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I don't think we are going to make any progress on this unless we can

get beyond a badness measure that says the best 5-limit temperament is

one that takes 49 generators before we get a single fifth (because it

has such teensy weensy errors).

This badness measure also says that meantone is only 7th best (or

thereabouts) and thinks that a temperament whose perfect fifth is 758

cents and whose major third is 442 cents is only slightly worse than

meantone (because it only needs 2 generators to get one of these

supposed 1:3:5 chords).

Does anyone really believe this stuff?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I don't think we are going to make any progress on this unless we can

> get beyond a badness measure that says the best 5-limit temperament is

> one that takes 49 generators before we get a single fifth (because it

> has such teensy weensy errors).

This is just like saying we should not regard 2460 as a super-good

5-limit scale because its errors are so small that it could make no practical difference if they were larger, and that given the choice between 53 tones and 2460, 53 seems much more practical. This misses the point, which is that 2460 is very, very good compared to other things *in its size range*. If you compare wildly different values of "g", you are getting into apples and elephants.

> This badness measure also says that meantone is only 7th best (or

> thereabouts) and thinks that a temperament whose perfect fifth is 758

> cents and whose major third is 442 cents is only slightly worse than

> meantone (because it only needs 2 generators to get one of these

> supposed 1:3:5 chords).

> Does anyone really believe this stuff?

Paul has pointed out that ultra-funky scales may have more possibilities than is at first apparent. Again, why compare apples with e coli?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> > I don't think we are going to make any progress on this unless we

can

> > get beyond a badness measure that says the best 5-limit

temperament is

> > one that takes 49 generators before we get a single fifth (because

it

> > has such teensy weensy errors).

>

> This is just like saying we should not regard 2460 as a super-good

> 5-limit scale because its errors are so small that it could make no

practical difference if they were larger, and that given the choice

between 53 tones and 2460, 53 seems much more practical. This misses

the point, which is that 2460 is very, very good compared to other

things *in its size range*. If you compare wildly different values of

"g", you are getting into apples and elephants.

>

> > This badness measure also says that meantone is only 7th best (or

> > thereabouts) and thinks that a temperament whose perfect fifth is

758

> > cents and whose major third is 442 cents is only slightly worse

than

> > meantone (because it only needs 2 generators to get one of these

> > supposed 1:3:5 chords).

>

> > Does anyone really believe this stuff?

>

> Paul has pointed out that ultra-funky scales may have more

possibilities than is at first apparent. Again, why compare apples

with e coli?

Why indeed? But that's exactly what you're doing. You only gave one

list, in which a single badness metric compares all the temperamants

against each other. You didn't give a list of e coli, a list of apples

and a list of elephants, but only the "32 best 5-limit linear

temperaments".

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

You didn't give a list of e coli, a list of apples

> and a list of elephants, but only the "32 best 5-limit linear

> temperaments".

Is it the subject line you object to?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> > I don't think we are going to make any progress on this unless we

can

> > get beyond a badness measure that says the best 5-limit

temperament is

> > one that takes 49 generators before we get a single fifth (because

it

> > has such teensy weensy errors).

>

> This is just like saying we should not regard 2460 as a super-good

> 5-limit scale because its errors are so small that it could make no

practical difference if they were larger, and that given the choice

between 53 tones and 2460, 53 seems much more practical. This misses

the point, which is that 2460 is very, very good compared to other

things *in its size range*. If you compare wildly different values of

"g", you are getting into apples and elephants.

>

> > This badness measure also says that meantone is only 7th best (or

> > thereabouts) and thinks that a temperament whose perfect fifth is

758

> > cents and whose major third is 442 cents is only slightly worse

than

> > meantone (because it only needs 2 generators to get one of these

> > supposed 1:3:5 chords).

>

> > Does anyone really believe this stuff?

>

> Paul has pointed out that ultra-funky scales may have more

possibilities than is at first apparent. Again, why compare apples

with e coli?

I assume we are making these lists of temperaments for people who are

considering making practical musical use of them (as opposed to

say theoretical mathematical, or practical engineering use). Such a

person, searching in a list entitled "5-limit linear temperaments",

can be presumed to want two things:

1. 5-limit harmony, and

2. temperament.

The former implies that s/he wants intervals that _sound_ like some

kind of approximation of ratios of 1, 3 and 5, and their octave

equivalents and inversions.

The latter implies that s/he doesn't want to have to deal with as many

pitches as would be required in a 5-limit _rational_ tuning giving the

same numbers of harmonies.

The two temperaments I singled out would be of no interest to such a

person (except as curiosities). The one, because it isn't 5-limit and

the other because it isn't a temperament, for any practical purpose.

The one has both its 4:5 approximation and its 3:4 approximation

sounding exactly the same as each other. They both sound like 7:9s.

The other requires _more_ notes than a 5-limit rational scale for any

reasonable number of 5-limit harmonies, assuming we use a contiguous

chain of generators.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> The one has both its 4:5 approximation and its 3:4 approximation

> sounding exactly the same as each other. They both sound like 7:9s.

> The other requires _more_ notes than a 5-limit rational scale for any

> reasonable number of 5-limit harmonies, assuming we use a contiguous

> chain of generators.

What you're saying is that the search was too broad, it seems. That could be rectified if there was general agreement it is so by the simple expedient of leaving off the extremes.

>What you're saying is that the search was too broad, it seems. That

>could be rectified if there was general agreement it is so by the

>simple expedient of leaving off the extremes.

I for one do not understand steps*cents for linear temperaments.

Shouldn't it be g*cents?

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> > The one has both its 4:5 approximation and its 3:4 approximation

> > sounding exactly the same as each other. They both sound like

7:9s.

> > The other requires _more_ notes than a 5-limit rational scale for

any

> > reasonable number of 5-limit harmonies, assuming we use a

contiguous

> > chain of generators.

>

> What you're saying is that the search was too broad, it seems. That

could be rectified if there was general agreement it is so by the

simple expedient of leaving off the extremes.

We've been over this before. That's exactly what I'm trying to

convince everyone to accept, not with sharp cutoffs, but gradual

rolloffs.

By choosing suitable values for the parameters k and p in

badness = gens * EXP((cents/k)^p)

we can start with a badness measure that gives exactly the same

ranking as your current measure, and then by tweaking these parameters

we can make those objectionable extreme cases fall off the bottom of

any length best-of list we choose to make.

I don't have time to check it at the moment, but I think if you set

p = 0.24 and k = 1 cent

you will get pretty much the same ranking as for gens^3 * cents.

In fact if you divide it by e (~=2.718) and cube it, you will get a

good approximation to your actual badness numbers. i.e.

gens^3 * cents

~=

(gens * EXP((cents/1)^0.24) / e)^3

but of course dividing by e and cubing doesn't change the ranking, so

they can be omitted. Then gradually increase p and k until both of

those objectionable temperaments (e-coli and elephant) just go off the

bottom of your list of 32 best and see how the rest of them are then

ranked. I guarantee it will make a lot more sense to most people, with

meantone much closer to the top for one thing, and you probably won't

need as many as 32 in the list to cover all the historical ones.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> You didn't give a list of e coli, a list of apples

> > and a list of elephants, but only the "32 best 5-limit linear

> > temperaments".

>

> Is it the subject line you object to?

No it's the badness measure. I actually _want_ a badness measure that

compares _all_ 5-limit temperaments irrespective of their e-coli-ness

or elephant-ness, but actually takes into account that e-coli and

elephants are inherently less interesting or useful than apples (but

in a continuous manner not a discrete one as the analogy of e-coli,

apples and elephants would suggest).

For 5-limit I'm currently using:

badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5)

Where wtd_rms_gens are weighted by log of odd limit.

I think this approaches your badness in the limit where the power (0.5

above) goes to zero and the 7.4 cents does something else (I forget

what).

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

> >What you're saying is that the search was too broad, it seems. That

> >could be rectified if there was general agreement it is so by the

> >simple expedient of leaving off the extremes.

>

> I for one do not understand steps*cents for linear temperaments.

> Shouldn't it be g*cents?

Yes Gene is using gens*cents for linear temperaments, or rather gens^n

* cents in general (different n>=1 for different limits). I don't like

abbreviating the number of generators to "g". That's being

unnecessarily obscure.

>>I for one do not understand steps*cents for linear temperaments.

>>Shouldn't it be g*cents?

>

>Yes Gene is using gens*cents for linear temperaments, or rather gens^n

>* cents in general (different n>=1 for different limits). I don't like

>abbreviating the number of generators to "g". That's being

>unnecessarily obscure.

Okay, thanks, this does answer my question -- he's using the number of

gens in one instance of the map, as opposed to the number in the et

that provides a near-optimal generator size, or something.

Though my particular suggestion was not only this; g is the rms of

gens in a map, or something. I'm still not clear exactly how it's

calculated, or if it's different from what Graham calls complexity.

For the record, my preferred complexity measure is...

(/ (- (max map) (min map)) (card map))

...but anything that levels the field for different limits is

fine, and Gene's already been using g, so...

-Carl

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

> Okay, thanks, this does answer my question -- he's using the number of

> gens in one instance of the map, as opposed to the number in the et

> that provides a near-optimal generator size, or something.

What's the difference?

> Though my particular suggestion was not only this; g is the rms of

> gens in a map, or something. I'm still not clear exactly how it's

> calculated, or if it's different from what Graham calls complexity.

It's a different measure of complexity.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5)

>

> Where wtd_rms_gens are weighted by log of odd limit.

This strikes me as completely ad hoc. Why not a non-fuzzy version, with a sharp cutoff?

>>Okay, thanks, this does answer my question -- he's using the number of

>>gens in one instance of the map, as opposed to the number in the et

>>that provides a near-optimal generator size, or something.

>

>What's the difference?

Meantone has a very compact 5-limit map. You only need 4 gens. In

listening tests I've preferred a generator close to that of 69-et,

though the rms optimum is closer to 31-et IIRC. In either case, why

should we penalize meantone because it takes 31 or 69 gens to yield

an et with the optimum generator?

>>Though my particular suggestion was not only this; g is the rms of

>>gens in a map, or something. I'm still not clear exactly how it's

>>calculated, or if it's different from what Graham calls complexity.

>

>It's a different measure of complexity.

What is different from what??

