I pointed out over on PostTonality that the 88CET is closely related to the Octafifths temperament; you might say it is Octafifths without the octave reduction. If Octafifths (with a generator around 8/109) has a generator small enough for this, so does Quartaminorthirds, with a generator arounf 7/108. Moreover, if the nonoctave people have not yet tried the secor, they are missing a bet. Slender, with a generator of about 4/125 gives something perversely related to 31-et, and we might even try Porcupine, Tertiathirds or Hemikleismic.

The Wendy Carlos alpha is related to an 11-limit temperament which appeared on my best-20 list; it is what you get by wedging 121/120,

126/125 and 176/175, [9,5,-3,7,-13,-30,-20,-21,-1,30], and we might call it Wendy or Alpha. Which is best?

I also looked at beta and gamma, but I didn't find much. Beta can be related to a cheeseball system obtainable as h75&h94, and it is possible to regard gamma as 5/171. Maybe Graham or Dave can point out something I am missing here.

>I also looked at beta and gamma, but I didn't find much. Beta

>can be related to a cheeseball system obtainable as h75&h94,

>and it is possible to regard gamma as 5/171. Maybe Graham or

>Dave can point out something I am missing here.

Would you expect that tempering the "octave" instead of the

"generator" at a given limit would lead to a different list

of optimal temperaments or optimal generator for a temperament?

Perhaps we should, since our lists tend to keep 2:1 on both

axes of the map (or at least with a very short period of ie's)

while not considering its temperament. That is, we often

discuss temperaments with 2:1 "octaves" and irrational or

dissonant "generators". Perhaps we should instead allow the

"octave" to be any size while applying a single weighted error

function to all consonances, including the 2:1.

Most weighting schemes would probably attract the good

generators of good temperaments to consonant intervals. It

might even be possible to always force one of the generators

to a pure 2:1 with a steep enough weighting. Rather than

guessing, it seems like a good place to plug in harmonic

entropy.

But why average (max, rms, etc.) complexity and error across

a map before weighting and calculating badness? Why not

weight per harmonic identity, then just sum to find badness at

the given limit? If you weight the error right, you shouldn't

have to weight the complexity.

Let g(x) be the graham complexity of identity x, and e(x) be

the weighted error of that identity. Then minimize

Sum [g(r) * e(r)]

where r goes over all the identities in the given limit. If a

minimum could be found, we would know the optimal generator of

the optimal temperament. Assuming a perfect error function,

which can't exist outside of perfect, deterministic neuroscience.

Given the way the error and graham complexity of identities

compound when covering a limit (some sort of consistency is

required by Gene's def. for linear temperaments, IIRC), per-

identity consideration should be enough. But one could imagine

weighting per-dyad, or even per-chord.

Or maybe I'm missing something?

-Carl

>Let g(x) be the graham complexity of identity x, and e(x) be

>the weighted error of that identity. Then minimize

>

>Sum [g(r) * e(r)]

>

>where r goes over all the identities in the given limit.

That's supposed to be...

Let g(x) be the graham complexity of identity x, and e(x) be

the weighted error of that identity. Then minimize

Sum [g(r) * e(r)]

over all maps, where r goes over the identities of the given map.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> Would you expect that tempering the "octave" instead of the

> "generator" at a given limit would lead to a different list

> of optimal temperaments or optimal generator for a temperament?

The problem with this is that it makes the question of what the consonances of the temperament are murky--we can't simply use everything in the odd limit for some odd n. Of course, we could simply create a set of intervals and then temper, or use n-limit

including evens.

Perhaps we should instead allow the

> "octave" to be any size while applying a single weighted error

> function to all consonances, including the 2:1.

Should we have 8-limit as well as 7-limit and 9-limit?

> But why average (max, rms, etc.) complexity and error across

> a map before weighting and calculating badness? Why not

> weight per harmonic identity, then just sum to find badness at

> the given limit? If you weight the error right, you shouldn't

> have to weight the complexity.

I'll return to this after I've had my breakfast coffee. :)

>The problem with this is that it makes the question of what the

>consonances of the temperament are murky--we can't simply use

>everything in the odd limit for some odd n. Of course, we could

>simply create a set of intervals and then temper, or use n-limit

>including evens.

Perhaps I'm not seeing it, but I don't think we need to change

our concept of limit.

>>But why average (max, rms, etc.) complexity and error across

>>a map before weighting and calculating badness? Why not

>>weight per harmonic identity, then just sum to find badness at

>>the given limit? If you weight the error right, you shouldn't

>>have to weight the complexity.

>

>I'll return to this after I've had my breakfast coffee. :)

That was a seriously late-night post, and hopefully it made

sense. For all I know you could already be calculating badness

this way.

For linear temperaments, we have a 2-D lattice of generators.

The map turns points on this lattice into points on the

(weighted, if you like) harmonic lattice, and back again. The

complexity of a pair of such points is the taxicab distance on

the lattice of generators, and the error is the taxicab distance

on the harmonic lattice.

You can define as many mapping points as you like. But if you

stick to consistent maps of the identities only, you should get

reasonable results for all the members of what we normally call

a limit.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >The problem with this is that it makes the question of what the

> >consonances of the temperament are murky--we can't simply use

> >everything in the odd limit for some odd n. Of course, we could

> >simply create a set of intervals and then temper, or use n-limit

> >including evens.

>

> Perhaps I'm not seeing it, but I don't think we need to change

> our concept of limit.

we certainly would, and could use "integer limit" as gene suggests, or

use product limit (tenney).

>>Perhaps I'm not seeing it, but I don't think we need to change

>>our concept of limit.

>

>we certainly would, and could use "integer limit" as gene

>suggests, or use product limit (tenney).

Maybe so, but I don't see why. I'm suggesting we think only of

the map, and let it do the walking. We get to pick what goes

in the map. Picking 2, 3, 5, 7 and calling it "7-limit" seems

fine to me.

>>If you weight the error right, you shouldn't have to weight

>>the complexity.

>

>I'll return to this after I've had my breakfast coffee. :)

Sorry, all I should have said is, * is communitive. So it's

really...

Sum ( raw-error(i) * graham-complexity(i) * weighting-factor(i) )

I'll wager a coke this eliminates the need for an averaging

function over the intervals of the limit. If so, it would

approximate traditional badness, and this could be checked.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> Sum ( raw-error(i) * graham-complexity(i) * weighting-factor(i) )

>

> I'll wager a coke this eliminates the need for an averaging

> function over the intervals of the limit. If so, it would

> approximate traditional badness, and this could be checked.

I've had my coffee, but still can't see the advantage of this system. We want to have a measure of absolute error independent of complexity in any case, do we not?

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>Perhaps I'm not seeing it, but I don't think we need to change

> >>our concept of limit.

> >

> >we certainly would, and could use "integer limit" as gene

> >suggests, or use product limit (tenney).

>

> Maybe so, but I don't see why. I'm suggesting we think only of

> the map, and let it do the walking. We get to pick what goes

> in the map. Picking 2, 3, 5, 7 and calling it "7-limit" seems

> fine to me.

you´re talking prime limit, which is fine for the mapping, as usual.

but for the optimization of the generator size, we need a list of

consonances to target.

>you´re talking prime limit, which is fine for the mapping,

>as usual.

I'm talking 'put whatever you want in the map'.

> but for the optimization of the generator size, we need a

> list of consonances to target.

Why not optimize the generator size for the map, and let

it target the consonances? Presumably because in some

tunings the errors for say 3 and 5 will cancel on consonances

like 5:3.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >you´re talking prime limit, which is fine for the mapping,

> >as usual.

>

> I'm talking 'put whatever you want in the map'.

>

> > but for the optimization of the generator size, we need a

> > list of consonances to target.

>

> Why not optimize the generator size for the map, and let

> it target the consonances? Presumably because in some

> tunings the errors for say 3 and 5 will cancel on consonances

> like 5:3.

i'm not following you, or where you differ from what's "standard"

around here . . . why don't you post a complete calculation for the

meantone case, or if you wish, some other, more contrived case . . .

>>Why not optimize the generator size for the map, and let

>>it target the consonances? Presumably because in some

>>tunings the errors for say 3 and 5 will cancel on consonances

>>like 5:3.

>

>i'm not following you, or where you differ from what's

"standard" around here . . .

As I say, I don't know how much differing from what's

standard. Calculations are seldom posted here at the

undergrad level.

As usual, I'm trying to figure things out by synthesizing

something and asking about it. When I hit something that

works, I keep it.

>why don't you post a complete calculation for the meantone

>case, or if you wish, some other, more contrived case . . .

Map for 5-limit meantone...

2 3 5

gen1 1 1 -2

gen2 0 1 4

Complexity for each identity...

2= 1

3= 2

5= 6

Let's weight by 1/base2log(i)...

2= 1.00

3= 1.26

5= 2.58

Now gen1 and gen2 are variables, and minimize...

error(2) + 1.26(error(3)) + 2.58(error(5))

I don't know how to do such a calculation, or even

if it's guarenteed to have a minimum. It would

give us minimum-badness generators, not minimum

error gens.

The log2(i) weighting is only off the top of my head.

One could imagine no weighting. One could imagine

weighting so steep we could find the optimal generators

for harmonic limit infinity.

If this does cause us to miss temperaments with good

composite consonances like 5:3, we can go back to

minimizing the error of all the intervals in the givin

limit, and keep the summed graham complexity.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>Why not optimize the generator size for the map, and let

> >>it target the consonances? Presumably because in some

> >>tunings the errors for say 3 and 5 will cancel on consonances

> >>like 5:3.

> >

> >i'm not following you, or where you differ from what's

> "standard" around here . . .

>

> As I say, I don't know how much differing from what's

> standard. Calculations are seldom posted here at the

> undergrad level.

well then you need to ask for clarification in such cases. there's no

reason anyone should be left behind.

> >why don't you post a complete calculation for the meantone

> >case, or if you wish, some other, more contrived case . . .

>

> Map for 5-limit meantone...

>

> 2 3 5

> gen1 1 1 -2

> gen2 0 1 4

hmm . . . gen1 is an octave, gen2 is a fifth . . . right?

> Complexity for each identity...

>

> 2= 1

> 3= 2

> 5= 6

defined how?

> Let's weight by 1/base2log(i)...

>

> 2= 1.00

> 3= 1.26

> 5= 2.58

> Now gen1 and gen2 are variables, and minimize...

>

> error(2) + 1.26(error(3)) + 2.58(error(5))

>

> I don't know how to do such a calculation, or even

> if it's guarenteed to have a minimum. It would

> give us minimum-badness generators, not minimum

> error gens.

i'm not sure what you're getting at. given the mapping, there's no

way to change the complexity, so we'd be holding complexity constant.

so isn't minimizing badness then the same thing as minimizing error?

>>Map for 5-limit meantone...

>>

>> 2 3 5

>>gen1 1 1 -2

>>gen2 0 1 4

>

>hmm . . . gen1 is an octave, gen2 is a fifth . . . right?

Right.

>>Complexity for each identity...

>>

>>2= 1

>>3= 2

>>5= 6

>

>defined how?

Those should be 2/1, 3/2, and 5/4. It's the taxicab distance

on the rectangular lattice of generators. Which I cooked up

as a generalization of Graham complexity for temperaments that

don't necessarily have octaves. How have you been calculating

Graham complexity for temperaments with more than one period

to an octave?

>>Let's weight by 1/base2log(i)...

>>

>>2= 1.00

>>3= 1.26

>>5= 2.58

>

>>Now gen1 and gen2 are variables, and minimize...

>>

>>error(2) + 1.26(error(3)) + 2.58(error(5))

>>

>>I don't know how to do such a calculation, or even

>>if it's guarenteed to have a minimum. It would

>>give us minimum-badness generators, not minimum

>>error gens.

>

>i'm not sure what you're getting at. given the mapping,

>there's no way to change the complexity, so we'd be

>holding complexity constant. so isn't minimizing

>badness then the same thing as minimizing error?

Doesn't the presence of the weighting factors change the

result?

How would you calculate complexity, error, badness, and

optimum generators for 5-limit meantone?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>Map for 5-limit meantone...

> >>

> >> 2 3 5

> >>gen1 1 1 -2

> >>gen2 0 1 4

> >

> >hmm . . . gen1 is an octave, gen2 is a fifth . . . right?

>

> Right.

>

> >>Complexity for each identity...

> >>

> >>2= 1

> >>3= 2

> >>5= 6

> >

> >defined how?

>

> Those should be 2/1, 3/2, and 5/4. It's the taxicab distance

> on the rectangular lattice of generators.

ok . . .

> Which I cooked up

> as a generalization of Graham complexity for temperaments that

> don't necessarily have octaves.

there seems to be a problem, in that by defining the generators as an

octave and a fifth, you get different numbers than by defining them

as an octave and a twelfth, say.

plus, graham complexity doesn't operate on a per-identity basis.

> How have you been calculating

> Graham complexity for temperaments with more than one period

> to an octave?

multiply the generator span of the otonal (or utonal) n-ad by the

number of periods per octave.

> >>Let's weight by 1/base2log(i)...

> >>

> >>2= 1.00

> >>3= 1.26

> >>5= 2.58

> >

> >>Now gen1 and gen2 are variables, and minimize...

> >>

> >>error(2) + 1.26(error(3)) + 2.58(error(5))

> >>

> >>I don't know how to do such a calculation, or even

> >>if it's guarenteed to have a minimum. It would

> >>give us minimum-badness generators, not minimum

> >>error gens.

> >

> >i'm not sure what you're getting at. given the mapping,

> >there's no way to change the complexity, so we'd be

> >holding complexity constant. so isn't minimizing

> >badness then the same thing as minimizing error?

>

> Doesn't the presence of the weighting factors change the

> result?

well, if you mean minimizing the expression above (assuming you meant

to the errors to be absolute values or squares), basically you've

just come up with a different weighting scheme for the error. not one

which i like, by the way, since you don't penalize the error in 3 and

the error in 5 for being in opposite directions; that is, you don't

take into account the error in 5:3.

> How would you calculate complexity, error, badness, and

> optimum generators for 5-limit meantone?

well, certainly woolhouse's derivation is known by everyone by now,

isn't it? that gives you the error and optimum generator, in the

equal-weighted RMS case. the complexity is a function of the mapping,

and can be defined in various ways (the graham complexity is 4), but

does not depend on the precise choice of generator. badness also has

several definitions -- log-flat badness is pretty much gene's

territory -- but is typically error times complexity to some power.

>>Which I cooked up as a generalization of Graham complexity for

>>temperaments that don't necessarily have octaves.

>

>there seems to be a problem, in that by defining the generators

>as an octave and a fifth, you get different numbers than by

>defining them as an octave and a twelfth, say.

I thought of that, but I thought also that as long as one always

uses the same set of targets across temperaments, one is ok.

Whaddya think?

>plus, graham complexity doesn't operate on a per-identity basis.

Indeed. That's part of my inquiry into the order of operations.

>>How have you been calculating Graham complexity for temperaments

>>with more than one period to an octave?

>

>multiply the generator span of the otonal (or utonal) n-ad by the

>number of periods per octave.

That's what I thought. How does this compare to the taxicab

approach? Say, for Pajara.

>>>>error(2) + 1.26(error(3)) + 2.58(error(5))

//

>>>i'm not sure what you're getting at. given the mapping,

>>>there's no way to change the complexity, so we'd be

>>>holding complexity constant. so isn't minimizing

>>>badness then the same thing as minimizing error?

>>

>>Doesn't the presence of the weighting factors change the

>>result?

>

>well, if you mean minimizing the expression above (assuming you

>meant to the errors to be absolute values or squares),

yep.

>basically you've just come up with a different weighting scheme

>for the error.

Ok.

>not one which i like, by the way, since you don't penalize the

>error in 3 and the error in 5 for being in opposite directions;

>that is, you don't take into account the error in 5:3.

That's what I said at the beginning of the thread. So how do

you do weighted error? Do you weight the error for an entire

limit by the limit, for intervals individually?

>>How would you calculate complexity, error, badness, and

>>optimum generators for 5-limit meantone?

>

>well, certainly woolhouse's derivation is known by everyone

>by now, isn't it? that gives you the error and optimum generator,

>in the equal-weighted RMS case. the complexity is a function of

>the mapping, and can be defined in various ways (the graham

>complexity is 4), but does not depend on the precise choice of

>generator. badness also has several definitions -- log-flat

>badness is pretty much gene's territory -- but is typically error

>times complexity to some power.

Okay, that's what I thought.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>Which I cooked up as a generalization of Graham complexity for

> >>temperaments that don't necessarily have octaves.

> >

> >there seems to be a problem, in that by defining the generators

> >as an octave and a fifth, you get different numbers than by

> >defining them as an octave and a twelfth, say.

>

> I thought of that, but I thought also that as long as one always

> uses the same set of targets across temperaments, one is ok.

> Whaddya think?

what about temperaments without octaves, or without fifths? and

anyway, why would keeping the same set of targets help? complexity

shouldn't be this arbitrary!

> >plus, graham complexity doesn't operate on a per-identity basis.

>

> Indeed. That's part of my inquiry into the order of operations.

?

> >>How have you been calculating Graham complexity for temperaments

> >>with more than one period to an octave?

> >

> >multiply the generator span of the otonal (or utonal) n-ad by the

> >number of periods per octave.

>

> That's what I thought. How does this compare to the taxicab

> approach? Say, for Pajara.

i'm unclear on what taxicab approach you mean. be patient with me, i

know this would be easier in person. but i have to go now.

> >>>>error(2) + 1.26(error(3)) + 2.58(error(5))

> //

> >>>i'm not sure what you're getting at. given the mapping,

> >>>there's no way to change the complexity, so we'd be

> >>>holding complexity constant. so isn't minimizing

> >>>badness then the same thing as minimizing error?

> >>

> >>Doesn't the presence of the weighting factors change the

> >>result?

> >

> >well, if you mean minimizing the expression above (assuming you

> >meant to the errors to be absolute values or squares),

>

> yep.

>

> >basically you've just come up with a different weighting scheme

> >for the error.

>

> Ok.

>

> >not one which i like, by the way, since you don't penalize the

> >error in 3 and the error in 5 for being in opposite directions;

> >that is, you don't take into account the error in 5:3.

>

> That's what I said at the beginning of the thread. So how do

> you do weighted error? Do you weight the error for an entire

> limit by the limit, for intervals individually?

i don't think anyone's been doing weighted error on this list. but if

you did, you'd minimize

f(w3*error(3),w5*error(5),w5*error(5:3))

where f is either RMS or MAD or MAX or whatever, and w3 is your

weight on ratios of 3, and w5 is your weight on ratios of 5.

>>I thought of that, but I thought also that as long as one always

>>uses the same set of targets across temperaments, one is ok.

>>Whaddya think?

//

>why would keeping the same set of targets help? complexity

>shouldn't be this arbitrary!

Because complexity is comparitive. The idea of a complete

otonal chord is not less arbitrary.

>>>multiply the generator span of the otonal (or utonal) n-ad by the

>>>number of periods per octave.

>>

>>That's what I thought. How does this compare to the taxicab

>>approach? Say, for Pajara.

>

>i'm unclear on what taxicab approach you mean. be patient with me,

>i know this would be easier in person. but i have to go now.

Oh, sure, dude. You still abroad? I've got to go now, too.

Just count the number of generators it takes, on the shortest

route in a rect. lattice, to get to the approx. of the target

interval.

>i don't think anyone's been doing weighted error on this list.

Oh, crap. What is it you've been pushing, then?

Weighted complexity?

>but if you did, you'd minimize

>

>f(w3*error(3),w5*error(5),w5*error(5:3))

>

>where f is either RMS or MAD or MAX or whatever, and w3 is your

>weight on ratios of 3, and w5 is your weight on ratios of 5.

Thanks. So my f is +, where you tend to use RMS. And I've got

complexity in the w's, which is a mistake, as I'll post about

shortly...

-Carl

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> > Which I cooked up

> > as a generalization of Graham complexity for temperaments that

> > don't necessarily have octaves.

>

> there seems to be a problem, in that by defining the generators as an

> octave and a fifth, you get different numbers than by defining them

> as an octave and a twelfth, say.

>

> plus, graham complexity doesn't operate on a per-identity basis.

All of which being why I came up with geometric complexity, which is invariant with respect to choice of generators and does not have this problem.

>All of which being why I came up with geometric complexity,

>which is invariant with respect to choice of generators and

>does not have this problem.

Unfortunately, you may be the only one on this list that

understands geometric complexity. :(

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>I thought of that, but I thought also that as long as one always

> >>uses the same set of targets across temperaments, one is ok.

> >>Whaddya think?

> //

> >why would keeping the same set of targets help? complexity

> >shouldn't be this arbitrary!

>

> Because complexity is comparitive. The idea of a complete

> otonal chord is not less arbitrary.

right, but whether a particular mapping is more complex than another

shouldn't be this arbitrary!

> >i don't think anyone's been doing weighted error on this list.

>

> Oh, crap. What is it you've been pushing, then?

> Weighted complexity?

that's been much more common, yes.

> >but if you did, you'd minimize

> >

> >f(w3*error(3),w5*error(5),w5*error(5:3))

> >

> >where f is either RMS or MAD or MAX or whatever, and w3 is your

> >weight on ratios of 3, and w5 is your weight on ratios of 5.

>

> Thanks. So my f is +,

are you sure? aren't there absolute values, in which case it's

equivalent to MAD? (or p=1, which gene doesn't want to consider)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> > > Which I cooked up

> > > as a generalization of Graham complexity for temperaments that

> > > don't necessarily have octaves.

> >

> > there seems to be a problem, in that by defining the generators

as an

> > octave and a fifth, you get different numbers than by defining

them

> > as an octave and a twelfth, say.

> >

> > plus, graham complexity doesn't operate on a per-identity basis.

>

> All of which being why I came up with geometric complexity, which

>is invariant with respect to choice of generators and does not have

>this problem.

can you demonstate the "problem" for other complexity measures, say

for meantone?

>>Because complexity is comparitive. The idea of a complete

>>otonal chord is not less arbitrary.

>

>right, but whether a particular mapping is more complex

>than another shouldn't be this arbitrary!

I'm lost. If you agree with that, then what's arbitrary?

>>Oh, crap. What is it you've been pushing, then?

>>Weighted complexity?

>

>that's been much more common, yes.

Ok. Here's my latest thinking, as promised.

Ideally we'd base everything on complete n-ads, with

harmonic entropy. Since that's not available, we'll look

at dyadic breakdowns.

If you use the concept of odd limits, and your best way

of measuring the error of an n-ad is to break it down

into dyads, you're basically saying that ratio containing

n is much different than any ratio containing at most n-2.

Thus, I suspect that my sum of abs-errors for each odd

identity up to the limit would make sense despite the fact

that for dyads like 5:3 the errors may cancel.

If we throw out odd-limit, however, we might be better off.

If there were a weighting that followed Tenney limit but

was steep enough to make near-perfect 2:1s a fact of life

and anything much beyond the 17-limit go away, we could

have individually-weighted errors and 'limit infinity'.

We should be able to search map space and assign generator

values from scratch. Pure 2:1 generators should definitely

not be assumed. Instead, we might use the appearence of

many near-octave generators as evidence the weighting is

right.

As far as my combining error and complexity before optimizing

generators, that was wrong. Moreover, combining them at all

is not for me. I'm not bound to ask, "What's the 'best' temp.

in size range x?". Rather, I might ask, "What's the most

accurate temperament in complexity range x?". Which is just

a sort on all possible temperaments, first by complexity, then

by accuracy. Which is how I set up Dave's 5-limit spreadsheet

after endlessly trying exponents in the badness calc. without

being able to get a sensicle ranking.

As for which complexity to use, we have the question of how to

define a map at 'limit infinity'. . . In the meantime, what

about standard n-limit complexity?

() Gene's geometric complexity sounds interesting (assuming

it's limit-specific...).

() The number of notes of the temperament needed to get all of

the n-limit dyads.

() The taxicab complexity of the n-limit commas on the harmonic

lattice. Or something that measured how much smaller the

average harmonic structure would be in the temperament than in

JI. This sort of formulation is probably best, and may in fact

be what Gene's geometric complexity does...

>>>f(w3*error(3),w5*error(5),w5*error(5:3))

>>>

>>>where f is either RMS or MAD or MAX or whatever, and w3 is

>>>your weight on ratios of 3, and w5 is your weight on ratios

>>>of 5.

>>

>>Thanks. So my f is +,

>

>are you sure? aren't there absolute values, in which case it's

>equivalent to MAD? (or p=1, which gene doesn't want to consider)

Yep, you're right. Though its choice was just an expedient

here, and it would be the MAD of just the identities, not of

all the dyads in the limit.

Last time we tested these things for all-the-dyads-in-a-chord,

I believe I preferred RMS. Which is not to say that MAD

shouldn't be included in the poptimal series.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>Because complexity is comparitive. The idea of a complete

> >>otonal chord is not less arbitrary.

> >

> >right, but whether a particular mapping is more complex

> >than another shouldn't be this arbitrary!

>

> I'm lost. If you agree with that, then what's arbitrary?

if you choose a different set of generators, you'll get a different

ranking for which mapping is more complex then which!

on the rest of this message, i'll have to get back to you later . . .

>if you choose a different set of generators, you'll get

>a different ranking for which mapping is more complex

>then which!

Doesn't a map uniquely determine its generators?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> >if you choose a different set of generators, you'll get

> >a different ranking for which mapping is more complex

> >then which!

> Doesn't a map uniquely determine its generators?

The problem is that the temperament does not uniquely determine the map.

>>Doesn't a map uniquely determine its generators?

>

>The problem is that the temperament does not uniquely

>determine the map.

What is a temperament, then, if not a map?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> >>Doesn't a map uniquely determine its generators?

> >

> >The problem is that the temperament does not uniquely

> >determine the map.

>

> What is a temperament, then, if not a map?

A regular temperament, when tuned, has values for the primes which map it to a group of lower rank. For instance, meantone might send

2 to 2, 3 to 5^(1/4) and 5 to 5. Starling, a 7-limit planar temperament, when tuned, might send 2 to 2, 3 to (375/14)^(1/3), 5 to 5 and 7 to (6125/18)^(1/3) (these are minimax tunings.) However, we want to define things more abstractly, and call the above 1/4-comma meantone, etc. not just "Meantone" or "Starling".

In the case of a linear temperament, we can assume one generator is an octave or portion thereof, which means that while we have not defined the temperament as a mapping, we've at least come close. In the case of a planar temperament like Starling, this won't work. We could define standardized mappings, such as the Hermite reduced mapping, but it becomes more artificial. My answer has been to define the temperament by its associated wedgie; another option is to reduce a basis for the kernel (the commas of the temperament) in a standard way, such as Tenney-Minkowski. The more abstract creature we get in this way is what I think of as the temperament; it does not, for instance, presume octave equivalence.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>Because complexity is comparitive. The idea of a complete

> >>otonal chord is not less arbitrary.

> >

> >right, but whether a particular mapping is more complex

> >than another shouldn't be this arbitrary!

>

> I'm lost. If you agree with that, then what's arbitrary?

>

> >>Oh, crap. What is it you've been pushing, then?

> >>Weighted complexity?

> >

> >that's been much more common, yes.

>

> Ok. Here's my latest thinking, as promised.

>

> Ideally we'd base everything on complete n-ads, with

> harmonic entropy. Since that's not available, we'll look

> at dyadic breakdowns.

>

> If you use the concept of odd limits, and your best way

> of measuring the error of an n-ad is to break it down

> into dyads, you're basically saying that ratio containing

> n is much different than any ratio containing at most n-2.

> Thus, I suspect that my sum of abs-errors for each odd

> identity up to the limit would make sense despite the fact

> that for dyads like 5:3 the errors may cancel.

i don't understand this reasoning at all. completely baffled, i am.

> If we throw out odd-limit, however, we might be better off.

> If there were a weighting that followed Tenney limit but

> was steep enough to make near-perfect 2:1s a fact of life

> and anything much beyond the 17-limit go away, we could

> have individually-weighted errors and 'limit infinity'.

but there would have to be an infinitely long map, or wedgie, or list

of unison vectors in order to define the temperament family.

> We should be able to search map space and assign generator

> values from scratch.

i don't understand this.

> Pure 2:1 generators should definitely

> not be assumed. Instead, we might use the appearence of

> many near-octave generators as evidence the weighting is

> right.

as gene expained, we can let the 2:1s fall as they may even with the

current framework. though the choice of what gets defined

as "generator" becomes arbitrary when the tuning actually has more

than one dimension of freedom to it, the actual parameters describing

the temperament's tuning remain perfectly well-defined.

> As far as my combining error and complexity before optimizing

> generators, that was wrong. Moreover, combining them at all

> is not for me. I'm not bound to ask, "What's the 'best' temp.

> in size range x?". Rather, I might ask, "What's the most

> accurate temperament in complexity range x?".

that's exactly how i've been looking at all this for the entire

history of this list -- witness my comments to dave defending gene's

log-flat badness measures; i took exactly this tack!

> Which is just

> a sort on all possible temperaments, first by complexity,

this is exactly how i proposed that we present the results in our

paper . . .

> then

> by accuracy.

well, you'll rarely have two temperaments with the same complexity,

though many complexity measures do put meantone and pelogic as

identical complexity, and my heuristic puts blackwood and porcupine

as identical complexity, for instance . . .

> Which is how I set up Dave's 5-limit spreadsheet

> after endlessly trying exponents in the badness calc. without

> being able to get a sensicle ranking.

well, at this point, it's easy enough to sort the 5-limit database by

complexity, at least complexity as defined by my heuristic:

/tuning/database?

method=reportRows&tbl=10&sortBy=3

or

/tuning/database?

method=reportRows&tbl=10&sortBy=8

> >>>f(w3*error(3),w5*error(5),w5*error(5:3))

> >>>

> >>>where f is either RMS or MAD or MAX or whatever, and w3 is

> >>>your weight on ratios of 3, and w5 is your weight on ratios

> >>>of 5.

> >>

> >>Thanks. So my f is +,

> >

> >are you sure? aren't there absolute values, in which case it's

> >equivalent to MAD? (or p=1, which gene doesn't want to consider)

>

> Yep, you're right. Though its choice was just an expedient

> here, and it would be the MAD of just the identities, not of

> all the dyads in the limit.

this again. how strange since the original context in which you

proposed MAD was, iirc, 15-equal. 15-equal has a minor third which is

less that 4 cents off just, giving its triads a "locked" quality

which you liked. if you keep the magnitudes of the errors on the

identities, but make them disagree in sign, the minor third will be

32 cents off just. is such a tuning really just as good as 15-equal?

try it!

>>If we throw out odd-limit, however, we might be better off.

>>If there were a weighting that followed Tenney limit but

>>was steep enough to make near-perfect 2:1s a fact of life

>>and anything much beyond the 17-limit go away, we could

>>have individually-weighted errors and 'limit infinity'.

>

>but there would have to be an infinitely long map, or wedgie,

>or list of unison vectors in order to define the temperament

>family.

Yeah, but we could approximate it with a finite map.

>>We should be able to search map space and assign generator

>>values from scratch.

>

>i don't understand this.

The number of possible maps I'm interested in isn't that

large. Gene didn't deny that a map uniquely defined its

generators (still working through how a list of commas

could be more fundamental than a map...).

>as gene expained, we can let the 2:1s fall as they may even

>with the current framework.

What is the current framework? How have we been searching

for new systems?

>though the choice of what gets defined as "generator" becomes

>arbitrary

I never gave any special significance to the generator

vs. the ie.

>>As far as my combining error and complexity before optimizing

>>generators, that was wrong. Moreover, combining them at all

>>is not for me. I'm not bound to ask, "What's the 'best' temp.

>>in size range x?". Rather, I might ask, "What's the most

>>accurate temperament in complexity range x?".

>

>that's exactly how i've been looking at all this for the entire

>history of this list -- witness my comments to dave defending

>gene's log-flat badness measures; i took exactly this tack!

How could it defend Gene's log-flat badness? It's utterly

opposed to it!

>>Which is just a sort on all possible temperaments, first by

>>complexity,

>

>this is exactly how i proposed that we present the results in

>our paper . . .

