]
> > It seems that to reduce a 5-d system to a 2-d mapping, one would need
> to use 3 commas. Obviously, the magic comma is one, but what are the
> other two? If there are two others, how were they chosen? If there are
> not two others (that is, if I misunderstand the table) how is the
> mapping derived?

One approach (not necessarily the one that was followed in this case, but it's possible) is to use combinations of ET's to find regular temperaments. Graham Breed has a page with a script that does this. In this case, take 22-ET and 41-ET. These are both temperaments that are consistent in the 11-limit, so you can take the 22-ET version of magic (with a 7-step generator) and the 41-ET version (with a 13-step generator) and combine them. A particular tuning for magic might have a generator closer to 7 steps of 22-ET, or closer to 13 steps of 41-ET, or anywhere in between.

Try this: http://x31eq.com/temper/twoet.html
type "22" in the "first et" box,
"41" in the "second et" box,
and "11" in the "odd limit" box.

If you look at a chart of ET's like the one on monz's Tonalsoft site (http://tonalsoft.com/enc/e/equal-temperament.aspx), and imagine this chart having two more dimensions, you can visualize 11-limit magic as the line connecting 22 and 41 on the chart. Any interval in 11-limit magic can be tuned at least as closely to just as 22-ET or 41-ET, and possibly even closer if the 22-ET and 41-ET deviations are in opposite directions. So it can be convenient to think of this as a "22&41 regular temperament" or 22&41-RT. Actually, most of the common temperaments can be identified in this way. Meantone can be 12&19, 12&19, 19&31, etc.

There's not a unique set of commas, since the sum or difference (in pitch space) of any two commas is another comma, but there are mathematical ways to derive the commas from the mapping or vice versa. There have been discussions on tuning-math about how to do that (which involves wedge products and is rather technical).

Hiya Bill,

Nobody else linked to this, so I thought I would

http://66.98.148.43/~xenharmo/commaseq.htm

-Carl

Thanks to all who replied --

So, it is true that to go from 11-limit JI (a 5-d system) to
a 2-generator representation (a 2-d system), three things are needed.
Herman pointed out that these can be commas
(as I was anticipating) but that they might also be ETs
(thanks Graham, for the on-line software!).

For the 11-limit magic family, Gene showed how to
get a set of commas that work, though
the final answer {100/99, 225/224, 245/243} is a bit odd
because it does not obviously contain the magic comma at all!
I guess this must be a result of the nonuniqueness of the
representation (i.e., there must be other trios of commas
that generate the same space and some of these representations
must contain the magic comma directly).

Gene then showed another tempering (the one using the Hermite comma)
which does contain the magic comma, along with two others...
This makes the procedures a lot clearer... and
I guess the bottom line is that *any* of these are legitimate
generalizations of magic to the 11-limit.

--Bill Sethares

On 7/19/07, Bill Sethares <sethares@ece.wisc.edu> wrote:
> I think this may be a question for Monzo or for Gene:
>
> I was looking at the tonalsoft site about the Magic family of tunings
>
> http://tonalsoft.com/enc/m/magic.aspx
>
> where there are versions of magic for 5-limit, 7-limit and 11-limit
> JI. My specific question is about the 11-limit information. If I
> understand this stuff correctly (and I may not) the 11-limit is based
> on integer powers of 2, 3, 5, 7, and 11, so it is a 5 dimensional
> system. In the "mapping" column, there is a 2-dimensional mapping
>
> [<1 0 2 -1 6|, <0 5 1 12 -8|]
>
> It seems that to reduce a 5-d system to a 2-d mapping, one would need
> to use 3 commas. Obviously, the magic comma is one, but what are the
> other two? If there are two others, how were they chosen? If there are
> not two others (that is, if I misunderstand the table) how is the
> mapping derived?

You are correct in thinking that there must be 3 independent commas,
but you are mistaken in thinking that the particular commas are
uniquely defined. A plane is defined by three points, but there is an
infinite number of triples of points that define the same plane, so
the question "What three points define this plane?" has no definite
answer.

As you say, one of them could be the magic comma 3125/3072, and the
others could be 225/224 and 100/99. Or one could be 3125/3072 and the
others could be 225/224 and 385/384. Or 3125/3072, 875/864, and
100/99.