Once again, I'll list my preferred map complexity measure in unambiguous

mathematical notation. Why don't you and Graham give yours for the

record, so Dave can tell us which one he likes best?

Carl's preferred map complexity measure:

(/ (- (max map) (min map)) (card map))

Gene's preferred map complexity measure:

________________________________________

Graham's preferred map complexity measure:

________________________________________

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> > badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5)

> >

> > Where wtd_rms_gens are weighted by log of odd limit.

>

> This strikes me as completely ad hoc.

Well it isn't. It is designed to fit what I've learned over the years

by corresponding with people on the tuning lists, regarding the

relative usefulness (or interest) in those 5-limit temperaments that

have been known for a long time. For example, I think meantone must be

in the top 3, or the badness measure is nonsense.

I recently adjusted the cents parameter upwards to take into account

Paul Erlich's suggestion that pelog just might exist in the real world

because it is a MOS of a rough 5-limit temperament.

I'd be happy to have lots of people play with those parameters to try

to make the list come out with the temperaments they are familiar

with, in the order they expect.

Another reason it isn't ad hoc. The perceptual "pain" caused by

mistuning is not directly proportional to the error in cents. Even the

best microtonal ear on the planet apparently experiences essentially

zero pain with a 0.5 cent mistuning. Most people aren't significantly

bothered by a 3c mistuning (depending on the interval and how long it

is sustained). But a 30 cent mistuning is so bad that a 40 cent one

could hardly be much worse.

So you could think of the

EXP((error/7.4_cents)^0.5)

part as the "mistuning pain", except that would leave the

"number-of-notes pain" as simply gens, which we both agree isn't

right.

If I take the "number-of-notes pain" as gens^2 then the "mistuning

pain" is

EXP((error/7.4_cents)^0.5) ^2

= EXP((error/1.85_cents)^0.5)

I also designed it so that it would have a choice of parameters that

gave a result very close to the "log flat" measure, so you could use

that as a starting point.

> Why not a non-fuzzy version, with a sharp cutoff?

Because that's not how musicians or composers relate to temperaments.

A temperament doesn't suddenly become of zero interest because it has

overstepped some sharp boundary of harmonic error or number of

generators.

By the way, I think my 3 previous messages in this thread arrived on

the list in the reverse order to the order I sent them. Thanks Yahoo.

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

> Once again, I'll list my preferred map complexity measure in

unambiguous

> mathematical notation. Why don't you and Graham give yours for the

> record, so Dave can tell us which one he likes best?

>

> Carl's preferred map complexity measure:

> (/ (- (max map) (min map)) (card map))

This isn't unambiguous mathematical notation. It's Lisp. Took me a

while to figure that out.

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

> Meantone has a very compact 5-limit map. You only need 4 gens. In

> listening tests I've preferred a generator close to that of 69-et,

> though the rms optimum is closer to 31-et IIRC. In either case, why

> should we penalize meantone because it takes 31 or 69 gens to yield

> an et with the optimum generator?

I don't; "g" has nothing to do with ets per se, and only measures

complexity.

> What is different from what??

Graham uses max error, and I use rms error.

> Once again, I'll list my preferred map complexity measure in

unambiguous

> mathematical notation. Why don't you and Graham give yours for the

> record, so Dave can tell us which one he likes best?

> Gene's preferred map complexity measure:

rms generator steps, times the number of periods in an octave. This

only works for linear temperaments, so I'm not that happy with it.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

For example, I think meantone must be

> in the top 3, or the badness measure is nonsense.

This is like saying 12-et must be in the top three. It's great in its

size range--if that happens to be the range you are interested in. If

it doesn't suit your requirements then it isn't great, whatever

number you come up with for it. What's top or not top depends on what

tone group you are looking at (5-limit, 7-limit?) and what sort of

accuracy you want.

Dave wrote...

>> Carl's preferred map complexity measure:

>> (/ (- (max map) (min map)) (card map))

>

>This isn't unambiguous mathematical notation. It's Lisp. Took me

>a while to figure that out.

It's generic prefix (Polish) notation with normal grouping by

parens. Graham knows scheme now, and you have a paper on the

lambda calculus on your web page, and Gene's a clever guy.

Gene wrote...

>>Meantone has a very compact 5-limit map. You only need 4 gens. In

>>listening tests I've preferred a generator close to that of 69-et,

>>though the rms optimum is closer to 31-et IIRC. In either case, why

>>should we penalize meantone because it takes 31 or 69 gens to yield

>>an et with the optimum generator?

>

>I don't; "g" has nothing to do with ets per se, and only measures

>complexity.

Good!

>>Once again, I'll list my preferred map complexity measure in

>>unambiguous mathematical notation. Why don't you and Graham

>>give yours for the record, so Dave can tell us which one he

>>likes best?

>>

>>Gene's preferred map complexity measure:

>

>rms generator steps, times the number of periods in an octave.

What made you go to rms? Isn't it over-kill?

>This only works for linear temperaments, so I'm not that happy with it.

What else do you want it to work for? Planar temperaments?

-Carl

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

> It's generic prefix (Polish) notation with normal grouping by

> parens. Graham knows scheme now, and you have a paper on the

> lambda calculus on your web page, and Gene's a clever guy.

Not clever enough to figure out why you want to use prefix notation.

> What made you go to rms? Isn't it over-kill?

It's less sensitive to outliers; if a temperament does a lot of

things well and some badly, it still get credit for it.

> >This only works for linear temperaments, so I'm not that happy

with it.

>

> What else do you want it to work for? Planar temperaments?

Of course.

Dave wrote...

>Another reason it isn't ad hoc. The perceptual "pain" caused by

>mistuning is not directly proportional to the error in cents. Even the

>best microtonal ear on the planet apparently experiences essentially

>zero pain with a 0.5 cent mistuning. Most people aren't significantly

>bothered by a 3c mistuning (depending on the interval and how long it

>is sustained). But a 30 cent mistuning is so bad that a 40 cent one

>could hardly be much worse.

Gene's "cents" are already rms, which we've long ago decided is the

best single error measure. Chopping off anything less than .5 is a

hack, and hopefully an un-necessary one.

Gene wrote...

>>For example, I think meantone must be in the top 3, or the badness

>>measure is nonsense.

>

>This is like saying 12-et must be in the top three. It's great in its

>size range-- if that happens to be the range you are interested in. If

>it doesn't suit your requirements then it isn't great, whatever

>number you come up with for it. What's top or not top depends on what

>tone group you are looking at (5-limit, 7-limit?) and what sort of

>accuracy you want.

Dave's just saying that you're not weighting the span of the map

enough, since he considers musical history to be a worthy badness

measure in its own right -- one that selected meantone, diminished,

augmented, over the infinity of temperaments bigger than schismic.

You want something that exposes the pattern in the series of

best temperaments.

It would be nice to see a list with a much stronger penalty

for size, but I can live with a flat measure with a sharp cutoff.

-Carl

>>It's generic prefix (Polish) notation with normal grouping by

>>parens. Graham knows scheme now, and you have a paper on the

>>lambda calculus on your web page, and Gene's a clever guy.

>

>Not clever enough to figure out why you want to use prefix notation.

It's trivial one way or the other, though I personally find it

easier to parse when dealing with ASCI.

>>What made you go to rms? Isn't it over-kill?

>

>It's less sensitive to outliers; if a temperament does a lot of

>things well and some badly, it still get credit for it.

That's exactly what I _don't_ want. I want to know the size of

the chain I need to complete my map. The only reason I divide

by (card map) is so I can compare temperaments at different

limits.

>>>This only works for linear temperaments, so I'm not that happy

>>>with it.

>>

>>What else do you want it to work for? Planar temperaments?

>

>Of course.

There's nothing I want more than to get to planar temperaments,

but we should finish with LTs first! I say this simply because

I suspect if we can't handle LTs, we don't stand a ghost of a

chance with PTs!

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> For example, I think meantone must be

> > in the top 3, or the badness measure is nonsense.

>

> This is like saying 12-et must be in the top three. It's great in

its

> size range--if that happens to be the range you are interested in.

So you're gonna make different lists for different size ranges, are

you?

I think meantone is one of the 3 best overall.

> If

> it doesn't suit your requirements then it isn't great, whatever

> number you come up with for it. What's top or not top depends on

what

> tone group you are looking at (5-limit, 7-limit?) and what sort of

> accuracy you want.

Of course I was meaning 5-limit for meantone. That's what this thread

is about.

I'd rather a single list that takes into account a typical tradeoff

between accuracy number of notes. You can then go looking for your

"size range" within that.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> So you're gonna make different lists for different size ranges, are

> you?

I've tried listing them in order of size, which makes sense to me, though I think Paul didn't like it. What do you think?

genewardsmith wrote:

> Graham uses max error, and I use rms error.

I use RMS error, but max width for complexity.

Graham

dkeenanuqnetau wrote:

> By choosing suitable values for the parameters k and p in

>

> badness = gens * EXP((cents/k)^p)

>

> we can start with a badness measure that gives exactly the same

> ranking as your current measure, and then by tweaking these parameters

> we can make those objectionable extreme cases fall off the bottom of

> any length best-of list we choose to make.

>

> I don't have time to check it at the moment, but I think if you set

> p = 0.24 and k = 1 cent

> you will get pretty much the same ranking as for gens^3 * cents.

Have you tried running <http://microtonal.co.uk/temper/linear.html> with

width*math.exp((error/1200.0)**0.24) as the figure of demerit? It might

be the wrong width, but it's worth a try.

Graham

Carl Lumma wrote:

> Once again, I'll list my preferred map complexity measure in unambiguous

> mathematical notation. Why don't you and Graham give yours for the

> record, so Dave can tell us which one he likes best?

>

> Carl's preferred map complexity measure:

> (/ (- (max map) (min map)) (card map))

That is ambiguous because you haven't defined what map or card mean.

> Graham's preferred map complexity measure:

This is the actual code:

complexity = self.getWidth(self.getWidestInterval(

consonances))*self.mapping[0][0]

Depending on what your map is, it may give the same results up to the 7

limit and ignoring card. Beyond that, the widest interval may not be a

column in the map. So I have a method for finding that, which is a bit

uglier. It either loops over all the consonances and takes the largest

absolute number of generator steps, or loops over the harmonics of a

tonality diamond and takes the max-min. That mapping[0][0] is the number

of periods to the equivalence interval because getWidth() doesn't take

account of that.