Cool.

>>then by accuracy.

>

>well, you'll rarely have two temperaments with the same

>complexity,

Funny, I don't see Dave's 5-limitTemp spreadsheet on his

website, but with Graham complexity, you do get a fair

number of collisions IIRC.

>well, at this point, it's easy enough to sort the 5-limit

>database by complexity, at least complexity as defined by

>my heuristic:

Too bad there's nothing explaining the heuristic. :(

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>If we throw out odd-limit, however, we might be better off.

> >>If there were a weighting that followed Tenney limit but

> >>was steep enough to make near-perfect 2:1s a fact of life

> >>and anything much beyond the 17-limit go away, we could

> >>have individually-weighted errors and 'limit infinity'.

> >

> >but there would have to be an infinitely long map, or wedgie,

> >or list of unison vectors in order to define the temperament

> >family.

>

> Yeah, but we could approximate it with a finite map.

hmm . . . aren't you then just talking about a finite limit of some

sort?

> >>We should be able to search map space and assign generator

> >>values from scratch.

> >

> >i don't understand this.

>

> The number of possible maps I'm interested in isn't that

> large. Gene didn't deny that a map uniquely defined its

> generators

not sure if this concept has been pinned down . . .

> (still working through how a list of commas

> could be more fundamental than a map...).

given a list of commas, you can determine the mapping, no matter how

you define your generators.

> >as gene expained, we can let the 2:1s fall as they may even

> >with the current framework.

>

> What is the current framework? How have we been searching

> for new systems?

well, i guess i really meant a *generalization* of the current

framework, using say an integer limit or product limit instead of an

odd limit to optimize (there, we've come full circle). gene has

pulled out a few examples, iirc.

we've been searching by commas, i believe -- i'll let gene answer

this more fully. graham has been searching by "+"ing ET maps to get

linear temperament maps -- something i'm not sure i can explain right

now.

> >though the choice of what gets defined as "generator" becomes

> >arbitrary

>

> I never gave any special significance to the generator

> vs. the ie.

that's not what i mean -- i mean, if you're dealing with a planar

temperament (which might simply be a linear temperament with

tweakable octaves) or something with higher dimension, there's no

unique choice of the basis of generators -- gene's used things such

as hermite reduction to make this arbitrary choice for him.

> >>As far as my combining error and complexity before optimizing

> >>generators, that was wrong. Moreover, combining them at all

> >>is not for me. I'm not bound to ask, "What's the 'best' temp.

> >>in size range x?". Rather, I might ask, "What's the most

> >>accurate temperament in complexity range x?".

> >

> >that's exactly how i've been looking at all this for the entire

> >history of this list -- witness my comments to dave defending

> >gene's log-flat badness measures; i took exactly this tack!

>

> How could it defend Gene's log-flat badness? It's utterly

> opposed to it!

hardly!! the idea is that, if you sort by complexity, using a log-

flat badness criterion guarantees that you'll have a similar number

of temperaments to look at within each complexity range, so the

complexity will increase rather smoothly in your list.

> Too bad there's nothing explaining the heuristic. :(

you can find old posts on this list explaining it, and other posts

which link to that explanation. the logic is complete, though the

mathematics of it is -- naturally -- heuristic in nature.

wallyesterpaulrus wrote:

> we've been searching by commas, i believe -- i'll let gene answer > this more fully. graham has been searching by "+"ing ET maps to get > linear temperament maps -- something i'm not sure i can explain right > now.

I wrote my method up a while back:

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

I've implemented a search by unison vectors as well. But it isn't as efficient if you use an arbitrarily large set of unison vectors, as you really should for an exhaustive search.

> that's not what i mean -- i mean, if you're dealing with a planar > temperament (which might simply be a linear temperament with > tweakable octaves) or something with higher dimension, there's no > unique choice of the basis of generators -- gene's used things such > as hermite reduction to make this arbitrary choice for him.

Even with linear temperaments, it can be difficult to make the choice unique. I had a lot of trouble with scales with a small period generated by unison vectors. The relative sizes of the possible generators can change after you optimize.

Graham

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> The number of possible maps I'm interested in isn't that

> large. Gene didn't deny that a map uniquely defined its

> generators (still working through how a list of commas

> could be more fundamental than a map...).

It is easy to define a canonical set of commas by insisting it be TM reduced; and thereby associate something (a set of commas) as a unique marker for the temperament. We can also do this with maps by for instance Hermite reducing the map. No one seems to like Hermite reduction much, so if you want to define a canonical map which is better the field of opportunity is open.

> >as gene expained, we can let the 2:1s fall as they may even

> >with the current framework.

>

> What is the current framework? How have we been searching

> for new systems?

I've been coming up with a set of wedgies, setting limits to error, complexity and log-flat badness, and filtering the list. This is, obviously, not the only way to do things, but it works, and has advantages.

> >>As far as my combining error and complexity before optimizing

> >>generators, that was wrong. Moreover, combining them at all

> >>is not for me. I'm not bound to ask, "What's the 'best' temp.

> >>in size range x?". Rather, I might ask, "What's the most

> >>accurate temperament in complexity range x?".

> >

> >that's exactly how i've been looking at all this for the entire

> >history of this list -- witness my comments to dave defending

> >gene's log-flat badness measures; i took exactly this tack!

>

> How could it defend Gene's log-flat badness? It's utterly

> opposed to it!

Eh? I go with Paul; this is the point of log-flat measures.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> --- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

> <clumma@y...> wrote:

> > >>If we throw out odd-limit, however, we might be better off.

> > >>If there were a weighting that followed Tenney limit but

> > >>was steep enough to make near-perfect 2:1s a fact of life

> > >>and anything much beyond the 17-limit go away, we could

> > >>have individually-weighted errors and 'limit infinity'.

> > >

> > >but there would have to be an infinitely long map, or wedgie,

> > >or list of unison vectors in order to define the temperament

> > >family.

> >

> > Yeah, but we could approximate it with a finite map.

>

> hmm . . . aren't you then just talking about a finite limit of some

> sort?

My Zeta function method in theory goes out to limit infinity, but gives much greater weight to smaller primes, 2 in particular.

> we've been searching by commas, i believe -- i'll let gene answer

> this more fully.

One can use various methods to produce a list of wedgies to test, and merge the lists.

graham has been searching by "+"ing ET maps to get

> linear temperament maps -- something i'm not sure i can explain right

> now.

I think plusing is the same as taking the wedge product. If you want

n-dimensional temperaments (where linear is 2, planar 3, etc.) then

you can wedge n et maps. You may also wedge pi(p)-n commas together for the same result, where p is the prime limit and pi(x) is the number theory function counting primes less than or equal to x.

>>The number of possible maps I'm interested in isn't that

>>large. Gene didn't deny that a map uniquely defined its

>>generators (still working through how a list of commas

>>could be more fundamental than a map...).

>

>It is easy to define a canonical set of commas by insisting

>it be TM reduced; and thereby associate something (a set of

>commas) as a unique marker for the temperament. We can also

>do this with maps by for instance Hermite reducing the map.

>No one seems to like Hermite reduction much, so if you want

>to define a canonical map which is better the field of

>opportunity is open.

Well, I don't know what a basis is, let alone a TM reduced

one, let alone a Hermite normal form.

>>>>As far as my combining error and complexity before optimizing

>>>>generators, that was wrong. Moreover, combining them at all

>>>>is not for me. I'm not bound to ask, "What's the 'best' temp.

>>>>in size range x?". Rather, I might ask, "What's the most

>>>>accurate temperament in complexity range x?".

>>>

>>>that's exactly how i've been looking at all this for the entire

>>>history of this list -- witness my comments to dave defending

>>>gene's log-flat badness measures; i took exactly this tack!

>>

>>How could it defend Gene's log-flat badness? It's utterly

>>opposed to it!

>

>Eh? I go with Paul; this is the point of log-flat measures.

Eh? Name your complexity range, and take the x most accurate

temperaments within it. Why do I need badness at all?

-Carl

>hmm . . . aren't you then just talking about a finite limit

>of some sort?

With a steep weighting and a promise that what we've left

out affects things less than x, sure. Maybe the Zeta

function stuff can do even better...

>>>>We should be able to search map space and assign

>>>>generator values from scratch.

>>>

>>>i don't understand this.

>>

>>The number of possible maps I'm interested in isn't that

>>large. Gene didn't deny that a map uniquely defined its

>>generators

>

>not sure if this concept has been pinned down . . .

Certainly it hasn't. For linear 5-limit temps, if I take

something like this...

2 3 5

gen1

gen2

...and start filling in numbers, either every choice of

numbers converges on a pair of exact sizes for gen1 and

gen2 under an error function of a certain class (say, one

to which RMS, and apparently not minimax, belongs), or it

doesn't. Am I to understand it doesn't?

>>(still working through how a list of commas

>>could be more fundamental than a map...).

>

>given a list of commas, you can determine the

>mapping, no matter how you define your generators.

Are you saying that the same set of commas could vanish

under two different maps, each with different gens? If

so, can you give an example?

>we've been searching by commas, i believe -- i'll let gene

>answer this more fully. graham has been searching by "+"ing

>ET maps to get linear temperament maps -- something i'm not

>sure i can explain right now.

I remember Graham using the +ing ets method. He claimed it

was pretty fast and comprehensive. I remember thinking it

was anything but pretty.

I don't understand wedgies, so...

If the above is true about commas, then complexity should be

defined in terms of commas, and we could search all sets of

simple commas...

>that's not what i mean -- i mean, if you're dealing with a

>planar temperament (which might simply be a linear

>temperament with tweakable octaves)

How can tweaking one of the generators of a linear temperament

turn it into a planar temperament? You need a third gen!

>or something with higher dimension, there's no unique choice

>of the basis of generators -- gene's used things such as

>hermite reduction to make this arbitrary choice for him.

What's a "basis of generators"?

>the idea is that, if you sort by complexity, using a log-

>flat badness criterion guarantees that you'll have a similar

>number of temperaments to look at within each complexity

>range, so the complexity will increase rather smoothly in

>your list.

You mean if I take the twenty "best" 5-limit temperaments

and sort by badness, the resulting list will alse be sorted

by complexity, then accuracy? There didn't seem to be any

exponents that would do this on Dave's spreadsheet, and I

thought I tried the critical exponent for log-flat temps.

>though the mathematics of it is -- naturally -- heuristic

>in nature.

?

AFAIK, a heuristic is an algorithm that attempts to search

only a fraction of a network yet still deliver results one

can have confidence in.

-Carl

Gene Ward Smith wrote:

> I think plusing is the same as taking the wedge product. If you want

> n-dimensional temperaments (where linear is 2, planar 3, etc.) then

> you can wedge n et maps. You may also wedge pi(p)-n commas together for the same result, where p is the prime limit and pi(x) is the number theory function counting primes less than or equal to x.

The dual/complement of the wedge product. But the wedge product can't distinguish torsion from contorsion.

I'm using the & operator for combining equal temperaments to get a linear temperament. The + operator can then be for adding equal temperaments to get another equal temperament. So h19&h31=meantone, but h19+h31=h50.

Graham

Carl Lumma wrote:

> If the above is true about commas, then complexity should be

> defined in terms of commas, and we could search all sets of

> simple commas...

What sets of simple commas do you propose to take? The files I have here show equivalences between second-order tonality diamonds. If you have p odd numbers in your ratios, the tonality diamond will have of the order of p**2 (p squared) ratios. The second order diamond is made by combining these, so that gives O(p**4) ratios. You're then setting pairs of these ratios to be equivalent, giving O(p**8) commas.

You then need to take combinations of pi(p)-1 commas. (That is one minus the number of primes in the ratios.) If you have many more commas than primes, that will go as the pi(p)-1st power. So the total number of wedgies you have to consider is O((p**8)**(pi(p)-1))

Lets simplify this by setting pi(p)~p. To get an understimate, I'll count it as pi(p) but still call it p. The complexity of finding p prime linear temperaments is then O(p**8(p-1))

In the 7-limit, there are only 4 primes, so the calculation is O(4**(8*3)) = O(4**24) = O(2.8e14) candidates (not that many, but that doesn't matter because we don't know how long each one will take).

In the 19-limit, there are 8 primes. So we need O(8**(8*7)) = O(8**56) = O(3.7e50)

If that really is 3.7e50 candidates, it's impossible. But even in comparison to the 7-limit case, it's huge. And (without much optimisation) I couldn't even do the 7-limit calculation!

For display purposes, I'm only showing commas between the first n*3 ratios from the second-order diamond, where n is the size of the second order diamond. Using that would mean only O(p**4(p-1)) candidates. So only O(8**28) in the 19-limit, or O(2e25). But this isn't good enough -- most of my top 19-limit temperaments don't have 7 such equivalences.

Whereas combing equal temperaments only gives O(n**2) calculations, where n is the number of ETs you consider. I find n=20 works well, requiring O(400) candidates. This is true in the 5-limit and also the 21-limit. I haven't heard of anybody doing a similar search with unison vectors in the 21-limit, or even suggesting ways to reduce the complexity.

Graham

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >Eh? I go with Paul; this is the point of log-flat measures.

>

> Eh? Name your complexity range, and take the x most accurate

> temperaments within it. Why do I need badness at all?

it's a much easier, and prettier, way to acheive this. you get the

distribution you want regardless of where you put the endpoints of

your complexity ranges.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> Are you saying that the same set of commas could vanish

> under two different maps, each with different gens? If

> so, can you give an example?

81:80

in terms of octave and fifth, the map is [1,0] [1,1] [4,0]

in terms of octave and twelfth, the map is [1,0] [0,1] [4,-4]

> If the above is true about commas, then complexity should be

> defined in terms of commas,

it's nice when there's only one comma. then the log of the numbers in

the comma (say, the log of the odd limit) is an excellent estimate of

complexity (it's what i call the heuristic complexity). if there's

more than one comma being tempered out, we need a notion of

the "angle" between the commas . . .

> >that's not what i mean -- i mean, if you're dealing with a

> >planar temperament (which might simply be a linear

> >temperament with tweakable octaves)

>

> How can tweaking one of the generators of a linear temperament

> turn it into a planar temperament? You need a third gen!

it's a planar temperament in the sense that there are two

independently tweakable generators, two independent dimensions needed

in order to define each pitch of the tuning.

> >or something with higher dimension, there's no unique choice

> >of the basis of generators -- gene's used things such as

> >hermite reduction to make this arbitrary choice for him.

>

> What's a "basis of generators"?

for the case of meantone, one example would be octave and fifth.

another example would be octave and twelfth. another example would be

fifth and fourth. another example would be major second and minor

second. each pair comprises a complete basis for the vector space of

pitches in the tuning.

> >the idea is that, if you sort by complexity, using a log-

> >flat badness criterion guarantees that you'll have a similar

> >number of temperaments to look at within each complexity

> >range, so the complexity will increase rather smoothly in

> >your list.

>

> You mean if I take the twenty "best" 5-limit temperaments

> and sort by badness, the resulting list will alse be sorted

> by complexity, then accuracy?

no. you use a badness cutoff simply to define the list of

temperaments in the first place. *then* you sort by complexity.

> >though the mathematics of it is -- naturally -- heuristic

> >in nature.

>

> ?

>

> AFAIK, a heuristic is an algorithm that attempts to search

> only a fraction of a network yet still deliver results one

> can have confidence in.

from yourdictionary.com:

heu·ris·tic

(click to hear the word) (hy-rstk)

adj.

Of or relating to a usually speculative formulation serving as a

guide in the investigation or solution of a problem: "The historian

discovers the past by the judicious use of such a heuristic device as

the 'ideal type'" (Karl J. Weintraub).

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> Gene Ward Smith wrote:

>

> > I think plusing is the same as taking the wedge product. If you

want

> > n-dimensional temperaments (where linear is 2, planar 3, etc.)

then

> > you can wedge n et maps. You may also wedge pi(p)-n commas

together for the same result, where p is the prime limit and pi(x) is

the number theory function counting primes less than or equal to x.

>

> The dual/complement of the wedge product. But the wedge product

can't

> distinguish torsion from contorsion.

> I'm using the & operator for combining equal temperaments to get a

> linear temperament.

that's the wedge product. i've never had any problems detecting when

torsion is or isn't present with it. when the gcd isn't 1, you have

torsion. right?

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> Whereas combing equal temperaments only gives O(n**2) calculations,

> where n is the number of ETs you consider. I find n=20 works well,

> requiring O(400) candidates. This is true in the 5-limit and also

the

> 21-limit. I haven't heard of anybody doing a similar search with

unison

> vectors in the 21-limit, or even suggesting ways to reduce the

complexity.

>

>

> Graham

hasn't gene done a 13-limit search . . . *not* by starting from ETs?

wallyesterpaulrus wrote:

> that's the wedge product. i've never had any problems detecting when > torsion is or isn't present with it. when the gcd isn't 1, you have > torsion. right?

The wedge product of a set of commas is the complement of the wedge product of a pair of equal mappings that define the same linear temperament. The two are different, and it matters when you do quantitative calculations, although Gene seems to get round this in a way I don't understand.

Graham

>>If the above is true about commas, then complexity should be

>>defined in terms of commas, and we could search all sets of

>>simple commas...

>

>What sets of simple commas do you propose to take?

All of them within taxicab radius r on the triangular harmonic

lattice. T(r), the number of tones within r, then seems to be

6r + T(r-1); T(0)=1 in the 5-limit. That seems to be roughly

O(r**2). The number of possible commas is than the 2-combin.

of this, which is again is roughly **2. So O(r**4) at the end

of the day.

Perhaps that first 2 is the number of dimensions on the lattice,

in which case the 15-limit would be O(r**14).

>The files I have here show equivalences between second-order

>tonality diamonds.

Which are? Tonality diamonds of the pitches in a tonality

diamond?

>If you have p odd numbers in your ratios, the tonality diamond

>will have of the order of p**2 (p squared) ratios. The second

>order diamond is made by combining these, so that gives O(p**4)

>ratios. You're then setting pairs of these ratios to be

>equivalent, giving O(p**8) commas.

Ah yeah, that's a lot of commas, and it seems a rather ad hoc

way to select them.

>You then need to take combinations of pi(p)-1 commas.

>(That is one minus the number of primes in the ratios.)

Right.

>If you have many more commas than primes, that will go as

>the pi(p)-1st power.

I'll take your word for it.

>So the total number of wedgies you have to consider is

>O((p**8)**(pi(p)-1)). Lets simplify this by setting pi(p)~p.

>To get an understimate, I'll count it as pi(p) but still

>call it p. The complexity of finding p prime linear

>temperaments is then O(p**8(p-1))

//

>In the 19-limit, there are 8 primes. So we need

>O(8**(8*7)) = O(8**56) = O(3.7e50). If that really

>is 3.7e50 candidates, it's impossible.

:(

>Whereas combing equal temperaments only gives O(n**2)

>calculations, where n is the number of ETs you consider.

>I find n=20 works well, requiring O(400) candidates.

>This is true in the 5-limit and also the 21-limit. I

>haven't heard of anybody doing a similar search with

>unison vectors in the 21-limit, or even suggesting ways

>to reduce the complexity.

There must be many ways to reduce the complexity of the

method I suggest. For example, rather than finding all

the pitches and taking combinations, we might set a max

comma size (as an interval) and step through ratios

involving up to r compoundings of the allowed factors

that are smaller than that bound.

-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> wallyesterpaulrus wrote:

>

> > that's the wedge product. i've never had any problems detecting

when

> > torsion is or isn't present with it. when the gcd isn't 1, you

have

> > torsion. right?

>

> The wedge product of a set of commas is the complement of the wedge

> product of a pair of equal mappings that define the same linear

> temperament. The two are different, and it matters when you do

> quantitative calculations, although Gene seems to get round this in

a

> way I don't understand.

well let's get these things understood!

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>If the above is true about commas, then complexity should be

> >>defined in terms of commas, and we could search all sets of

> >>simple commas...

> >

> >What sets of simple commas do you propose to take?

>

> All of them within taxicab radius r on the triangular harmonic

> lattice. T(r), the number of tones within r, then seems to be

> 6r + T(r-1); T(0)=1 in the 5-limit. That seems to be roughly

> O(r**2). The number of possible commas is than the 2-combin.

> of this, which is again is roughly **2. So O(r**4) at the end

> of the day.

this is silly. all you need to do to get all these commas to come up

as *tones* is to double your radius. so 2*O(r**2) will do -- the O

(r**4) method is extremely redundant.

> Perhaps that first 2 is the number of dimensions on the lattice,

> in which case the 15-limit would be O(r**14).

if we're looking for an efficient search, we'd definitely use a prime-

limit lattice, not one with redundant points!

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> --- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

> <clumma@y...> wrote:

> > >>If the above is true about commas, then complexity should be

> > >>defined in terms of commas, and we could search all sets of

> > >>simple commas...

> > >

> > >What sets of simple commas do you propose to take?

> >

> > All of them within taxicab radius r on the triangular harmonic

> > lattice. T(r), the number of tones within r, then seems to be

> > 6r + T(r-1); T(0)=1 in the 5-limit. That seems to be roughly

> > O(r**2). The number of possible commas is than the 2-combin.

> > of this, which is again is roughly **2. So O(r**4) at the end

> > of the day.

>

> this is silly. all you need to do to get all these commas to come

up

> as *tones* is to double your radius. so 2*O(r**2) will do -- the O

> (r**4) method is extremely redundant.

and, in fact, kees van prooijen does exactly this search on tones,

here:

http://www.kees.cc/tuning/perbl.html

and keeps the smallest, second-smallest, and third-smallest commas

for each possible r, and comes up with this list:

>>Are you saying that the same set of commas could vanish

>>under two different maps, each with different gens? If

>>so, can you give an example?

>

>81:80

>

>in terms of octave and fifth, the map is [1,0] [1,1] [4,0]

>

>in terms of octave and twelfth, the map is [1,0] [0,1] [4,-4]

Hmm. How bad are such cases? Could we take them out at the

end?

>>If the above is true about commas, then complexity should be

>>defined in terms of commas,

>

>it's nice when there's only one comma. then the log of the

>numbers in the comma (say, the log of the odd limit) is an

>excellent estimate of complexity (it's what i call the

>heuristic complexity).

That's what I call taxicab complexity, I think.

Isn't it also the case that the same temperament may be defined

by different lists of commas?

>if there's more than one comma being tempered out, we need

>a notion of the "angle" between the commas . . .

Please explain.

> > How can tweaking one of the generators of a linear temperament

> > turn it into a planar temperament? You need a third gen!

>

> it's a planar temperament in the sense that there are two

> independently tweakable generators, two independent dimensions

> needed in order to define each pitch of the tuning.

You assume there was an untweakable one in the 5-limit case?

Bah!

And in the 'now it's planar' case, you would no longer have an

untweakable one. Something's got to give!

>>What's a "basis of generators"?

>

>for the case of meantone, one example would be octave and fifth.

>another example would be octave and twelfth. another example

>would be fifth and fourth. another example would be major second

>and minor second. each pair comprises a complete basis for the

>vector space of pitches in the tuning.

Thanks. There is also, I gather, such a thing as a basis of

commas? TM (TM stands for?) reduction applies to commas only,

right?

> > You mean if I take the twenty "best" 5-limit temperaments

> > and sort by badness, the resulting list will alse be sorted

> > by complexity, then accuracy?

>

> no. you use a badness cutoff simply to define the list of

> temperaments in the first place.

That's the same as taking the 20 "best" temperaments.

>*then* you sort by complexity.

Aha! This isn't better than taking the 20 simplest temperaments

and sorting by accuracy.

-Carl

>and, in fact, kees van prooijen does exactly this search

>on tones, here:

>

> http://www.kees.cc/tuning/perbl.html

Man, it's been a while since I looked at Kees' site. This

is a goldmine!

>and keeps the smallest, second-smallest, and third-smallest

>commas for each possible r, and comes up with this list:

>

> http://www.kees.cc/tuning/s235.html

That's badass.

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>Are you saying that the same set of commas could vanish

> >>under two different maps, each with different gens? If

> >>so, can you give an example?

> >

> >81:80

> >

> >in terms of octave and fifth, the map is [1,0] [1,1] [4,0]

> >

> >in terms of octave and twelfth, the map is [1,0] [0,1] [4,-4]

>

> Hmm. How bad are such cases? Could we take them out at the

> end?

i have no idea what you're asking. bad?

> >>If the above is true about commas, then complexity should be

> >>defined in terms of commas,

> >

> >it's nice when there's only one comma. then the log of the

> >numbers in the comma (say, the log of the odd limit) is an

> >excellent estimate of complexity (it's what i call the

> >heuristic complexity).

>

> That's what I call taxicab complexity, I think.

not quite. for one thing, read this:

http://www.kees.cc/tuning/lat_perbl.html

including the link to my observations.

> Isn't it also the case that the same temperament may be defined

> by different lists of commas?

>

> >if there's more than one comma being tempered out, we need

> >a notion of the "angle" between the commas . . .

>

> Please explain.

search for "straightess" in these archives . . .

> > > How can tweaking one of the generators of a linear temperament

> > > turn it into a planar temperament? You need a third gen!

> >

> > it's a planar temperament in the sense that there are two

> > independently tweakable generators, two independent dimensions

> > needed in order to define each pitch of the tuning.

>

> You assume there was an untweakable one in the 5-limit case?

> Bah!

are you saying the octave should never be assumed to be exactly 1200

cents?

> And in the 'now it's planar' case, you would no longer have an

> untweakable one. Something's got to give!

i'm not following you.

> >>What's a "basis of generators"?

> >

> >for the case of meantone, one example would be octave and fifth.

> >another example would be octave and twelfth. another example

> >would be fifth and fourth. another example would be major second

> >and minor second. each pair comprises a complete basis for the

> >vector space of pitches in the tuning.

>

> Thanks. There is also, I gather, such a thing as a basis of

> commas? TM (TM stands for?)

tenney-minkowski. tenney is the metric being minimized, and minkowski

provided a basis-reduction algorithm applicable to such a case.

> reduction applies to commas only,

> right?

right.

> > > You mean if I take the twenty "best" 5-limit temperaments

> > > and sort by badness, the resulting list will alse be sorted

> > > by complexity, then accuracy?

> >

> > no. you use a badness cutoff simply to define the list of

> > temperaments in the first place.

>

> That's the same as taking the 20 "best" temperaments.

well, if your badness cutoff, extreme error cutoff, and extreme

complexity cutoff leave you with 20 inside, and if such a clunky

tripartite criterion is what you define as "best".

> >*then* you sort by complexity.

>

> Aha! This isn't better than taking the 20 simplest temperaments

> and sorting by accuracy.

of course it is. most of the 20 simplest temperaments are garbage.

don't you want the several best temperaments in each complexity

range, going up to complexity higher than you could ever use?

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >and, in fact, kees van prooijen does exactly this search

> >on tones, here:

> >

> > http://www.kees.cc/tuning/perbl.html

>

> Man, it's been a while since I looked at Kees' site. This

> is a goldmine!

>

> >and keeps the smallest, second-smallest, and third-smallest

> >commas for each possible r, and comes up with this list:

> >

> > http://www.kees.cc/tuning/s235.html

>

> That's badass.

>

> -C.

and, since there are only two coordinates in the 5-limit, the comma

search (and thus the linear temperament search) requires only O(r^2)

operations -- kees apparently took r to be on the order of 10^5, much

more than anyone could ever possibly need :)

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> search for "straightess" in these archives . . .

oops -- i meant "straightness"!

Carl Lumma wrote:

> Perhaps that first 2 is the number of dimensions on the lattice,

> in which case the 15-limit would be O(r**14).

Yes, it's the volume of a hypersphere. So it climbs dramatically with the number of dimensions, which is one less the number of primes (or odd numbers if you're being masochistic). For 8 dimensions, you still get O((r**7)**7) = O(r**49) candidate wedgies.

>>The files I have here show equivalences between second-order

>>tonality diamonds.

>

> Which are? Tonality diamonds of the pitches in a tonality

> diamond?

Yes

> Ah yeah, that's a lot of commas, and it seems a rather ad hoc

> way to select them.

The advantage of commas taken from nth order tonality diamond is that you can predict how good a temperament can be involved in. The size of the comma in cents divided by the order of the tonality diamond is the best possible minimax error.

> There must be many ways to reduce the complexity of the

> method I suggest. For example, rather than finding all

> the pitches and taking combinations, we might set a max

> comma size (as an interval) and step through ratios

> involving up to r compoundings of the allowed factors

> that are smaller than that bound.

Yes, you could, for example, take only commas less than 10 cents. That means only 10/1200 or 1/120 of the commas stay. So you save two orders of magnitude from you set of commas, but no more. I've found the code for my search, and I was using this kind of optimization.

The big reduction in complexity would come from reducing the number of combinations of commas you need to take. Ideally some way of predicting that a small set of commas (all good on their own) will never give a good temperament by adding more.

There are certainly a lot of commas that can be relevant. Below are all 42 the equivalences from my list of 19-limit temperaments (generated from 20 equal temperaments). Some may not be unique unison vectors, but whatever, they aren't sufficient to generate most of the linear temperaments anyway. There are 27 million combinations of 7 from 42. 7 from 28 already gives over a million combinations.

22:19 =~ 15:13

16:15 =~ 17:16

15:11 =~ 26:19

135:128 =~ 19:18

65:48 =~ 19:14

20:19 =~ 19:18

24:19 =~ 19:15

39:32 =~ 128:105

14:13 =~ 128:119

45:32 =~ 128:91

256:195 =~ 21:16

256:221 =~ 22:19

65:64 =~ 64:63

128:117 =~ 35:32

17:13 =~ 64:49

13:10 =~ 64:49

21:20 =~ 20:19

40:33 =~ 17:14

48:35 =~ 256:187

18:17 =~ 128:121

40:39 =~ 49:48

128:117 =~ 12:11

11:9 =~ 39:32

256:221 =~ 22:19

25:24 =~ 133:128

22:17 =~ 128:99

128:91 =~ 7:5

64:57 =~ 28:25

32:27 =~ 13:11

256:209 =~ 11:9

19:18 =~ 128:121

28:25 =~ 143:128

18:17 =~ 17:16

9:8 =~ 64:57

171:128 =~ 4:3

28:27 =~ 133:128

128:95 =~ 27:20

32:27 =~ 19:16

24:19 =~ 81:64

135:128 =~ 20:19

165:128 =~ 128:99

135:128 =~ 19:18 =~ 128:121

Graham

> > Hmm. How bad are such cases? Could we take them out at the

> > end?

>

> i have no idea what you're asking. bad?

Can you find them at the end of a map-space search, and take

them out?

>>That's what I call taxicab complexity, I think.

>

> not quite. for one thing, read this:

>

> http://www.kees.cc/tuning/lat_perbl.html

>

> including the link to my observations.

//

> > >if there's more than one comma being tempered out, we need

> > >a notion of the "angle" between the commas . . .

> >

> > Please explain.

>

> search for "straightess" in these archives . . .

Working...

> > You assume there was an untweakable one in the 5-limit case?

> > Bah!

>

> are you saying the octave should never be assumed to be

> exactly 1200 cents?

Yes, yes, and... yes.

>>And in the 'now it's planar' case, you would no longer have an

>>untweakable one. Something's got to give!

>

> i'm not following you.

If you insist on there being an untweakable generator, you won't

have it if you take a linear temperament, tweak the octave, and

call it planar.

> > Thanks. There is also, I gather, such a thing as a basis of

> > commas? TM (TM stands for?)

>

> tenney-minkowski. tenney is the metric being minimized, and

> minkowski provided a basis-reduction algorithm applicable to

> such a case.

>

> > reduction applies to commas only,

> > right?

>

> right.

Thanks again. So if reduction is necc., it means that a

temperament can be described by two different lists of commas,

right? This means we'll have the same problem searching comma

space as we did map space. So wedgies are our last hope.

> > > no. you use a badness cutoff simply to define the list of

> > > temperaments in the first place.

> >

> > That's the same as taking the 20 "best" temperaments.

>

> well, if your badness cutoff, extreme error cutoff, and extreme

> complexity cutoff leave you with 20 inside, and if such a clunky

> tripartite criterion is what you define as "best".