Or one of the commas doesn't even have to be the magic comma. If you
take 225/224, 875/864, and 100/99, the system produced by tempering
those out is exactly the same system as the one you get by tempering
out any of those other triples.

What's really going on here is that each of an infinite set of related
commas (called the "kernel") is tempered out. In this case, the kernel
has dimension three, so you need at least three commas to "span" it.
It's possible to have three commas that don't span it (for example,
100/99, 10000/9801, and 1000000/970299, because they're all powers of
100/99), but it's easy to find three commas that do span the kernel. I
think they're related by unimodular matrices or something.

Gene has defined a way to get a unique set (or rather sequence) or
commas for a given temperament, called the Hermite comma sequence
(http://66.98.148.43/~xenharmo/commaseq.htm). It always ends up
looking at least as simple as any other set of commas, and they have
the added benifit that chopping of the sequence gives you a
lower-prime-limit version of the system. For example, the sequence
3125/3072, 225/224, 100/99 has this property (I'm not sure it's *the*
Hermite comma sequence, but it has the prime limit property), so the
system defined by just 3125/3072 is 5-limit magic, and 3125/3072 and
225/224 define a 7-limit system.

Keenan

--- In tuning@yahoogroups.com, "Bill Sethares" <sethares@...> wrote:

> Gene then showed another tempering (the one using the Hermite comma)
> which does contain the magic comma, along with two others...
> This makes the procedures a lot clearer... and
> I guess the bottom line is that *any* of these are legitimate
> generalizations of magic to the 11-limit.

I wouldn't call it a tempering myself. One way to think of it is as a
way of notating 11-limit JI in a way where the reduction to magic is
perspicuous. The magic comma, tempered out, gives the 5-limit version
of magic. We can write any magic interval by means of a 5-limit
interval, with magic comma equivalencies. In particular (shown by the
5-part of the factorization, 5^(-1), of the magic comma) we can
express it in terms of just 2 and 5. Then the next two commas of the
Hermite sequence are of the form |x x x 1 0> and |x x x 0 1>, so you
see how they are a basis for extending the notation to the 7 and then
the 11-limit. This is the usual pattern with Hermite comma sequences,
though sometimes the extension needs to be done another way because
thr +-1 exponents are lacking.

Hence, an 11-limit interval is written 2^a 5^b c1^c c2^d c3^e, where
c1, c2, c3 are the three commas of the Hermite sequence. The
corresponding 2-5 interval is just 2^a 5^b, and that interval tuned
to 11-limit magic simply results from your tuning of 2 and 5. Which
you already knew.

Bill Sethares wrote:
> Thanks to all who replied -- > > So, it is true that to go from 11-limit JI (a 5-d system) to
> a 2-generator representation (a 2-d system), three things are needed.
> Herman pointed out that these can be commas
> (as I was anticipating) but that they might also be ETs
> (thanks Graham, for the on-line software!).

No. In general you only need 2 ETs to get the 2-generator representation (what we call a rank 2 temperament). That makes it a more concise (but still not unique) way to specify the temperament compared to the list of commas.

> For the 11-limit magic family, Gene showed how to > get a set of commas that work, though
> the final answer {100/99, 225/224, 245/243} is a bit odd
> because it does not obviously contain the magic comma at all!
> I guess this must be a result of the nonuniqueness of the
> representation (i.e., there must be other trios of commas
> that generate the same space and some of these representations
> must contain the magic comma directly).

Gene's giving the simplest set of commas that work. It's natural that the only 5-limit comma for a system isn't that simple. You get much more freedom the more commas you play with.

> Gene then showed another tempering (the one using the Hermite comma)
> which does contain the magic comma, along with two others...
> This makes the procedures a lot clearer... and
> I guess the bottom line is that *any* of these are legitimate
> generalizations of magic to the 11-limit.

Historically speaking, magic was identified because it came at the top of a 9-limit search. The software didn't require commas, and still doesn't have a routine for finding commas that gives sensible results for magic. The consistent 22&41 mapping is the obvious way to extend it to the 11-limit.

Graham