Graham

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

> Dave wrote...

> >Another reason it isn't ad hoc. The perceptual "pain" caused by

> >mistuning is not directly proportional to the error in cents. Even

the

> >best microtonal ear on the planet apparently experiences

essentially

> >zero pain with a 0.5 cent mistuning. Most people aren't

significantly

> >bothered by a 3c mistuning (depending on the interval and how long

it

> >is sustained). But a 30 cent mistuning is so bad that a 40 cent one

> >could hardly be much worse.

>

> Gene's "cents" are already rms, which we've long ago decided is the

> best single error measure.

Whether it is rms or max-absolute is irrelevant to my argument. I'm

happy to use minimum rms error as the input to badness (though I think

it's good to know the minimum max-absolute as well). If you're

suggesting that using rms somehow removes the need to apply a

nonlinear "pain" function, then you're mistaken. You've put "cents" in

scare quotes above as if you think the units aren't really cents when

it's rms. They are.

> Chopping off anything less than .5 is a

> hack, and hopefully an un-necessary one.

It is definitely unnecessary and I do not propose to chop anything

off. Gene's use of straight cents gives far too much credit to

temperaments that have very small sub-half-cent errors. It is allowed

to compensate so much for large gens that a temperament with a

complexity of 35.5 gens (rms) can be considered the best 5-limit

temperament! If you were to take a 0.5 cent threshold into account in

a discontinuous manner, you would treat any temperament whose rms

error was less than 0.5 c as if its error _was_ 0.5 c. But I'm not

proposing we do that. My pain function is non-linear but smooth. A

temperament still gets some credit for being sub-half-cent, just not

so much.

> Gene wrote...

> >>For example, I think meantone must be in the top 3, or the badness

> >>measure is nonsense.

> >

> >This is like saying 12-et must be in the top three.

I don't have a problem with that, of course 12-ET is in the top 3 ETs

for 5-limit.

> It's great in its

> >size range-- if that happens to be the range you are interested in.

If

> >it doesn't suit your requirements then it isn't great, whatever

> >number you come up with for it. What's top or not top depends on

what

> >tone group you are looking at (5-limit, 7-limit?) and what sort of

> >accuracy you want.

>

> Dave's just saying that you're not weighting the span of the map

> enough, since he considers musical history to be a worthy badness

> measure in its own right -- one that selected meantone, diminished,

> augmented, over the infinity of temperaments bigger than schismic.

That's a good way of putting it, except I wouldn't say that Gene's not

weighting the complexity enough, I'd say he's failing to level off

(asymptote) with sub-cent errors and not weighting super-20-cent

errors enough.

> You want something that exposes the pattern in the series of

> best temperaments.

>

> It would be nice to see a list with a much stronger penalty

> for size, but I can live with a flat measure with a sharp cutoff.

You don't _have_ to live with it.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>

wrote:

>

> > So you're gonna make different lists for different size ranges,

are

> > you?

>

> I've tried listing them in order of size, which makes sense to me,

though I think Paul didn't like it. What do you think?

I'd rather they were listed in increasing order of "badness", assuming

"badness" actually means something, like badness. Then if I'm looking

for the best temperament whose error is in a particular range of sizes

I'll just go down the list until I find the first one _in_ that range.

The same goes for any other property that might be important to me at

the time, like having a half-octave period or having a single

generator making the perfect fifth.

If you think it's ok to have a badness measure that does not allow

comparison between temperaments whose errors are in different size

ranges, would you also think it ok to have one that didn't allow

comparison between say temperaments whose generators were in different

size ranges. i.e. you couldn't compare temperaments generated by

approximate fourths with those generated by approximate thirds.

If you agree that this would be a somewhat defective badness measure,

then you should know that this is how I view the badness measure you

are using.

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

> Have you tried running <http://microtonal.co.uk/temper/linear.html>

with

> width*math.exp((error/1200.0)**0.24) as the figure of demerit? It

might

> be the wrong width, but it's worth a try.

That's awesome! I had no idea it was that easy!

You might mention on that page that "error" is in octaves, not cents.

Or better still make "error" be in cents.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> > badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5)

> >

> > Where wtd_rms_gens are weighted by log of odd limit.

>

> This strikes me as completely ad hoc. Why not a non-fuzzy version,

with a sharp cutoff?

If you insist on sharp cutoffs (but why aren't _they_ too ad hoc?) I

think you'd be quite safe to ignore temperaments whose rms complexity

is more than 12 gens or whose rms error is more than 30 cents. That

will get rid of all of what I consider junk from your list (all the

unnamed ones, limmal, fourth-thirds and the two that end in "decal").

However, it won't solve the problem that the remaining temperaments

will not be sensibly ranked by complexity^3*error, and it won't solve

the problem that you are missing some temperaments that are IMHO at

least as good as those that would remain.

These temperaments are apparently rejected for the crime of having

medium errors (around 5 cents) and medium complexity (around 6 gens).

badness = complexity^3*error favours the extremes over these.

The four I know of have an octave period. Two of these might never be

generated by your algorithm, since you seemed to want to deny that

they were anything but a repeat of meantone last time they were

mentioned, despite the fact that they have different size MOS and work

in different ETs.

Here are the missing four.

Mapping Gen

to 3,5 (cents) Description MOS or ET sizes (improper)

------------------------------------------------------------------

[ 4, 9] 176.3 minimal diesic 6 7 (13 20 27) 34 (41 75 109 ...)

[-4, 3] 126.2 16875/16384 9 10 19 (29 48 67 86 105 ...)

[ 2, 8] 348.1 half meantone-fifth 7 (10 17 24) 31 (38 69 100)

[-2,-8] 251.9 half meantone-fourth 5 (9 14) 19 (24 43 62) 81 (100

...)

It turns out that complexity*EXP((error/1_cent)^0.24) does not produce

the same ranking as complexity^3*error. Even as gentle a rollof as

that sends most of the junk to where it belongs and puts meantone on

the top of the list where _it_ belongs.

I've made a spreadsheet so you can play with sorting Gene's list (plus

the four above) according to Gene's badness, my badness with variable

parameters, or any badness you care to calculate from the given

information.

http://dkeenan.com/Music/5LimitTemp.xls.zip

14KB zipped Excel spreadsheet with macros

Gene,

Could you humor me and temporarily use badness =

complexity*EXP((error/7.4_cents)^0.5)

in your program and see if we get any others that come out better than

pelogic by this measure but aren't in the list that started this

thread (or my recent spreadsheet).

There may be some with fractional-octave periods that I haven't found.

Also these should just squeak in.

Mapping Gen

to 3,5 (cents) Description MOS ET sizes

-------------------------------------------------------------------

[ 12, 10] 158.5 c half kleismic-minor-third 7 8 15 (23 38) 53 (68 121

...)

[-12,-10] 441.5 c half kleismic-major-sixth (5) 8 11 19 (30 49 68) 87

106

>Whether it is rms or max-absolute is irrelevant to my argument. I'm

>happy to use minimum rms error as the input to badness (though I think

>it's good to know the minimum max-absolute as well). If you're

>suggesting that using rms somehow removes the need to apply a

>nonlinear "pain" function, then you're mistaken. You've put "cents" in

>scare quotes above as if you think the units aren't really cents when

>it's rms. They are.

That's true. If you want pain, it should just be ms.

>> Chopping off anything less than .5 is a

>> hack, and hopefully an un-necessary one.

>

>It is definitely unnecessary and I do not propose to chop anything

>off. Gene's use of straight cents gives far too much credit to

>temperaments that have very small sub-half-cent errors. It is allowed

>to compensate so much for large gens that a temperament with a

>complexity of 35.5 gens (rms) can be considered the best 5-limit

>temperament! If you were to take a 0.5 cent threshold into account in

>a discontinuous manner, you would treat any temperament whose rms

>error was less than 0.5 c as if its error _was_ 0.5 c. But I'm not

>proposing we do that. My pain function is non-linear but smooth. A

>temperament still gets some credit for being sub-half-cent, just not

>so much.

((rms_error/7.4_cents)^0.5)

Well, this just has too many constants for my taste.

>That's a good way of putting it, except I wouldn't say that Gene's not

>weighting the complexity enough, I'd say he's failing to level off

>(asymptote) with sub-cent errors and not weighting super-20-cent

>errors enough.

I think I'd rather have a smooth pain function, like ms, and a

stronger exponent on complexity.

>> You want something that exposes the pattern in the series of

>> best temperaments.

>>

>> It would be nice to see a list with a much stronger penalty

>> for size, but I can live with a flat measure with a sharp cutoff.

>

>You don't _have_ to live with it.

True. But it will take more learning and more dissatisfaction with

Gene and Graham than I'm currently experiencing to make me cook my

own list.

-Carl

>I'd rather they were listed in increasing order of "badness", assuming

>"badness" actually means something, like badness. Then if I'm looking

>for the best temperament whose error is in a particular range of sizes

>I'll just go down the list until I find the first one _in_ that range.

>The same goes for any other property that might be important to me at

>the time, like having a half-octave period or having a single

>generator making the perfect fifth.

I could be wrong, but I don't think you can have it both ways. If

you want small temperaments to be better, your list will be finite.

That's what happened with steps^3cents and ets (right, Gene?).

I think I can feel a two-list paper cooking. First, we say, "in some

meaningful sense, 2971 (or whatever) is as good as 31... there's a

periodicity here... here's a list of the best 20 temperaments up to

10,000 (or something)...". Then, "but increasing the exponent on

steps to yield a finite list meaningful for physical instruments such

as guitars and pianos, we have ...".

-Carl

>http://dkeenan.com/Music/5LimitTemp.xls.zip

>14KB zipped Excel spreadsheet with macros

Hooray! :)

-Carl

>> Carl's preferred map complexity measure:

>> (/ (- (max map) (min map)) (card map))

>

>That is ambiguous because you haven't defined what map or card mean.

That's true. Actually, card is standard set theory stuff.

I just need to define map. I was originally only referring

to the map of the generator, but as Gene points out, you need

to take into account if your other generator is a fraction

of your ie. I'll see if I can't schlep something together.