Are you saying a badness cutoff is not sufficient to give a

finite list of temperaments?

-Carl

>

>

>Yes, it's the volume of a hypersphere. So it climbs dramatically with >the number of dimensions, which is one less the number of primes (or odd >numbers if you're being masochistic). For 8 dimensions, you still get >O((r**7)**7) = O(r**49) candidate wedgies.

>

Oops! That should be O((r**7)**6) or O(r**42) and 6 from 42 only gives 5 million or so combinations. I made the same mistake in the earlier post: there are n-2 commas needed to define a linear temperament using n primes.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> Carl Lumma wrote:

> Lets simplify this by setting pi(p)~p.

The prime number theorem says pi(x)~x/(ln(x)-1).

> Whereas combing equal temperaments only gives O(n**2) calculations,

> where n is the number of ETs you consider. I find n=20 works well,

> requiring O(400) candidates.

Once you take wedgies, you should have fewer candidates.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> The wedge product of a set of commas is the complement of the wedge

> product of a pair of equal mappings that define the same linear

> temperament. The two are different, and it matters when you do

> quantitative calculations, although Gene seems to get round this in a

> way I don't understand.

The two may be identified via Poincare duality, so I identify them. In practice, I reorder the terms of the wedge product I get from wedging commas so that it is the same list as the wedge product I get from wedging ets.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> > > Hmm. How bad are such cases? Could we take them out at the

> > > end?

> >

> > i have no idea what you're asking. bad?

>

> Can you find them at the end of a map-space search, and take

> them out?

what do you want to take out? from what?

> >>That's what I call taxicab complexity, I think.

> >

> > not quite. for one thing, read this:

> >

> > http://www.kees.cc/tuning/lat_perbl.html

> >

> > including the link to my observations.

> //

what does "//" mean?

> If you insist on there being an untweakable generator, you won't

> have it if you take a linear temperament, tweak the octave, and

> call it planar.

correct. the untweakable generator has been tweaked. is that all?

> > > reduction applies to commas only,

> > > right?

> >

> > right.

>

> Thanks again. So if reduction is necc., it means that a

> temperament can be described by two different lists of commas,

> right?

right, although for single-comma temperaments, only one choice leaves

you without torsion.

> This means we'll have the same problem searching comma

> space as we did map space. So wedgies are our last hope.

the problem i was pointing out with map space, i think, was that the

arbitrariness of the set of generators means your complexity ranking

(if it's just based on the numbers of the map) will be meaningless.

> > > > no. you use a badness cutoff simply to define the list of

> > > > temperaments in the first place.

> > >

> > > That's the same as taking the 20 "best" temperaments.

> >

> > well, if your badness cutoff, extreme error cutoff, and extreme

> > complexity cutoff leave you with 20 inside, and if such a clunky

> > tripartite criterion is what you define as "best".

>

> Are you saying a badness cutoff is not sufficient to give a

> finite list of temperaments?

exactly. in *every* complexity range you have about the same number

of temperaments with log-flat badness lower than some cutoff -- and

there are an infinite number of non-overlapping complexity ranges.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> >

> >

> >Yes, it's the volume of a hypersphere. So it climbs dramatically

with

> >the number of dimensions, which is one less the number of primes

(or odd

> >numbers if you're being masochistic). For 8 dimensions, you still

get

> >O((r**7)**7) = O(r**49) candidate wedgies.

> >

> Oops! That should be O((r**7)**6) or O(r**42) and 6 from 42 only

gives

> 5 million or so combinations.

still vastly redundant.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

>

> tenney-minkowski. tenney is the metric being minimized, and minkowski

> provided a basis-reduction algorithm applicable to such a case.

He supplied a criterion; there seem to be no algorithms better than brute force.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> >

> > tenney-minkowski. tenney is the metric being minimized, and

minkowski

> > provided a basis-reduction algorithm applicable to such a case.

>

> He supplied a criterion; there seem to be no algorithms better than

brute force.

but it's a criterion guaranteed to give a unique answer, right?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> but it's a criterion guaranteed to give a unique answer, right?

It's unique.

>>Can you find them at the end of a map-space search, and take

>>them out?

>

>what do you want to take out?

Can I identify the duplicate temperaments?

>from what?

A search of all possible maps.

> what does "//" mean?

Cut. I've been using it since I've been on these lists.

>correct. the untweakable generator has been tweaked. is that all?

Assuming 2:1 reduction makes me squirm in my chair, is all.

Plentiful near-2:1s should emerge from the search if the criteria

are right.

>>Thanks again. So if reduction is necc., it means that a

>>temperament can be described by two different lists of commas,

>>right?

>

>right, although for single-comma temperaments, only one choice

>leaves you without torsion.

Thanks yet again.

>>This means we'll have the same problem searching comma

>>space as we did map space. So wedgies are our last hope.

>

>the problem i was pointing out with map space, i think, was

>that the arbitrariness of the set of generators means your

>complexity ranking (if it's just based on the numbers of the

>map) will be meaningless.

Oh. Now I get it! You're right. But doesn't the same

problem occur with different commatic representations, when

defining complexity off the commas?

>>Are you saying a badness cutoff is not sufficient to give a

>>finite list of temperaments?

>

>exactly. in *every* complexity range you have about the same

>number of temperaments with log-flat badness lower than some

>cutoff -- and there are an infinite number of non-overlapping

>complexity ranges.

Oh. I guess I need some examples, then, of most of the simple

temperaments that are garbage...

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>Can you find them at the end of a map-space search, and take

> >>them out?

> >

> >what do you want to take out?

>

> Can I identify the duplicate temperaments?

duplicate? any temperament can be generated from an infinite number

of possible basis vectors.

> >correct. the untweakable generator has been tweaked. is that all?

>

> Assuming 2:1 reduction makes me squirm in my chair, is all.

> Plentiful near-2:1s should emerge from the search if the criteria

> are right.

if the criteria include mapping to, and minimizing error from, 2:1,

then of course a near 2:1 will emerge in each temperament.

> >>This means we'll have the same problem searching comma

> >>space as we did map space. So wedgies are our last hope.

> >

> >the problem i was pointing out with map space, i think, was

> >that the arbitrariness of the set of generators means your

> >complexity ranking (if it's just based on the numbers of the

> >map) will be meaningless.

>

> Oh. Now I get it! You're right. But doesn't the same

> problem occur with different commatic representations, when

> defining complexity off the commas?

not if you define complexity right!

> >>Are you saying a badness cutoff is not sufficient to give a

> >>finite list of temperaments?

> >

> >exactly. in *every* complexity range you have about the same

> >number of temperaments with log-flat badness lower than some

> >cutoff -- and there are an infinite number of non-overlapping

> >complexity ranges.

>

> Oh. I guess I need some examples, then, of most of the simple

> temperaments that are garbage...

what are the 20 simplest 5-limit intervals? now set each of these to

be the commatic unison vector, and what temperaments do you get? gene

has studied a few of them because they actually had low badness --

but the error is so high that the generators themselves sometimes

provide better approximations to a consonance than the interval the

consonance is mapped to!

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> --- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

> <clumma@y...> wrote:

> > >>Can you find them at the end of a map-space search, and take

> > >>them out?

> > >

> > >what do you want to take out?

> >

> > Can I identify the duplicate temperaments?

>

> duplicate? any temperament can be generated from an infinite number

> of possible basis vectors.

oops -- i meant sets of possible basis vectors, assuming the

temperament is 2-d or more.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> > Oops! That should be O((r**7)**6) or O(r**42) and 6 from 42 only

> gives

> > 5 million or so combinations.

>

> still vastly redundant.

One possibility would be to choose a comma list with a badness cutoff, where the badness was the badness of the corresponding codimension one, or one-comma, temperament. I think I'll try that unless someone has a better suggestion.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> > --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>

wrote:

>

> > > Oops! That should be O((r**7)**6) or O(r**42) and 6 from 42

only

> > gives

> > > 5 million or so combinations.

> >

> > still vastly redundant.

>

> One possibility would be to choose a comma list with a badness

>cutoff, where the badness was the badness of the corresponding

>codimension one, or one-comma, temperament.

you'd get a lot of temperaments that are worse than a lot you woudn't

get, because of the straightness thing.

>> Can I identify the duplicate temperaments?

>

> duplicate? any temperament can be generated from an infinite

> number of possible basis vectors.

"The results of a search of all possible maps is bound to return

pairs, trios, etc. of maps that represent the same temperament.

Can we find them in the mess of results?" Was all I was asking!

>>Assuming 2:1 reduction makes me squirm in my chair, is all.

>>Plentiful near-2:1s should emerge from the search if the criteria

>>are right.

>

>if the criteria include mapping to, and minimizing error from,

>2:1, then of course a near 2:1 will emerge in each temperament.

Yup, that's what I've been saying alright.

> > Oh. Now I get it! You're right. But doesn't the same

> > problem occur with different commatic representations, when

> > defining complexity off the commas?

>

> not if you define complexity right!

Aha!

> > >>Are you saying a badness cutoff is not sufficient to give a

> > >>finite list of temperaments?

> > >

> > >exactly. in *every* complexity range you have about the same

> > >number of temperaments with log-flat badness lower than some

> > >cutoff -- and there are an infinite number of non-overlapping

> > >complexity ranges.

> >

> > Oh. I guess I need some examples, then, of most of the simple

> > temperaments that are garbage...

>

> what are the 20 simplest 5-limit intervals? now set each of

> these to be the commatic unison vector, and what temperaments

> do you get?

Perhaps we could enforce "validity", and maybe also Kees'

'complexity validity'.

-Carl

[I wrote...]

>Perhaps we could enforce "validity", and maybe also Kees'

>'complexity validity'.

I guess I should have called that 'expressibility validity'.

-Carl

> > >it's nice when there's only one comma. then the log of the

> > >numbers in the comma (say, the log of the odd limit) is an

> > >excellent estimate of complexity (it's what i call the

> > >heuristic complexity).

> >

> > That's what I call taxicab complexity, I think.

>

> not quite. for one thing, read this:

>

> http://www.kees.cc/tuning/lat_perbl.html

>

> including the link to my observations.

All I get from this is that it depends whether one uses a

triangular or rectangular lattice. I must be missing

something...

-----

> > >if there's more than one comma being tempered out, we need

> > >a notion of the "angle" between the commas . . .

> >

> > Please explain.

>

> search for "straightess" in these archives . . .

For a given block, notes can be transposed by unison vectors.

This changes the shape of the block. Does it change its

straightness?

Don't follow why the less straight blocks are supposed to

be less interesting.

-----

Here's what I found on the heuristic. Last time I asked,

you referred me to this message:

/tuning-math/messages/2491?expand=1

Which I can't follow at all. Which column is the heuristic,

what are the other columns, and what are their values expected

to do (go down or up...)?

I also found this blurb:

>the heuristics are only formulated for the one-unison-vector

>case (e.g., 5-limit linear temperaments), and no one has bothered

>to figure out the metric that makes it work exactly (though it

>seems like a tractable math problem). but they do seem to work

>within a factor of two for the current "step" and "cent"

>functions. "step" is approximately proportional to log(d),

>and "cent" is approximately proportional to (n-d)/(d*log(d)).

Why are they called "step" and "cent"? How were they derrived?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >> Can I identify the duplicate temperaments?

> >

> > duplicate? any temperament can be generated from an infinite

> > number of possible basis vectors.

>

> "The results of a search of all possible maps is bound to return

> pairs, trios, etc. of maps that represent the same temperament.

> Can we find them in the mess of results?" Was all I was asking!

yes, since you can calculate the wedgie of each. but clearly this is

a terrible way to go about the search. how would you even delimit it?

> >>Assuming 2:1 reduction makes me squirm in my chair, is all.

> >>Plentiful near-2:1s should emerge from the search if the criteria

> >>are right.

> >

> >if the criteria include mapping to, and minimizing error from,

> >2:1, then of course a near 2:1 will emerge in each temperament.

>

> Yup, that's what I've been saying alright.

and this kind of optimization has been done a few times, for example

by gene.

> > > >>Are you saying a badness cutoff is not sufficient to give a

> > > >>finite list of temperaments?

> > > >

> > > >exactly. in *every* complexity range you have about the same

> > > >number of temperaments with log-flat badness lower than some

> > > >cutoff -- and there are an infinite number of non-overlapping

> > > >complexity ranges.

> > >

> > > Oh. I guess I need some examples, then, of most of the simple

> > > temperaments that are garbage...

> >

> > what are the 20 simplest 5-limit intervals? now set each of

> > these to be the commatic unison vector, and what temperaments

> > do you get?

>

> Perhaps we could enforce "validity",

?

> and maybe also Kees'

> 'complexity validity'.

??

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> > > >it's nice when there's only one comma. then the log of the

> > > >numbers in the comma (say, the log of the odd limit) is an

> > > >excellent estimate of complexity (it's what i call the

> > > >heuristic complexity).

> > >

> > > That's what I call taxicab complexity, I think.

> >

> > not quite. for one thing, read this:

> >

> > http://www.kees.cc/tuning/lat_perbl.html

> >

> > including the link to my observations.

>

> All I get from this is that it depends whether one uses a

> triangular or rectangular lattice. I must be missing

> something...

it shows that the 'expressibility' metric is not quite the same as

the taxicab metric on the isosceles-triangular lattice. you need to

use scalene triangles of a certain type, which i don't think is what

you were thinking when you wrote "taxicab complexity" above.

> > > >if there's more than one comma being tempered out, we need

> > > >a notion of the "angle" between the commas . . .

> > >

> > > Please explain.

> >

> > search for "straightess" in these archives . . .

>

> For a given block, notes can be transposed by unison vectors.

> This changes the shape of the block. Does it change its

> straightness?

straightness applies to a set of unison vectors. different sets of

unison vectors can define the same temperament. a temperament may

look good on the basis of being defined by good unison vectors. but

in fact you may end up with a terrible temperament if the unison

vectors point in approximately the same direction.

> Here's what I found on the heuristic. Last time I asked,

> you referred me to this message:

>

> /tuning-math/messages/2491?expand=1

>

> Which I can't follow at all.

well, let me help you then.

> Which column is the heuristic,

column V is proportional to the heuristic error, and Y is

proportional to the heuristic complexity.

> what are the other columns,

U is the rms error, W is the ratio of the two error measures. X is

the complexity (weighted rms of generators-per-consonance at that

point i believe), and Z is the ratio of the two complexity measures.

> and what are their values expected

> to do (go down or up...)?

W and Z are expected to remain relatively constant.

> I also found this blurb:

>

> >the heuristics are only formulated for the one-unison-vector

> >case (e.g., 5-limit linear temperaments), and no one has bothered

> >to figure out the metric that makes it work exactly (though it

> >seems like a tractable math problem). but they do seem to work

> >within a factor of two for the current "step" and "cent"

> >functions. "step" is approximately proportional to log(d),

> >and "cent" is approximately proportional to (n-d)/(d*log(d)).

>

> Why are they called "step" and "cent"? How were they derrived?

that's what gene used to call them. "step" is simply complexity,

and "cent" is simply rms error.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

but they do seem to work

> > >within a factor of two for the current "step" and "cent"

> > >functions. "step" is approximately proportional to log(d),

> > >and "cent" is approximately proportional to (n-d)/(d*log(d)).

> >

> > Why are they called "step" and "cent"? How were they derrived?

>

> that's what gene used to call them. "step" is simply complexity,

> and "cent" is simply rms error.

We could put this together, and get "bad" as heuristically as

bad = (n-d)*log(d)^2/d

We might also do multiple linear regression analysis on log badness vs log(n-d), loglog(d) and log(d) and see what we got.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

>

> but they do seem to work

> > > >within a factor of two for the current "step" and "cent"

> > > >functions. "step" is approximately proportional to log(d),

> > > >and "cent" is approximately proportional to (n-d)/(d*log(d)).

> > >

> > > Why are they called "step" and "cent"? How were they derrived?

> >

> > that's what gene used to call them. "step" is simply complexity,

> > and "cent" is simply rms error.

>

> We could put this together, and get "bad" as heuristically as

>

> bad = (n-d)*log(d)^2/d

of course -- where the second "d" should really be odd limit.

> We might also do multiple linear regression analysis on log badness

>vs log(n-d), loglog(d) and log(d) and see what we got.

i'm not following. i do multiple linear regression analysis every

day, so please clarify!

> > We could put this together, and get "bad" as heuristically as

> >

> > bad = (n-d)*log(d)^2/d

>

> of course -- where the second "d" should really be odd limit.

"d" does not mean denominator?

> > We might also do multiple linear regression analysis on log badness

> >vs log(n-d), loglog(d) and log(d) and see what we got.

> i'm not following. i do multiple linear regression analysis every

> day, so please clarify!

I'm not following what you're not following.

I find that log(d) is a good heuristic for geometric 5-limit complexity, at least in the comma range of sizes. A linear regression

of log(complexity) vs log(d) gives c ~ .991685*log(d)^.986763, which is awfully close to log(d); in fact some of the time the geometric complexity is exactly log(d). A model using log|n-d|, loglog(d), and log(d) (each d being the denominator, and n the numerator) gives

log(c) ~ -.009541*log|n-d|+.009534*log(d)+.967642*loglog(d)-.0090687

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

>

> > > We could put this together, and get "bad" as heuristically as

> > >

> > > bad = (n-d)*log(d)^2/d

> >

> > of course -- where the second "d" should really be odd limit.

>

> "d" does not mean denominator?

it can, or it can mean odd limit of the ratio (in other words, the

ratio is a "Ratio Of d") -- the difference is slight.

the second and third "d", i meant to say.

> > > We might also do multiple linear regression analysis on log

badness

> > >vs log(n-d), loglog(d) and log(d) and see what we got.

>

> > i'm not following. i do multiple linear regression analysis every

> > day, so please clarify!

>

> I'm not following what you're not following.

>

> I find that log(d) is a good heuristic for geometric 5-limit

>complexity, at least in the comma range of sizes.

in all ranges, i think you'll find.

> A linear regression

> of log(complexity) vs log(d) gives c ~ .991685*log(d)^.986763,

oh, so you mean to use *your* complexity/badness/whatever as the

dependent variable in the regression!

clearly, though, you understand the heuristic.

>which is awfully close to log(d); in fact some of the time the

>geometric complexity is exactly log(d). A model using log|n-d|,

>loglog(d), and log(d) (each d being the denominator, and n the

>numerator) gives

>

> log(c) ~ -.009541*log|n-d|+.009534*log(d)+.967642*loglog(d)-.0090687

where c is complexity . . .

don't forget to report standard error ranges for your coefficient

estimates!

>>"The results of a search of all possible maps is bound to return

>>pairs, trios, etc. of maps that represent the same temperament.

>>Can we find them in the mess of results?" Was all I was asking!

>

>yes, since you can calculate the wedgie of each. but clearly this

>is a terrible way to go about the search.

Well, well by now I of course agree.

>how would you even delimit it?

I won't ask...

>>Perhaps we could enforce "validity",

>

> ?

That's Gene's name for a concept you said was equivalent to

the condition that all steps of a block be larger than its

unison vectors.

>>and maybe also Kees' 'complexity validity'.

>

> ??

See later message.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>"The results of a search of all possible maps is bound to return

> >>pairs, trios, etc. of maps that represent the same temperament.

> >>Can we find them in the mess of results?" Was all I was asking!

> >

> >yes, since you can calculate the wedgie of each. but clearly this

> >is a terrible way to go about the search.

>

> Well, well by now I of course agree.

>

> >how would you even delimit it?

>

> I won't ask...

>

> >>Perhaps we could enforce "validity",

> >

> > ?

>

> That's Gene's name for a concept you said was equivalent to

> the condition that all steps of a block be larger than its

> unison vectors.

there are no blocks here, just temperaments.

Gene Ward Smith wrote:

> The prime number theorem says pi(x)~x/(ln(x)-1).

Oh, I think I had that backwards.

>>Whereas combing equal temperaments only gives O(n**2) calculations, >>where n is the number of ETs you consider. I find n=20 works well, >>requiring O(400) candidates. > > Once you take wedgies, you should have fewer candidates.

Maybe, but isn't taking wedgies as hard as finding the maps? How do you avoid calculating a huge number of wedge products?

(Oh, n=20 gives exactly 190 candidates)

Graham

>it shows that the 'expressibility' metric is not quite the same

>as the taxicab metric on the isosceles-triangular lattice. you

>need to use scalene triangles of a certain type, which i don't

>think is what you were thinking when you wrote "taxicab

>complexity" above.

That's true; I just count the rungs. Any lattice that's

'topologically' (?) equivalent to the triangular lattice

will do.

>straightness applies to a set of unison vectors. different sets

>of unison vectors can define the same temperament. a temperament

>may look good on the basis of being defined by good unison

>vectors. but in fact you may end up with a terrible temperament

>if the unison vectors point in approximately the same direction.

Why would it be terrible?

>>/tuning-math/messages/2491?expand=1

//

>>Which column is the heuristic,

>

>column V is proportional to the heuristic error, and Y is

>proportional to the heuristic complexity.

>

>>what are the other columns,

>

>U is the rms error, W is the ratio of the two error measures. X

>is the complexity (weighted rms of generators-per-consonance at

>that point i believe), and Z is the ratio of the two complexity

>measures.

>

>>and what are their values expected

>>to do (go down or up...)?

>

>W and Z are expected to remain relatively constant.

Thanks again! Now everything is clear. Except how you

derrived the heuristics!

Seriously man, one expository blurb would save you from having

to do this for each person who's interested, and for me again

when I've forgotten it in 6 months. :~)

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >it shows that the 'expressibility' metric is not quite the same

> >as the taxicab metric on the isosceles-triangular lattice. you

> >need to use scalene triangles of a certain type, which i don't

> >think is what you were thinking when you wrote "taxicab

> >complexity" above.

>

> That's true; I just count the rungs. Any lattice that's

> 'topologically' (?) equivalent to the triangular lattice

> will do.

you mean approximately?

> >straightness applies to a set of unison vectors. different sets

> >of unison vectors can define the same temperament. a temperament

> >may look good on the basis of being defined by good unison

> >vectors. but in fact you may end up with a terrible temperament

> >if the unison vectors point in approximately the same direction.

>

> Why would it be terrible?

heuristically speaking,

in most cases the *difference* between the unison vectors will be of

similar or greater magnitude in terms of JI comma interval size as

the unison vectors themselves, but since the angle is very small,

this *difference* vector will be very short (i.e., low complexity). a

comma of a given JI interval size will lead to much higher error if

tempered out over much fewer consonant rungs (i.e., if it's very

short) than if it's tempered out over more consonant rungs (i.e., if

it's long). therefore, you may end up with a temperament with much

larger error than you would have expected given your original pair of

unison vectors.

> Thanks again! Now everything is clear. Except how you

> derrived the heuristics!

i did post the derivation a while back, probably before i even used

the word "heuristic", but if you search for "heuristic", i think

you'll find a post that links to the derivation post.

> Seriously man, one expository blurb would save you from having

> to do this for each person who's interested, and for me again

> when I've forgotten it in 6 months. :~)

i welcome suggestions or just make your own blurb, and let's put this

on a webpage somewhere.

> > A linear regression

> > of log(complexity) vs log(d) gives c ~ .991685*log(d)^.986763,

>

> oh, so you mean to use *your* complexity/badness/whatever as the

> dependent variable in the regression!

>

> clearly, though, you understand the heuristic.

The heuristic for complexity works particularly well as an estimate for geometric complexity. I did a regression analysis (which you could no doubt do better, if you have stat software) which also confirmed that your heuristic for error seems to have the right coefficients. The result is that your estimates look like an excellent way of quickly filtering a list of commas for geometric complexity, rms error, and geometic badness, which could then be used to compute wedgies. If we had estimates for the wedgies that would be even better.

> don't forget to report standard error ranges for your coefficient

> estimates!

I don't have a program which gives me residuals or F-ratios or anything which one really wants to do this right. You can probably do it better; just remember to take the log of both sides.

--- wallyesterpaulrus wrote:

> straightness applies to a set of unison vectors. different sets of

> unison vectors can define the same temperament. a temperament may

> look good on the basis of being defined by good unison vectors. but

> in fact you may end up with a terrible temperament if the unison

> vectors point in approximately the same direction.

Then that's what you need to reduce the complexity of the search. I

can't find a quantitative definition of "straightness" in the

archives. What is it? I presume it works for insufficient sets of

unison vectors.

Graham

>>That's true; I just count the rungs. Any lattice that's

>>'topologically' (?) equivalent to the triangular lattice

>>will do.

>

>you mean approximately?

? I originally meant that counting the rungs between a dyad

and the origin would be approximately the same as the log of

the odd limit of the dyad...

lnoddlimit 7limittaxicab ratio

15:8 2.7 2 1.35

5:3 1.6 1 1.6

105:64 4.7 3 1.6

225:224 5.4 4 1.35

>>>but in fact you may end up with a terrible temperament

>>>if the unison vectors point in approximately the same

>>>direction.

>>

>>Why would it be terrible?

>

>heuristically speaking,

>

>in most cases the *difference* between the unison vectors

>will be of similar or greater magnitude in terms of JI comma

>interval size as the unison vectors themselves, but since

>the angle is very small, this *difference* vector will be

>very short (i.e., low complexity). a comma of a given JI

>interval size will lead to much higher error if tempered out

>over much fewer consonant rungs (i.e., if it's very short)

>than if it's tempered out over more consonant rungs (i.e.,

>if it's long). therefore, you may end up with a temperament

>with much larger error than you would have expected given

>your original pair of unison vectors.

Right, the difference vector has to vanish, too. Ok. What

I don't get is, for a given temperament, can I change the

straightness by changing the unison vector representation?

If so, this means that badness is not fixed for a given

temperament...

Also, can I change the straightness by transposing pitches

by uvs?

Finally, is "commatic basis" an acceptable synonym for

"kernel"?

>i did post the derivation a while back, probably before i

>even used the word "heuristic", but if you search for

>"heuristic", i think you'll find a post that links to the

>derivation post.

I think I remember it coming out, but I couldn't find it

today in my searches. I did try.

>i welcome suggestions or just make your own blurb, and

>let's put this on a webpage somewhere.

Maybe the original exposition can just be updated a bit, and

then monz or I could host it, certainly.

-C.

--- In tuning-math@yahoogroups.com, "Graham Breed <graham@m...>"

<graham@m...> wrote:

> --- wallyesterpaulrus wrote:

> Then that's what you need to reduce the complexity of the search. I

> can't find a quantitative definition of "straightness" in the

> archives.

i haven't come up with one yet.

> I presume it works for insufficient sets of

> unison vectors.

insufficient? how can a set of unison vectors be insufficient?

wallyesterpaulrus wrote:

> insufficient? how can a set of unison vectors be insufficient?

One unison vector is insufficient to define a 7-limit linear temperament, two or three unison vectors are insufficient to define a 21-limit linear temperament. For an arbitrary search to be practicable, there has to be a way of rejecting sets of three unison vectors because you know they can't give a good linear temperament. That would reduce the search to more like O(n**3) in the number of unison vectors instead of O(n**6).

Graham

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

> Right, the difference vector has to vanish, too. Ok. What

> I don't get is, for a given temperament, can I change the

> straightness by changing the unison vector representation?

yes.

> If so, this means that badness is not fixed for a given

> temperament...

that's not true. since both the defining unison vectors *and* the

straightness change, the badness can (and will) remain constant.

> Also, can I change the straightness by transposing pitches

> by uvs?

this is meaningless, as we're talking about temperaments, not

irregular finite periodicity blocks. we're talking either equal

temperaments or infinite regular tunings.

> Finally, is "commatic basis" an acceptable synonym for

> "kernel"?

no. every vector that can be generated from the basis belongs to the

kernel, but not every set of n members of the kernel (even if they're

linearly independent) is a basis (since you may get torsion).

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> wallyesterpaulrus wrote:

>

> > insufficient? how can a set of unison vectors be insufficient?

>

> For an arbitrary search to be practicable,

> there has to be a way of rejecting sets of three unison vectors

because

> you know they can't give a good linear temperament.

then you need a 3-d generalization of "straightness". i bet if i went

and learned grassmann algebra i'd be able to get a better grasp on

all this.

> > Right, the difference vector has to vanish, too. Ok. What

> > I don't get is, for a given temperament, can I change the

> > straightness by changing the unison vector representation?

>

> yes.

Ok.

> > If so, this means that badness is not fixed for a given

> > temperament...

>

> that's not true. since both the defining unison vectors *and*

> the straightness change, the badness can (and will) remain

> constant.

Then how can it become "terrible"?

> > Also, can I change the straightness by transposing pitches

> > by uvs?

>

> this is meaningless, as we're talking about temperaments, not

> irregular finite periodicity blocks. we're talking either equal

> temperaments or infinite regular tunings.

Straightness certainly sounds like it can be defined on an

untempered block.

> > Finally, is "commatic basis" an acceptable synonym for

> > "kernel"?

>

> no. every vector that can be generated from the basis belongs

> to the kernel, but not every set of n members of the kernel

> (even if they're linearly independent) is a basis (since you may

> get torsion).

So is it possible to have more than one kernel representation

for a temperament?

-Carl

wallyesterpaulrus wrote:

> then you need a 3-d generalization of "straightness". i bet if i went > and learned grassmann algebra i'd be able to get a better grasp on > all this.

Yes, probably. Vectors are straighter the smaller the area of the parallelogram they describe. The wedge product is a generalisation of area, as it tends towards the determinant. So hopefully we can do something with the intermediate wedge products. But I don't know what :(

Graham

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> > > Right, the difference vector has to vanish, too. Ok. What

> > > I don't get is, for a given temperament, can I change the

> > > straightness by changing the unison vector representation?

> >

> > yes.

>

> Ok.

>

> > > If so, this means that badness is not fixed for a given

> > > temperament...

> >

> > that's not true. since both the defining unison vectors *and*

> > the straightness change, the badness can (and will) remain

> > constant.

>

> Then how can it become "terrible"?

if you change to a "straighter" pair of unison vectors, one or both

of them will have to be a lot shorter, thus less distribution of

error and a worse temperament.

> > > Also, can I change the straightness by transposing pitches

> > > by uvs?

> >

> > this is meaningless, as we're talking about temperaments, not

> > irregular finite periodicity blocks. we're talking either equal

> > temperaments or infinite regular tunings.

>

> Straightness certainly sounds like it can be defined on an

> untempered block.

well, that's not the context in which it's been discussed, and thus

not the context into which your questions about it were posed.

> > > Finally, is "commatic basis" an acceptable synonym for

> > > "kernel"?

> >

> > no. every vector that can be generated from the basis belongs

> > to the kernel, but not every set of n members of the kernel

> > (even if they're linearly independent) is a basis (since you may

> > get torsion).

>

> So is it possible to have more than one kernel representation

> for a temperament?

no. the kernel is an infinite set of vectors, and is unique to the

temperament.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> Maybe the original exposition can just be updated a bit, and

> then monz or I could host it, certainly.

You might want to add to

complexity ~ log(d)

error ~ log(n-d)/(d log(d))

a badness heursitic of

badness ~ log(n-d) log(d)^e / d

where e = pi(prime limit)-1 = number of odd primes in limit.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> wallyesterpaulrus wrote:

>

> > then you need a 3-d generalization of "straightness". i bet if i went

> > and learned grassmann algebra i'd be able to get a better grasp on

> > all this.

>

> Yes, probably. Vectors are straighter the smaller the area of the

> parallelogram they describe. The wedge product is a generalisation of

> area, as it tends towards the determinant. So hopefully we can do

> something with the intermediate wedge products. But I don't know what :(

The geometic complexity is a measure of the length of the wedge product of the commas.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:

> The geometic complexity is a measure of the length of the wedge product of the commas.

Of commas, taken mod 2 (note classes or whatever you want to call it.)

I'm not sure what you want here--if all of the commas point in about the same direction, do you mean with or without 2?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>

wrote:

> > wallyesterpaulrus wrote:

> >

> > > then you need a 3-d generalization of "straightness". i bet if

i went

> > > and learned grassmann algebra i'd be able to get a better grasp

on

> > > all this.

