error-free badness

>>I say the most powerful maps are the ones with the smallest
>>numbers in them. Sum of abs value would work.
>
>Or, maybe the sum of the abs values of the max and min numbers in
>the map, for a given limit ...

This latter measure is just the length of the chain of generators
needed to play the mapped chord, and has been suggested and used
by Dave Keenan several times.

>Which is better?

Probably Dave's.

>(or divided by the card of the map, if
>you want to compare across limits).

I remember thinking something was wrong with this, but now I
can't think what it was. It would be nice if we could flatten
this measure across limits.

Anywho, does anyone else agree on the utility of this measure?
If the approximations are acceptable, this is very much like
badness. What I'm thinking is, instead of searching comma space
or generator space, why not search map space? For values of
Dave's measure that are reasonable, there aren't that many
possibilities, and each one specifies its own optimum generator.
You find it, calculate the errors, rank them by badness, and
you're done.

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Anywho, does anyone else agree on the utility of this measure?
> If the approximations are acceptable, this is very much like
> badness. What I'm thinking is, instead of searching comma space
> or generator space, why not search map space? For values of
> Dave's measure that are reasonable, there aren't that many
> possibilities, and each one specifies its own optimum generator.
> You find it, calculate the errors, rank them by badness, and
> you're done.

How do you speed up the error-calculation part?

I've been thinking about working directly from wedgies; one way would
involve using even limits--that is, for example the 10-limit, only without octave equivalence. Any comments on that?

>> Anywho, does anyone else agree on the utility of this measure?
>> If the approximations are acceptable, this is very much like
>> badness. What I'm thinking is, instead of searching comma space
>> or generator space, why not search map space? For values of
>> Dave's measure that are reasonable, there aren't that many
>> possibilities, and each one specifies its own optimum generator.
>> You find it, calculate the errors, rank them by badness, and
>> you're done.
>
>How do you speed up the error-calculation part?

I imagine the error-calc would be hard to get below a certain
obvious footprint. So I think we need to get the number of
things to check down. If we search by generators, assume one
of them is a 2:1, and desire a tenth of a cent accuracy, there
are 12,000 temperaments to check. I think there are fewer
7-limit maps spanning 20 generators or less.

>I've been thinking about working directly from wedgies

Cool. Not that I know what a wedgie is, or how to do a
wedge product. Yet.

>one way would involve using even limits--that is, for example the
>10-limit, only without octave equivalence. Any comments on that?

Well, it wouldn't be my preferred way, but it could be okay.

-Carl

In-Reply-To: <200203040803.g2483r130861@satyr.host4u.net>
Carl Lumma wrote:

> Anywho, does anyone else agree on the utility of this measure?
> If the approximations are acceptable, this is very much like
> badness. What I'm thinking is, instead of searching comma space
> or generator space, why not search map space? For values of
> Dave's measure that are reasonable, there aren't that many
> possibilities, and each one specifies its own optimum generator.
> You find it, calculate the errors, rank them by badness, and
> you're done.

If you mean the octave-equivalent map, you first need an algorithm to get
a full description of the temperament from that. It's more difficult than
you might think, but shouldn't be as difficult as Paul and Gene are
suggesting. You might need to guess generators within a half-period. You
can assume the number of periods to the equivalence interval is the GCD of
the map. That won't work with torsion, but you can always enforce GIGO.

The map, period and example generator are enough to fully define a
temperament. In some cases different generators give different
octave-specific mappings that are equally good. So pick one. All you
have to do is try enough generators that you'll know you got the best one.

Now, the problem with this method is that it is complex to calculate.
Take a calculation that would give Miracle -- all 11-limit temperaments
with a complexity no greater than 22. Relative to the lowest-mapped
prime, each prime can take any value between 0 and 22. There are 4
entries in the map, one of which is fixed. So the number of options is
22*22*22 or 22**3. For the total number, each prime could be the
lowest-mapped, which will give slightly less than 4*22**3 options. That
is 42,592 different temperaments which have to be considered. Certainly
possible, but this wasn't a problem anyway.