Now, it's off to brunch!

-C.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> If you insist on sharp cutoffs (but why aren't _they_ too ad hoc?)

Because we know what they mean--we haven't hidden assumptions in a function; we've got it right out in the open.

I

> think you'd be quite safe to ignore temperaments whose rms complexity

> is more than 12 gens or whose rms error is more than 30 cents.

Well, I went up to 50 gens and 50 cents, which seems to be a lot of your complaint.

> The four I know of have an octave period. Two of these might never be

> generated by your algorithm, since you seemed to want to deny that

> they were anything but a repeat of meantone last time they were

> mentioned, despite the fact that they have different size MOS and work

> in different ETs.

And you can't get from one note to the other using 5-limit consonances, so that they aren't authentic 5-limit temperaments. Why complain about me introducing "junk" and then insist on this? Toss 'em, and consider them again at higher limits, where they make sense.

> Here are the missing four.

>

> Mapping Gen

> to 3,5 (cents) Description MOS or ET sizes (improper)

> ------------------------------------------------------------------

> [ 4, 9] 176.3 minimal diesic 6 7 (13 20 27) 34 (41 75 109 ...)

> [-4, 3] 126.2 16875/16384 9 10 19 (29 48 67 86 105 ...)

Theese both have a badness under 1000; do you think a search in the range g < 12, rms < 30 and badness < 1000 would be a good idea? It seems you are saying that would be more relevant.

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

> I could be wrong, but I don't think you can have it both ways. If

> you want small temperaments to be better, your list will be finite.

> That's what happened with steps^3cents and ets (right, Gene?).

You can tweak it a little and not get a finite list, but only a little, and log-flat seems like the right place for an infinite list.

Dave's objections in good measure are that he doesn't *want* an infinite list, with or without microtemperaments on the actual list, because he doesn't want an infinity of micro-micro-micro-temperaments with no practical meaning being theoretically wonderful according to some measure. To me, that shows the measure is working, to Dave, that it's broken.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>

wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

>

> For example, I think meantone must be

> > in the top 3, or the badness measure is nonsense.

>

> This is like saying 12-et must be in the top three. It's great in its

> size range--if that happens to be the range you are interested

in. If

> it doesn't suit your requirements then it isn't great, whatever

> number you come up with for it. What's top or not top depends

on what

> tone group you are looking at (5-limit, 7-limit?) and what sort of

> accuracy you want.

dave is trying to make an important subjective decision for

musicians. gene, by insisting on a log-flat measure, best

permits the musician to make this decision for him/herself.

going too far in both directions doesn't hurt anyone. gene, you

may note, has given his lists in order of complexity (or similar,

but it should be in order of complexity), but not as an overall

ranking. an overall ranking is pretty meaningless outside of a

single musician's desiderata.

so i'm completely with gene on this one.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>

wrote:

> > What made you go to rms? Isn't it over-kill?

>

> It's less sensitive to outliers; if a temperament does a lot of

> things well and some badly, it still get credit for it.

and it seems that the rms measure even agrees with the

heuristic reasonably well -- though i'd prefer a weighted

measure, as i've mentioned before. dave keenan has given the

exact formula for the weighting i want.

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

> That's exactly what I _don't_ want. I want to know the size of

> the chain I need to complete my map.

why insist so vehemently on completeness? you can do

wonderful musical things with an incomplete map.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> --- In tuning-math@y..., "genewardsmith"

<genewardsmith@j...> wrote:

> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> > > --- In tuning-math@y..., "genewardsmith"

<genewardsmith@j...>

> wrote:

> >

> > > So you're gonna make different lists for different size

ranges,

> are

> > > you?

> >

> > I've tried listing them in order of size, which makes sense to

me,

> though I think Paul didn't like it. What do you think?

>

> I'd rather they were listed in increasing order of "badness",

assuming

> "badness" actually means something, like badness. Then if

I'm looking

> for the best temperament whose error is in a particular range

of sizes

> I'll just go down the list until I find the first one _in_ that range.

sounds like a terrible idea.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>

wrote:

> And you can't get from one note to the other using 5-limit

>consonances, so that they aren't authentic 5-limit

>temperaments. Why complain about me introducing "junk" and

>then insist on this? Toss 'em, and consider them again at

>higher limits, where they make sense.

i vote for just making a brief mention of the phenomenon, as

anyone can easily calculate them (and their badness values)

from the "real" ones if desired.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>

wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> > --- In tuning-math@y..., "genewardsmith"

<genewardsmith@j...> wrote:

>

> > So you're gonna make different lists for different size ranges,

are

> > you?

>

> I've tried listing them in order of size, which makes sense to

>me, though I think Paul didn't like it.

hmm? listing in order of complexity? works wonderfully. no need

to make separate lists, dave.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> dave is trying to make an important subjective decision for

> musicians.

This subjective/objective false-dichotomy again? Haven't read enough

Wilber yet? I'd like us to agree on a badness measure that best

represents the collective subjective experiences of lots of musicians.

In the same manner as my favourite definition of "just".

> gene, by insisting on a log-flat measure, best

> permits the musician to make this decision for him/herself.

> going too far in both directions doesn't hurt anyone.

How can you say that!?

I just showed that Gene's list failed to include at least two

temperaments that are _way_ more interesting and useful than

"fourth-thirds" and the one with 49 gens to the fifth. Namely

minimal diesic and the unnamed one with the [-4,3] map and the 126.2

cent generator. They were discriminated against purely because they

were middle-of-the-road in error and complexity.

No matter how far Gene goes with his badness cutoff, I will always be

able to show that he has left off temperaments which any sane musician

would find to be much better than some of those he has included,

unless his error and complexity cutoffs can be adjusted (ad hoc) to

avoid it.

> gene, you

> may note, has given his lists in order of complexity (or similar,

> but it should be in order of complexity), but not as an overall

> ranking. an overall ranking is pretty meaningless outside of a

> single musician's desiderata.

Some overall rankings are a lot more meaningful than others. If you're

going to give a single list and you're going to put the badness

numbers in there, then why _wouldn't_ readers assume they were

applicable overall? I don't think that, not putting the list in

badness order, will be enough.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

> > I'd rather they were listed in increasing order of "badness",

> assuming

> > "badness" actually means something, like badness. Then if

> I'm looking

> > for the best temperament whose error is in a particular range

> of sizes

> > I'll just go down the list until I find the first one _in_ that

range.

>

> sounds like a terrible idea.

Would you mind saying why?

I'm just as likely to be looking for the best temperament whose

generator size is in a particular range or whose period is a

particular fraction of an octave, or whose rms error is in a

particular range, as I am to be looking for one whose number of gens

is in a particular range. Why favour any one of these (and thereby

make the others much more difficult to find) by sorting the list on

it?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > dave is trying to make an important subjective decision for

> > musicians.

>

> This subjective/objective false-dichotomy again? Haven't read

enough

> Wilber yet? I'd like us to agree on a badness measure that

best

> represents the collective subjective experiences of lots of

musicians.

> In the same manner as my favourite definition of "just".

ok. let our paper have two sets of lists then.

> > gene, by insisting on a log-flat measure, best

> > permits the musician to make this decision for him/herself.

> > going too far in both directions doesn't hurt anyone.

>

> How can you say that!?

>

> I just showed that Gene's list failed to include at least two

> temperaments that are _way_ more interesting and useful

than

> "fourth-thirds" and the one with 49 gens to the fifth.

not in their respective ranges of complexity -- and there were

other reasons for excluding "fourth-thirds", as i recall.

> > gene, you

> > may note, has given his lists in order of complexity (or

similar,

> > but it should be in order of complexity), but not as an overall

> > ranking. an overall ranking is pretty meaningless outside of a

> > single musician's desiderata.

>

> Some overall rankings are a lot more meaningful than others.

If you're

> going to give a single list and you're going to put the badness

> numbers in there,

*don't* put the badness numbers in there. the important part of

the paper should be about how temperaments work and why

they're important, not about how to *rank* them!

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

> > > I'd rather they were listed in increasing order of "badness",

> > assuming

> > > "badness" actually means something, like badness. Then

if

> > I'm looking

> > > for the best temperament whose error is in a particular

range

> > of sizes

> > > I'll just go down the list until I find the first one _in_ that

> range.

> >

> > sounds like a terrible idea.

>

> Would you mind saying why?

because the systems of similar complexity should be next to

each other, so a musician who's interested in a particular

complexity range can immediately compare and contrast the

systems in that range.

> I'm just as likely to be looking for the best temperament whose

> generator size is in a particular range

why? seems overly specific. who is likely to have that as their

priority?

> or whose period is a

> particular fraction of an octave,

ditto.

temperaments are just mappings of ji. it's more important,

compositionally, *which* commas vanish than what the

generator is or what the period is. of course a fifth generator is

preferable and that's why i cling to my unpopular idea of a

*weighted* complexity calculation.

> or whose rms error is in a

> particular range,

you can always introduce more error by other means. really one

wants the tunings that *minimize* the error within the musician's

preferred sphere of 'complexity'.

> Why favour any one of these (and thereby

> make the others much more difficult to find) by sorting the list

on

> it?

hopefully this makes it clear. maybe this is why you didn't

understand my liking of log-flat badness -- it's because i've

assumed this is how you'd present the tunings.

>> That's exactly what I _don't_ want. I want to know the size of

>> the chain I need to complete my map.

>

>why insist so vehemently on completeness? you can do

>wonderful musical things with an incomplete map.

I agree, but then it's just a good temperament at a different

limit. If we use the concept of limit, then we should. If

a _few_ temperaments come up as really good except for one

identity, they could be mentioned in a footnote or something.

Otherwise, folks interested in stuff like 7:9:11 can go to

Graham's site.

-Carl

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

> >> That's exactly what I _don't_ want. I want to know the size of

> >> the chain I need to complete my map.

> >

> >why insist so vehemently on completeness? you can do

> >wonderful musical things with an incomplete map.

>

> I agree, but then it's just a good temperament at a different

> limit.

not necessarily.

> If we use the concept of limit, then we should.

?

i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7},

{3,5,7}, and {2,3,5,7}. isn't that right?

>> I agree, but then it's just a good temperament at a different

>> limit.

>

>not necessarily.

Example?

>> If we use the concept of limit, then we should.

>

>?