> >

> > Yes, probably. Vectors are straighter the smaller the area of

the

> > parallelogram they describe. The wedge product is a

generalisation of

> > area, as it tends towards the determinant. So hopefully we can

do

> > something with the intermediate wedge products. But I don't know

what :(

>

> The geometic complexity is a measure of the length of the wedge

>product of the commas.

or the length of the comma, if there's only one.

so the main difference is that you're using a euclidean metric (for

geometric complexity), while i'm using a taxicab one (for heuristic

complexity).

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> I'm not sure what you want here--if all of the commas point in

>about the same direction, do you mean with or without 2?

i'm thinking without.

wallyesterpaulrus wrote:

> so the main difference is that you're using a euclidean metric (for > geometric complexity), while i'm using a taxicab one (for heuristic > complexity).

I take it you're adding either the moduli or squares of the coefficients of the exterior element (wedge product of vector)? Well, that's easy enough. And the more complex the better, because that covers the complexity of the original unison vectors and their straightness. If we've already chosen unison vectors that are small pitch intervals, do we have a badness measure?

Graham

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith

> <genewardsmith@j...>" <genewardsmith@j...> wrote:

>

> > I'm not sure what you want here--if all of the commas point in

> >about the same direction, do you mean with or without 2?

>

> i'm thinking without.

One method which might come to the same thing as "straightness" in effect is to take two commas, and combine to get a codimension 2 wedgie. Produce a list of these by taking the best (here you run into geometric badness) and then wedge these with another comma, and so forth.

Gene Ward Smith wrote:

> One method which might come to the same thing as "straightness" in effect is to take two commas, and combine to get a codimension 2 wedgie. Produce a list of these by taking the best (here you run into geometric badness) and then wedge these with another comma, and so forth.

So is "geometric badness" simplicity or complexity of the exterior element?

Graham

> > > that's not true. since both the defining unison vectors *and*

> > > the straightness change, the badness can (and will) remain

> > > constant.

> >

> > Then how can it become "terrible"?

>

> if you change to a "straighter" pair of unison vectors, one or

> both of them will have to be a lot shorter, thus less distribution

> of error and a worse temperament.

I was trying to point out that badness here has failed to reflect

your opinion of the temperament.

>>>>Also, can I change the straightness by transposing pitches

>>>>by uvs?

>>>

>>>this is meaningless, as we're talking about temperaments, not

>>>irregular finite periodicity blocks. we're talking either equal

>>>temperaments or infinite regular tunings.

>>

>>Straightness certainly sounds like it can be defined on an

>>untempered block.

>

>well, that's not the context in which it's been discussed, and

>thus not the context into which your questions about it were posed.

That's true, but I couldn't see making a separate post. I'm

trying to understand your concept of straightness.

>no. the kernel is an infinite set of vectors, and is unique to the

>temperament.

Whew. Got it!

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> > > > that's not true. since both the defining unison vectors *and*

> > > > the straightness change, the badness can (and will) remain

> > > > constant.

> > >

> > > Then how can it become "terrible"?

> >

> > if you change to a "straighter" pair of unison vectors, one or

> > both of them will have to be a lot shorter, thus less distribution

> > of error and a worse temperament.

>

> I was trying to point out that badness here has failed to reflect

> your opinion of the temperament.

how so?

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> wallyesterpaulrus wrote:

>

> > so the main difference is that you're using a euclidean metric

(for

> > geometric complexity), while i'm using a taxicab one (for

heuristic

> > complexity).

>

> I take it you're adding either the moduli or squares of the

coefficients

> of the exterior element (wedge product of vector)?

i don't even know what that means.

> Well, that's easy

> enough. And the more complex the better, because that covers the

> complexity of the original unison vectors and their straightness.

If

> we've already chosen unison vectors that are small pitch intervals,

do

> we have a badness measure?

not following.

>>I was trying to point out that badness here has failed

>>to reflect your opinion of the temperament.

>

> how so?

You said the temperament got worse, but the badness

remained constant. -C.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>I was trying to point out that badness here has failed

> >>to reflect your opinion of the temperament.

> >

> > how so?

>

> You said the temperament got worse,

shortening the unison vectors makes the temperament worse, but in a

given temperament, this would be counteracted by an increase in

straighness, which makes the temperament better. overall, the measure

must remain fixed for a given temperament, otherwise it's meaningless.

>>in most cases the *difference* between the unison vectors

>>will be of similar or greater magnitude in terms of JI comma

>>interval size as the unison vectors themselves, but since

>>the angle is very small, this *difference* vector will be

>>very short (i.e., low complexity). a comma of a given JI

>>interval size will lead to much higher error if tempered out

>>over much fewer consonant rungs (i.e., if it's very short)

>>than if it's tempered out over more consonant rungs (i.e.,

>>if it's long). therefore, you may end up with a temperament

>>with much larger error than you would have expected given

>>your original pair of unison vectors.

>

>Right, the difference vector has to vanish, too. Ok. What

>I don't get is, for a given temperament, can I change the

>straightness by changing the unison vector representation?

>If so, this means that badness is not fixed for a given

>temperament...

>

>>that's not true. since both the defining unison vectors *and*

>>the straightness change, the badness can (and will) remain

>>constant.

>

>Then how can it [the temperament] become "terrible"?

>

>>if you change to a "straighter" pair of unison vectors,

Wait a minute -- straightness goes up or down with the angle

between the vectors? I thought up.

>>one or both of them will have to be a lot shorter, thus less

>>distribution of error and a worse temperament.

I don't follow the 'shorter' bit. The only thing I thought

straightness did was make the difference vector more complex.

>shortening the unison vectors makes the temperament worse, but

>in a given temperament, this would be counteracted by an

>increase in straighness, which makes the temperament better.

You lost me.

>overall, the measure must remain fixed for a given temperament,

>otherwise it's meaningless.

If by "the measure", you mean badness, I agree.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>in most cases the *difference* between the unison vectors

> >>will be of similar or greater magnitude in terms of JI comma

> >>interval size as the unison vectors themselves, but since

> >>the angle is very small, this *difference* vector will be

> >>very short (i.e., low complexity). a comma of a given JI

> >>interval size will lead to much higher error if tempered out

> >>over much fewer consonant rungs (i.e., if it's very short)

> >>than if it's tempered out over more consonant rungs (i.e.,

> >>if it's long). therefore, you may end up with a temperament

> >>with much larger error than you would have expected given

> >>your original pair of unison vectors.

> >

> >Right, the difference vector has to vanish, too. Ok. What

> >I don't get is, for a given temperament, can I change the

> >straightness by changing the unison vector representation?

> >If so, this means that badness is not fixed for a given

> >temperament...

> >

> >>that's not true. since both the defining unison vectors *and*

> >>the straightness change, the badness can (and will) remain

> >>constant.

> >

> >Then how can it [the temperament] become "terrible"?

> >

> >>if you change to a "straighter" pair of unison vectors,

>

> Wait a minute -- straightness goes up or down with the angle

> between the vectors? I thought up.

yep -- until you reach 90 degrees (or whatever the equivalent is in

the metric you're using).

> >>one or both of them will have to be a lot shorter, thus less

> >>distribution of error and a worse temperament.

>

> I don't follow the 'shorter' bit. The only thing I thought

> straightness did was make the difference vector more complex.

yes, or if you're decreasing from >90 degrees to 90 degrees, it's the

sum vector that gets more complex.

> >shortening the unison vectors makes the temperament worse, but

> >in a given temperament, this would be counteracted by an

> >increase in straighness, which makes the temperament better.

>

> You lost me.

well, there are probably too many counterfactuals here. why don't we

start again with any examples from the archives which came up in

connection with straightness. your choice.

> >overall, the measure must remain fixed for a given temperament,

> >otherwise it's meaningless.

>

> If by "the measure", you mean badness, I agree.

good!

>>>shortening the unison vectors makes the temperament worse, but

>>>in a given temperament, this would be counteracted by an

>>>increase in straighness, which makes the temperament better.

>>

>>You lost me.

>

>well, there are probably too many counterfactuals here. why don't

>we start again with any examples from the archives which came up

>in connection with straightness. your choice.

Good idea. I did find some talk between you and Gene, referring

to results I couldn't find. :( At one point, you mention low-

badness blocks of one note... If you pick the example, I'm happy

to attempt to build the block and find the alternate versions...

no promises, though, since I'm new to this.

If we get a good example or two, I'll collect everything and

give you a URL. Bless you, monz, for doing this sort of thing.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>>shortening the unison vectors makes the temperament worse, but

> >>>in a given temperament, this would be counteracted by an

> >>>increase in straighness, which makes the temperament better.

> >>

> >>You lost me.

> >

> >well, there are probably too many counterfactuals here. why don't

> >we start again with any examples from the archives which came up

> >in connection with straightness. your choice.

>

> Good idea. I did find some talk between you and Gene, referring

> to results I couldn't find. :( At one point, you mention low-

> badness blocks of one note... If you pick the example, I'm happy

> to attempt to build the block and find the alternate versions...

> no promises, though, since I'm new to this.

what are we talking about, anyway? (no offence)

>>>>>shortening the unison vectors makes the temperament worse,

>>>>>but in a given temperament, this would be counteracted by

>>>>>an increase in straighness, which makes the temperament

>>>>>better.

>>>>

>>>>You lost me.

>>>

>>>well, there are probably too many counterfactuals here. why

>>>don't we start again with any examples from the archives

>>>which came up in connection with straightness. your choice.

>>

>>Good idea. I did find some talk between you and Gene,

>>referring to results I couldn't find. :( At one point, you

>>mention low-badness blocks of one note... If you pick the

>>example, I'm happy to attempt to build the block and find the

>>alternate versions... no promises, though, since I'm new to

>>this.

>

>what are we talking about, anyway? (no offence)

I'm trying to figure out what you meant in the top paragraph,

there...

>>>>>shortening the unison vectors makes the temperament worse,

If we heuristically ignore the sizes, then yes.

>>>>>but in a given temperament, this would be counteracted by

>>>>>an increase in straighness, which makes the temperament

>>>>>better.

??? If straightness is maximal when the uvs are maximally

orthogonal, how does this mean the uvs have gotten shorter?

I thought it was a *decrease* in straightness that made the

difference/sum vector shorten, making the temperament *worse*.

Ultimately, I'm trying to figure out how changing the

straigtness can make a temperament "worse" but keep badness

constant. Either badness is broken or it doesn't!

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma

<clumma@y...>" <clumma@y...> wrote:

> >>>>>shortening the unison vectors makes the temperament

worse,

> >>>>>but in a given temperament, this would be counteracted

by

> >>>>>an increase in straighness, which makes the

temperament

> >>>>>better.

> >>>>

> >>>>You lost me.

> >>>

> >>>well, there are probably too many counterfactuals here. why

> >>>don't we start again with any examples from the archives

> >>>which came up in connection with straightness. your

choice.

> >>

> >>Good idea. I did find some talk between you and Gene,

> >>referring to results I couldn't find. :( At one point, you

> >>mention low-badness blocks of one note... If you pick the

> >>example, I'm happy to attempt to build the block and find the

> >>alternate versions... no promises, though, since I'm new to

> >>this.

> >

> >what are we talking about, anyway? (no offence)

>

> I'm trying to figure out what you meant in the top paragraph,

> there...

if you have a set of unison vectors A and a set of unison vectors

B, where both sets have the same straightness, and vectors in A

and B are about the same size intervals in JI, but the vectors in B

are shorter than those in A, then the temperament defined by B

will have larger error than the temperament defined by A.

but if the straightness is different, and the vectors are the same

length, then it is the straighter set that will have lower error.

does the paragraph make more sense now?

> >>>>>shortening the unison vectors makes the temperament

worse,

>

> If we heuristically ignore the sizes, then yes.

>

> >>>>>but in a given temperament, this would be counteracted

by

> >>>>>an increase in straighness, which makes the

temperament

> >>>>>better.

>

> ??? If straightness is maximal when the uvs are maximally

> orthogonal, how does this mean the uvs have gotten shorter?

any basis for the temperament that uses much longer unison

vectors will have to have much less straightness, because the

area/volume/etc. enclosed by the UVs must remain constant.

otherwise, you have torsion.

>>??? If straightness is maximal when the uvs are maximally

>>orthogonal, how does this mean the uvs have gotten shorter?

>

>any basis for the temperament that uses much longer unison

>vectors will have to have much less straightness, because the

>area/volume/etc. enclosed by the UVs must remain constant.

>otherwise, you have torsion.

So is this right:

Straightness...LengthUVs...Length+/-UV...Badness

Down...........Up..........Down..........Same

Up.............Down........Up............Same

?

If so, you can't say that changing the straightness of a

given temperament makes it "worse". But am I correct

that across temperaments, 'crooked' ones will tend to be

worse (assuming we've already considered the LengthUVs)?

-Carl

>>>the heuristics are only formulated for the one-unison-vector

>>>case (e.g., 5-limit linear temperaments), and no one has

>>>bothered to figure out the metric that makes it work exactly

>>>(though it seems like a tractable math problem). but they do

>>>seem to work within a factor of two for the current "step"

>>>and "cent" functions. "step" is approximately proportional to

>>>log(d), and "cent" is approximately proportional to

>>>(n-d)/(d*log(d)).

>>

>>Why are they called "step" and "cent"? How were they derrived?

>

>that's what gene used to call them. "step" is simply complexity,

>and "cent" is simply rms error.

Now, look here. Maybe this was obvious to everyone but me, but

a single paragraph on the derrivation each of these would have

saved us much heartache...

"There are complexity and error heuristics. They approximate

many different complexity and error functions (resp.) of

temperaments in which one comma is tempered out, through simple

math on the ratio, n/d (in lowest terms, n > d) representing

the comma that is tempered out.

"The complexity heuristic is log(d). It works because ratios

sharing denominator d are confined to a certain radius on the

harmonic lattice. blah blah blah

"The error heuristic is |n-d|/d*log(d). It works because it

reflects the size of the comma, per the number of consonant

intervals over which it must vanish. That is, ratios whose

n is far bigger than d are larger, and you'll recognize the

complexity heuristic underneath. blah blah blah

"To apply the heuristics to temperaments where more than one

comma vanishes, we might consider each of them in turn, but

we must be careful to include the difference/sum vector,

because it too must vanish. A concept called "straightness"

measures the angle between the commas on the harmonic lattice,

and therefore the relative length of the difference/sum vector.

blah blah blah"

Just an example, and probably contains errors. I must fly

to lunch!

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>??? If straightness is maximal when the uvs are maximally

> >>orthogonal, how does this mean the uvs have gotten shorter?

> >

> >any basis for the temperament that uses much longer unison

> >vectors will have to have much less straightness, because the

> >area/volume/etc. enclosed by the UVs must remain constant.

> >otherwise, you have torsion.

>

> So is this right:

>

> Straightness...LengthUVs...Length+/-UV...Badness

> Down...........Up..........Down..........Same

> Up.............Down........Up............Same

>

> ?

if you replace "badness" with "error", it's right.

> If so, you can't say that changing the straightness of a

> given temperament makes it "worse". But am I correct

> that across temperaments, 'crooked' ones will tend to be

> worse (assuming we've already considered the LengthUVs)?

if you're using TM reduction or something similar to define the

basis, then yes, this will be the general trend.

sorry for the heartache, folk(s). save this post!

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>>the heuristics are only formulated for the one-unison-vector

> >>>case (e.g., 5-limit linear temperaments), and no one has

> >>>bothered to figure out the metric that makes it work exactly

> >>>(though it seems like a tractable math problem). but they do

> >>>seem to work within a factor of two for the current "step"

> >>>and "cent" functions. "step" is approximately proportional to

> >>>log(d), and "cent" is approximately proportional to

> >>>(n-d)/(d*log(d)).

> >>

> >>Why are they called "step" and "cent"? How were they derrived?

> >

> >that's what gene used to call them. "step" is simply complexity,

> >and "cent" is simply rms error.

>

> Now, look here. Maybe this was obvious to everyone but me, but

> a single paragraph on the derrivation each of these would have

> saved us much heartache...

>

> "There are complexity and error heuristics. They approximate

> many different complexity and error functions (resp.) of

> temperaments in which one comma is tempered out, through simple

> math on the ratio, n/d (in lowest terms, n > d) representing

> the comma that is tempered out.

>

> "The complexity heuristic is log(d). It works because ratios

> sharing denominator d are confined to a certain radius on the

> harmonic lattice. blah blah blah

>

> "The error heuristic is |n-d|/d*log(d). It works because it

> reflects the size of the comma, per the number of consonant

> intervals over which it must vanish. That is, ratios whose

> n is far bigger than d are larger, and you'll recognize the

> complexity heuristic underneath. blah blah blah

>

> "To apply the heuristics to temperaments where more than one

> comma vanishes, we might consider each of them in turn, but

> we must be careful to include the difference/sum vector,

> because it too must vanish. A concept called "straightness"

> measures the angle between the commas on the harmonic lattice,

> and therefore the relative length of the difference/sum vector.

> blah blah blah"

>

>

> Just an example, and probably contains errors. I must fly

> to lunch!

>

> -Carl

sorry for the heartache, folk(s). save this post!

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>>the heuristics are only formulated for the one-unison-vector

> >>>case (e.g., 5-limit linear temperaments), and no one has

> >>>bothered to figure out the metric that makes it work exactly

> >>>(though it seems like a tractable math problem). but they do

> >>>seem to work within a factor of two for the current "step"

> >>>and "cent" functions. "step" is approximately proportional to

> >>>log(d), and "cent" is approximately proportional to

> >>>(n-d)/(d*log(d)).

> >>

> >>Why are they called "step" and "cent"? How were they derrived?

> >

> >that's what gene used to call them. "step" is simply complexity,

> >and "cent" is simply rms error.

>

> Now, look here. Maybe this was obvious to everyone but me, but

> a single paragraph on the derrivation each of these would have

> saved us much heartache...

>

> "There are complexity and error heuristics. They approximate

> many different complexity and error functions (resp.) of

> temperaments in which one comma is tempered out, through simple

> math on the ratio, n/d (in lowest terms, n > d) representing

> the comma that is tempered out.

>

> "The complexity heuristic is log(d). It works because ratios

> sharing denominator d are confined to a certain radius on the

> harmonic lattice. blah blah blah

>

> "The error heuristic is |n-d|/d*log(d). It works because it

> reflects the size of the comma, per the number of consonant

> intervals over which it must vanish. That is, ratios whose

> n is far bigger than d are larger, and you'll recognize the

> complexity heuristic underneath. blah blah blah

>

> "To apply the heuristics to temperaments where more than one

> comma vanishes, we might consider each of them in turn, but

> we must be careful to include the difference/sum vector,

> because it too must vanish. A concept called "straightness"

> measures the angle between the commas on the harmonic lattice,

> and therefore the relative length of the difference/sum vector.

> blah blah blah"

>

>

> Just an example, and probably contains errors. I must fly

> to lunch!

>

> -Carl

> > So is this right:

> >

> > Straightness...LengthUVs...Length+/-UV...Badness

> > Down...........Up..........Down..........Same

> > Up.............Down........Up............Same

> >

> > ?

>

> if you replace "badness" with "error", it's right.

I should have noted that this was for a given temperament,

not for all temperaments, though I take it you took it

that way.

So what should the badness column be?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> > > So is this right:

> > >

> > > Straightness...LengthUVs...Length+/-UV...Badness

> > > Down...........Up..........Down..........Same

> > > Up.............Down........Up............Same

> > >

> > > ?

> >

> > if you replace "badness" with "error", it's right.

>

> I should have noted that this was for a given temperament,

> not for all temperaments, though I take it you took it

> that way.

yes, because of the last column being all "same".

> So what should the badness column be?

well, i guess that's all "same" too!

> > > > Straightness...LengthUVs...Length+/-UV...Badness

> > > > Down...........Up..........Down..........Same

> > > > Up.............Down........Up............Same

> > > >

> > > > ?

> > >

> > > if you replace "badness" with "error", it's right.

> >

> > I should have noted that this was for a given temperament,

> > not for all temperaments, though I take it you took it

> > that way.

>

> yes, because of the last column being all "same".

>

> > So what should the badness column be?

>

> well, i guess that's all "same" too!

So if error is the same and badness is the same, then

complexity is the same (which I suppose makes since,

if the volume of the block is not to change). So is it

safe to conclude that straightness is important for

heuristically searching temperaments, but not for

choosing a commatic basis for a given temperament?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> > > > > Straightness...LengthUVs...Length+/-UV...Badness

> > > > > Down...........Up..........Down..........Same

> > > > > Up.............Down........Up............Same

> > > > >

> > > > > ?

> > > >

> > > > if you replace "badness" with "error", it's right.

> > >

> > > I should have noted that this was for a given temperament,

> > > not for all temperaments, though I take it you took it

> > > that way.

> >

> > yes, because of the last column being all "same".

> >

> > > So what should the badness column be?

> >

> > well, i guess that's all "same" too!

>

> So if error is the same and badness is the same, then

> complexity is the same (which I suppose makes since,

> if the volume of the block is not to change). So is it

> safe to conclude that straightness is important for

> heuristically searching temperaments,

i think so.

> but not for

> choosing a commatic basis for a given temperament?

the commatic basis with the shortest uvs will generally be the

straightest one.

carl, here's an old message where i explained the error heuristic:

and you can see that gene, in his reply, was the one who actually

suggested the word "heuristic" in connection with this . . .

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>>the heuristics are only formulated for the one-unison-vector

> >>>case (e.g., 5-limit linear temperaments), and no one has

> >>>bothered to figure out the metric that makes it work exactly

> >>>(though it seems like a tractable math problem). but they do

> >>>seem to work within a factor of two for the current "step"

> >>>and "cent" functions. "step" is approximately proportional to

> >>>log(d), and "cent" is approximately proportional to

> >>>(n-d)/(d*log(d)).

> >>

> >>Why are they called "step" and "cent"? How were they derrived?

> >

> >that's what gene used to call them. "step" is simply complexity,

> >and "cent" is simply rms error.

>

> Now, look here. Maybe this was obvious to everyone but me, but

> a single paragraph on the derrivation each of these would have

> saved us much heartache...

>

> "There are complexity and error heuristics. They approximate

> many different complexity and error functions (resp.) of

> temperaments in which one comma is tempered out, through simple

> math on the ratio, n/d (in lowest terms, n > d) representing

> the comma that is tempered out.

>

> "The complexity heuristic is log(d). It works because ratios

> sharing denominator d are confined to a certain radius on the

> harmonic lattice. blah blah blah

>

> "The error heuristic is |n-d|/d*log(d). It works because it

> reflects the size of the comma, per the number of consonant

> intervals over which it must vanish. That is, ratios whose

> n is far bigger than d are larger, and you'll recognize the

> complexity heuristic underneath. blah blah blah

>

> "To apply the heuristics to temperaments where more than one

> comma vanishes, we might consider each of them in turn, but

> we must be careful to include the difference/sum vector,

> because it too must vanish. A concept called "straightness"

> measures the angle between the commas on the harmonic lattice,

> and therefore the relative length of the difference/sum vector.

> blah blah blah"

>

>

> Just an example, and probably contains errors. I must fly

> to lunch!

>

> -Carl

> carl, here's an old message where i explained the error

> heuristic:

>

> /tuning-math/message/1437

Great, thanks! I hadn't seen this, as "heuristic" doesn't

appear in it.

> and you can see that gene, in his reply, was the one who

> actually suggested the word "heuristic" in connection

> with this . . .

I do see that... you were already using the term for the

complexity heuristic at that time, right?

I understand everything but a few details...

>log(n) + log(d) (hence approx. proportional to log(d)

Is it any more proportional to log(d) than log(n) in this

case? Since n~=d?

>w=log(n/d)

Got that.

>w~=n/d-1

How do you get this from that?

>w~=(n-d)/d

Ditto.

-Carl

Carl Lumma wrote:

>>log(n) + log(d) (hence approx. proportional to log(d

> > Is it any more proportional to log(d) than log(n) in this

> case? Since n~=d?

No, and the spreadsheet sorted by d is also sorted by n. And in that case it would have been easier to go straight to log(n*d).

>>w=log(n/d)

> > > Got that.

> > >>w~=n/d-1

> > > How do you get this from that?

It's the first order approximaton where n/d ~= 1. See (8) in

http://mathworld.wolfram.com/NaturalLogarithm.html

or check on your calculator.

>>w~=(n-d)/d

> > Ditto.

That's subtracting fractions. Did you do fractions at school?

4/3 - 1 = 4/3 - 3/3 = (4-3)/3 = 1/3

5/4 - 1 = 5/4 - 4/4 = (5-4)/4 = 1/4

n/d - 1 = n/d - d/d = (n-d)/d

Graham

>>>log(n) + log(d) (hence approx. proportional to log(d

>>

>> Is it any more proportional to log(d) than log(n) in this

>> case? Since n~=d?

>

>No, and the spreadsheet sorted by d is also sorted by n.

So it could just as well be (n-d)/(d*log(n))?

>And in that case it would have been easier to go straight

>to log(n*d).

Straight to where (do you see log(n*d))?

>>>w=log(n/d)

//

>>>w~=n/d-1

>>

>>

>>How do you get this from that?

Oh, (n/d)-1, not n/(d-1).

>It's the first order approximaton where n/d ~= 1. See (8) in

>http://mathworld.wolfram.com/NaturalLogarithm.html

The Mercator series?? And all the stuff on this page applies

only to ln, not log in general (which is what I assume Paul

meant), right?

>>>w~=(n-d)/d

>>

>>Ditto.

>

>That's subtracting fractions. Did you do fractions at school?

Yes; I was still seeing n/(d-1).

-Carl

Carl Lumma wrote:

> So it could just as well be (n-d)/(d*log(n))?

This is approximately log(n/d)/log(n)

The order will be reversed. If you can calculate it, see if it holds the ordering. With numerators it's easy as they're already in the database.

>>And in that case it would have been easier to go straight

>>to log(n*d).

> > Straight to where (do you see log(n*d))?

log(n*d) = log(n) + log(d)

The intervals start out in prime factor notation. So log(n*d) can be calculated as log(2)*abs(x_1) + log(3)*abs(x_2) + log(5)*abs(x_3) + log(7)*abs(x_4) where x_i is the ith prime component of n/d.

> The Mercator series?? And all the stuff on this page applies

> only to ln, not log in general (which is what I assume Paul

> meant), right?

log2(x) = log(x)/log(2). In this case we're only estimating the complexity, so the common factor if 1/log(2) isn't important.

For estimating comma sizes using mental arithmetic, remembering that 1200/log(2) is approximately 1730 comes in handy. 1730/80 = 21.625, so a syntonic commma's around 22 cents.

Graham

>>>And in that case it would have been easier to go straight

>>>to log(n*d).

>>

>>Straight to where (do you see log(n*d))?

>

>log(n*d) = log(n) + log(d)

Of course... so we're coming from log(n*d), not going to it.

>>The Mercator series?? And all the stuff on this page applies

>>only to ln, not log in general (which is what I assume Paul

>>meant), right?

>

>log2(x) = log(x)/log(2). In this case we're only estimating

>the complexity, so the common factor if 1/log(2) isn't important.

Ok, but I still don't get how the "Mercator series" shown in (8)

dictates the rules for this approximation.

-Carl

Carl Lumma wrote:

> Ok, but I still don't get how the "Mercator series" shown in (8)

> dictates the rules for this approximation.

Oh, I thought you had followed that.

It's usually called the Taylor series. I don't know what Mercator's got to do with it. But anyway it's

ln(1+x) = x - x**2/2 + x**3/3 + ...

where x is small. Because x is small, x**2 must be even smaller, so you can use the first order approximation

ln(1+x) =~ x

In this case, 1+x is n/d, so x = n/d - 1

ln(n/d) =~ n/d - 1

If we really wanted the logarithm to base 2, that'd be

log2(n/d) = ln(n/d)/ln(2) =~ (n/d - 1)/ln(2)

or

log2(n/d) ~ n/d - 1

where ~ is "roughly proportional to" which is all we need to know. And the same's true whatever base logarithm you use.

Graham

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> > carl, here's an old message where i explained the error

> > heuristic:

> >

> > /tuning-math/message/1437

>

> Great, thanks! I hadn't seen this, as "heuristic" doesn't

> appear in it.

>

> > and you can see that gene, in his reply, was the one who

> > actually suggested the word "heuristic" in connection

> > with this . . .

>

> I do see that... you were already using the term for the

> complexity heuristic at that time, right?

no, gene introduced the word "heuristic".

> I understand everything but a few details...

>

> >log(n) + log(d) (hence approx. proportional to log(d)

>

> Is it any more proportional to log(d) than log(n) in this

> case?

no.

> Since n~=d?

yes.

i prefer log(odd limit) over either log(n) or log(d).

> >w=log(n/d)

>

> Got that.

>

> >w~=n/d-1

>

> How do you get this from that?

standard taylor series approximation for log . . . if x is close to

1, then log(x) is close to x-1 (since the derivative of log(x) near

x=1 is 1/1 = 1).

> >w~=(n-d)/d

>

> Ditto.

arithmetic. n/d - 1 = n/d - d/d = (n-d)/d.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>>log(n) + log(d) (hence approx. proportional to log(d

> >>

> >> Is it any more proportional to log(d) than log(n) in this

> >> case? Since n~=d?

> >

> >No, and the spreadsheet sorted by d is also sorted by n.

>

> So it could just as well be (n-d)/(d*log(n))?

a very different sorting. that would be heuristic error, not

heuristic complexity. of course it's a very different sorting, since

knowing log(n) or log(d) tells you nothing about (n-d).

> >And in that case it would have been easier to go straight

> >to log(n*d).

>

> Straight to where (do you see log(n*d))?

meaning the sorting by log(n*d) would be virtually identical to the

sorting by log(n) or by log(d).

>

> >It's the first order approximaton where n/d ~= 1. See (8) in

> >http://mathworld.wolfram.com/NaturalLogarithm.html

>

> The Mercator series?? And all the stuff on this page applies

> only to ln, not log in general (which is what I assume Paul

> meant), right?

i meant ln. i always use matlab, in which "log" means ln.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>>And in that case it would have been easier to go straight

> >>>to log(n*d).

> >>

> >>Straight to where (do you see log(n*d))?

> >

> >log(n*d) = log(n) + log(d)

>

> Of course... so we're coming from log(n*d), not going to it.

what do you mean, we're coming from log(n*d)??

>>>> Is it any more proportional to log(d) than log(n) in this

>>>> case? Since n~=d?

>>>

>>>No, and the spreadsheet sorted by d is also sorted by n.

>>

>>So it could just as well be (n-d)/(d*log(n))?

>

>a very different sorting. that would be heuristic error, not

>heuristic complexity. of course it's a very different sorting,

>since knowing log(n) or log(d) tells you nothing about (n-d).

? I was asking if the *error* heuristic could become

(n-d)/(d*log(n)) if we substituted log(n) for log(d).

>>>It's the first order approximaton where n/d ~= 1. See (8) in

>>>http://mathworld.wolfram.com/NaturalLogarithm.html

>>

>>The Mercator series?? And all the stuff on this page applies

>>only to ln, not log in general (which is what I assume Paul

>>meant), right?

>

>i meant ln. i always use matlab, in which "log" means ln.

Oh. How odd.

-Carl

>>>>>And in that case it would have been easier to go straight

>>>>>to log(n*d).

>>>>

>>>>Straight to where (do you see log(n*d))?

>>>

>>>log(n*d) = log(n) + log(d)

>>

>>Of course... so we're coming from log(n*d), not going to it.

>

>what do you mean, we're coming from log(n*d)??

In your original message, you *start from* "is proportional to

log(n) + log(d)" and *arrive at* "Hence the amount of tempering

implied by the unison vector is approx. proportional to

(n-d)/(d*log(d))".

-Carl

> > I do see that... you were already using the term for the

> > complexity heuristic at that time, right?

>

> no, gene introduced the word "heuristic".

K.

> > >w=log(n/d)

> >

> > Got that.

> >

> > >w~=n/d-1

> >

> > How do you get this from that?

>

> standard taylor series approximation for log . . . if x is

> close to 1, then log(x) is close to x-1 (since the derivative

> of log(x) near x=1 is 1/1 = 1).