My method of taking pairs of ETs only needs to consider the best 20 ETs to
get miracle and a load of more complex temperaments. That means it only
has to consider 400 linear temperaments. Enumerating all mappings
requires 100 times as many. On top of that, it's

The general formula, for n dimensions, is n*complexity**(n-1). My top
17-limit temperament has a complexity of 66. To get this by guessing maps
you'd need to consider 6*66**5 = 7,513,995,456 different options, which is
huge. My program gets good results (may miss some) from only those 400
linear temperaments.

Okay, the inconsistent list does give one with a complexity of 34. To get
that, you'd need 6*34**5 = 272,612,544 which is better than before. But
it must be less efficient than a modified version of my program that
considers all versions of inconsistent temperaments.

An alternative approach which is certainly worth trying (and Gene may have
tried it) is to take all mappings for the ETs you're looking at. That
should be quadratic time, same as the ET-pair algorithm, and you can
probably get away with looking at fewer ETs. If a temperament doesn't
include a fairly good ET with fewer than 100 notes, you could consider
that to be badness in itself.

That's partly the logic behind my algorithm. One of the good things about
miracle is that it unifies three 11-limit consistent ETs. If an LT
doesn't even cover two almost-consistent ETs, I probably won't be
interested in it.

Graham

>but you can always enforce GIGO.

GIGO?

>Enumerating all mappings requires 100 times as many.

I'll take your word for it.

>If a temperament doesn't include a fairly good ET with fewer than
>100 notes, you could consider that to be badness in itself.

Could you go over the your reasoning here?

-Carl

In-Reply-To: <200203050710.g257Ad819253@satyr.host4u.net>
Carl Lumma wrote:

> >but you can always enforce GIGO.
>
> GIGO?

Garbage In, Garbage Out

> >If a temperament doesn't include a fairly good ET with fewer than
> >100 notes, you could consider that to be badness in itself.
>
> Could you go over the your reasoning here?

It goes back to constant structures. It's useful to think about a linear
temperament as being like an equal temperament, but with the scale steps
unequal. Miracle's good for this, because you can think of it as
approximating 31, 41 or 72-equal. With Blackjack, you have to be a lot
more aware of the inequality of the intervals, and things like a 9:8 being
smaller than a 10:9 or however it works out. So you think of it as a
subset of 31-equal, or whatever.

Temperaments that aren't mappings of consistent ETs are going to be like
Blackjack forever -- no consistent ET you can get a handle on. You can't
train musicians to think in terms of so many equal steps. It's harder to
guess the relative sizes of two intervals if you can't count scale steps.

If the first consistent ET has more than 100 notes, that's also harder to
think of, but you can always raise the bar to fit your brain. And you can
allow slightly inconsistent temperaments like 46 in, which is an option my
program already has.

Beyond all this, I'm thinking about how we can add melodic criteria to the
search. For example, magic isn't so good in practice because you need 19
notes to get a sensible MOS. There's nothing close to proper in the 7+/-2
range. Miracle has the decimal scale almost in that range, so it works a
lot better. The Pelog-type temperament has a 7 note MOS, and so is
favoured despite having a relatively poor 5-limit approximation. These
small scales will tend not to be consistent, but searching for any
remotely sensible mapping with between 5 and 12 notes might be productive.

Graham

--- In tuning-math@y..., graham@m... wrote:

> Beyond all this, I'm thinking about how we can add melodic criteria
to the
> search. For example, magic isn't so good in practice because you
need 19
> notes to get a sensible MOS. There's nothing close to proper in
the 7+/-2
> range. Miracle has the decimal scale almost in that range, so it
works a
> lot better.

don't forget mohajira!