What it says.

>i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7},

>{3,5,7}, and {2,3,5,7}. isn't that right?

Oh- I didn't know. Okay then, you're not using limit. So you should

really have no reason to list these temperaments, barring the example

requested above.

-Carl

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> ok. let our paper have two sets of lists then.

OK. The mathematicians list and the musicians list. But if we're not

including any badness measure and we're not listing the temperaments

in order of any badness measure, then that may not be necessary. I've

just given cutoffs for Gene, so that he is guaranteed to include my

list in his (assuming my list doesn't grow when Gene finds out what

went wrong with his attempt to generate my list).

> > > gene, by insisting on a log-flat measure, best

> > > permits the musician to make this decision for him/herself.

> > > going too far in both directions doesn't hurt anyone.

> >

> > How can you say that!?

> >

> > I just showed that Gene's list failed to include at least two

> > temperaments that are _way_ more interesting and useful

> than

> > "fourth-thirds" and the one with 49 gens to the fifth.

>

> not in their respective ranges of complexity

That's not relevant. My point is that it _does_ hurt someone. It hurts

the person interested in middle-of-the-road temperaments.

> -- and there were

> other reasons for excluding "fourth-thirds", as i recall.

OK. Just substitute "pelogic" for "fourth-thirds" in the above.

> *don't* put the badness numbers in there. the important part of

> the paper should be about how temperaments work and why

> they're important, not about how to *rank* them!

OK. I can accept that.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Would you mind saying why?

Supposing you saw a list which rated calculators, portable PCs, desktops, workstations, servers, mainframes and supercomputers with a numerical score, which combined price with performance. Would it make much sense to find cheap $15 calculaors next to supercomputers?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

> wrote:

> > Would you mind saying why?

>

> because the systems of similar complexity should be next to

> each other, so a musician who's interested in a particular

> complexity range can immediately compare and contrast the

> systems in that range.

>

> > I'm just as likely to be looking for the best temperament whose

> > generator size is in a particular range

>

> why? seems overly specific. who is likely to have that as their

> priority?

e.g. Folks considering making a LT guitar and wanting to keep close to

standard open string tuning would favour a generator that was a

fourth.

Fourth/fifth generators are favoured in general.

Or someone might be specifically looking for generators that will be

steps of the MOS.

That reminds me, we must list _all_ possible generators less than an

octave, for each temperament. i.e gen, period-gen, period+gen,

2*period-gen, etc.

> > or whose period is a

> > particular fraction of an octave,

>

> ditto.

e.g. Someone might prefer not to have a temperament with multiple

chains, because dealing with that can be an added complexity in

itself.

> temperaments are just mappings of ji. it's more important,

> compositionally, *which* commas vanish than what the

> generator is or what the period is.

It may be so to you, but you can't predict that everyone will find it

so.

> of course a fifth generator is

> preferable and that's why i cling to my unpopular idea of a

> *weighted* complexity calculation.

I support that idea, but that won't help me with actually _finding_

the best fourth/fifth generators in the list.

> > or whose rms error is in a

> > particular range,

>

> you can always introduce more error by other means. really one

> wants the tunings that *minimize* the error within the musician's

> preferred sphere of 'complexity'.

I have done this very thing already, looking thru graham's lists for

microtemperaments suitable for fretting a guitar for some JI scale. I

wanted errors in the range of 2 to 3 cents, no more no less. Keeping

the error above 2 c served to limit the complexity, below 3c limited

the mistuning pain.

>

> > Why favour any one of these (and thereby

> > make the others much more difficult to find) by sorting the list

> on

> > it?

>

> hopefully this makes it clear.

Well a little. But it doesn't matter. If the only way we can move

forward is to agree not to list them in order of _any_ badness,

then I'm happy for them to be listed in order of complexity.

> maybe this is why you didn't

> understand my liking of log-flat badness -- it's because i've

> assumed this is how you'd present the tunings.

Er no. I still don't understand it, except as a mathematicians cutoff

vs a musicians cutoff.

Anyway, it doesn't matter now, if everyone else agrees to the

following.

1. No badness shown.

2. Listed in order of increasing complexity, which is:

neutral thirds

augmented

meantone

pelogic

diminished

porcupine

small diesic

diaschismic

kleismic

16875/16384

twin meantone

half meantone-fifth

half meantone-fourth

wuerschmidt

minimal diesic

1990656/1953125

tiny diesic

schismic

orwell

twin kleismic

half kleismic-minor-third

half kleismic-major-sixth

amt

semisuper

parakleismic

hemithird

and maybe a few others

3. Smith badness < 861

rms complexity < 13.2 gens

rms error < 28.9 cents

You can of course round the numbers up so it doesn't look so contrived

provided that doesn't add more than 2 or 3 extra temperaments.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7},

> {3,5,7}, and {2,3,5,7}. isn't that right?

That's ok by me. But if we're listing these separately (which seems

the right thing to me), then the maximum interval width (which is also

the width of the complete chord - Carl's measure) seems more relevant

to me than the rms interval width.

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

> >> I agree, but then it's just a good temperament at a different

> >> limit.

> >

> >not necessarily.

>

> Example?

orwell with 9 or 13 notes. still 11-limit.

and what if the subset of 'primes' you're thinking about is not

really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> That's not relevant. My point is that it _does_ hurt someone. It

hurts

> the person interested in middle-of-the-road temperaments.

>

> > -- and there were

> > other reasons for excluding "fourth-thirds", as i recall.

>

> OK. Just substitute "pelogic" for "fourth-thirds" in the above.

then i'd definitely disagree with you.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> > Would you mind saying why?

>

> Supposing you saw a list which rated calculators, portable PCs,

desktops, workstations, servers, mainframes and supercomputers with a

numerical score, which combined price with performance. Would it make

much sense to find cheap $15 calculaors next to supercomputers?

>

If the analogy is say

dollars <-> complexity

megaflop.gigabytes <-> 1/error

then badness would be in dollars per megaflop.gigabyte.

Well if that's what your badness is, 1/value-for-money, then yes there

might well be a calculator next to a supercomputer.

Anyway. I'm willing to accept a list ordered by complexity provided

smith-badness is not shown anywhere.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Well if that's what your badness is, 1/value-for-money, then yes

there

> might well be a calculator next to a supercomputer.

I meant to add that that seems perfectly logical to me.

> Anyway. I'm willing to accept a list ordered by complexity provided

> smith-badness is not shown anywhere.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> then the maximum interval width (which is also

> the width of the complete chord - Carl's measure) seems more

relevant

> to me than the rms interval width.

and shouldn't it be the maximum or rms *weighted* interval

width?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > i thought the paper was going to concern {2,3,5}, {2,3,7},

{2,5,7},

> > {3,5,7}, and {2,3,5,7}. isn't that right?

>

> That's ok by me. But if we're listing these separately (which

>seems

> the right thing to me),

yes,

> then the maximum interval width (which is also

> the width of the complete chord - Carl's measure) seems more

> relevant

> to me than the rms interval width.

why is that?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> Anyway. I'm willing to accept a list ordered by complexity

provided

> smith-badness is not shown anywhere.

phew. a consensus emerges?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

> wrote:

>

> > That's not relevant. My point is that it _does_ hurt someone. It

> hurts

> > the person interested in middle-of-the-road temperaments.

> >

> > > -- and there were

> > > other reasons for excluding "fourth-thirds", as i recall.

> >

> > OK. Just substitute "pelogic" for "fourth-thirds" in the above.

>

> then i'd definitely disagree with you.

Perhaps that got a bit muddled. I mean that using a smith badness

cutoff without small enough cutoffs on error and complexity, I would

always find temperaments that any sane musician would consider better

than either pelogic or 49-gens-to-the-fifth. That would be

disadvantaging someone.

Anyway, if Gene uses the right cutoffs on eror and complexity this is

now a moot point.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

> wrote:

> > then the maximum interval width (which is also

> > the width of the complete chord - Carl's measure) seems

more

> relevant

> > to me than the rms interval width.

>

> and shouldn't it be the maximum or rms *weighted* interval

> width?

by the way, my now-famous heursitic for complexity would sort

the 5-limit temperaments by the size of the numerators (or

denominators, or n*d) of the commas. kees van prooijen

webpages seem to suggest clearly that this should be

expressible as a distance measure of some kind. no one else

seems interested in pursuing this obversation, however :(

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> and what if the subset of 'primes' you're thinking about is not

> really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?

{2,3,5/3} defines the same subgroup as {2,3,5}, but {2,3,7/5} is a good example of a subgroup which doesn't fit the missing prime paradigm. I posted a list of such subgroups a while back.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

> > wrote:

> >

> > > That's not relevant. My point is that it _does_ hurt someone.

It

> > hurts

> > > the person interested in middle-of-the-road temperaments.

> > >

> > > > -- and there were

> > > > other reasons for excluding "fourth-thirds", as i recall.

> > >

> > > OK. Just substitute "pelogic" for "fourth-thirds" in the above.

> >

> > then i'd definitely disagree with you.

>

> Perhaps that got a bit muddled. I mean that using a smith

badness

> cutoff without small enough cutoffs on error and complexity,

you mean large enough?

> I would

> always find temperaments

you mean with even larger error and complexity?

> that any sane musician would consider better

> than either pelogic or 49-gens-to-the-fifth.

you lost me.

> That would be

> disadvantaging someone.

>

> Anyway, if Gene uses the right cutoffs on eror and complexity

this is

> now a moot point.

what are the right cutoffs? i favor going well into uninteresting

territory on both ends, to demonstrate how the algorithm

functions (once it's truly an algorithm -- it seems it may be

missing some things currently). best to be explicit about it!

>Supposing you saw a list which rated calculators, portable PCs, desktops,

>workstations, servers, mainframes and supercomputers with a numerical

>score, which combined price with performance. Would it make much sense to

>find cheap $15 calculaors next to supercomputers?

That depends on the price/performance function!

You could be saying so yourself, in fact. There's nothing in your

analogy keeping price to what you want it to be. Dave might think

price is like size (complexity).

-Carl

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> by the way, my now-famous heursitic for complexity would sort

> the 5-limit temperaments by the size of the numerators (or

> denominators, or n*d) of the commas. kees van prooijen

> webpages seem to suggest clearly that this should be

> expressible as a distance measure of some kind. no one else

> seems interested in pursuing this obversation, however :(

Height functions can be thought of as distance measures, but I'm not getting your point.