Cool.

> > >w~=(n-d)/d

> >

> > Ditto.

>

> arithmetic. n/d - 1 = n/d - d/d = (n-d)/d.

Yeah, I thought the 1 was in the denominator.

-C.

>>Ok, but I still don't get how the "Mercator series" shown in (8)

>>dictates the rules for this approximation.

>

>Oh, I thought you had followed that.

>

>It's usually called the Taylor series. I don't know what

>Mercator's got to do with it.

On the mathworld page, it says "the Mercator series gives a

Taylor series for the natural logarithm", and in fact makes

it look like the Taylor series is the Mercator series.

>But anyway it's

>

> ln(1+x) = x - x**2/2 + x**3/3 + ...

>

> where x is small. Because x is small, x**2 must be even

> smaller, so you can use the first order approximation

>

> ln(1+x) =~ x

>

> In this case, 1+x is n/d, so x = n/d - 1

>

> ln(n/d) =~ n/d - 1

Thanks!

-Carl

>>It's usually called the Taylor series. I don't know what

>>Mercator's got to do with it.

>

>On the mathworld page, it says "the Mercator series gives a

>Taylor series for the natural logarithm", and in fact makes

>it look like the Taylor series is the Mercator series.

...by the way the page is formatted.

Turns out that the Mercator series is what we're talking

about; a special case of the Taylor series.

http://mathworld.wolfram.com/TaylorSeries.html

"A Taylor series is a series expansion of a function about

a point. ... "

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> On the mathworld page, it says "the Mercator series gives a

> Taylor series for the natural logarithm", and in fact makes

> it look like the Taylor series is the Mercator series.

It is in this case--it's a special name for this series alone, sort of like "Gregory/Leibniz" for the arctangent (or the special value when x=1, namely pi/4.)

This is "our" Mercator, of the Mercator comma, incidentally, and not the map Mercator.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>>> Is it any more proportional to log(d) than log(n) in this

> >>>> case? Since n~=d?

> >>>

> >>>No, and the spreadsheet sorted by d is also sorted by n.

> >>

> >>So it could just as well be (n-d)/(d*log(n))?

> >

> >a very different sorting. that would be heuristic error, not

> >heuristic complexity. of course it's a very different sorting,

> >since knowing log(n) or log(d) tells you nothing about (n-d).

>

> ? I was asking if the *error* heuristic could become

> (n-d)/(d*log(n)) if we substituted log(n) for log(d).

oh . . . yeah sure. these are almost identical. i use odd limit for

both occurences of n or d in the denominator.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

> >>>>>And in that case it would have been easier to go straight

> >>>>>to log(n*d).

> >>>>

> >>>>Straight to where (do you see log(n*d))?

> >>>

> >>>log(n*d) = log(n) + log(d)

> >>

> >>Of course... so we're coming from log(n*d), not going to it.

> >

> >what do you mean, we're coming from log(n*d)??

>

> In your original message, you *start from* "is proportional to

> log(n) + log(d)" and *arrive at* "Hence the amount of tempering

> implied by the unison vector is approx. proportional to

> (n-d)/(d*log(d))".

>

> -Carl

right . . . well if you temper the octave too, then you'd use the

tenney metric which is indeed log(n) + log(d) . . . but typically

we've not been tempering the octave, so it's the van prooijen metric,

or log of the "odd limit" of the ratio, which is really relevant

here . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

<clumma@y...> wrote:

>

> > Maybe the original exposition can just be updated a bit, and

> > then monz or I could host it, certainly.

>

> You might want to add to

>

> complexity ~ log(d)

>

> error ~ log(n-d)/(d log(d))

>

> a badness heursitic of

>

> badness ~ log(n-d) log(d)^e / d

>

> where e = pi(prime limit)-1 = number of odd primes in limit.

gene, you too got the error heuristic wrong, it's

error ~ |n-d|/(d log(d))

and what kind of temperaments was this badness heuristic meant to

apply to?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>

wrote:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith

> <genewardsmith@j...>" <genewardsmith@j...> wrote:

> > --- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"

> <clumma@y...> wrote:

> >

> > > Maybe the original exposition can just be updated a bit, and

> > > then monz or I could host it, certainly.

> >

> > You might want to add to

> >

> > complexity ~ log(d)

> >

> > error ~ log(n-d)/(d log(d))

> >

> > a badness heursitic of

> >

> > badness ~ log(n-d) log(d)^e / d

> >

> > where e = pi(prime limit)-1 = number of odd primes in limit.

>

> gene, you too got the error heuristic wrong, it's

>

> error ~ |n-d|/(d log(d))

>

> and what kind of temperaments was this badness heuristic meant to

> apply to?

if i correct the error and use 5-limit linear temperaments (of course

you meant single-comma temperaments, duh), and thus use e=2, and cut

off the numerator and denominator at about 10^50, but don't cut off

for error (i just insist the size of the comma is under 600 cents), i

get the following for lowest badness:

numerator denominator

( 1 1)

2.92300327466181e+048 2.92297733949268e+048 atomic

32805 32768 schismic

1.77635683940025e+034 1.77630864952823e+034 pirate

81 80 meantone

4.5035996273705e+017 4.50283905890997e+017 monzismic

4 3 -

25 24 dicot

1.7179869184e+047 1.7179250691067e+047 raider

15625 15552 kleismic

7629394531250 7625597484987 ennealimmal

9.01016235351562e+015 9.00719925474099e+015 kwazy

16 15 father

6 5 -

5 4 -

9 8 -

10 9 -

274877906944 274658203125 semithirds

128 125 augmented

3.81520424476946e+029 3.814697265625e+029 senior

2048 2025 diaschismic

1600000 1594323 amity

27 25 beep

1.16450459770592e+023 1.16415321826935e+023 whoosh

250 243 porcupine

1.62285243890121e+032 1.62259276829213e+032 fortune

5.00315450989997e+016 5e+016 minortone

1076168025 1073741824 UNNAMED!!!!!!!!

6115295232 6103515625 semisuper

78732 78125 semisixths

3125 3072 magic

393216 390625 würschmidt

2109375 2097152 orwell

135 128 pelogic

10485760000 10460353203 vulture

68719476736000 68630377364883 tricot

4.44089209850063e+035 4.44002166576103e+035 egads

1224440064 1220703125 parakleismic

2.23007451985306e+043 2.22975839456296e+043 gross

19073486328125 19042491875328 enneadecal

648 625 diminished

20000 19683 tetracot

256 243 blackwood

2.47588007857076e+027 2.47471500188112e+027 astro

6561 6400 -

32 27 -

2.02824096036517e+035 2.02755595904453e+035 -

531441 524288 aristoxenean

2.95431270655083e+021 2.95147905179353e+021 counterschismic

31381059609 31250000000 -

5.82076609134674e+023 5.81595589965365e+023 -

4294967296 4271484375 escapade

75 64 -

16875 16384 negri

27 20 -

95367431640625 95105071448064 -

2.25283995449392e+034 2.25179981368525e+034 -

32 25 -

25 18 -

129140163 128000000 -

125 108 -

390625000 387420489 -

2.9557837600708e+020 2.95147905179353e+020 vavoom

625 576 -

35303692060125 35184372088832 -

3.4359738368e+030 3.43368382029251e+030 -

67108864 66430125 misty

244140625 241864704 -

etc. etc. etc. etc. etc. etc. etc. etc. etc. etc. etc. etc. etc.

gene, did 1076168025:1073741824 not make your geometric badness

cutoff, or did i mistakenly skip over it when i was working from your

list? 67108864:66430125 made it onto your list, does that have lower

geometric badness? if so, why is 1076168025:1073741824 so unusual

from the point of view of heuristic vs. geometric badness?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> 1076168025 1073741824 UNNAMED!!!!!!!!

Unnamed since it is a schisma squared.

>> > You might want to add to

>> >

>> > complexity ~ log(d)

>> >

>> > error ~ log(n-d)/(d log(d))

>> >

>> > a badness heursitic of

>> >

>> > badness ~ log(n-d) log(d)^e / d

>> >

>> > where e = pi(prime limit)-1 = number of odd primes in limit.

>>

>> gene, you too got the error heuristic wrong, it's

>>

>> error ~ |n-d|/(d log(d))

>>

>> and what kind of temperaments was this badness heuristic meant to

>> apply to?

>

>if i correct the error

Giving (n-d)log(d)^e / d ?

I don't get the point of the log(d)^e term.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>

wrote:

>

>

> > 1076168025 1073741824

UNNAMED!!!!!!!!

>

> Unnamed since it is a schisma squared.

OOPS!!!!!!!! (i can feel that torsion in my gut)

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> > You might want to add to

> >> >

> >> > complexity ~ log(d)

> >> >

> >> > error ~ log(n-d)/(d log(d))

> >> >

> >> > a badness heursitic of

> >> >

> >> > badness ~ log(n-d) log(d)^e / d

> >> >

> >> > where e = pi(prime limit)-1 = number of odd primes in limit.

> >>

> >> gene, you too got the error heuristic wrong, it's

> >>

> >> error ~ |n-d|/(d log(d))

> >>

> >> and what kind of temperaments was this badness heuristic meant

to

> >> apply to?

> >

> >if i correct the error

>

> Giving (n-d)log(d)^e / d ?

>

> I don't get the point of the log(d)^e term.

>

> -Carl

that gives you a log-flat badness measure.

>> >> > a badness heursitic of

>> >> >

>> >> > badness ~ log(n-d) log(d)^e / d

>> >> >

>> >> > where e = pi(prime limit)-1 = number of odd primes in limit.

>> >>

>> >> gene, you too got the error heuristic wrong, it's

>> >>

>> >> error ~ |n-d|/(d log(d))

//

>> >if i correct the error

>>

>> Giving (n-d)log(d)^e / d ?

>>

>> I don't get the point of the log(d)^e term.

>>

>> -Carl

>

>that gives you a log-flat badness measure.

Aha. Can we get results with e = 7 (19-limit)?

You said I just needed to penalize complexity, but:

() Wouldn't this ruin the log-flatness?

() Here you are restricting yourself to the 5-limit!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> >> > a badness heursitic of

> >> >> >

> >> >> > badness ~ log(n-d) log(d)^e / d

> >> >> >

> >> >> > where e = pi(prime limit)-1 = number of odd primes in limit.

> >> >>

> >> >> gene, you too got the error heuristic wrong, it's

> >> >>

> >> >> error ~ |n-d|/(d log(d))

> //

> >> >if i correct the error

> >>

> >> Giving (n-d)log(d)^e / d ?

yes, if n>d.

> >> I don't get the point of the log(d)^e term.

> >>

> >> -Carl

> >

> >that gives you a log-flat badness measure.

>

> Aha. Can we get results with e = 7 (19-limit)?

sure, but then we'd be talking about 6-dimensional temperaments. i

*might* attempt the calculation if you give me a nice low cutoff for

numerator and denominator . . .

> You said I just needed to penalize complexity,

penalize it more, yes.

> but:

>

> () Wouldn't this ruin the log-flatness?

no, it would get you closer to it.

> () Here you are restricting yourself to the 5-limit!

yes, though a few of the commas are 3-limit too. so?

>> >> Giving (n-d)log(d)^e / d ?

>

>yes, if n>d.

I thought you might say that!

>> >> I don't get the point of the log(d)^e term.

>> >>

>> >> -Carl

>> >

>> >that gives you a log-flat badness measure.

>>

>> Aha. Can we get results with e = 7 (19-limit)?

>

>sure, but then we'd be talking about 6-dimensional temperaments.

Saints preserve us!

>i *might* attempt the calculation if you give me a nice low cutoff

>for numerator and denominator . . .

Well, 10^50 would send my code to the sun. I could probably do this

with an imperative style, but since you apparently already have done

so, I thought I'd ask you.

What I don't get is why upping the prime limit from 5 to 19 would

make it any harder. The way I'd do it, is for each d < 10^50, run

n until n/d > 600 cents, kicking out any ratios where n*d has a

factor greater than 19. The factoring algorithm I'm using walks

up from 2, so aborting it after 19 or 5 wouldn't make much difference.

>> You said I just needed to penalize complexity,

>

>penalize it more, yes.

>

>> but:

>>

>> () Wouldn't this ruin the log-flatness?

>

>no, it would get you closer to it.

You mean without the log(d)^e term? Because if that term gives

flatness, and then I put an exponent on d, wouldn't I be ruining

the flatness?

>> () Here you are restricting yourself to the 5-limit!

>

>yes, though a few of the commas are 3-limit too. so?

I asked if these searches could be done without a restriction

on prime-limit, and you said yes.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> >> Giving (n-d)log(d)^e / d ?

> >

> >yes, if n>d.

>

> I thought you might say that!

>

> >> >> I don't get the point of the log(d)^e term.

> >> >>

> >> >> -Carl

> >> >

> >> >that gives you a log-flat badness measure.

> >>

> >> Aha. Can we get results with e = 7 (19-limit)?

> >

> >sure, but then we'd be talking about 6-dimensional temperaments.

>

> Saints preserve us!

>

> >i *might* attempt the calculation if you give me a nice low cutoff

> >for numerator and denominator . . .

>

> Well, 10^50 would send my code to the sun. I could probably do this

> with an imperative style, but since you apparently already have done

> so, I thought I'd ask you.

i've only done it for prime limit 5.

> What I don't get is why upping the prime limit from 5 to 19 would

> make it any harder. The way I'd do it, is for each d < 10^50, run

> n until n/d > 600 cents, kicking out any ratios where n*d has a

> factor greater than 19. The factoring algorithm I'm using walks

> up from 2, so aborting it after 19 or 5 wouldn't make much

>difference.

ok, so why don't you do it? (seriously -- my factoring algorithm

refuses numbers higher than 2^32). see if you can reproduce my 5-

limit results first.

> >> You said I just needed to penalize complexity,

> >

> >penalize it more, yes.

> >

> >> but:

> >>

> >> () Wouldn't this ruin the log-flatness?

> >

> >no, it would get you closer to it.

>

> You mean without the log(d)^e term?

i mean, you were using complexity * error, and that didn't penalize

complexity enough, while a higher power on complexity would.

> Because if that term gives

> flatness, and then I put an exponent on d, wouldn't I be ruining

> the flatness?

>

> >> () Here you are restricting yourself to the 5-limit!

> >

> >yes, though a few of the commas are 3-limit too. so?

>

> I asked if these searches could be done without a restriction

> on prime-limit, and you said yes.

i don't think i would have been referring to the same search. the

exponent on complexity in the log-flat badness formula, at least

according to gene, depends on the prime limit.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> What I don't get is why upping the prime limit from 5 to 19 would

> make it any harder.

i did it this way:

>> What I don't get is why upping the prime limit from 5 to 19 would

>> make it any harder. The way I'd do it, is for each d < 10^50, run

>> n until n/d > 600 cents, kicking out any ratios where n*d has a

>> factor greater than 19. The factoring algorithm I'm using walks

>> up from 2, so aborting it after 19 or 5 wouldn't make much

>> difference.

>

>ok, so why don't you do it? (seriously -- my factoring algorithm

>refuses numbers higher than 2^32). see if you can reproduce my 5-

>limit results first.

Ok, maybe later tonight/this morning. But how'd you do 10^50 if

you can't factor above 2^32?

>> >> You said I just needed to penalize complexity,

>> >

>> >penalize it more, yes.

>> >

>> >> but:

>> >>

>> >> () Wouldn't this ruin the log-flatness?

>> >

>> >no, it would get you closer to it.

>>

>> You mean without the log(d)^e term?

>

>i mean, you were using complexity * error, and that didn't penalize

>complexity enough, while a higher power on complexity would.

Ok.

>> Because if that term gives

>> flatness, and then I put an exponent on d, wouldn't I be ruining

>> the flatness?

>>

>> >> () Here you are restricting yourself to the 5-limit!

>> >

>> >yes, though a few of the commas are 3-limit too. so?

>>

>> I asked if these searches could be done without a restriction

>> on prime-limit, and you said yes.

>

>i don't think i would have been referring to the same search. the

>exponent on complexity in the log-flat badness formula, at least

>according to gene, depends on the prime limit.

You were referring to |n-d|/d. I get it now.

-Carl

>> What I don't get is why upping the prime limit from 5 to 19 would

>> make it any harder.

>

>i did it this way:

>

>http://www.kees.cc/tuning/perbl.html

This doesn't make a method clear to me. Could you go over it,

and/or post your code?

How long did it take you to do n=10^50? I've been running n=5000

for the last hour, and I'm still waiting.

Here's what I do...

For each d <= n, I run n up to 600 cents. That's well less than

n(n+1)/2 ratios -- shouldn't be a problem for n=5000. I check if

the ratio's in lowest-terms; if not, I throw it out. I then

compute its prime-limit; if > p (runtime param, 5 in this case),

I throw it out. I then compute the badness, using this handy-dandy

formula for pi(x)...

;; Minac (unpublished, proof in Ribenboim 1995, p. 181).

;; pi(x)= (sum j=2...x) (floor ((j-1)! + 1)/j - (floor (j-1)!/j)).

I believe all these tests to be quite cheap. Anywho, I then place

the ratio in a running list of the best r (runtime param, in this

case 10) ratios.

I really don't know what other optimizations to make. Anyway,

here are the results for n=1000, p=5, r=10, which takes about a

minute...

(badness primelimit ratio)

((0.24002696783739128 5 81/80)

(0.25993019270997947 3 9/8)

(0.2938674846231569 3 256/243)

(0.3662040962227033 3 4/3)

(0.42083442285788536 5 25/24)

(0.4804530139182014 5 5/4)

(0.4889023927793147 5 16/15)

(0.5180580787960469 5 6/5)

(0.5364217603611476 5 10/9)

(0.5595027250997308 5 128/125))

...they look roughly consistent with your results. p=7 takes no

longer, but returns the same top-10 results as above. In res.

10-20 we do see some 7-limit ratios...

(0.6103401603711721 3 32/27)

(0.7075223009389495 7 225/224)

(0.828892926073675 5 27/25)

(0.8692019265111186 5 250/243)

(0.9004848978857011 7 126/125)

(0.9587113625980545 7 7/6)

(1.0526168298843461 7 8/7)

(1.1047033190174127 3 81/64)

(1.1288769832608407 7 64/63)

(1.2029906627249671 7 50/49))

-Carl

>>> |n-d|log(d)^e / d

() Is e here supposed to be based on the prime limit of the

particular ratio, or the prime limit of the group of ratios?

() log-flat is supposed to give us an equal number of results

in all size ranges? I tend to think this allows too much

complexity in the small ratios.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> What I don't get is why upping the prime limit from 5 to 19 would

> >> make it any harder. The way I'd do it, is for each d < 10^50,

run

> >> n until n/d > 600 cents, kicking out any ratios where n*d has a

> >> factor greater than 19. The factoring algorithm I'm using walks

> >> up from 2, so aborting it after 19 or 5 wouldn't make much

> >> difference.

> >

> >ok, so why don't you do it? (seriously -- my factoring algorithm

> >refuses numbers higher than 2^32). see if you can reproduce my 5-

> >limit results first.

>

> Ok, maybe later tonight/this morning. But how'd you do 10^50 if

> you can't factor above 2^32?

i started with the factors!

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> What I don't get is why upping the prime limit from 5 to 19 would

> >> make it any harder.

> >

> >i did it this way:

> >

> >http://www.kees.cc/tuning/perbl.html

>

> This doesn't make a method clear to me. Could you go over it,

> and/or post your code?

i did it with command-line instructions. first create your 2-d array

of 5-limit ratios (the 5-limit lattice) and then calculate the

numerator and denominator of each ratio . . .

> How long did it take you to do n=10^50?

the computer time was negligible.

> I then

> compute its prime-limit; if > p (runtime param, 5 in this case),

> I throw it out. I then compute the badness, using this handy-dandy

> formula for pi(x)...

>

> ;; Minac (unpublished, proof in Ribenboim 1995, p. 181).

> ;; pi(x)= (sum j=2...x) (floor ((j-1)! + 1)/j - (floor (j-1)!/j)).

you're misunderstanding something. the highest prime in the ratio is

not necessarily the prime limit in which it will be used. witness the

pythagorean limma and pythagorean comma in the 5-limit case, for

example.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >>> |n-d|log(d)^e / d

>

> () Is e here supposed to be based on the prime limit of the

> particular ratio, or the prime limit of the group of ratios?

the prime limit of the lattice you're tempering.

> () log-flat is supposed to give us an equal number of results

> in all size ranges?

when they're defined logarithmically, yes.

> I tend to think this allows too much

> complexity in the small ratios.

so you want to penalize complexity even more? go for it.

>> I then

>> compute its prime-limit; if > p (runtime param, 5 in this case),

>> I throw it out. I then compute the badness, using this handy-dandy

>> formula for pi(x)...

>>

>> ;; Minac (unpublished, proof in Ribenboim 1995, p. 181).

>> ;; pi(x)= (sum j=2...x) (floor ((j-1)! + 1)/j - (floor (j-1)!/j)).

>

>you're misunderstanding something. the highest prime in the ratio is

>not necessarily the prime limit in which it will be used. witness the

>pythagorean limma and pythagorean comma in the 5-limit case, for

>example.

//

>> >>> |n-d|log(d)^e / d

>>

>> () Is e here supposed to be based on the prime limit of the

>> particular ratio, or the prime limit of the group of ratios?

>

>the prime limit of the lattice you're tempering.

Aha! Well then, I certainly don't need to compute pi(x) more

than once.

There's no apparent speedup, but at least these should be correct...

(primelimit=5 max-d=1000, res=10)

(badness primelimit ratio)

((0.24002696783739128 5 81/80)

(0.4023163202708607 3 4/3)

(0.42083442285788536 5 25/24)

(0.4804530139182014 5 5/4)

(0.4889023927793147 5 16/15)

(0.5180580787960469 5 6/5)

(0.5364217603611476 5 10/9)

(0.5405096406579765 3 9/8)

(0.5595027250997308 5 128/125)

(0.828892926073675 5 27/25))

(primelimit=7 max-d=1000, res=20)

(badness primelimit ratio)

((0.4419896533813025 3 4/3)

(0.6660493039778589 5 5/4)

(0.7075223009389495 7 225/224)

(0.8337823128571303 5 6/5)

(0.9004848978857011 7 126/125)

(0.9587113625980545 7 7/6)

(1.0518045661034596 5 81/80)

(1.0526168298843461 7 8/7)

(1.1239582004626365 3 9/8)

(1.1288769832608407 7 64/63)

(1.1786390756834733 5 10/9)

(1.2029906627249671 7 50/49)

(1.20863712518982 7 49/48)

(1.284039331328201 7 36/35)

(1.312860066467948 7 15/14)

(1.3239722230853748 5 16/15)

(1.3259689601439075 7 28/27)

(1.3374344495057697 5 25/24)

(1.3442467614814364 7 21/20)

(1.3641655968558717 7 245/243))

>> () log-flat is supposed to give us an equal number of results

>> in all size ranges?

>

>when they're defined logarithmically, yes.

Does this also give an equal number of results in all

complexity ranges? 'Cause that seems more intuitive to me.

>> I tend to think this allows too much

>> complexity in the small ratios.

>

>so you want to penalize complexity even more? go for it.

Prosibly. If I can get this working. Looks like the start-from-

factors approach is the only workable solution. There's no

native array in scheme, but I'm sure you can make them with vectors

of vectors.

Weirdly, though, my method is apparently O(n^2) no matter what the

limit, whereas your lattice-based method is O(n^2) at the 5-limit,

O(n^3) at the 7-limit, etc. No wait, my n is max-d, and your n is

radius on the lattice. Not the same. Did you just up the radius

until max-d was exceeded somewhere?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> () log-flat is supposed to give us an equal number of results

> >> in all size ranges?

> >

> >when they're defined logarithmically, yes.

>

> Does this also give an equal number of results in all

> complexity ranges? 'Cause that seems more intuitive to me.

by 'size', i meant complexity, or size of the numerator and

denominator. i didn't mean cents size of the comma in JI!

> Weirdly, though, my method is apparently O(n^2) no matter what the

> limit, whereas your lattice-based method is O(n^2) at the 5-limit,

> O(n^3) at the 7-limit, etc. No wait, my n is max-d, and your n is

> radius on the lattice. Not the same. Did you just up the radius

> until max-d was exceeded somewhere?

i made sure it was large enough to enclose kees' figure. note the

relationship with his work mentioned here:

>> >> () log-flat is supposed to give us an equal number of results

>> >> in all size ranges?

>> >

>> >when they're defined logarithmically, yes.

>>

>> Does this also give an equal number of results in all

>> complexity ranges? 'Cause that seems more intuitive to me.

>

>by 'size', i meant complexity, or size of the numerator and

>denominator. i didn't mean cents size of the comma in JI!

Oh. I guess I'm satisfied, then.

>> Weirdly, though, my method is apparently O(n^2) no matter what the

>> limit, whereas your lattice-based method is O(n^2) at the 5-limit,

>> O(n^3) at the 7-limit, etc. No wait, my n is max-d, and your n is

>> radius on the lattice. Not the same. Did you just up the radius

>> until max-d was exceeded somewhere?

>

>i made sure it was large enough to enclose kees' figure. note the

>relationship with his work mentioned here:

>

>http://www.sonic-arts.org/dict/heuristic-complexity.htm

...

>heuristic complexity

>

>for a unison vector n/d, which is a ratio of R, the heuristic for

>complexity is log(R).

>

>the complexity heuristic is identical to kees van prooijen's

>'expressibility' measure

Huh? For n/d, I thought it was just log(d). What's R here, odd-

or prime-limit?

Further, isn't there something on kees' page about your meassure

and and his being subtly different?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >heuristic complexity

> >

> >for a unison vector n/d, which is a ratio of R, the heuristic for

> >complexity is log(R).

> >

> >the complexity heuristic is identical to kees van prooijen's

> >'expressibility' measure

>

> Huh? For n/d, I thought it was just log(d). What's R here, odd-

> or prime-limit?

ratio of R, related to odd limit (click on "ratio of" for the

definition). for a small (in cents) comma, d is a very good

approximation of R.

> Further, isn't there something on kees' page about your meassure

> and and his being subtly different?

that was a different measure of mine, namely taxicab distance on the

isosceles triangular lattice.

>> >heuristic complexity

>> >

>> >for a unison vector n/d, which is a ratio of R, the heuristic for

>> >complexity is log(R).

>> >

>> >the complexity heuristic is identical to kees van prooijen's

>> >'expressibility' measure

>>

>> Huh? For n/d, I thought it was just log(d). What's R here, odd-

>> or prime-limit?

>

>ratio of R, related to odd limit (click on "ratio of" for the

>definition). for a small (in cents) comma, d is a very good

>approximation of R.

>

>> Further, isn't there something on kees' page about your meassure

>> and and his being subtly different?

>

>that was a different measure of mine, namely taxicab distance on the

>isosceles triangular lattice.

Sweet. Tx.

-Carl

>>> >the complexity heuristic is identical to kees van prooijen's

>>> >'expressibility' measure

>>> Isn't there something on kees' page about your meassure

>>> and and his being subtly different?

>>that was a different measure of mine, namely taxicab distance on the

>>isosceles triangular lattice.

>...but I still don't understand what's wrong with an

>octave-equiv. rect. lattice.

Doesn't this

http://www.kees.cc/tuning/erl_perbl.html

show that Kees' expressibility is an rectangular, octave-equivalent

measure?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Doesn't this

>

> http://www.kees.cc/tuning/erl_perbl.html

>

> show that Kees' expressibility is an rectangular, octave-equivalent

> measure?

The other way to define it is that given a monzo |a0 a1 ... an> we first

form |a0 log2(3)a1 log2(5)a2 ... log2(p)an> = |b0 ... bn>, and then

the so-called "expressibility" will be

(|b1| + |b2| + ... + |bn| + |b1+b2+...+bn|)/2

From this it is clear that things cannot be described as rectangular,

even to the extent that this sort of terminology applies at all in

non-Euclidean contexts.

>> Doesn't this

>>

>> http://www.kees.cc/tuning/erl_perbl.html

>>

>> show that Kees' expressibility is an rectangular, octave-equivalent

>> measure?

>

>The other way to define it is that given a monzo |a0 a1 ... an> we

>first form |a0 log2(3)a1 log2(5)a2 ... log2(p)an> = |b0 ... bn>, and

>then the so-called "expressibility" will be

>

>(|b1| + |b2| + ... + |bn| + |b1+b2+...+bn|)/2

>

>From this it is clear that things cannot be described as rectangular,

>even to the extent that this sort of terminology applies at all in

>non-Euclidean contexts.

I would call a taxicab distance measure weighted/unweighted,

rectangular/triangular if there is a sphere (centered on a lattice

point with with an arbitrary radius) that encloses all of the

lattice points of taxicab distance <= x and no or few other lattice

points.

I think you get rectangular in this sense if you factor n*d,

whether weighted or unweighted. Am I wrong?

Unweighted triangular is max ((n1+n2...) (d1+d2...)), where

n1... and d1... are the exponents in the prime factorizations of

n and d. Am I wrong?

Weighted triangular is trickier. I believe the formula is

>Given a Fokker-style interval vector (I1, I2, . . . In):

>

>1. Go to the rightmost nonzero exponent; add the product of its

>absolute value with the log of its base to the total.

>2. Use that exponent to cancel out as many exponents of the opposite

>sign as possible, starting to its immediate left and working right;

>discard anything remaining of that exponent.

> Example: starting with, say, (4 2 -3), we would add 3 lg(7) to

> our total, then cancel the -3 against the 2, then the remaining

> -1 against the 4, leaving (3 0 0). OTOH, starting with

> (-2 3 5), we would add 5 lg(7) to our total, then cancel 2 of

> the 5 against the -2 and discard the remainder, leaving (0 3 0).

>3. If any nonzero exponents remain, go back to step one, otherwise

>stop.

This gives Paul's "taxicab isosceles" measure from the link above.

Kees shows it on the lattice at

http://kees.cc/tuning/lat_perbl.html

and it doesn't look spherical, but that's possibly because he's

still got the axes at 90deg.

So I take it back that expressibility is rectangluar. In fact

I can't see how it has anything to do with lattice distance, unless

we have a limit-infinity lattice. I made this criticism in 99, I

think, when it first came up.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >>> >the complexity heuristic is identical to kees van prooijen's

> >>> >'expressibility' measure

>

> >>> Isn't there something on kees' page about your meassure

> >>> and and his being subtly different?

>

> >>that was a different measure of mine, namely taxicab distance on

the

> >>isosceles triangular lattice.

>

> >...but I still don't understand what's wrong with an

> >octave-equiv. rect. lattice.

>

> Doesn't this

>

> http://www.kees.cc/tuning/erl_perbl.html

>

> show that Kees' expressibility is an rectangular, octave-equivalent

> measure?

>

> -Carl

No, absolutely not. I don't know what gave you that impression. When

the lattice is rectangular, Kees expressibility is given by a

hexagonal norm of skewed shape (while a rectangular measure of

complexity would be given by a rectangular norm here). When the

hexagon is made regular, the lattice ends up looking like this (see

the second-to-last lattice here):

http://www.kees.cc/tuning/lat_perbl.html

Recall that expressibility is not a taxicab metric but is a hexagonal-

norm metric. So there is no paradox (contra Kees); 5/4 and 5/3 lie on

the same regular hexagon centered on 1/1, and thus they get the

same "distance" according to the regular-hexagon metric.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> Doesn't this

> >>

> >> http://www.kees.cc/tuning/erl_perbl.html

> >>

> >> show that Kees' expressibility is an rectangular, octave-

equivalent

> >> measure?

> >

> >The other way to define it is that given a monzo |a0 a1 ... an> we

> >first form |a0 log2(3)a1 log2(5)a2 ... log2(p)an> = |b0 ... bn>,

and

> >then the so-called "expressibility" will be

> >

> >(|b1| + |b2| + ... + |bn| + |b1+b2+...+bn|)/2

> >

> >From this it is clear that things cannot be described as

rectangular,

> >even to the extent that this sort of terminology applies at all in

> >non-Euclidean contexts.