> The Pelog-type temperament has a 7 note MOS, and so is
> favoured despite having a relatively poor 5-limit approximation.
These
> small scales will tend not to be consistent,

?

> but searching for any
> remotely sensible mapping with between 5 and 12 notes might be
productive.

i agree (i think).

Thanks for the gloss. Processing.

>Beyond all this, I'm thinking about how we can add melodic criteria to
>the search. For example, magic isn't so good in practice because you
>need 19 notes to get a sensible MOS. There's nothing close to proper
>in the 7+/-2 range. Miracle has the decimal scale almost in that range,
>so it works a lot better. The Pelog-type temperament has a 7 note MOS,
>and so is favoured despite having a relatively poor 5-limit
>approximation. These small scales will tend not to be consistent, but
>searching for any remotely sensible mapping with between 5 and 12 notes
>might be productive.

I think this would be a mistake. It's a fine idea to say what the
smallest MOS is that contains the map, as you do now, but our idea
of what makes an LT good is far better-formed than our idea of what
makes a melody good. For example, kleismic has an MOS at 10, or
something, which is improper and not good melodically, but there's
an 8-tone non-MOS chain which is not MOS and works melodically.

-Carl

>but there's an 8-tone non-MOS chain which is not MOS and works
>melodically.

"is proper" was supposed to be in there somewhere

-Carl

In-Reply-To: <a63p45+tjdn@eGroups.com>
Me:
> > The Pelog-type temperament has a 7 note MOS, and so is
> > favoured despite having a relatively poor 5-limit approximation.
> These
> > small scales will tend not to be consistent,

Paul:
> ?

The smallest 11-limit consistent ET is 22. If we're going to choose
temperaments from ET mappings around 7+/-2, they can't be consistent when
we get beyond the 9-limit.

Graham

In-Reply-To: <200203060123.g261NaN04450@satyr.host4u.net>
Carl Lumma wrote:

> I think this would be a mistake. It's a fine idea to say what the
> smallest MOS is that contains the map, as you do now, but our idea
> of what makes an LT good is far better-formed than our idea of what
> makes a melody good. For example, kleismic has an MOS at 10, or
> something, which is improper and not good melodically, but there's
> an 8-tone non-MOS chain which is not MOS and works melodically.

If an LT's being used for melody, our idea of what makes it good can't be
so well formed after all. Isn't it something we should be thinking about?

Graham

>> I think this would be a mistake. It's a fine idea to say what the
>> smallest MOS is that contains the map, as you do now, but our idea
>> of what makes an LT good is far better-formed than our idea of what
>> makes a melody good. For example, kleismic has an MOS at 10, or
>> something, which is improper and not good melodically, but there's
>> an 8-tone non-MOS chain which is not MOS and works melodically.
>
>If an LT's being used for melody, our idea of what makes it good can't
>be so well formed after all. Isn't it something we should be thinking
>about?

Paul likes tetrachordality, I like propriety, and there are reasons to
talk about MOS. Let's show these properties, but not rank by them:

() There's a singular mathematical beauty about complexity vs. good
approximations that we want to expose, and which is completely
independent of anything melodic.

() Some of the melodic properties mentioned above are not invariants
of a map. Propriety can change depending on the number of tones, and
as Dave has pointed out, it can change wildly with changes to the
generator size.

-Carl

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a63p45+tjdn@e...>
> Me:
> > > The Pelog-type temperament has a 7 note MOS, and so is
> > > favoured despite having a relatively poor 5-limit
approximation.
> > These
> > > small scales will tend not to be consistent,
>
> Paul:
> > ?
>
> The smallest 11-limit consistent ET is 22. If we're going to
choose
> temperaments from ET mappings around 7+/-2, they can't be
consistent when
> we get beyond the 9-limit.

this is a really strange way of thinking about this. i guess we're on
totally different wavelengths. maybe i'm stuck thinking of
temperaments in terms of commas -- it seems to be less 'slippery'
way, anyhow.