>If the analogy is say

>

>dollars <-> complexity

>megaflop.gigabytes <-> 1/error

I swear I did not read this before I sent my reply. :)

-Carl

>> i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7},

>> {3,5,7}, and {2,3,5,7}. isn't that right?

>

>That's ok by me. But if we're listing these separately (which seems

>the right thing to me), then the maximum interval width (which is also

>the width of the complete chord - Carl's measure) seems more relevant

>to me than the rms interval width.

That is a good one, but mine is that divided by the number of elements

in your chord (I said "card map", which is the same if you define map

like me, and remember to mulitply by periods in your interval of

equivalence at the end), which would be 3, 3, 3, 3, and 4 above,

respectively.

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>

wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

>

> > by the way, my now-famous heursitic for complexity would

sort

> > the 5-limit temperaments by the size of the numerators (or

> > denominators, or n*d) of the commas. kees van prooijen

> > webpages seem to suggest clearly that this should be

> > expressible as a distance measure of some kind. no one

else

> > seems interested in pursuing this obversation, however :(

>

> Height functions can be thought of as distance measures, but

>I'm not getting your point.

a _lattice_ distance function.

>orwell with 9 or 13 notes. still 11-limit.

I show orwell with a 9-tone MOS, and this map:

[0 7 -3 8 2]

[1 0 0 0 0]

'zthat right? That covers 10 gens in the 5-limit, 11-gens in

the 7-limit, and 11 gens in the 11-limit. You're saying...

[0 -3 2]

[1 0 0]

[1 5 11]

...only takes 5 notes? I know I said "limit", but I meant it

figuratively, not literally. It's an abuse of terminology to

call [1 5 11] the "[1 5 11]-limit", for example, but that's what

I meant.

So if your paper's really going to cover all these (by that I

mean have a list for each one), I'd suggest ranking by

complexity, with a sharp cutoff. Just show the most accurate

three temperaments for each integer of complexity up to 15 or

so. You could seed this with whatever badness measure you wanted,

as long as you let it go up to 500. By then, even the kind that

Dave can always add wouldn't make it up into my list

You can tell I don't care about sharp cutoffs. :)

>and what if the subset of 'primes' you're thinking about is not

>really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?

Fine. (I don't understand what you thought I thought... Maybe

I'm still thinking it! :)

-Carl

>e.g. Folks considering making a LT guitar and wanting to keep close to

>standard open string tuning would favour a generator that was a

>fourth.

Ah, mere mention of the days of Keenan GuitarFrettings^TM make me

shudder. In 6 months, will Dave be cooking up linear-tempered

guitars... or maybe by then, planar-tempered guitars? Mmmmmmm....

Two things where I've always thought Dave Keenan kicks butt: notation

systems, and guitar frettings. Well, I can't really speak to the

latter. Paul- should the people at home start cracking eachother's

skulls open and feasting on the sweet goo inside?

(simpsons reference -- Answer: Yes, Kent.)

Did you ever manage to automate the process at all, Dave?

-Carl

>> wrote:

>>> then the maximum interval width (which is also

>>> the width of the complete chord - Carl's measure) seems more

>>> relevant to me than the rms interval width.

>>

>> and shouldn't it be the maximum or rms *weighted* interval

>> width?

>

>by the way, my now-famous heursitic for complexity would sort

>the 5-limit temperaments by the size of the numerators (or

>denominators, or n*d) of the commas.

Sounds interesting, but I'll have to read up again on your

heuristic a bit to even know. I have the original post here

somewhere, I think....

-Carl

>> and what if the subset of 'primes' you're thinking about is not

>> really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?

>

>{2,3,5/3} defines the same subgroup as {2,3,5}, but {2,3,7/5} is a

>good example of a subgroup which doesn't fit the missing prime

>paradigm. I posted a list of such subgroups a while back.

Hmmm. You once told me it had to be primes. Does this have anything

to do with that?

-Carl

>>Height functions can be thought of as distance measures, but

>>I'm not getting your point.

>

>a _lattice_ distance function.

Does that mean a taxicab one? Whatever it is, if it can do

n*d, who cares?

-Carl

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

> >>Height functions can be thought of as distance measures,

but

> >>I'm not getting your point.

> >

> >a _lattice_ distance function.

>

> Does that mean a taxicab one? Whatever it is, if it can do

> n*d, who cares?

i care because it gives us the natural lattice-based complexity

measure for linear temperaments in the 7-limit and higher,

where a single comma won't do, and a sort of 'angle' between

the two is at work.

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

> (simpsons reference -- Answer: Yes, Kent.)

And people let their children watch that crap.

> Did you ever manage to automate the process at all, Dave?

Only parts of it and not in a user friendly way.

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

> Hmmm. You once told me it had to be primes. Does this have anything

> to do with that?

What had to be primes? The generators? Nope.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> i care because it gives us the natural lattice-based complexity

> measure for linear temperaments in the 7-limit and higher,

> where a single comma won't do, and a sort of 'angle' between

> the two is at work.

Any metric on R^n allows you to define lattices, so I don't see what your angle is.

>> (simpsons reference -- Answer: Yes, Kent.)

>

>And people let their children watch that crap.

Oh, sorry Dave. I forgot you weren't a fan.

I'll stop... it wasn't cool anyway.

-Carl

>>Hmmm. You once told me it had to be primes. Does this have anything

>>to do with that?

>

>What had to be primes? The generators? Nope.

The identities (column vectors of the map, I think).

-Carl

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

> wrote:

> > then the maximum interval width (which is also

> > the width of the complete chord - Carl's measure) seems more

> relevant

> > to me than the rms interval width.

>

> and shouldn't it be the maximum or rms *weighted* interval

> width?

You also asked why max abs? I'll answer that first.

One of the reasons given for using rms was that if there was an

outlier the LT still got credit for the intervals it did well. But if

we're looking at say {1,3,5,7} then if _any_ {1,3,5,7} interval is

very wide, I don't think it is a good {1,3,5,7} temperament. Let it be

found to be a good {2,5,7} temperament or whatever.

As far as the weighting by odd-limit thing goes. I'm reasonably in

favour. But it's a very musician-type "subjective" thing to do and not

at all a mathematician type thing. And I'm not prepared to argue over

it, mainly because I expect the good temperaments will still make it

on the list without it. Also, I have no easy way of finding optimum

generators (which you'd have to do all over again before you can find

the error). Gene would have to do it.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

> you lost me.

Never mind. It doesn't matter now.

> > Anyway, if Gene uses the right cutoffs on eror and complexity

> this is

> > now a moot point.

>

> what are the right cutoffs? i favor going well into uninteresting

> territory on both ends, to demonstrate how the algorithm

> functions (once it's truly an algorithm -- it seems it may be

> missing some things currently). best to be explicit about it!

As I posted twice before:

Smith badness < 861

rms complexity < 13.2 gens

rms error < 28.9 cents

You can of course round the numbers up so it doesn't look so

contrived, provided that doesn't add more than 2 or 3 extra

temperaments.

I wrote...

>I show orwell with a 9-tone MOS, and this map:

>

>[0 7 -3 8 2]

>[1 0 0 0 0]

>

>'zthat right?

Whoops, I think it should be more something like this:

[2] [0 1]

[3] [7 0]

[5] [-3 2]

[7] [8 1]

[11] [2 3]

Is the column out front a way to write that? Certainly we wouldn't

put the identities inside the map, as I first tried?

Gene, when you showed me this, you gave:

[0 1]

[1 1]

[4 0]

for meantone. but shouldn't the last row be:

[4 2]

Graham's script gives:

[0 1]

[1 0]

[4 -4]

-Carl

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

> >orwell with 9 or 13 notes. still 11-limit.

>

> I show orwell with a 9-tone MOS,

and 13 too . . .

and this map:

>

> [0 7 -3 8 2]

> [1 0 0 0 0]

>

> 'zthat right? That covers 10 gens in the 5-limit, 11-gens in

> the 7-limit, and 11 gens in the 11-limit. You're saying...

>

> [0 -3 2]

> [1 0 0]

> [1 5 11]

>

> ...only takes 5 notes?

you lost me.

>I know I said "limit", but I meant it

> figuratively, not literally. It's an abuse of terminology to

> call [1 5 11] the "[1 5 11]-limit", for example, but that's what

> I meant.

is [2 3 7/5] a limit too?

>

> So if your paper's really going to cover all these (by that I

> mean have a list for each one), I'd suggest ranking by

> complexity, with a sharp cutoff. Just show the most accurate

> three temperaments for each integer of complexity up to 15 or

> so.

this is flat badness. i think log-flat is better, and it's what gene's

using.

>>I know I said "limit", but I meant it

>> figuratively, not literally. It's an abuse of terminology to

>> call [1 5 11] the "[1 5 11]-limit", for example, but that's what

>> I meant.

>

>is [2 3 7/5] a limit too?

In the sense that you can give a list for it or tell the reader

to try Graham's script with input X, so that you don't have to

use rms of the "highest interval width" (Graham's term for the

length of the chain).

>> So if your paper's really going to cover all these (by that I

>> mean have a list for each one), I'd suggest ranking by

>> complexity, with a sharp cutoff. Just show the most accurate

>> three temperaments for each integer of complexity up to 15 or

>> so.

>

>this is flat badness.

Which badness formulas produce flat?

>i think log-flat is better, and it's what

>gene's using.

Ok. Is steps^n*cents log-flat for any n > 1?

-Carl

Dave wrote:

> orwell

From George Orwell?

Manuel

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>

> > wrote:

> > > then the maximum interval width (which is also

> > > the width of the complete chord - Carl's measure) seems

more

> > relevant

> > > to me than the rms interval width.

> >

> > and shouldn't it be the maximum or rms *weighted* interval

> > width?

>

> You also asked why max abs? I'll answer that first.

>

> One of the reasons given for using rms was that if there was

an

> outlier the LT still got credit for the intervals it did well. But if

> we're looking at say {1,3,5,7} then if _any_ {1,3,5,7} interval is

> very wide, I don't think it is a good {1,3,5,7} temperament. Let it

be

> found to be a good {2,5,7} temperament or whatever.

it might not -- again, what if it's really a good {2, 3, 7/5}

temperament?