>

> I would call a taxicab distance measure weighted/unweighted,

> rectangular/triangular if there is a sphere (centered on a lattice

> point with with an arbitrary radius) that encloses all of the

> lattice points of taxicab distance <= x and no or few other lattice

> points.

>

> I think you get rectangular in this sense

You gave only one if clause above; you didn't explain how one "gets"

or doesn't "get" rectangular.

> Unweighted triangular is max ((n1+n2...) (d1+d2...)), where

> n1... and d1... are the exponents in the prime factorizations of

> n and d. Am I wrong?

You probably don't want to use prime factorization here, but rather

Hahn's "odd factorization". Also below.

> Weighted triangular is trickier. I believe the formula is

> >Given a Fokker-style interval vector (I1, I2, . . . In):

> >

> >1. Go to the rightmost nonzero exponent; add the product of its

> >absolute value with the log of its base to the total.

> >2. Use that exponent to cancel out as many exponents of the

opposite

> >sign as possible, starting to its immediate left and working right;

> >discard anything remaining of that exponent.

> > Example: starting with, say, (4 2 -3), we would add 3 lg(7) to

> > our total, then cancel the -3 against the 2, then the

remaining

> > -1 against the 4, leaving (3 0 0). OTOH, starting with

> > (-2 3 5), we would add 5 lg(7) to our total, then cancel 2 of

> > the 5 against the -2 and discard the remainder, leaving (0 3

0).

> >3. If any nonzero exponents remain, go back to step one, otherwise

> >stop.

> This gives Paul's "taxicab isosceles" measure from the link above.

> Kees shows it on the lattice at

> http://kees.cc/tuning/lat_perbl.html

> and it doesn't look spherical, but that's possibly because he's

> still got the axes at 90deg.

It's the second lattice on that page. I don't see any 90 degree

angles there!

> So I take it back that expressibility is rectangluar. In fact

> I can't see how it has anything to do with lattice distance, unless

> we have a limit-infinity lattice. I made this criticism in 99, I

> think, when it first came up.

And it took me years to address it, but I eventually did. If you give

up both taxicab and Euclidean metrics, and use a hexagonal norm

instead, then Kees expressibility can be made exactly identical to

lattice distance, no matter how few or many dimensions (primes minus

1) there are. This norm is a regular hexagon if you use the lengths

and angles in the second-to-last lattice on that page

(http://kees.cc/tuning/lat_perbl.html).

>Recall that expressibility is not a taxicab metric but is a hexagonal-

>norm metric. So there is no paradox (contra Kees); 5/4 and 5/3 lie on

>the same regular hexagon centered on 1/1, and thus they get the

>same "distance" according to the regular-hexagon metric.

That's what I'd forgotten. Why is hexagon better than taxicab?

-Carl

>> I would call a taxicab distance measure weighted/unweighted,

>> rectangular/triangular if there is a sphere (centered on a lattice

>> point with with an arbitrary radius) that encloses all of the

>> lattice points of taxicab distance <= x and no or few other lattice

>> points.

>>

>> I think you get rectangular in this sense

>

>You gave only one if clause above; you didn't explain how one "gets"

>or doesn't "get" rectangular.

If such a sphere exists on an un/weighted tri/rectangular lattice

the distance measure is called un/weighted tri/rectangular. Makes

sense to me.

>> Unweighted triangular is max ((n1+n2...) (d1+d2...)), where

>> n1... and d1... are the exponents in the prime factorizations of

>> n and d. Am I wrong?

>

>You probably don't want to use prime factorization here, but rather

>Hahn's "odd factorization". Also below.

For unweighted I usually prefer prime factorization. I'm not

trying to outdo harmonic entropy here in terms of a dissonance

measure, I'm just trying to get a reasonable notion of lattice

distance.

>> Weighted triangular is trickier. I believe the formula is

//

>> This gives Paul's "taxicab isosceles" measure from the link above.

>> Kees shows it on the lattice at

>> http://kees.cc/tuning/lat_perbl.html

>> and it doesn't look spherical, but that's possibly because he's

>> still got the axes at 90deg.

>

>It's the second lattice on that page. I don't see any 90 degree

>angles there!

I don't see the unit shell there either.

>> So I take it back that expressibility is rectangluar. In fact

>> I can't see how it has anything to do with lattice distance, unless

>> we have a limit-infinity lattice. I made this criticism in 99, I

>> think, when it first came up.

>

>And it took me years to address it, but I eventually did. If you give

>up both taxicab and Euclidean metrics, and use a hexagonal norm

>instead, then Kees expressibility can be made exactly identical to

>lattice distance, no matter how few or many dimensions (primes minus

>1) there are. This norm is a regular hexagon if you use the lengths

>and angles in the second-to-last lattice on that page

>(http://kees.cc/tuning/lat_perbl.html).

Is the motivation for this that odd limit seems to be such a good

measure of octave-equivalent dissonance?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >Recall that expressibility is not a taxicab metric but is a

hexagonal-

> >norm metric. So there is no paradox (contra Kees); 5/4 and 5/3 lie

on

> >the same regular hexagon centered on 1/1, and thus they get the

> >same "distance" according to the regular-hexagon metric.

>

> That's what I'd forgotten. Why is hexagon better than taxicab?

It's not "better", it's just what works here.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> I would call a taxicab distance measure weighted/unweighted,

> >> rectangular/triangular if there is a sphere (centered on a

lattice

> >> point with with an arbitrary radius) that encloses all of the

> >> lattice points of taxicab distance <= x and no or few other

lattice

> >> points.

> >>

> >> I think you get rectangular in this sense

> >

> >You gave only one if clause above; you didn't explain how

one "gets"

> >or doesn't "get" rectangular.

>

> If such a sphere exists on an un/weighted tri/rectangular lattice

> the distance measure is called un/weighted tri/rectangular. Makes

> sense to me.

It seems circular to me. Start with the weighting part. A weighted

taxicab measure means to me that some rungs are longer than others.

So the sphere will naturally accomodate fewer rungs in the longer

direction, which seems to mean you "get" weighted according to the

above. But if you started with unweighted, the lengths would be

equal, so you'd "get" unweighted of course . . . (?)

> >> Unweighted triangular is max ((n1+n2...) (d1+d2...)), where

> >> n1... and d1... are the exponents in the prime factorizations of

> >> n and d. Am I wrong?

> >

> >You probably don't want to use prime factorization here, but

rather

> >Hahn's "odd factorization". Also below.

>

> For unweighted I usually prefer prime factorization. I'm not

> trying to outdo harmonic entropy here in terms of a dissonance

> measure, I'm just trying to get a reasonable notion of lattice

> distance.

I think Hahn gives a reasonable notion of lattice distance, and it's

based on odd factorization and is unweighted. Replacing odd with

prime just seems to reduce the reasonableness of the lattice

distances.

> >> Weighted triangular is trickier. I believe the formula is

> //

> >> This gives Paul's "taxicab isosceles" measure from the link

above.

> >> Kees shows it on the lattice at

> >> http://kees.cc/tuning/lat_perbl.html

> >> and it doesn't look spherical, but that's possibly because he's

> >> still got the axes at 90deg.

> >

> >It's the second lattice on that page. I don't see any 90 degree

> >angles there!

>

> I don't see the unit shell there either.

It's shown on the fourth graphic on that page.

> >> So I take it back that expressibility is rectangluar. In fact

> >> I can't see how it has anything to do with lattice distance,

unless

> >> we have a limit-infinity lattice. I made this criticism in 99, I

> >> think, when it first came up.

> >

> >And it took me years to address it, but I eventually did. If you

give

> >up both taxicab and Euclidean metrics, and use a hexagonal norm

> >instead, then Kees expressibility can be made exactly identical to

> >lattice distance, no matter how few or many dimensions (primes

minus

> >1) there are. This norm is a regular hexagon if you use the

lengths

> >and angles in the second-to-last lattice on that page

> >(http://kees.cc/tuning/lat_perbl.html).

>

> Is the motivation for this that odd limit seems to be such a good

> measure of octave-equivalent dissonance?

Partly. I wonder what Kees has to say?

>> > Recall that expressibility is not a taxicab metric but is a

>> > hexagonal-norm metric. So there is no paradox (contra Kees);

>> > 5/4 and 5/3 lie on the same regular hexagon centered on 1/1,

>> > and thus they get the same "distance" according to the

>> > regular-hexagon metric.

>>

>> That's what I'd forgotten. Why is hexagon better than taxicab?

>

>It's not "better", it's just what works here.

Can you describe the desiderata making up "here"?

-Carl

>> >> I would call a taxicab distance measure weighted/unweighted,

>> >> rectangular/triangular if there is a sphere (centered on a

>> >> lattice point with with an arbitrary radius) that encloses

>> >> all of the lattice points of taxicab distance <= x and no or

>> >> few other lattice points.

//

>> >You gave only one if clause above; you didn't explain how

>> >one "gets" or doesn't "get" rectangular.

>>

>> If such a sphere exists on an un/weighted tri/rectangular lattice

>> the distance measure is called un/weighted tri/rectangular. Makes

>> sense to me.

>

>It seems circular to me. Start with the weighting part. A weighted

>taxicab measure means to me that some rungs are longer than others.

>So the sphere will naturally accomodate fewer rungs in the longer

>direction, which seems to mean you "get" weighted according to the

>above. But if you started with unweighted, the lengths would be

>equal, so you'd "get" unweighted of course . . . (?)

You're right, it doesn't work for weighting. But for tri/rect,

I still think it does.

>> For unweighted I usually prefer prime factorization. I'm not

>> trying to outdo harmonic entropy here in terms of a dissonance

>> measure, I'm just trying to get a reasonable notion of lattice

>> distance.

>

>I think Hahn gives a reasonable notion of lattice distance, and it's

>based on odd factorization and is unweighted. Replacing odd with

>prime just seems to reduce the reasonableness of the lattice

>distances.

This doesn't work for arbitrary commas with no a priori harmonic

limit, since everything would just be 1.

>> >> Weighted triangular is trickier. I believe the formula is

>> //

>> >> This gives Paul's "taxicab isosceles" measure from the link

>> >> above.

>> >> Kees shows it on the lattice at

>> >> http://kees.cc/tuning/lat_perbl.html

>> >> and it doesn't look spherical, but that's possibly because he's

>> >> still got the axes at 90deg.

>> >

>> >It's the second lattice on that page. I don't see any 90 degree

>> >angles there!

>>

>> I don't see the unit shell there either.

>

>It's shown on the fourth graphic on that page.

Looks like the axes are at 90 degrees to me!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> > Recall that expressibility is not a taxicab metric but is a

> >> > hexagonal-norm metric. So there is no paradox (contra Kees);

> >> > 5/4 and 5/3 lie on the same regular hexagon centered on 1/1,

> >> > and thus they get the same "distance" according to the

> >> > regular-hexagon metric.

> >>

> >> That's what I'd forgotten. Why is hexagon better than taxicab?

> >

> >It's not "better", it's just what works here.

>

> Can you describe the desiderata making up "here"?

"Distance" = Kees expressibility.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> I would call a taxicab distance measure weighted/unweighted,

> >> >> rectangular/triangular if there is a sphere (centered on a

> >> >> lattice point with with an arbitrary radius) that encloses

> >> >> all of the lattice points of taxicab distance <= x and no or

> >> >> few other lattice points.

> //

> >> >You gave only one if clause above; you didn't explain how

> >> >one "gets" or doesn't "get" rectangular.

> >>

> >> If such a sphere exists on an un/weighted tri/rectangular lattice

> >> the distance measure is called un/weighted tri/rectangular.

Makes

> >> sense to me.

> >

> >It seems circular to me. Start with the weighting part. A weighted

> >taxicab measure means to me that some rungs are longer than

others.

> >So the sphere will naturally accomodate fewer rungs in the longer

> >direction, which seems to mean you "get" weighted according to the

> >above. But if you started with unweighted, the lengths would be

> >equal, so you'd "get" unweighted of course . . . (?)

>

> You're right, it doesn't work for weighting. But for tri/rect,

> I still think it does.

The same argument applies. With the triangular lattice, you

have "shortcuts" the taxicab can take, and the sphere best

approximates the hexagonal balls when the geometry is actually

triangular. For rectangular, the same argument leads to a rectangular

geometry.

> >> For unweighted I usually prefer prime factorization. I'm not

> >> trying to outdo harmonic entropy here in terms of a dissonance

> >> measure, I'm just trying to get a reasonable notion of lattice

> >> distance.

> >

> >I think Hahn gives a reasonable notion of lattice distance, and

it's

> >based on odd factorization and is unweighted. Replacing odd with

> >prime just seems to reduce the reasonableness of the lattice

> >distances.

>

> This doesn't work for arbitrary commas with no a priori harmonic

> limit, since everything would just be 1.

Huh? How did commas get into this? Everything would just be 1?? I

must be on crack . . .

> >> >> Weighted triangular is trickier. I believe the formula is

> >> //

> >> >> This gives Paul's "taxicab isosceles" measure from the link

> >> >> above.

> >> >> Kees shows it on the lattice at

> >> >> http://kees.cc/tuning/lat_perbl.html

> >> >> and it doesn't look spherical, but that's possibly because

he's

> >> >> still got the axes at 90deg.

> >> >

> >> >It's the second lattice on that page. I don't see any 90 degree

> >> >angles there!

> >>

> >> I don't see the unit shell there either.

> >

> >It's shown on the fourth graphic on that page.

>

> Looks like the axes are at 90 degrees to me!

Ha ha. Kees superimposed a horizontal and vertical axis on

the "shell" diagram. So what? That doesn't mean that either the shell

or the coresponding lattice have any 90 angles in them!

:)

>> >> > Recall that expressibility is not a taxicab metric but is a

>> >> > hexagonal-norm metric. So there is no paradox (contra Kees);

>> >> > 5/4 and 5/3 lie on the same regular hexagon centered on 1/1,

>> >> > and thus they get the same "distance" according to the

>> >> > regular-hexagon metric.

>> >>

>> >> That's what I'd forgotten. Why is hexagon better than taxicab?

>> >

>> >It's not "better", it's just what works here.

>>

>> Can you describe the desiderata making up "here"?

>

>"Distance" = Kees expressibility.

If the only reason for using hexagons is that they correspond to

the measure you like, it hardly justifies liking the measure.

But I think you've already said elsewhere in this thread that that

was based on psychoacoustics. Anyway, now may be an excellent time

for me to finally review that long IM we had about hexagonal

contours in projective tuningspace (or whatever)....

-Carl

>> >> >> I would call a taxicab distance measure weighted/unweighted,

>> >> >> rectangular/triangular if there is a sphere (centered on a

>> >> >> lattice point with with an arbitrary radius) that encloses

>> >> >> all of the lattice points of taxicab distance <= x and no or

>> >> >> few other lattice points.

//

>> You're right, it doesn't work for weighting. But for tri/rect,

>> I still think it does.

>

>With the triangular lattice, you have "shortcuts" the taxicab can

>take, and the sphere best approximates the hexagonal balls

And isn't unweighted triangular (Hahn diameter) the hexagonal

ball measure?

Given the agreement Gene's found between Euclidean distance and

unweighted Hahn diameter, I tend to think I'm right here.

>when the geometry is actually triangular. For rectangular, the

>same argument leads to a rectangular geometry.

Nah, there are better sphere approximations on the rect. lattice

than rectangles.

>> >> For unweighted I usually prefer prime factorization.

//

>> >I think Hahn gives a reasonable notion of lattice distance,

>> >and it's based on odd factorization and is unweighted.

//

>> This doesn't work for arbitrary commas with no a priori harmonic

>> limit, since everything would just be 1.

>

>Huh? How did commas get into this? Everything would just be 1?? I

>must be on crack . . .

I just realized the subject line is different, but I thought I

explained myself in the "the thing to do" message, and (after

the original message of this subject) in the "exploring badness"

thread.

Everything would be 1 because we assume the entire comma is

intended to be consonant and (since we're not weighting) assign

it length 1.

>> >> >> Weighted triangular is trickier. I believe the formula is

>> >> //

>> >> >> This gives Paul's "taxicab isosceles" measure from the link

>> >> >> above.

>> >> >> Kees shows it on the lattice at

>> >> >> http://kees.cc/tuning/lat_perbl.html

>> >> >> and it doesn't look spherical, but that's possibly because

>he's

>> >> >> still got the axes at 90deg.

>> >> >

>> >> >It's the second lattice on that page. I don't see any 90 degree

>> >> >angles there!

>> >>

>> >> I don't see the unit shell there either.

>> >

>> >It's shown on the fourth graphic on that page.

>>

>> Looks like the axes are at 90 degrees to me!

>

>Ha ha. Kees superimposed a horizontal and vertical axis on

>the "shell" diagram. So what? That doesn't mean that either the shell

>or the coresponding lattice have any 90 angles in them!

>

>:)

Well, it sure fooled me, since that's how he draws the axes on

every other lattice on his site.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> And isn't unweighted triangular (Hahn diameter) the hexagonal

> ball measure?

>

> Given the agreement Gene's found between Euclidean distance and

> unweighted Hahn diameter, I tend to think I'm right here.

The big difference is between weighted and unweighted. If for monzos

we define the unweighted p-distance to be

|| |a1 ... an> ||_p = ((|a1|^p + ... + |an|^p + |a1+...+an|)/2)^(1/p)

then the unweighted 2-distance is symmetrical Euclidean, and the

1-distance is the unweighted version of Kees distance. The

infinity-distance isn't Hahn distance, but it is related and in many

cases will give the same answer.

> >Ha ha. Kees superimposed a horizontal and vertical axis on

> >the "shell" diagram. So what? That doesn't mean that either the shell

> >or the coresponding lattice have any 90 angles in them!

Given that the lattices are not Euclidean, they can't have any 90

angles. Angles don't exist for Kees, Tenney, Hahn etc. lattices.

>> And isn't unweighted triangular (Hahn diameter) the hexagonal

>> ball measure?

>>

>> Given the agreement Gene's found between Euclidean distance and

>> unweighted Hahn diameter, I tend to think I'm right here.

>

>The big difference is between weighted and unweighted. If for monzos

>we define the unweighted p-distance to be

>

>|| |a1 ... an> ||_p = ((|a1|^p + ... + |an|^p + |a1+...+an|)/2)^(1/p)

>

>then the unweighted 2-distance is symmetrical Euclidean, and the

>1-distance is the unweighted version of Kees distance. The

>infinity-distance isn't Hahn distance, but it is related and in many

>cases will give the same answer.

Neat formula.

Just for my records, you've previously given Hahn distance as:

||(a,b,c)||_Hahn = max(|a|,|b|,|c|,|b+c|,|a+c|,|a+b|,|a+b+c|)

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Neat formula.

> Just for my records, you've previously given Hahn distance as:

>

> ||(a,b,c)||_Hahn = max(|a|,|b|,|c|,|b+c|,|a+c|,|a+b|,|a+b+c|)

Right, whereas the infinity-distance would be

max(|a|,|b|,|c|,|a+b+c|)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

>

> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> > Neat formula.

> > Just for my records, you've previously given Hahn distance as:

> >

> > ||(a,b,c)||_Hahn = max(|a|,|b|,|c|,|b+c|,|a+c|,|a+b|,|a+b+c|)

>

> Right, whereas the infinity-distance would be

>

> max(|a|,|b|,|c|,|a+b+c|)

However, be it noted that we can define not only

||(a,b,c)||_p = ((|a|^p+|b|^p+|c|^p+|a+b+c|^p)/2)^(1/p)

but also

||(a,b,c)||_q = ((|a|^q+|b|^q+|c|^q+|a+b|^q+|b+c|^q+|c+a|^q+

|a+b+c|^q)/4)^(1/q)

It happens that *both* p=2 and q=2 give the symmetric Euclidean norm,

and q=infinity gives the Hahn norm.

>If for monzos we define the unweighted p-distance to be

>

>|| |a1 ... an> ||_p = ((|a1|^p + ... + |an|^p + |a1+...+an|)/2)^(1/p)

>

>then the unweighted 2-distance is symmetrical Euclidean, and the

>1-distance is the unweighted version of Kees distance. The

>infinity-distance isn't Hahn distance, but it is related and in many

>cases will give the same answer.

//

>||(a,b,c)||_q = ((|a|^q+|b|^q+|c|^q+|a+b|^q+|b+c|^q+|c+a|^q+

>|a+b+c|^q)/4)^(1/q)

>

>It happens that *both* p=2 and q=2 give the symmetric Euclidean norm,

>and q=infinity gives the Hahn norm.

It looks like q=1 will still give unweighted Kees, so ||_q looks

better. Aside from that it's harder to compute when the monzo

gets longer, is there any reason to use ||_p?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> > Recall that expressibility is not a taxicab metric but is a

> >> >> > hexagonal-norm metric. So there is no paradox (contra Kees);

> >> >> > 5/4 and 5/3 lie on the same regular hexagon centered on 1/1,

> >> >> > and thus they get the same "distance" according to the

> >> >> > regular-hexagon metric.

> >> >>

> >> >> That's what I'd forgotten. Why is hexagon better than

taxicab?

> >> >

> >> >It's not "better", it's just what works here.

> >>

> >> Can you describe the desiderata making up "here"?

> >

> >"Distance" = Kees expressibility.

>

> If the only reason for using hexagons is that they correspond to

> the measure you like, it hardly justifies liking the measure.

Of course! I liked the measure long before I found the hexagons.

> But I think you've already said elsewhere in this thread that that

> was based on psychoacoustics. Anyway, now may be an excellent time

> for me to finally review that long IM we had about hexagonal

> contours in projective tuningspace (or whatever)....

That's related to the dual picture to this one . . .

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> >> I would call a taxicab distance measure

weighted/unweighted,

> >> >> >> rectangular/triangular if there is a sphere (centered on a

> >> >> >> lattice point with with an arbitrary radius) that encloses

> >> >> >> all of the lattice points of taxicab distance <= x and no

or

> >> >> >> few other lattice points.

> //

> >> You're right, it doesn't work for weighting. But for tri/rect,

> >> I still think it does.

> >

> >With the triangular lattice, you have "shortcuts" the taxicab can

> >take, and the sphere best approximates the hexagonal balls

>

> And isn't unweighted triangular (Hahn diameter) the hexagonal

> ball measure?

In the very lowest limits, it can be, depending on what shape

hexagons you mean and how you're constructing the lattice.

> Given the agreement Gene's found between Euclidean distance and

> unweighted Hahn diameter, I tend to think I'm right here.

Huh? I have no idea how this fits in, or how it contradicts what I

said. Your statement above didn't bring the ratios on the lattice at

all, so something like Hahn seems a separate consideration.

>> >when the geometry is actually triangular. For rectangular, the

>> >same argument leads to a rectangular geometry.

>

>> Nah, there are better sphere approximations on the rect. lattice

>> than rectangles.

The relevant approximations would be close to regular octahedra in

the 3D case. If you want to get even closer to spheres, it seems you

have to throw taxicab out the window. But your original statement was

all about taxicab measures!

>> >> >> For unweighted I usually prefer prime factorization.

> //

>> >> >I think Hahn gives a reasonable notion of lattice distance,

>> >> >and it's based on odd factorization and is unweighted.

> //

>> >> This doesn't work for arbitrary commas with no a priori harmonic

>> >> limit, since everything would just be 1.

> >

>> >Huh? How did commas get into this? Everything would just be 1?? I

>> >must be on crack . . .

>

>> I just realized the subject line is different, but I thought I

>> explained myself in the "the thing to do" message, and (after

>> the original message of this subject) in the "exploring badness"

>> thread.

>

>> Everything would be 1 because we assume the entire comma is

>> intended to be consonant and (since we're not weighting) assign

> it length 1.

I think you're trying to imply that everything is consonant when

there's no a priori harmonic limit. But "odd factorization" in the

sense required here makes no sense without one.

> >> >> >> Weighted triangular is trickier. I believe the formula is

> >> >> //

> >> >> >> This gives Paul's "taxicab isosceles" measure from the link

> >> >> >> above.

> >> >> >> Kees shows it on the lattice at

> >> >> >> http://kees.cc/tuning/lat_perbl.html

> >> >> >> and it doesn't look spherical, but that's possibly because

> >he's

> >> >> >> still got the axes at 90deg.

> >> >> >

> >> >> >It's the second lattice on that page. I don't see any 90

degree

> >> >> >angles there!

> >> >>

> >> >> I don't see the unit shell there either.

> >> >

> >> >It's shown on the fourth graphic on that page.

> >>

> >> Looks like the axes are at 90 degrees to me!

> >

> >Ha ha. Kees superimposed a horizontal and vertical axis on

> >the "shell" diagram. So what? That doesn't mean that either the

shell

> >or the coresponding lattice have any 90 angles in them!

> >

> >:)

>

> Well, it sure fooled me,

Why? Kees clearly indicates which lattice each shell corresponds to.

> since that's how he draws the axes on

> every other lattice on his site.

What are you talking about? The 2nd and 3rd lattices on this page,

for example, clearly don't have any 90 degree angles in them; when he

draws the shells for these lattices, he superimposes a vertical and

horizontal line just for reference (since the transformations

matrices tell you how to switch the lattice between one set of x and

y axes and another).

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

>

> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> > And isn't unweighted triangular (Hahn diameter) the hexagonal

> > ball measure?

> >

> > Given the agreement Gene's found between Euclidean distance and

> > unweighted Hahn diameter, I tend to think I'm right here.

>

> The big difference is between weighted and unweighted. If for monzos

> we define the unweighted p-distance to be

>

> || |a1 ... an> ||_p = ((|a1|^p + ... + |an|^p + |a1+...+an|)/2)^

(1/p)

>

> then the unweighted 2-distance is symmetrical Euclidean, and the

> 1-distance is the unweighted version of Kees distance. The

> infinity-distance isn't Hahn distance, but it is related and in many

> cases will give the same answer.

>

> > >Ha ha. Kees superimposed a horizontal and vertical axis on

> > >the "shell" diagram. So what? That doesn't mean that either the

shell

> > >or the coresponding lattice have any 90 angles in them!

>

> Given that the lattices are not Euclidean, they can't have any 90

> angles. Angles don't exist for Kees, Tenney, Hahn etc. lattices.

Kees depicted the very same lattice, and the very same shell, using

several different conventions of how to embed it in familiar 2D

space. For each of these, he displays the shell against a vertical

and horizontal axis at 90 degree angles. This gives the

transformation matrices he's talking about direct meaning. But the

vertical and horizontal axes don't have to signify anything in

particular for any of these conventions.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >If for monzos we define the unweighted p-distance to be

> >

> >|| |a1 ... an> ||_p = ((|a1|^p + ... + |an|^p + |a1+...+an|)/2)^

(1/p)

> >

> >then the unweighted 2-distance is symmetrical Euclidean, and the

> >1-distance is the unweighted version of Kees distance. The

> >infinity-distance isn't Hahn distance, but it is related and in

many

> >cases will give the same answer.

> //

> >||(a,b,c)||_q = ((|a|^q+|b|^q+|c|^q+|a+b|^q+|b+c|^q+|c+a|^q+

> >|a+b+c|^q)/4)^(1/q)

> >

> >It happens that *both* p=2 and q=2 give the symmetric Euclidean

norm,

> >and q=infinity gives the Hahn norm.

>

> It looks like q=1 will still give unweighted Kees,

What does that mean? In my view, the Kees lattice doesn't even have

rungs, so I don't know what it would mean for them all to be the same

length -- or, if you picked some and made them the same, why this

wouldn't be the same as some already-named thing. Unweighted Kees is

an oxymoron to me but what does it mean to you?

>> >> >> >> I would call a taxicab distance measure

>> >> >> >> rectangular/triangular if there is a sphere (centered on a

>> >> >> >> lattice point with with an arbitrary radius) that encloses

>> >> >> >> all of the lattice points of taxicab distance <= x and no

>> >> >> >> or few other lattice points.

//

>> Given the agreement Gene's found between Euclidean distance and

>> unweighted Hahn diameter, I tend to think I'm right here.

>

>Huh? I have no idea how this fits in, or how it contradicts what I

>said. Your statement above didn't bring the ratios on the lattice at

>all, so something like Hahn seems a separate consideration.

I'm not sure what "statement above" you're referring to. The one

at the top of this message seems to be the one in contention, and

mentions "lattice points" which are "ratios on the lattice".

Gene has reported that ||_Hahn and Euclidean norm shells differ by

few notes in a variety of harmonic limits, which is all I'm saying

at the top.

>>> >when the geometry is actually triangular. For rectangular, the

>>> >same argument leads to a rectangular geometry.

>>

>>> Nah, there are better sphere approximations on the rect. lattice

>>> than rectangles.

>

>The relevant approximations would be close to regular octahedra in

>the 3D case. If you want to get even closer to spheres, it seems you

>have to throw taxicab out the window. But your original statement was

>all about taxicab measures!

Howabout: The closest you can get with unit lengths to spheres on the

triangular lattice is to count ratios of consonances as a single

step. The closest you can get with unit lengths to spheres on the

rectangular latice is to count ratios of consonances as two steps.

>>> >> >I think Hahn gives a reasonable notion of lattice distance,

>>> >> >and it's based on odd factorization and is unweighted.

>> //

>>> >> This doesn't work for arbitrary commas with no a priori harmonic

>>> >> limit, since everything would just be 1.

//

>>> Everything would be 1 because we assume the entire comma is

>>> intended to be consonant and (since we're not weighting) assign

>> it length 1.

>

>I think you're trying to imply that everything is consonant when

>there's no a priori harmonic limit. But "odd factorization" in the

>sense required here makes no sense without one.

Exactly.

>> >> >> >> Weighted triangular is trickier. I believe the formula is

>> >> >> //

>> >> >> >> This gives Paul's "taxicab isosceles" measure from the link

>> >> >> >> above.

>> >> >> >> Kees shows it on the lattice at

>> >> >> >> http://kees.cc/tuning/lat_perbl.html

>> >> >> >> and it doesn't look spherical, but that's possibly because

>> >> >> >> he's still got the axes at 90deg.

>> >> >> >

>> >> >> >It's the second lattice on that page. I don't see any 90

>> >> >> >degree angles there!

>> >> >>

>> >> >> I don't see the unit shell there either.

>> >> >

>> >> >It's shown on the fourth graphic on that page.

>> >>

>> >> Looks like the axes are at 90 degrees to me!

>> >

>> >Ha ha. Kees superimposed a horizontal and vertical axis on

>> >the "shell" diagram. So what? That doesn't mean that either the

>> >shell or the coresponding lattice have any 90 angles in them!

>> >

>> >:)

>>

>> Well, it sure fooled me,

>

>Why? Kees clearly indicates which lattice each shell corresponds to.

Let's go back...

> >It seems circular to me. Start with the weighting part. A

> >weighted taxicab measure means to me that some rungs are longer

> >than others. So the sphere will naturally accomodate fewer

> >rungs in the longer direction, which seems to mean you "get"

> >weighted according to the above. But if you started with

> >unweighted, the lengths would be equal, so you'd "get" unweighted

> >of course . . . (?)

Faced with this, I took back that it worked for weighted. But it

does, and in fact now I read this to be agreeing with me.

-Carl

>> >||(a,b,c)||_q = ((|a|^q+|b|^q+|c|^q+|a+b|^q+|b+c|^q+|c+a|^q+

>> >|a+b+c|^q)/4)^(1/q)

>> >

>> >It happens that *both* p=2 and q=2 give the symmetric Euclidean

>> >norm and q=infinity gives the Hahn norm.

>>

>> It looks like q=1 will still give unweighted Kees,

>

>What does that mean? In my view, the Kees lattice doesn't even have

>rungs, so I don't know what it would mean for them all to be the same

>length -- or, if you picked some and made them the same, why this

>wouldn't be the same as some already-named thing. Unweighted Kees is

>an oxymoron to me but what does it mean to you?

Gene defined it in that message.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> >> >> I would call a taxicab distance measure

> >> >> >> >> rectangular/triangular if there is a sphere (centered

on a

> >> >> >> >> lattice point with with an arbitrary radius) that

encloses

> >> >> >> >> all of the lattice points of taxicab distance <= x and

no

> >> >> >> >> or few other lattice points.