>

> As far as the weighting by odd-limit thing goes. I'm reasonably

in

> favour. But it's a very musician-type "subjective" thing to do and

not

> at all a mathematician type thing. And I'm not prepared to argue

over

> it, mainly because I expect the good temperaments will still

make it

> on the list without it. Also, I have no easy way of finding

optimum

> generators (which you'd have to do all over again before you

can find

> the error).

huh? the optimum generator would be exactly the same as

before, since we're not weighting the 'cents' part of it, just the

'steps' part of it . . .

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> it might not -- again, what if it's really a good {2, 3, 7/5}

> temperament?

Ok let's stick with rms error. We seem to have the most agreement on

that.

> huh? the optimum generator would be exactly the same as

> before, since we're not weighting the 'cents' part of it, just the

> 'steps' part of it . . .

You're right. I got confused. So ask gene to do a run with weighted

rms gens to see if anything new comes up.

Oh and Manuel, Yes that's from George Orwell. The generator for that

temperament is essentially 19/84 oct. Gene's idea.

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

> >>I know I said "limit", but I meant it

> >> figuratively, not literally. It's an abuse of terminology to

> >> call [1 5 11] the "[1 5 11]-limit", for example, but that's what

> >> I meant.

> >

> >is [2 3 7/5] a limit too?

>

> In the sense that you can give a list for it or tell the reader

> to try Graham's script with input X, so that you don't have to

> use rms of the "highest interval width" (Graham's term for the

> length of the chain).

so do you want to propose a modification of the list of cases to be

studied in our paper?

> >> So if your paper's really going to cover all these (by that I

> >> mean have a list for each one), I'd suggest ranking by

> >> complexity, with a sharp cutoff. Just show the most accurate

> >> three temperaments for each integer of complexity up to 15 or

> >> so.

> >

> >this is flat badness.

>

> Which badness formulas produce flat?

this is a question gene can surely answer (if only to say, 'none').

> >i think log-flat is better, and it's what

> >gene's using.

>

> Ok. Is steps^n*cents log-flat for any n > 1?

no. n depends on the number of independent intervals in the map,

i.e., the number of 'terms' in the 'limit'.

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:

> Dave wrote:

>

> > orwell

>

> From George Orwell?

gene calls it that because the generator is about 19/84 octave.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > it might not -- again, what if it's really a good {2, 3, 7/5}

> > temperament?

>

> Ok let's stick with rms error. We seem to have the most agreement

on

> that.

>

> > huh? the optimum generator would be exactly the same as

> > before, since we're not weighting the 'cents' part of it, just

the

> > 'steps' part of it . . .

>

> You're right. I got confused. So ask gene to do a run with weighted

> rms gens to see if anything new comes up.

ok. gene, once again, this means that in the 'gens' calculation, the

number of generators in the 3:1 should be multiplied by log(3), the

number of generators in the 5:3 should be multiplied by log(5), the

number of generators in the 5:1 should be multiplied by log(5). this

will cause temperaments generated by the fifth to look better than

they currently do, relative to those that aren't. this is important

since augmented (especially) and diminished (a little less so) are

far harder for the ear to understand than meantone, even when all are

*tuned* in 12-equal, and the badness values would no longer put

meantone as the 'best'.

>> In the sense that you can give a list for it or tell the reader

>> to try Graham's script with input X, so that you don't have to

>> use rms of the "highest interval width" (Graham's term for the

>> length of the chain).

>

>so do you want to propose a modification of the list of cases to be

>studied in our paper?

I'm just saying I think rms(complexity) smooths over information

I want. You objected that mean(complexity) might be unfair to

temperaments that only did one thing badly. I suggested such

temperaments be listed separately, as doing everything well over

a restricted set of identities.

If I were to say some temperaments are good over some list of

identities, you'd expect I meant all the identities, wouldn't

you? Isn't the idea behind a "limit" that you need the lower

identities to make the higher ones work (not that I think this

is correct)?

>> >i think log-flat is better, and it's what

>> >gene's using.

>>

>> Ok. Is steps^n*cents log-flat for any n > 1?

>

>no. n depends on the number of independent intervals in the map,

>i.e., the number of 'terms' in the 'limit'.

Oh!! Howso?

-Carl

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

> >> In the sense that you can give a list for it or tell the reader

> >> to try Graham's script with input X, so that you don't have to

> >> use rms of the "highest interval width" (Graham's term for the

> >> length of the chain).

> >

> >so do you want to propose a modification of the list of cases to

be

> >studied in our paper?

>

> I'm just saying I think rms(complexity) smooths over information

> I want. You objected that mean(complexity) might be unfair to

> temperaments that only did one thing badly. I suggested such

> temperaments be listed separately, as doing everything well over

> a restricted set of identities.

by identities, you really mean all the consonant intervals, right?

well, i think if you look on a case-by-case basis, you won't be able

to object to what rms does. because only doing 'one thing badly'

isn't really possible in this context. find an example, if you can!

> >> >i think log-flat is better, and it's what

> >> >gene's using.

> >>

> >> Ok. Is steps^n*cents log-flat for any n > 1?

> >

> >no. n depends on the number of independent intervals in the map,

> >i.e., the number of 'terms' in the 'limit'.

>

> Oh!! Howso?

gene derived this from diophantine approximation theory.

>by identities, you really mean all the consonant intervals, right?

Right.

>well, i think if you look on a case-by-case basis, you won't be able

>to object to what rms does. because only doing 'one thing badly'

>isn't really possible in this context. find an example, if you can!

Well, you just gave Orwell, so let's look at it. I still have no

idea about the map. My best guess was:

[2] [0 1]

[3] [7 0]

[5] [-3 2]

[7] [8 1]

[11] [2 3]

Which disagrees with Gene:

>[ 0 1]

>[ 7 0]

>[-3 3]

Going by my map:

max int width card(map) me g

5-limit 10 2 5 7.257

11-limit 11 4 2.75 ??

I have no idea how Gene got 7.257, so I can't fill

g in for the 11-limit.

Still don't know why we can't include 9 in the map.

>>>> Ok. Is steps^n*cents log-flat for any n > 1?

>>>

>>>no. n depends on the number of independent intervals in the map,

>>>i.e., the number of 'terms' in the 'limit'.

>>

>>Oh!! Howso?

>

>gene derived this from diophantine approximation theory.

I remember now.

-Carl

>Going by my map:

>

> max int width card(map) me g

>5-limit 10 2 5 7.257

>11-limit 11 4 2.75 ??

>

>I have no idea how Gene got 7.257, so I can't fill

>g in for the 11-limit.

Dave's getting 7.3 too, with this:

SQRT((E13^2+F13^2+(E13-F13)^2)/3)*1200/L13

Anybody care to explain why this isn't total rubbish? Putting

both the individual gens per idenitity (E13 and F13) and the

total width of the chain (E13-F13) together into the rms calc???

-Carl

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

> >Going by my map:

> >

> > max int width card(map) me g

> >5-limit 10 2 5 7.257

> >11-limit 11 4 2.75 ??

> >

> >I have no idea how Gene got 7.257, so I can't fill

> >g in for the 11-limit.

>

> Dave's getting 7.3 too, with this:

>

> SQRT((E13^2+F13^2+(E13-F13)^2)/3)*1200/L13

>

> Anybody care to explain why this isn't total rubbish? Putting

> both the individual gens per idenitity (E13 and F13) and the

> total width of the chain (E13-F13) together into the rms calc???

carl, if 'identity' is defined as 'consonant interval', then the

*only* thing going in here is the individual gens per identity.

that's all. E13-F13 is the major sixth or minor third.

>carl, if 'identity' is defined as 'consonant interval', then the

>*only* thing going in here is the individual gens per identity.

>that's all. E13-F13 is the major sixth or minor third.

Oh. Well, I'm not sure how that's significant, since in regular

temperaments it will always be the difference of the 3 and 5

mappings.

-Carl

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> ok. gene, once again, this means that in the 'gens' calculation, the

> number of generators in the 3:1 should be multiplied by log(3), the

> number of generators in the 5:3 should be multiplied by log(5), the

> number of generators in the 5:1 should be multiplied by log(5).

And to make the result meaningful (i.e. comparable to the unweighted

values) then after you take the RMS of these weighted values you

should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).

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

> Dave's getting 7.3 too, with this:

>

> SQRT((E13^2+F13^2+(E13-F13)^2)/3)*1200/L13

>

> Anybody care to explain why this isn't total rubbish? Putting

> both the individual gens per idenitity (E13 and F13) and the

> total width of the chain (E13-F13) together into the rms calc???

That's unweighted rms complexity which is

SQRT((gens(1:3)^2+gens(1:5)^2+gens(3:5)^2)/3)*1200/period

where period is in cents and

gens(3:5) = gens(1:5)-gens(1:3), where the gens are signed quantities,

not absolute values. So it's not necessarily the total width.

Max-absolute complexity (your total width) is

MAX(ABS(gens(1:3),ABS(gens(1:5),ABS(gens(3:5))*1200/period

which is equivalent to

ABS(MAX(gens(1:3),gens(1:5))-MIN(gens(1:3),gens(1:5)))*1200/period

By expanding gen(3:5)^2, rms complexity could be calculated as

SQRT((gens(1:3)^2+gens(1:5)^2-gens(1:3)*gens(1:5))*2/3)*1200/period

but who cares.

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

> >carl, if 'identity' is defined as 'consonant interval', then the

> >*only* thing going in here is the individual gens per identity.

> >that's all. E13-F13 is the major sixth or minor third.

>

> Oh. Well, I'm not sure how that's significant, since in regular

> temperaments it will always be the difference of the 3 and 5

> mappings.

right -- so what's your objection?

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

>> >carl, if 'identity' is defined as 'consonant interval', then the

>> >*only* thing going in here is the individual gens per identity.

>> >that's all. E13-F13 is the major sixth or minor third.

>>

>> Oh. Well, I'm not sure how that's significant, since in regular

>> temperaments it will always be the difference of the 3 and 5

>> mappings.

>

>right -- so what's your objection?

None anymore, except that people aren't considering max-absolute

complexity and ms error, with an exponent on complexity that makes

the list finite or one that makes it log-flat supplemented by a

sharp cutoff.