> //

> >> Given the agreement Gene's found between Euclidean distance and

> >> unweighted Hahn diameter, I tend to think I'm right here.

> >

> >Huh? I have no idea how this fits in, or how it contradicts what I

> >said. Your statement above didn't bring the ratios on the lattice

at

> >all, so something like Hahn seems a separate consideration.

>

> I'm not sure what "statement above" you're referring to. The one

> at the top of this message seems to be the one in contention, and

> mentions "lattice points" which are "ratios on the lattice".

>

> Gene has reported that ||_Hahn and Euclidean norm shells differ by

> few notes in a variety of harmonic limits, which is all I'm saying

> at the top.

It is? Can you clarify your point then?

>> >>> >when the geometry is actually triangular. For rectangular, the

>> >>> >same argument leads to a rectangular geometry.

> >>

>> >>> Nah, there are better sphere approximations on the rect.

>lattice

>> >>> than rectangles.

> >

>> >The relevant approximations would be close to regular octahedra

in

>> >the 3D case. If you want to get even closer to spheres, it seems

you

>> >have to throw taxicab out the window. But your original statement

was

>> >all about taxicab measures!

>

>> Howabout: The closest you can get with unit lengths to spheres on

the

>> triangular lattice is to count ratios of consonances as a single

>> step.

If the triangular lattice is equilateral and has (all of the)

consonances for rungs, then this gets pretty close, though of course

Euclidean gets closer.

> The closest you can get with unit lengths to spheres on the

> rectangular latice is to count ratios of consonances as two steps.

Two steps? This makes no sense to me. Can you elaborate on your

thinking here?

> >>> >> >I think Hahn gives a reasonable notion of lattice distance,

> >>> >> >and it's based on odd factorization and is unweighted.

> >> //

> >>> >> This doesn't work for arbitrary commas with no a priori

harmonic

> >>> >> limit, since everything would just be 1.

> //

> >>> Everything would be 1 because we assume the entire comma is

> >>> intended to be consonant and (since we're not weighting) assign

> >> it length 1.

> >

> >I think you're trying to imply that everything is consonant when

> >there's no a priori harmonic limit. But "odd factorization" in the

> >sense required here makes no sense without one.

>

> Exactly.

:-p

> >> >> >> >> Weighted triangular is trickier. I believe the formula

is

> >> >> >> //

> >> >> >> >> This gives Paul's "taxicab isosceles" measure from the

link

> >> >> >> >> above.

> >> >> >> >> Kees shows it on the lattice at

> >> >> >> >> http://kees.cc/tuning/lat_perbl.html

> >> >> >> >> and it doesn't look spherical, but that's possibly

because

> >> >> >> >> he's still got the axes at 90deg.

> >> >> >> >

> >> >> >> >It's the second lattice on that page. I don't see any 90

> >> >> >> >degree angles there!

> >> >> >>

> >> >> >> I don't see the unit shell there either.

> >> >> >

> >> >> >It's shown on the fourth graphic on that page.

> >> >>

> >> >> Looks like the axes are at 90 degrees to me!

> >> >

> >> >Ha ha. Kees superimposed a horizontal and vertical axis on

> >> >the "shell" diagram. So what? That doesn't mean that either the

> >> >shell or the coresponding lattice have any 90 angles in them!

> >> >

> >> >:)

> >>

> >> Well, it sure fooled me,

> >

> >Why? Kees clearly indicates which lattice each shell corresponds

to.

>

> Let's go back...

>

> > >It seems circular to me. Start with the weighting part. A

> > >weighted taxicab measure means to me that some rungs are longer

> > >than others. So the sphere will naturally accomodate fewer

> > >rungs in the longer direction, which seems to mean you "get"

> > >weighted according to the above. But if you started with

> > >unweighted, the lengths would be equal, so you'd "get" unweighted

> > >of course . . . (?)

>

> Faced with this, I took back that it worked for weighted. But it

> does, and in fact now I read this to be agreeing with me.

OK, maybe we need to rephrase what we're talking about so that it

means the same thing to both of us. I certainly believe in the

principle of embedding these lattices in Euclidean space such that

their balls come as close to being spherical as possible. This

dictates a rectangular geometry for octave-specific Tenney, an

equilateral triangular geometry for equal-weighted Hahn (breaks down

beyond 7-limit), and a geometry without rungs but conforming to the

third full lattice diagram on the Kees page we've been looking at for

expressibility or "odd limit" (really the smallest odd limit the

interval belongs to).

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >||(a,b,c)||_q = ((|a|^q+|b|^q+|c|^q+|a+b|^q+|b+c|^q+|c+a|^q+

> >> >|a+b+c|^q)/4)^(1/q)

> >> >

> >> >It happens that *both* p=2 and q=2 give the symmetric Euclidean

> >> >norm and q=infinity gives the Hahn norm.

> >>

> >> It looks like q=1 will still give unweighted Kees,

> >

> >What does that mean? In my view, the Kees lattice doesn't even

have

> >rungs, so I don't know what it would mean for them all to be the

same

> >length -- or, if you picked some and made them the same, why this

> >wouldn't be the same as some already-named thing. Unweighted Kees

is

> >an oxymoron to me but what does it mean to you?

>

> Gene defined it in that message.

>

> -Carl

Gene, how do you respond? I don't see how "unweighted Kees" could

possibly make any sense, so I look to you to fill me in.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> OK, maybe we need to rephrase what we're talking about so that it

> means the same thing to both of us. I certainly believe in the

> principle of embedding these lattices in Euclidean space such that

> their balls come as close to being spherical as possible.

Are you taking about introducing two norms, a Euclidean and a

noneuclidean one?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Gene, how do you respond? I don't see how "unweighted Kees" could

> possibly make any sense, so I look to you to fill me in.

We can define the "p-norm" on 7-limit monzo classes as

|| |* a b c> ||_p = ((|a|^p + |b|^p + |c|^p + |a+b+c|^p)/2)^(1/p)

Then || ||_2 is the symmetrical Euclidean norm, and || ||_1 is the

unweighted Kees norm. The reason for calling it that is that if

a, b and c are weighted by log2(p) factors, then what you get is in

fact the Kees norm. Hence, the Kees norm can be described as a

weighted p-norm.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

>

> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>

wrote:

>

> > OK, maybe we need to rephrase what we're talking about so that it

> > means the same thing to both of us. I certainly believe in the

> > principle of embedding these lattices in Euclidean space such

that

> > their balls come as close to being spherical as possible.

>

> Are you taking about introducing two norms, a Euclidean and a

> noneuclidean one?

Not really, I'm just talking about visually depicting these lattices

in such a way that the norms we want to use with them come out

looking as spherical as possible.

I see no reason we can't understand them as living in Euclidean space

and then use various custom notions of "distance" to represent the

complexity measures we care about; for example, the third lattice on

the Kees page we keep looking at can be associated with a little

vehicle that is constrained to be able to move in only the

six "cardinal" directions at 60 degree increments from one another;

the distance that this vehicle has to travel to get from one lattice

point to another corresponds to Kees expressibility (due to the ball

becoming a regular hexagon in this depiction).

In the Tenney case, there's less motivation to do so, since the

taxicab distance is so straightforward and remains the same

regardless of what the angles are. To give the sense of this, I

suggested imagining a Tenney lattice where the angle between the axes

is constantly changing; the quantities we care about are invariants

in this world.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

>

> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>

wrote:

>

> > Gene, how do you respond? I don't see how "unweighted Kees" could

> > possibly make any sense, so I look to you to fill me in.

>

> We can define the "p-norm" on 7-limit monzo classes as

>

> || |* a b c> ||_p = ((|a|^p + |b|^p + |c|^p + |a+b+c|^p)/2)^(1/p)

>

> Then || ||_2 is the symmetrical Euclidean norm, and || ||_1 is the

> unweighted Kees norm. The reason for calling it that is that if

> a, b and c are weighted by log2(p) factors, then what you get is in

> fact the Kees norm. Hence, the Kees norm can be described as a

> weighted p-norm.

But don't we have another name for this (what you're

calling "unweighted Kees norm") already?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> I see no reason we can't understand them as living in Euclidean space

> and then use various custom notions of "distance" to represent the

> complexity measures we care about...

You can, but then technically you don't get a Tenney, Hahn etc.

lattice. There are advantages at times to having these be honest

lattices, but it isn't too helpful if you want to picture them.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > We can define the "p-norm" on 7-limit monzo classes as

> >

> > || |* a b c> ||_p = ((|a|^p + |b|^p + |c|^p + |a+b+c|^p)/2)^(1/p)

> But don't we have another name for this (what you're

> calling "unweighted Kees norm") already?

None that I know of. A better name for this p-norm, and also

the "q-norm"

|| |* a b c> ||_q = ((|a|^q + |b|^q + |c|^q +

|a+b|^q + |b+c|^q + |c+a|^q + |a+b+c|^q)/4)^(1/q)

would be a good start.

>> >> >> >> >> I would call a taxicab distance measure

>> >> >> >> >> rectangular/triangular if there is a sphere (centered

>> >> >> >> >> on a lattice point with with an arbitrary radius) that

>> >> >> >> >> encloses all of the lattice points of taxicab

>> >> >> >> >> distance <= x and no or few other lattice points.

//

>> Gene has reported that ||_Hahn and Euclidean norm shells differ by

>> few notes in a variety of harmonic limits, which is all I'm saying

>> at the top.

>

>It is?

Well, that's the triangular case.

>Can you clarify your point then?

I thought you'd asked what I meant by triangular and rectangular.

>>> >>> Nah, there are better sphere approximations on the rect.

>>> >>> lattice than rectangles.

>> >

>>> >The relevant approximations would be close to regular octahedra

>>> >in the 3D case.

Yes, I think that's right.

>>> If you want to get even closer to spheres, it seems you have

>>> to throw taxicab out the window. But your original statement

>>> was all about taxicab measures!

Where's the problem? I'm defining "triangular" to be 'as close

as you can get to Euclidiean with taxicab on a triangular lattice'.

>>> Howabout: The closest you can get with unit lengths to spheres on

>>> the triangular lattice is to count ratios of consonances as a

>>> single step.

>

>If the triangular lattice is equilateral and has (all of the)

>consonances for rungs, then this gets pretty close, though of course

>Euclidean gets closer.

Euclidean *is* "spheres".

>> The closest you can get with unit lengths to spheres on the

>> rectangular latice is to count ratios of consonances as two steps.

>

>Two steps? This makes no sense to me. Can you elaborate on your

>thinking here?

If you count them (ie 5:3) as two steps on the rectangular lattice,

it looks like the nth shell of that agrees with the Euclidean shell

of radius n. Though it does look like you can count them as one

step and make it agree with the Euclidean shell of radius n*sqrt(2).

>> Let's go back...

>>

>> > >It seems circular to me. Start with the weighting part. A

>> > >weighted taxicab measure means to me that some rungs are longer

>> > >than others. So the sphere will naturally accomodate fewer

>> > >rungs in the longer direction, which seems to mean you "get"

>> > >weighted according to the above. But if you started with

>> > >unweighted, the lengths would be equal, so you'd "get" unweighted

>> > >of course . . . (?)

>>

>> Faced with this, I took back that it worked for weighted. But it

>> does, and in fact now I read this to be agreeing with me.

>

>OK, maybe we need to rephrase what we're talking about so that it

>means the same thing to both of us. I certainly believe in the

>principle of embedding these lattices in Euclidean space such that

>their balls come as close to being spherical as possible. This

>dictates a rectangular geometry for octave-specific Tenney, an

>equilateral triangular geometry for equal-weighted Hahn

Now we're in business. But doesn't it dictate a rectangular

geometry for any measure where rungs represent factors (not ratios),

and a triangular measure where the reverse is true? (And have

nothing to do with octaves or the particular weighting?)

>(breaks down beyond 7-limit),

Yes, I remember something about this. Something about A3 = D3...

but I can't find anything more on this at mathworld or google.

>and a geometry without rungs but conforming to the third full

>lattice diagram on the Kees page we've been looking at for

>expressibility or "odd limit" (really the smallest odd limit the

>interval belongs to).

With hexagonal contours, gotcha. Does this depend on limit?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

>

> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>

wrote:

>

> > > We can define the "p-norm" on 7-limit monzo classes as

> > >

> > > || |* a b c> ||_p = ((|a|^p + |b|^p + |c|^p + |a+b+c|^p)/2)^(1/p)

>

> > But don't we have another name for this (what you're

> > calling "unweighted Kees norm") already?

>

> None that I know of.

So this isn't either L_1 or L_inf in the symmetrical (equilateral

triangular) lattice? How do they differ?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

I think we're good up to here . . .

> >> The closest you can get with unit lengths to spheres on the

> >> rectangular latice is to count ratios of consonances as two

steps.

> >

> >Two steps? This makes no sense to me. Can you elaborate on your

> >thinking here?

>

> If you count them (ie 5:3) as two steps on the rectangular lattice,

This can be a ratio of consonances but so can, say, 9:5, which is the

ratio between 5/6 and 3/2; and so can, say, 5:4, which is one step

(right?) . . .

> it looks like the nth shell of that agrees with the Euclidean shell

> of radius n.

You're saying you can get a perfect sphere? Wow. I'd love to see this

demostrated in more detail.

> Though it does look like you can count them as one

> step and make it agree with the Euclidean shell of radius n*sqrt(2).

Again, I'd love a demonstration. That would blow my mind.

> >OK, maybe we need to rephrase what we're talking about so that it

> >means the same thing to both of us. I certainly believe in the

> >principle of embedding these lattices in Euclidean space such that

> >their balls come as close to being spherical as possible. This

> >dictates a rectangular geometry for octave-specific Tenney, an

> >equilateral triangular geometry for equal-weighted Hahn

>

> Now we're in business.

Whew!

> But doesn't it

What's "it"?

> dictate a rectangular

> geometry for any measure where rungs represent factors (not ratios),

> and a triangular measure where the reverse is true?

The Kees lattice, for one, is best understood with no rungs

whatsoever, so you can't *begin* by asking what the rungs represent

for a given measure. Since the other measures can be also be

understood in terms of a ball of particular shape rather that in

terms of taxicab distance, I could say the same for all of them. We

can do away with rungs altogether and still come up with the

configurations of ratios that best "spherify" the balls; that's what

I had in mind above.

> (And have

> nothing to do with octaves or the particular weighting?)

My statement above to which you replied "Now we're in business" has

everything to do with octaves and the particular weightings.

> >(breaks down beyond 7-limit),

>

> Yes, I remember something about this. Something about A3 = D3...

> but I can't find anything more on this at mathworld or google.

Huh? I don't know if we've jumped the tracks here. Beyond the 7-

limit, no Euclidean lattice can represent each of the consonances,

and only the consonances, by an equal and minimal length. Think about

ratios of 9 . . .

> >and a geometry without rungs but conforming to the third full

> >lattice diagram on the Kees page we've been looking at for

> >expressibility or "odd limit" (really the smallest odd limit the

> >interval belongs to).

>

> With hexagonal contours, gotcha. Does this depend on limit?

For higher prime limits, you need more dimensions, and the hexagon

becomes a rhombic dodecahedron and then its higher-dimensional

analogues . . . but if you can accept higher-dimensional Euclidean

spaces, this doesn't break down beyond the 7-limit the way the Hahn

construction does . . .

>> >> The closest you can get with unit lengths to spheres on the

>> >> rectangular latice is to count ratios of consonances as two

>> >> steps.

>> >

>> >Two steps? This makes no sense to me. Can you elaborate on your

>> >thinking here?

>>

>> If you count them (ie 5:3) as two steps on the rectangular lattice,

>

>This can be a ratio of consonances but so can, say, 9:5, which is the

>ratio between 5/6 and 3/2; and so can, say, 5:4, which is one step

>(right?) . . .

Depends on what's consonant. In an octave-specific formulation,

5:4 is three steps... Howabout if we swap "is to count fractional

factors (as opposed to integer factors) as two steps" in the above.

>> it looks like the nth shell of that agrees with the Euclidean shell

>> of radius n.

>

>You're saying you can get a perfect sphere? Wow. I'd love to see this

>demostrated in more detail.

>

>> Though it does look like you can count them as one

>> step and make it agree with the Euclidean shell of radius n*sqrt(2).

>

>Again, I'd love a demonstration. That would blow my mind.

We're obviously talking about different things. I'm superimposing

a sphere on a lattice (a Euclidean object centered on a lattice point)

and then asking what lattice points it encloses. Yes, at least in

2-D, n-taxicab on a rect. lattice agrees perfectly with radius-n

Euclidean (I assume this works in higher-D since I'm under the

impression that 90deg axes are always orthogonal). I originally

said no such spheres existed for taxicab of fractional-factor

taxicab on a rectangular lattice. However, n*sqrt(2) spheres

apparently do the trick.

>> >OK, maybe we need to rephrase what we're talking about so that it

>> >means the same thing to both of us. I certainly believe in the

>> >principle of embedding these lattices in Euclidean space such that

>> >their balls come as close to being spherical as possible. This

>> >dictates a rectangular geometry for octave-specific Tenney, an

>> >equilateral triangular geometry for equal-weighted Hahn

>>

>> Now we're in business.

>

>Whew!

>

>> But doesn't it

>

>What's "it"?

The business we're now in. :)

>> dictate a rectangular

>> geometry for any measure where rungs represent factors (not ratios),

>> and a triangular measure where the reverse is true?

>

>The Kees lattice, for one, is best understood with no rungs

>whatsoever, so you can't *begin* by asking what the rungs represent

>for a given measure.

Sure. Let's not worry about Kees for now.

>Since the other measures can be also be understood in terms of a

>ball of particular shape rather that in terms of taxicab distance,

>I could say the same for all of them. We can do away with rungs

>altogether and still come up with the configurations of ratios

>that best "spherify" the balls; that's what I had in mind above.

Great. I'd love to see more of this.

>> (And have

>> nothing to do with octaves or the particular weighting?)

>

>My statement above to which you replied "Now we're in business" has

>everything to do with octaves and the particular weightings.

How could this possibly be so?

>> >(breaks down beyond 7-limit),

>>

>> Yes, I remember something about this. Something about A3 = D3...

>> but I can't find anything more on this at mathworld or google.

>

>Huh? I don't know if we've jumped the tracks here. Beyond the 7-

>limit, no Euclidean lattice

I don't know what a Euclidean lattice is.

>can represent each of the consonances,

>and only the consonances, by an equal and minimal length. Think about

>ratios of 9 . . .

All of the stuff in my "exploring badness" msg. and since has assumed

prime limits, btw (aside from log-weighted versions, which come out

the same). Have you been thinking odd limits?

>> >and a geometry without rungs but conforming to the third full

>> >lattice diagram on the Kees page we've been looking at for

>> >expressibility or "odd limit" (really the smallest odd limit the

>> >interval belongs to).

>>

>> With hexagonal contours, gotcha. Does this depend on limit?

>

>For higher prime limits, you need more dimensions, and the hexagon

>becomes a rhombic dodecahedron and then its higher-dimensional

>analogues . . . but if you can accept higher-dimensional Euclidean

>spaces, this doesn't break down beyond the 7-limit the way the Hahn

>construction does . . .

Huh.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> The closest you can get with unit lengths to spheres on the

> >> >> rectangular latice is to count ratios of consonances as two

> >> >> steps.

> >> >

> >> >Two steps? This makes no sense to me. Can you elaborate on your

> >> >thinking here?

> >>

> >> If you count them (ie 5:3) as two steps on the rectangular

lattice,

> >

> >This can be a ratio of consonances but so can, say, 9:5, which is

the

> >ratio between 5/6 and 3/2; and so can, say, 5:4, which is one step

> >(right?) . . .

>

> Depends on what's consonant. In an octave-specific formulation,

> 5:4 is three steps... Howabout if we swap "is to count fractional

> factors (as opposed to integer factors) as two steps" in the above.

I don't get it -- how do you define fractional factors, and what are

we swapping this for?

> >> it looks like the nth shell of that agrees with the Euclidean

shell

> >> of radius n.

> >

> >You're saying you can get a perfect sphere? Wow. I'd love to see

this

> >demostrated in more detail.

> >

> >> Though it does look like you can count them as one

> >> step and make it agree with the Euclidean shell of radius n*sqrt

(2).

> >

> >Again, I'd love a demonstration. That would blow my mind.

>

> We're obviously talking about different things. I'm superimposing

> a sphere on a lattice (a Euclidean object centered on a lattice

point)

> and then asking what lattice points it encloses.

Yes, we're talking about the same thing.

> Yes, at least in

> 2-D, n-taxicab on a rect. lattice agrees perfectly with radius-n

> Euclidean

Of course it doesn't. If the sphere (circle) is big enough, it won't

be able to contain the same lattice points as the diamond which is

the taxicab "shell". Perhaps you're only looking at very small

circles?

> (I assume this works in higher-D since I'm under the

> impression that 90deg axes are always orthogonal).

:)

> I originally

> said no such spheres existed for taxicab of fractional-factor

> taxicab on a rectangular lattice.

I must be misunderstanding "fractional-factor taxicab" -- what does

that mean?

> However, n*sqrt(2) spheres

> apparently do the trick.

How did you arrive at this startling conclusion?

> >> >OK, maybe we need to rephrase what we're talking about so that

it

> >> >means the same thing to both of us. I certainly believe in the

> >> >principle of embedding these lattices in Euclidean space such

that

> >> >their balls come as close to being spherical as possible. This

> >> >dictates a rectangular geometry for octave-specific Tenney, an

> >> >equilateral triangular geometry for equal-weighted Hahn

> >>

> >> Now we're in business.

> >

> >Whew!

> >

> >> But doesn't it

> >

> >What's "it"?

>

> The business we're now in. :)

>

> >> dictate a rectangular

> >> geometry for any measure where rungs represent factors (not

ratios),

> >> and a triangular measure where the reverse is true?

> >

> >The Kees lattice, for one, is best understood with no rungs

> >whatsoever, so you can't *begin* by asking what the rungs

represent

> >for a given measure.

>

> Sure. Let's not worry about Kees for now.

>

> >Since the other measures can be also be understood in terms of a

> >ball of particular shape rather that in terms of taxicab distance,

> >I could say the same for all of them. We can do away with rungs

> >altogether and still come up with the configurations of ratios

> >that best "spherify" the balls; that's what I had in mind above.

>

> Great. I'd love to see more of this.

I don't know what else to say to convince you that the answer to your

question above is "no".

> >> (And have

> >> nothing to do with octaves or the particular weighting?)

> >

> >My statement above to which you replied "Now we're in business"

has

> >everything to do with octaves and the particular weightings.

>

> How could this possibly be so?

Because the complexity measures in question have everything to do

with them.

> >> >(breaks down beyond 7-limit),

> >>

> >> Yes, I remember something about this. Something about A3 = D3...

> >> but I can't find anything more on this at mathworld or google.

> >

> >Huh? I don't know if we've jumped the tracks here. Beyond the 7-

> >limit, no Euclidean lattice

>

> I don't know what a Euclidean lattice is.

No lattice that exists in a Euclidean space.

> >can represent each of the consonances,

> >and only the consonances, by an equal and minimal length. Think

about

> >ratios of 9 . . .

>

> All of the stuff in my "exploring badness" msg. and since has

assumed

> prime limits, btw (aside from log-weighted versions, which come out

> the same). Have you been thinking odd limits?

In one sense, no -- ratios of 9 exist whether or not anyone thinks

odd limits. In another sense, "odd limit" -- rather, the minimum odd

limit that a particular ratio could belong to -- is supposed to be

the determinant of lattice distance in both the Hahn and Kees cases.

> >> >and a geometry without rungs but conforming to the third full

> >> >lattice diagram on the Kees page we've been looking at for

> >> >expressibility or "odd limit" (really the smallest odd limit

the

> >> >interval belongs to).

> >>

> >> With hexagonal contours, gotcha. Does this depend on limit?

> >

> >For higher prime limits, you need more dimensions, and the hexagon

> >becomes a rhombic dodecahedron and then its higher-dimensional

> >analogues . . . but if you can accept higher-dimensional Euclidean

> >spaces, this doesn't break down beyond the 7-limit the way the

Hahn

> >construction does . . .

>

> Huh.

Mm-hm.

>> >> >> The closest you can get with unit lengths to spheres on the

>> >> >> rectangular latice is to count ratios of consonances as two

>> >> >> steps.

>> >> >

>> >> >Two steps? This makes no sense to me. Can you elaborate on your

>> >> >thinking here?

>> >>

>> >> If you count them (ie 5:3) as two steps on the rectangular

>> >> lattice,

>> >

>> >This can be a ratio of consonances but so can, say, 9:5, which is

>> >the ratio between 5/6 and 3/2; and so can, say, 5:4, which is one

>> >step (right?) . . .

>>

>> Depends on what's consonant. In an octave-specific formulation,

>> 5:4 is three steps... Howabout if we swap "is to count fractional

>> factors (as opposed to integer factors) as two steps" in the above.

>

>I don't get it -- how do you define fractional factors,

Rational numbers with a non-1 numerator and denominator.

>and what are

>we swapping this for?

Find "is to count" in the quoted text at top.

>> We're obviously talking about different things. I'm superimposing

>> a sphere on a lattice (a Euclidean object centered on a lattice

>> point) and then asking what lattice points it encloses.

>

>Yes, we're talking about the same thing.

>

>> Yes, at least in

>> 2-D, n-taxicab on a rect. lattice agrees perfectly with radius-n

>> Euclidean

>

>Of course it doesn't. If the sphere (circle) is big enough, it won't

>be able to contain the same lattice points as the diamond which is

>the taxicab "shell". Perhaps you're only looking at very small

>circles?

Maybe so...

>> (I assume this works in higher-D since I'm under the

>> impression that 90deg axes are always orthogonal).

>

>:)

Is this wrong?

>> I originally

>> said no such spheres existed for taxicab of fractional-factor

>> taxicab on a rectangular lattice.

>

>I must be misunderstanding "fractional-factor taxicab" -- what does

>that mean?

5/3 = 1

>> However, n*sqrt(2) spheres apparently do the trick.

>

>How did you arrive at this startling conclusion?

Since the Euclidean distance to fractional factors is sqrt(2).

But maybe I'm not looking at big enough circles...

>> >> >OK, maybe we need to rephrase what we're talking about so that

>> >> >it means the same thing to both of us. I certainly believe in

>> >> >the principle of embedding these lattices in Euclidean space

>> >> >such that their balls come as close to being spherical as

>> >> >possible. This dictates a rectangular geometry for octave-

>> >> >specific Tenney, an equilateral triangular geometry for equal-

>> >> >weighted Hahn

//

>> >Since the other measures can be also be understood in terms of a

>> >ball of particular shape rather that in terms of taxicab distance,

>> >I could say the same for all of them. We can do away with rungs

>> >altogether and still come up with the configurations of ratios

>> >that best "spherify" the balls; that's what I had in mind above.

>>

>> Great. I'd love to see more of this.

>

>I don't know what else to say to convince you that the answer to your

>question above is "no".

I meant, I'd love to know the ball shapes.

>> >> (And have

>> >> nothing to do with octaves or the particular weighting?)

>> >

>> >My statement above to which you replied "Now we're in business"

>> >has everything to do with octaves and the particular weightings.

>>

>> How could this possibly be so?

>

>Because the complexity measures in question have everything to do

>with them.

Yes, but forgetting about psychoacoustics just for the moment...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> >> The closest you can get with unit lengths to spheres on the

> >> >> >> rectangular latice is to count ratios of consonances as two

> >> >> >> steps.

> >> >> >

> >> >> >Two steps? This makes no sense to me. Can you elaborate on

your

> >> >> >thinking here?

> >> >>

> >> >> If you count them (ie 5:3) as two steps on the rectangular

> >> >> lattice,

> >> >

> >> >This can be a ratio of consonances but so can, say, 9:5, which

is

> >> >the ratio between 5/6 and 3/2; and so can, say, 5:4, which is

one

> >> >step (right?) . . .

> >>

> >> Depends on what's consonant. In an octave-specific formulation,

> >> 5:4 is three steps... Howabout if we swap "is to count

fractional

> >> factors (as opposed to integer factors) as two steps" in the

above.

> >

> >I don't get it -- how do you define fractional factors,

>

> Rational numbers with a non-1 numerator and denominator.

So 81:80 would be two steps?

> >and what are

> >we swapping this for?

>

> Find "is to count" in the quoted text at top.

OK . . . I'm still not grasping your construction . . .

> >> We're obviously talking about different things. I'm

superimposing

> >> a sphere on a lattice (a Euclidean object centered on a lattice

> >> point) and then asking what lattice points it encloses.

> >

> >Yes, we're talking about the same thing.

> >

> >> Yes, at least in

> >> 2-D, n-taxicab on a rect. lattice agrees perfectly with radius-n

> >> Euclidean

> >

> >Of course it doesn't. If the sphere (circle) is big enough, it

won't

> >be able to contain the same lattice points as the diamond which is

> >the taxicab "shell". Perhaps you're only looking at very small

> >circles?

>

> Maybe so...

>

> >> (I assume this works in higher-D since I'm under the

> >> impression that 90deg axes are always orthogonal).

> >

> >:)

>

> Is this wrong?

90 degree axes are always orthogonal, yes, but I don't see how that

supports your assumption.

>> >> I originally

>> >> said no such spheres existed for taxicab of fractional-factor

>> >> taxicab on a rectangular lattice.

> >

>> >I must be misunderstanding "fractional-factor taxicab" -- what

does

>> >that mean?

>

> 5/3 = 1

What is that supposed to mean? A major sixth is a unison?

> >> However, n*sqrt(2) spheres apparently do the trick.

> >

> >How did you arrive at this startling conclusion?

>

> Since the Euclidean distance to fractional factors is sqrt(2).

> But maybe I'm not looking at big enough circles...

Clearly.

> >> >> >OK, maybe we need to rephrase what we're talking about so

that

> >> >> >it means the same thing to both of us. I certainly believe in

> >> >> >the principle of embedding these lattices in Euclidean space

> >> >> >such that their balls come as close to being spherical as

> >> >> >possible. This dictates a rectangular geometry for octave-

> >> >> >specific Tenney, an equilateral triangular geometry for

equal-

> >> >> >weighted Hahn

> //

> >> >Since the other measures can be also be understood in terms of a

> >> >ball of particular shape rather that in terms of taxicab

distance,

> >> >I could say the same for all of them. We can do away with rungs

> >> >altogether and still come up with the configurations of ratios

> >> >that best "spherify" the balls; that's what I had in mind above.

> >>

> >> Great. I'd love to see more of this.

> >

> >I don't know what else to say to convince you that the answer to

your

> >question above is "no".

>

> I meant, I'd love to know the ball shapes.

For 3-limit Tenney HD, you get equilateral diamonds (squares rotated

by 45 degrees). For 5-limit Tenney HD, as you acknowledged before,

you get regular octahedra. For 5-limit equal-weighted Hahn diameter,

you get regular hexagons. For 5-limit Kees expressibility or "minimum

odd limit" (using the configuration of lattice points shown in the

third lattice on that Kees page), you get regular hexagons again. The

7-limit and higher-limit extensions to this are straighforward,

except that equal-weighted Hahn breaks down with the first odd

composite.

> >> >> (And have

> >> >> nothing to do with octaves or the particular weighting?)

> >> >

> >> >My statement above to which you replied "Now we're in business"

> >> >has everything to do with octaves and the particular weightings.

> >>

> >> How could this possibly be so?

> >

> >Because the complexity measures in question have everything to do

> >with them.

>

> Yes, but forgetting about psychoacoustics just for the moment...

Forget that lattice distance is supposed to correspond to some

measure of ratio complexity? Then we're back to the "circular"

scenario where the lattice defines its own geometry again.

Sorry it's taken me so long to reply. I've come down with the flu.

First time since I was a kid. Nasty stuff. I'm still running a

fever, but I'll try...

>> >Of course it doesn't. If the sphere (circle) is big enough, it

>> >won't be able to contain the same lattice points as the diamond

>> >which is the taxicab "shell". Perhaps you're only looking at

>> >very small circles?

>>

>> Maybe so...