-Carl

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > ok. gene, once again, this means that in the 'gens' calculation,

the

> > number of generators in the 3:1 should be multiplied by log(3),

the

> > number of generators in the 5:3 should be multiplied by log(5),

the

> > number of generators in the 5:1 should be multiplied by log(5).

>

> And to make the result meaningful (i.e. comparable to the

unweighted

> values) then after you take the RMS of these weighted values you

> should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).

why do you want them to be comparable to the unweighted values?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > > ok. gene, once again, this means that in the 'gens' calculation,

> the

> > > number of generators in the 3:1 should be multiplied by log(3),

> the

> > > number of generators in the 5:3 should be multiplied by log(5),

> the

> > > number of generators in the 5:1 should be multiplied by log(5).

> >

> > And to make the result meaningful (i.e. comparable to the

> unweighted

> > values) then after you take the RMS of these weighted values you

> > should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).

>

> why do you want them to be comparable to the unweighted values?

Obviously it doesn't matter as far as choosing lists, but I like for a

human (e.g. me) to be able to look at the error values and have them

mean something. i.e. to actually be in cents. So when you see 5 you

know kinda what a 5c mistuning sounds like.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > > > ok. gene, once again, this means that in the 'gens'

calculation,

> > the

> > > > number of generators in the 3:1 should be multiplied by log

(3),

> > the

> > > > number of generators in the 5:3 should be multiplied by log

(5),

> > the

> > > > number of generators in the 5:1 should be multiplied by log

(5).

> > >

> > > And to make the result meaningful (i.e. comparable to the

> > unweighted

> > > values) then after you take the RMS of these weighted values

you

> > > should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).

> >

> > why do you want them to be comparable to the unweighted values?

>

> Obviously it doesn't matter as far as choosing lists, but I like

for a

> human (e.g. me) to be able to look at the error values and have

them

> mean something. i.e. to actually be in cents. So when you see 5 you

> know kinda what a 5c mistuning sounds like.

but the units here are not cents, they're gens. what does 5 gens

sound like? faggeddabbouddit.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> but the units here are not cents, they're gens. what does 5 gens

> sound like? faggeddabbouddit.

Oh dear I keep doing that don't I. Must be the Alzheimer's. :-)

In-Reply-To: <a6jdeq+58k4@eGroups.com>

paulerlich wrote:

> ok. gene, once again, this means that in the 'gens' calculation, the

> number of generators in the 3:1 should be multiplied by log(3), the

> number of generators in the 5:3 should be multiplied by log(5), the

> number of generators in the 5:1 should be multiplied by log(5). this

> will cause temperaments generated by the fifth to look better than

> they currently do, relative to those that aren't. this is important

> since augmented (especially) and diminished (a little less so) are

> far harder for the ear to understand than meantone, even when all are

> *tuned* in 12-equal, and the badness values would no longer put

> meantone as the 'best'.

When was it decided these temperaments were "far harder for the ear to

understand"? Even if so, augmented is more complex than meantone if you

measure by the simplest MOS to contain a consonant chord (8 compared to 5

notes).

Still, for whatever reason you want to privilege fifths. The rule you

give, generalized to higher limits, will favour simpler ratios in general.

That'll give the usual bias towards temperaments that work with the

limit lower than the one you asked for. And it'll miss temperaments where

a simple interval is close enough to use in modulations, but isn't the

defined approximation to 3:2.

Graham

In-Reply-To: <a6iupi+ksb1@eGroups.com>

Dave K:

> > One of the reasons given for using rms was that if there was

> an

> > outlier the LT still got credit for the intervals it did well. But if

> > we're looking at say {1,3,5,7} then if _any_ {1,3,5,7} interval is

> > very wide, I don't think it is a good {1,3,5,7} temperament. Let it

> be

> > found to be a good {2,5,7} temperament or whatever.

Paul:

> it might not -- again, what if it's really a good {2, 3, 7/5}

> temperament?

There shouldn't be anything special about 1.3.5 and 1.3.7 compared to

1.3.7:5 as defining chords. So far my script only works with the former,

but don't make any plans assuming that will always be the case. It should

even be possible to automatically find and evaluate the simplest subset if

that's all you want.

Graham

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

> In-Reply-To: <a6jdeq+58k4@e...>

> paulerlich wrote:

>

> > ok. gene, once again, this means that in the 'gens' calculation,

the

> > number of generators in the 3:1 should be multiplied by log(3),

the

> > number of generators in the 5:3 should be multiplied by log(5),

the

> > number of generators in the 5:1 should be multiplied by log(5).

this

> > will cause temperaments generated by the fifth to look better

than

> > they currently do, relative to those that aren't. this is

important

> > since augmented (especially) and diminished (a little less so)

are

> > far harder for the ear to understand than meantone, even when all

are

> > *tuned* in 12-equal, and the badness values would no longer put

> > meantone as the 'best'.

>

> When was it decided these temperaments were "far harder for the ear

to

> understand"? Even if so, augmented is more complex than meantone

if you

> measure by the simplest MOS to contain a consonant chord (8

compared to 5

> notes).

augmented is 6, diminished is 8 . . .

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > > ok. gene, once again, this means that in the 'gens' calculation,

> the

> > > number of generators in the 3:1 should be multiplied by log(3),

> the

> > > number of generators in the 5:3 should be multiplied by log(5),

> the

> > > number of generators in the 5:1 should be multiplied by log(5).

> this

> > > will cause temperaments generated by the fifth to look better

> than

> > > they currently do, relative to those that aren't.

Paul, don't you mean "divided by". log(3) is smaller than log(5) so if

you want to favour fifths ...

Here's the formula I'm using:

SQRT(((gens(1:3)/LN(3))^2+(gens(1:5)/LN(5))^2+(gens(3:5)/LN(5))^2)/(1/

LN(3)^2+1/LN(5)^2+1/LN(5)^2))*1200/period_in_cents

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > > > ok. gene, once again, this means that in the 'gens'

calculation,

> > the

> > > > number of generators in the 3:1 should be multiplied by log

(3),

> > the

> > > > number of generators in the 5:3 should be multiplied by log

(5),

> > the

> > > > number of generators in the 5:1 should be multiplied by log

(5).

> > this

> > > > will cause temperaments generated by the fifth to look better

> > than

> > > > they currently do, relative to those that aren't.

>

> Paul, don't you mean "divided by". log(3) is smaller than log(5) so

if

> you want to favour fifths ...

oh yeah, i meant divided by . . . i said it the right way last time,

about two months ago . . .

and i don't buy graham's objections one bit. yes, it favors those

systems which are *aspects* of lower-limit systems -- but very often

you get more than one such aspect per lower-limit system! this is a

very positive feature, not a bug -- it sits nicely with a view of

higher-limit systems often evolving out of lower-limit ones. and of

course it reflects musical reality much better.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> oh yeah, i meant divided by . . . i said it the right way last time,

> about two months ago . . .

So if were to do yet another search, what should be the limits?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

>

> > oh yeah, i meant divided by . . . i said it the right way last

time,

> > about two months ago . . .

>

> So if were to do yet another search, what should be the limits?

what do you mean (who and what limits)?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> what do you mean (who and what limits)?

I don't want to do another search, this time weighted, unless you tell me what you want to look at.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

>

> > what do you mean (who and what limits)?

>

> I don't want to do another search, this time weighted, unless you

>tell me what you want to look at.

i think dave needs to answer this question (or repeat his answer).

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> >

> > > what do you mean (who and what limits)?

> >

> > I don't want to do another search, this time weighted, unless you

> >tell me what you want to look at.

>

> i think dave needs to answer this question (or repeat his answer).

well, failing that, just do it with the same bounds as before, or

still more inclusive bounds if you prefer, and with the applied

badness measure given for each entry.

after that's done, we should strive for consensus and move on to the

7-limit.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>

wrote:

> > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > >

> > > > what do you mean (who and what limits)?

> > >

> > > I don't want to do another search, this time weighted, unless

you

> > >tell me what you want to look at.

> >

> > i think dave needs to answer this question (or repeat his answer).

Yes I have already posted them, and they are also in my spreadsheet.

But here they are again. 35 cents rms, 12 weighted-rms generators (or

10 for the shorter list) and max 625 for Gene's badness (using

weighted complexity).

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Yes I have already posted them, and they are also in my spreadsheet.

> But here they are again. 35 cents rms, 12 weighted-rms generators (or

> 10 for the shorter list) and max 625 for Gene's badness (using

> weighted complexity).

Here's some examples of weighted badness, using weighted complexity. Can you check your limits in terms of this?

81/80: 103.257

15625/15552: 187.010

250/243: 463.641

78732/78125: 525.371

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> > Yes I have already posted them, and they are also in my

spreadsheet.

> > But here they are again. 35 cents rms, 12 weighted-rms generators

(or

> > 10 for the shorter list) and max 625 for Gene's badness (using

> > weighted complexity).

>

> Here's some examples of weighted badness, using weighted complexity.

Can you check your limits in terms of this?

>

> 81/80: 103.257

> 15625/15552: 187.010

> 250/243: 463.641

> 78732/78125: 525.371

In terms of these, the badness limit would need to be around 915. You

could have checked them yourself against my spreadsheet.

http://dkeenan.com/Music/5LimitTemp.xls.zip

Are you able to read Excel spreadsheets?

Somehow your badnesses are a factor of 1.46131 times those I

calculate. How could that be? Are you still using complexity^3*error.

Here are the weighted-rms complexities and errors I have for those

temperaments (in case you can't read the spreadsheet). In the order

above.

error complexity

4.218 2.559

1.030 4.991

7.976 3.414

1.157 6.772

So Gene,

How's the 5-limit list coming along.

Also do you have any objections or suggestions for Graham's or my

'standard generators and mapping' proposals. Surely we want canonical

forms that are musically meaningful and convenient where possible.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> So Gene,

>

> How's the 5-limit list coming along.

It isn't; I've been writing my paper instead.

> Also do you have any objections or suggestions for Graham's or my

> 'standard generators and mapping' proposals. Surely we want canonical

> forms that are musically meaningful and convenient where possible.

I modifed my proposal and got it to work, so I suppose I should explain that. The next few days are teaching days, but then we have spring break.