>>

>> >> (I assume this works in higher-D since I'm under the

>> >> impression that 90deg axes are always orthogonal).

>> >

>> >:)

>>

>> Is this wrong?

>

>90 degree axes are always orthogonal, yes, but I don't see how that

>supports your assumption.

If it were true, I assume it would remain true in higher-D.

>>> >> I originally said no such spheres existed for taxicab of

>>> >> fractional-factor taxicab on a rectangular lattice.

>> >

>>> >I must be misunderstanding "fractional-factor taxicab" -- what

>>> >does that mean?

Just noticed there's a typo here -- delete "for taxicab of".

>> 5/3 = 1

>

>What is that supposed to mean? A major sixth is a unison?

A major sixth has length 1.

>> >> However, n*sqrt(2) spheres apparently do the trick.

>> >

>> >How did you arrive at this startling conclusion?

>>

>> Since the Euclidean distance to fractional factors is sqrt(2).

>> But maybe I'm not looking at big enough circles...

>

>Clearly.

Ok.

>> >> >Since the other measures can be also be understood in terms of

>> >> >a ball of particular shape rather that in terms of taxicab

>> >> >distance, I could say the same for all of them. We can do away

>> >> >with rungs altogether and still come up with the configurations

>> >> >of ratios that best "spherify" the balls; that's what I had in

>> >> >mind above.

//

>> I'd love to know the ball shapes.

>

>For 3-limit Tenney HD, you get equilateral diamonds (squares rotated

>by 45 degrees). For 5-limit Tenney HD, as you acknowledged before,

>you get regular octahedra.

Yes.

>For 5-limit equal-weighted Hahn diameter, you get regular hexagons.

Knew that.

>For 5-limit Kees expressibility or "minimum odd limit" (using the

>configuration of lattice points shown in the third lattice on that

>Kees page), you get regular hexagons again.

Exactly what sort of lattice is this?

>The 7-limit and higher-limit extensions to this are straighforward,

>except that equal-weighted Hahn breaks down with the first odd

>composite.

How so? Does this depend on that Hahn typically used odd

limits (I've been using a prime-limit approach)?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> Sorry it's taken me so long to reply. I've come down with the flu.

> First time since I was a kid. Nasty stuff. I'm still running a

> fever, but I'll try...

Hope you feel better. I'll try to go easy on you . . . :)

> >>> >> I originally said no such spheres existed for taxicab of

> >>> >> fractional-factor taxicab on a rectangular lattice.

> >> >

> >>> >I must be misunderstanding "fractional-factor taxicab" -- what

> >>> >does that mean?

>

> Just noticed there's a typo here -- delete "for taxicab of".

>

> >> 5/3 = 1

> >

> >What is that supposed to mean? A major sixth is a unison?

>

> A major sixth has length 1.

OK. But intervals that aren't "fractional factors" can also get

length one, right?

> >> >> >Since the other measures can be also be understood in terms

of

> >> >> >a ball of particular shape rather that in terms of taxicab

> >> >> >distance, I could say the same for all of them. We can do

away

> >> >> >with rungs altogether and still come up with the

configurations

> >> >> >of ratios that best "spherify" the balls; that's what I had

in

> >> >> >mind above.

> //

> >> I'd love to know the ball shapes.

> >

> >For 3-limit Tenney HD, you get equilateral diamonds (squares

rotated

> >by 45 degrees). For 5-limit Tenney HD, as you acknowledged before,

> >you get regular octahedra.

>

> Yes.

>

> >For 5-limit equal-weighted Hahn diameter, you get regular hexagons.

>

> Knew that.

>

> >For 5-limit Kees expressibility or "minimum odd limit" (using the

> >configuration of lattice points shown in the third lattice on that

> >Kees page), you get regular hexagons again.

>

> Exactly what sort of lattice is this?

It is what it is. I don't know what else to say. But I certainly

would have loved to know of this much earlier.

> >The 7-limit and higher-limit extensions to this are

straighforward,

> >except that equal-weighted Hahn breaks down with the first odd

> >composite.

>

> How so? Does this depend on that Hahn typically used odd

> limits (I've been using a prime-limit approach)?

Yes, so that in the 9-limit and beyond, ratios of 9 can still have

length 1, etc. I thought you mentioned "odd factorization" so I'm

surprised if you now say that you've been using a primes-only version

of Hahn's algorithm . . . (?)

>> Sorry it's taken me so long to reply. I've come down with the flu.

>> First time since I was a kid. Nasty stuff. I'm still running a

>> fever, but I'll try...

>

>Hope you feel better. I'll try to go easy on you . . . :)

Probably wasn't the flu after all. I feel better, but bloody

diarrhea has set in. Waiting on the results of a stool culture.

Sorry if that's too much information. They don't seem to think

it's terribly serious.

>> >>> >> I originally said no such spheres existed for taxicab of

>> >>> >> fractional-factor taxicab on a rectangular lattice.

>> >> >

>> >>> >I must be misunderstanding "fractional-factor taxicab" -- what

>> >>> >does that mean?

>>

>> Just noticed there's a typo here -- delete "for taxicab of".

>>

>> >> 5/3 = 1

>> >

>> >What is that supposed to mean? A major sixth is a unison?

>>

>> A major sixth has length 1.

>

>OK. But intervals that aren't "fractional factors" can also get

>length one, right?

(I've answered this affirmatively elsewhere.)

>> >For 5-limit Kees expressibility or "minimum odd limit" (using the

>> >configuration of lattice points shown in the third lattice on that

>> >Kees page), you get regular hexagons again.

>>

>> Exactly what sort of lattice is this?

>

>It is what it is. I don't know what else to say. But I certainly

>would have loved to know of this much earlier.

Yes, I'm grokking it now. For some reason Kees assumes the two

5-limit axes should be equally scaled. And indeed it seems that

way, since they have the same expressibility! IS this the paradox

he refers to, and how have you decided it doesn't exist?

Why do you like it better than your isosceles lattice?

>> >The 7-limit and higher-limit extensions to this are

>> >straighforward, except that equal-weighted Hahn breaks

>> >down with the first odd composite.

>>

>> How so? Does this depend on that Hahn typically used odd

>> limits (I've been using a prime-limit approach)?

>

>Yes, so that in the 9-limit and beyond, ratios of 9 can still have

>length 1, etc. I thought you mentioned "odd factorization" so I'm

>surprised if you now say that you've been using a primes-only version

>of Hahn's algorithm . . . (?)

Where did I mention odd factorization? Only when I was claiming

it's the same as prime factorization under log weighting. But

I've been using strictly prime limit thinking this entire thread.

So was your claim the generalized FCC lattice dosen't work

in > 3-D related to pure geometry, or some expectation based

on odd limit?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> Sorry it's taken me so long to reply. I've come down with the

flu.

> >> First time since I was a kid. Nasty stuff. I'm still running a

> >> fever, but I'll try...

> >

> >Hope you feel better. I'll try to go easy on you . . . :)

>

> Probably wasn't the flu after all. I feel better, but bloody

> diarrhea has set in. Waiting on the results of a stool culture.

> Sorry if that's too much information. They don't seem to think

> it's terribly serious.

I hope not. I suffered with food poisoning for 10 days (including an

appendectomy) before it was discovered. Good luck!

> >> >>> >> I originally said no such spheres existed for taxicab of

> >> >>> >> fractional-factor taxicab on a rectangular lattice.

> >> >> >

> >> >>> >I must be misunderstanding "fractional-factor taxicab" --

what

> >> >>> >does that mean?

> >>

> >> Just noticed there's a typo here -- delete "for taxicab of".

> >>

> >> >> 5/3 = 1

> >> >

> >> >What is that supposed to mean? A major sixth is a unison?

> >>

> >> A major sixth has length 1.

> >

> >OK. But intervals that aren't "fractional factors" can also get

> >length one, right?

>

> (I've answered this affirmatively elsewhere.)

OK, so where does that leave us?

> >> >For 5-limit Kees expressibility or "minimum odd limit" (using

the

> >> >configuration of lattice points shown in the third lattice on

that

> >> >Kees page), you get regular hexagons again.

> >>

> >> Exactly what sort of lattice is this?

> >

> >It is what it is. I don't know what else to say. But I certainly

> >would have loved to know of this much earlier.

>

> Yes, I'm grokking it now. For some reason Kees assumes the two

> 5-limit axes should be equally scaled.

Huh?

> And indeed it seems that

> way, since they have the same expressibility!

What do you mean? 3 doesn't have the same expressibility as 5.

> IS this the paradox

> he refers to, and how have you decided it doesn't exist?

He refers to 5:4 and 5:3 looking like different distances in the

lattice. But that's not correct, since the relevant distance measure

is one that only allows motions (of whatever length) along six

equally-spaced directions (60 degrees apart) -- not along the rungs

Kees draws -- or equivalently, one that measures the size of a

regular hexagon with the origin at the center and the destination on

the outer edge.

> Why do you like it better than your isosceles lattice?

It's better for a prime limit of 5 and the new paradigm where

consonance/dissonance is shades-of-gray rather than black-and-white,

because all the (hexagonally-determined) distances correspond to

complexity (expressibility), no matter how complex the ratios. This

is why I believe the rungs should be deleted. If ratios of 9, in

particular, are to be considered something other than mere

dissonances in a 5-prime-limit tuning, then you want to represent

them as shorter than ratios of 15. This is exactly what this lattice,

with a hexagonal-norm metric, accomplishes perfectly. I should make a

diagram with lots of ratios on it . . .

> >> >The 7-limit and higher-limit extensions to this are

> >> >straighforward, except that equal-weighted Hahn breaks

> >> >down with the first odd composite.

> >>

> >> How so? Does this depend on that Hahn typically used odd

> >> limits (I've been using a prime-limit approach)?

> >

> >Yes, so that in the 9-limit and beyond, ratios of 9 can still have

> >length 1, etc. I thought you mentioned "odd factorization" so I'm

> >surprised if you now say that you've been using a primes-only

version

> >of Hahn's algorithm . . . (?)

>

> Where did I mention odd factorization? Only when I was claiming

> it's the same as prime factorization under log weighting. But

> I've been using strictly prime limit thinking this entire thread.

So you're not actually using Hahn's algorithm as he proposed it?

> So was your claim the generalized FCC lattice dosen't work

> in

> 3-D related to pure geometry,

No, if I even said such a thing.

> or some expectation based

> on odd limit?

I don't know what you mean by "expectation", but clearly it fails for

the Hahn case, where ratios of 9 must be given length 1.

>I hope not. I suffered with food poisoning for 10 days (including an

>appendectomy) before it was discovered. Good luck!

I remembered that!

-Carl

Turned out to be food poisoning (Campylobacter). Antibiotics

to the rescue!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> Turned out to be food poisoning (Campylobacter).

Exactly what I had -- sucks when it goes undiagnosed for a while . . .

> Antibiotics

> to the rescue!

Good luck again!!

Good to know you're still live and kicking, my incurable naturalist

colleague!

Cordially,

Ozan

----- Original Message -----

From: "Carl Lumma" <ekin@lumma.org>

To: <tuning-math@yahoogroups.com>

Sent: 14 Kasï¿½m 2005 Pazartesi 23:04

Subject: [tuning-math] my health

> Turned out to be food poisoning (Campylobacter). Antibiotics

> to the rescue!

>

> -Carl

>

>

>> Turned out to be food poisoning (Campylobacter).

>

>Exactly what I had -- sucks when it goes undiagnosed for a while . . .

Dude, the stuff's a real killer. I'm lucky I didn't wait for

the stool culture (72 hours) to finish -- I demanded antibiotics

on Friday. I'm hip to antibiotic overuse, but I rarely get sick

so I knew something serious was happening and I could feel I

wasn't making progress on my own.

>> Antibiotics to the rescue!

>

>Good luck again!!

I'm basically all better now, except having to rebuild my

strength and be careful with keeping everything clean (still

shedding going on according to the Doc).

-Carl

>> >> >>> >> I originally said no such spheres existed for taxicab of

>> >> >>> >> fractional-factor taxicab on a rectangular lattice.

>> >> >> >

>> >> >>> >I must be misunderstanding "fractional-factor taxicab" --

>> >> >>> >what does that mean?

>> >>

>> >> Just noticed there's a typo here -- delete "for taxicab of".

>> >>

>> >> >> 5/3 = 1

>> >> >

>> >> >What is that supposed to mean? A major sixth is a unison?

>> >>

>> >> A major sixth has length 1.

>> >

>> >OK. But intervals that aren't "fractional factors" can also get

>> >length one, right?

>>

>> (I've answered this affirmatively elsewhere.)

>

>OK, so where does that leave us?

I'm not sure.

'be nice to have a list of complexity measures, showing their ball

shapes on different lattices in 1- through 3-D (or higher)...

lattice type................Triangu. Rect.

dimensions..................1 2 3... 1 2 3...

taxicab

odd limit

Tenney height

symmetric Euclidean

The weighted versions of these would be

"isosceles"

expressibility

Tenney HD

?

>> >> >For 5-limit Kees expressibility shown in the third lattice on

>> >> >that Kees page), you get regular hexagons again.

>> >>

>> >> Exactly what sort of lattice is this?

>> >

>> >It is what it is. I don't know what else to say. But I certainly

>> >would have loved to know of this much earlier.

>>

>> Yes, I'm grokking it now. For some reason Kees assumes the two

>> 5-limit axes should be equally scaled.

>

>Huh?

"Now we seem to hit a paradox, because the measure of a 5/4 (blue)

and a 5/3 (green) should be the same"

>> And indeed it seems that

>> way, since they have the same expressibility!

>

>What do you mean? 3 doesn't have the same expressibility as 5.

5/3 and 5/4 do.

>> IS this the paradox

>> he refers to, and how have you decided it doesn't exist?

>

>He refers to 5:4 and 5:3 looking like different distances in the

>lattice. But that's not correct, since the relevant distance measure

>is one that only allows motions (of whatever length) along six

>equally-spaced directions (60 degrees apart) -- not along the rungs

>Kees draws -- or equivalently, one that measures the size of a

>regular hexagon with the origin at the center and the destination

>on the outer edge.

Huh. Is there such a motion of length n from the unison to

both 5:3 and 5:4?

>> Why do you like it better than your isosceles lattice?

>

>It's better for a prime limit of 5 and the new paradigm where

>consonance/dissonance is shades-of-gray rather than black-and-white,

>because all the (hexagonally-determined) distances correspond to

>complexity (expressibility), no matter how complex the ratios. This

>is why I believe the rungs should be deleted. If ratios of 9, in

>particular, are to be considered something other than mere

>dissonances in a 5-prime-limit tuning, then you want to represent

>them as shorter than ratios of 15. This is exactly what this lattice,

>with a hexagonal-norm metric, accomplishes perfectly. I should make a

>diagram with lots of ratios on it . . .

Hmm...

>> >> >The 7-limit and higher-limit extensions to this are

>> >> >straighforward, except that equal-weighted Hahn breaks

>> >> >down with the first odd composite.

>> >>

>> >> How so? Does this depend on that Hahn typically used odd

>> >> limits (I've been using a prime-limit approach)?

>> >

>> >Yes, so that in the 9-limit and beyond, ratios of 9 can still have

>> >length 1, etc. I thought you mentioned "odd factorization" so I'm

>> >surprised if you now say that you've been using a primes-only

>> >version of Hahn's algorithm . . . (?)

>>

>> Where did I mention odd factorization? Only when I was claiming

>> it's the same as prime factorization under log weighting. But

>> I've been using strictly prime limit thinking this entire thread.

>

>So you're not actually using Hahn's algorithm as he proposed it?

Right.

>> So was your claim the generalized FCC lattice dosen't work

>> in 3-D related to pure geometry,

>

>No, if I even said such a thing.

That was supposed to be "> 3-D" not "in 3-D".

>> or some expectation based on odd limit?

>

>I don't know what you mean by "expectation", but clearly it fails for

>the Hahn case, where ratios of 9 must be given length 1.

That's an expectation.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Dude, the stuff's a real killer. I'm lucky I didn't wait for

> the stool culture (72 hours) to finish -- I demanded antibiotics

> on Friday.

Sometimes that is wise, I went into an energency room once, and the ER

doctor said he didn't know what I had, but it looked like it was

probably mono or strep throat. Whatever it was, it was bad, and he was

going to give me an antibiotic right now, because my throat looked

like raw liver.

It turned out to be mono *and* strep throat.

>> Dude, the stuff's a real killer. I'm lucky I didn't wait for

>> the stool culture (72 hours) to finish -- I demanded antibiotics

>> on Friday.

>

>Sometimes that is wise, I went into an energency room once, and the ER

>doctor said he didn't know what I had, but it looked like it was

>probably mono or strep throat. Whatever it was, it was bad, and he was

>going to give me an antibiotic right now, because my throat looked

>like raw liver.

>

>It turned out to be mono *and* strep throat.

Heh! -C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> 'be nice to have a list of complexity measures, showing their ball

> shapes on different lattices in 1- through 3-D (or higher)...

Yes, I'd like to draw the relevant pictures soon.

>

> lattice type................Triangu. Rect.

> dimensions..................1 2 3... 1 2 3...

>

> taxicab

> odd limit

> Tenney height

> symmetric Euclidean

>

> The weighted versions of these would be

>

> "isosceles"

> expressibility

> Tenney HD

> ?

Makes no sense to me, so I'll ignore it :)

> >> >> >For 5-limit Kees expressibility shown in the third lattice on

> >> >> >that Kees page), you get regular hexagons again.

> >> >>

> >> >> Exactly what sort of lattice is this?

> >> >

> >> >It is what it is. I don't know what else to say. But I

certainly

> >> >would have loved to know of this much earlier.

> >>

> >> Yes, I'm grokking it now. For some reason Kees assumes the two

> >> 5-limit axes should be equally scaled.

> >

> >Huh?

>

> "Now we seem to hit a paradox, because the measure of a 5/4 (blue)

> and a 5/3 (green) should be the same"

Right, that's what I discussed below. I don't consider these to be

the two 5-limit axes, though.

> >> And indeed it seems that

> >> way, since they have the same expressibility!

> >

> >What do you mean? 3 doesn't have the same expressibility as 5.

>

> 5/3 and 5/4 do.

Yes -- that's what I discussed below.

> >> IS this the paradox

> >> he refers to, and how have you decided it doesn't exist?

> >

> >He refers to 5:4 and 5:3 looking like different distances in the

> >lattice. But that's not correct, since the relevant distance

measure

> >is one that only allows motions (of whatever length) along six

> >equally-spaced directions (60 degrees apart) -- not along the

rungs

> >Kees draws -- or equivalently, one that measures the size of a

> >regular hexagon with the origin at the center and the destination

> >on the outer edge.

>

> Huh. Is there such a motion of length n from the unison to

> both 5:3 and 5:4?

There are an infinite number of such motions to either, and they're

all the same length (assuming you don't backtrack in any of the six

directions).

> >> Why do you like it better than your isosceles lattice?

> >

> >It's better for a prime limit of 5 and the new paradigm where

> >consonance/dissonance is shades-of-gray rather than black-and-

white,

> >because all the (hexagonally-determined) distances correspond to

> >complexity (expressibility), no matter how complex the ratios.

This

> >is why I believe the rungs should be deleted. If ratios of 9, in

> >particular, are to be considered something other than mere

> >dissonances in a 5-prime-limit tuning, then you want to represent

> >them as shorter than ratios of 15. This is exactly what this

lattice,

> >with a hexagonal-norm metric, accomplishes perfectly. I should

make a

> >diagram with lots of ratios on it . . .

>

> Hmm...

As in you understand, or you don't?

> >> >> >The 7-limit and higher-limit extensions to this are

> >> >> >straighforward, except that equal-weighted Hahn breaks

> >> >> >down with the first odd composite.

> >> >>

> >> >> How so? Does this depend on that Hahn typically used odd

> >> >> limits (I've been using a prime-limit approach)?

> >> >

> >> >Yes, so that in the 9-limit and beyond, ratios of 9 can still

have

> >> >length 1, etc. I thought you mentioned "odd factorization" so

I'm

> >> >surprised if you now say that you've been using a primes-only

> >> >version of Hahn's algorithm . . . (?)

> >>

> >> Where did I mention odd factorization? Only when I was claiming

> >> it's the same as prime factorization under log weighting. But

> >> I've been using strictly prime limit thinking this entire thread.

> >

> >So you're not actually using Hahn's algorithm as he proposed it?

>

> Right.

>

> >> So was your claim the generalized FCC lattice dosen't work

> >> in 3-D related to pure geometry,

> >

> >No, if I even said such a thing.

>

> That was supposed to be "> 3-D" not "in 3-D".

I got that.

> >> or some expectation based on odd limit?

> >

> >I don't know what you mean by "expectation", but clearly it fails

for

> >the Hahn case, where ratios of 9 must be given length 1.

>

> That's an expectation.

What do you mean, an expectation?

>> lattice type................Triangu. Rect.

>> dimensions..................1 2 3... 1 2 3...

>>

>> taxicab

>> odd limit

>> Tenney height

>> symmetric Euclidean

>>

>> The weighted versions of these would be

>>

>> "isosceles"

>> expressibility

>> Tenney HD

>> ?

>

>Makes no sense to me, so I'll ignore it :)

The entire chart? The entries would be things like

"rhombic dodecahedron", under each of the six (in this case)

columns (lattice types) for each measure.

Certainly you're not balking at the term "isosceles", which

we've been using to refer to your weighted-rungs thing you

originally thought would be the same as expressibility.

You just finished lecturing me on the difference between

Tenney height and Tenney HD.

"Odd limit" should perhaps be called "Ratio of".

What else?

>> >> Yes, I'm grokking it now. For some reason Kees assumes the two

>> >> 5-limit axes should be equally scaled.

>> >

>> >Huh?

>>

>> "Now we seem to hit a paradox, because the measure of a 5/4 (blue)

>> and a 5/3 (green) should be the same"

>

>Right, that's what I discussed below. I don't consider these to be

>the two 5-limit axes, though.

Obviously.

>> >Kees refers to 5:4 and 5:3 looking like different distances in

>> >the lattice. But that's not correct, since the relevant distance

>> >measure is one that only allows motions (of whatever length)

>> >along six equally-spaced directions (60 degrees apart) -- not

>> >along the rungs Kees draws -- or equivalently, one that measures

>> >the size of a regular hexagon with the origin at the center and

>> >the destination on the outer edge.

>>

>> Huh. Is there such a motion of length n from the unison to

>> both 5:3 and 5:4?

>

>There are an infinite number of such motions to either, and they're

>all the same length (assuming you don't backtrack in any of the six

>directions).

Wow. Sounds good.

>> >> Why do you like it better than your isosceles lattice?

>> >

>> >It's better for a prime limit of 5 and the new paradigm where

>> >consonance/dissonance is shades-of-gray rather than black-and-

>> >white, because all the (hexagonally-determined) distances

>> >correspond to complexity (expressibility), no matter how

>> >complex the ratios. This is why I believe the rungs should

>> >be deleted. If ratios of 9, in particular, are to be considered

>> >something other than mere dissonances in a 5-prime-limit tuning,

>> >then you want to represent them as shorter than ratios of 15.

>> >This is exactly what this lattice, with a hexagonal-norm metric,

>> >accomplishes perfectly. I should make a diagram with lots of

>> >ratios on it . . .

>>

>> Hmm...

>

>As in you understand, or you don't?

Don't. I'm not sure what paradigm you're referring to.

The weighted paradigm?

I do take it you're saying allowed motions to 9 are of length

log(9)?

>> >> >> >equal-weighted Hahn breaks

>> >> >> >down with the first odd composite.

>> >> >>

>> >> >> How so? Does this depend on that Hahn typically used odd

>> >> >> limits (I've been using a prime-limit approach)?

>> >>

>> >> it fails for the Hahn case, where ratios of 9 must be given

>> >> length 1.

>>

>> That's an expectation.

>

>What do you mean, an expectation?

If I construct a 4-D triangular lattice with four prime

numbers, are all the ratios of these primes and the primes

themselves, and no other intervals, mapped to single rungs?

If so, the "Hahn approach" does not break down above 3-D.

It breaks down when you don't define limits the way I do,

and the only problem is an expectation on your or Hahn's

part. If not, it's a problem of geometry, and I need a

new lattice (of course I'm already doing this numerically

with no apparent problems).

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> lattice type................Triangu. Rect.

> >> dimensions..................1 2 3... 1 2 3...

> >>

> >> taxicab

> >> odd limit

> >> Tenney height

> >> symmetric Euclidean

> >>

> >> The weighted versions of these would be

> >>

> >> "isosceles"

> >> expressibility

> >> Tenney HD

> >> ?

> >

> >Makes no sense to me, so I'll ignore it :)

>

> The entire chart? The entries would be things like

> "rhombic dodecahedron", under each of the six (in this case)

> columns (lattice types) for each measure.

I'd rather use only one lattice type for each measure; the one that

makes the balls of that measure most symmetrical.

> Certainly you're not balking at the term "isosceles", which

> we've been using to refer to your weighted-rungs thing you

> originally thought would be the same as expressibility.

Especially if you *do* use multiple lattice types for each complexity

measure, it's a bad idea to use names for complexity measures which

sound more like names of lattice types.

And although you can make a lattice where expressibility corresponds

to a distance measure, you can't do that for odd limit or Tenney

height. Can you do it for taxicab? Makes no sense. Can you do it for

symmetric Euclidean? Makes no sense.

> You just finished lecturing me on the difference between

> Tenney height and Tenney HD.

>

> "Odd limit" should perhaps be called "Ratio of".

>

> What else?

>

> >> >> Yes, I'm grokking it now. For some reason Kees assumes the

two

> >> >> 5-limit axes should be equally scaled.

> >> >

> >> >Huh?

> >>

> >> "Now we seem to hit a paradox, because the measure of a 5/4

(blue)

> >> and a 5/3 (green) should be the same"

> >

> >Right, that's what I discussed below. I don't consider these to be

> >the two 5-limit axes, though.

>

> Obviously.

>

> >> >Kees refers to 5:4 and 5:3 looking like different distances in

> >> >the lattice. But that's not correct, since the relevant distance

> >> >measure is one that only allows motions (of whatever length)

> >> >along six equally-spaced directions (60 degrees apart) -- not

> >> >along the rungs Kees draws -- or equivalently, one that measures

> >> >the size of a regular hexagon with the origin at the center and

> >> >the destination on the outer edge.

> >>

> >> Huh. Is there such a motion of length n from the unison to

> >> both 5:3 and 5:4?

> >

> >There are an infinite number of such motions to either, and

they're

> >all the same length (assuming you don't backtrack in any of the

six

> >directions).

>

> Wow. Sounds good.

And you only need to move in two of these direction for any

particular interval.

> >> >> Why do you like it better than your isosceles lattice?

> >> >

> >> >It's better for a prime limit of 5 and the new paradigm where

> >> >consonance/dissonance is shades-of-gray rather than black-and-

> >> >white, because all the (hexagonally-determined) distances

> >> >correspond to complexity (expressibility), no matter how

> >> >complex the ratios. This is why I believe the rungs should

> >> >be deleted. If ratios of 9, in particular, are to be considered

> >> >something other than mere dissonances in a 5-prime-limit tuning,

> >> >then you want to represent them as shorter than ratios of 15.

> >> >This is exactly what this lattice, with a hexagonal-norm metric,

> >> >accomplishes perfectly. I should make a diagram with lots of

> >> >ratios on it . . .

> >>

> >> Hmm...

> >

> >As in you understand, or you don't?

>

> Don't.

Think of a set of nested, concentric regular hexagons. These

represent increasing "distances". This is the notion of "distance" we

need here.

> I'm not sure what paradigm you're referring to.

> The weighted paradigm?

No, just the paradigm where we drop the assumption of a "consonance

limit", and things like consistency are out the window.

> I do take it you're saying allowed motions to 9 are of length

> log(9)?

Yes; if you can't see that on the diagram as it is, I'll make a new

one for you.

> >> >> >> >equal-weighted Hahn breaks

> >> >> >> >down with the first odd composite.

> >> >> >>

> >> >> >> How so? Does this depend on that Hahn typically used odd

> >> >> >> limits (I've been using a prime-limit approach)?

> >> >>

> >> >> it fails for the Hahn case, where ratios of 9 must be given

> >> >> length 1.

> >>

> >> That's an expectation.

> >

> >What do you mean, an expectation?

>

> If I construct a 4-D triangular lattice with four prime

> numbers, are all the ratios of these primes and the primes

> themselves, and no other intervals, mapped to single rungs?

Yes.

> If so, the "Hahn approach" does not break down above 3-D.

Yes it does, if you ever read anything Hahn wrote.

> It breaks down when you don't define limits the way I do,

> and the only problem is an expectation on your or Hahn's

> part.

Ha!

> If not,

If not what?

> it's a problem of geometry, and I need a

> new lattice (of course I'm already doing this numerically

> with no apparent problems).

>

> -Carl

>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> If I construct a 4-D triangular lattice

This is what Gene calls the "symmetrical" lattice, right?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > I do take it you're saying allowed motions to 9 are of length

> > log(9)?

>

> Yes; if you can't see that on the diagram as it is, I'll make a new

> one for you.

Actually, for the case of 9 (9:8, 9:4, 16:9 etc.), which appears to be

what you're asking about, it's trivial to see it. Since it's located at

an angle zero degrees from the horizontal, there's only one allowed

motion -- a straight line. And since the length from 1/1 to 3/2 is log

(3), the length from 1/1 to 9/4 is log(3)+log(3)=log(9).

>> >> lattice type................Triangu. Rect.

>> >> dimensions..................1 2 3... 1 2 3...

>> >>

>> >> taxicab

>> >> odd limit

>> >> Tenney height

>> >> symmetric Euclidean

>> >>

>> >> The weighted versions of these would be

>> >>

>> >> "isosceles"

>> >> expressibility

>> >> Tenney HD

>> >> ?

>> >

>> >Makes no sense to me, so I'll ignore it :)

>>

>> The entire chart? The entries would be things like

>> "rhombic dodecahedron", under each of the six (in this case)

>> columns (lattice types) for each measure.

>

>I'd rather use only one lattice type for each measure; the one that

>makes the balls of that measure most symmetrical.

One way to find that out is to read the ball shapes off a chart

like this!

>And although you can make a lattice where expressibility corresponds

>to a distance measure, you can't do that for odd limit //

>Can you do it for symmetric Euclidean? Makes no sense.

Why not? It's only a difference of a log.

What do you mean by "distance measure"? Crow flies? Surely

Euclidean distance as Gene's defined it measures crow's distance

on some lattice.

>> >> >> Why do you like it better than your isosceles lattice?

>> >> >

>> >> >It's better for a prime limit of 5 and the new paradigm where

>> >> >consonance/dissonance is shades-of-gray rather than black-and-

>> >> >white, because all the (hexagonally-determined) distances

>> >> >correspond to complexity (expressibility), no matter how

>> >> >complex the ratios. This is why I believe the rungs should

>> >> >be deleted. If ratios of 9, in particular, are to be considered

>> >> >something other than mere dissonances in a 5-prime-limit tuning,

>> >> >then you want to represent them as shorter than ratios of 15.

>> >> >This is exactly what this lattice, with a hexagonal-norm metric,

>> >> >accomplishes perfectly. I should make a diagram with lots of

>> >> >ratios on it . . .

>> >>

>> >> Hmm...

>> >

>> >As in you understand, or you don't?

>>

>> Don't.

>

>Think of a set of nested, concentric regular hexagons. These

>represent increasing "distances". This is the notion of "distance" we

>need here.

I have the picture, it's an example of 'grey-area dissonance and

ratios of 9' that I'm missing.

>> I'm not sure what paradigm you're referring to.

>> The weighted paradigm?

>

>No, just the paradigm where we drop the assumption of a "consonance

>limit",

How long has this been going on?

>> I do take it you're saying allowed motions to 9 are of length

>> log(9)?

>

>Yes; if you can't see that on the diagram as it is, I'll make a new

>one for you.

It's hard to visualize regular hexagons over that lattices skewed

ones.

>> If so, the "Hahn approach" does not break down above 3-D.

>

>Yes it does, if you ever read anything Hahn wrote.

Hahn's code showed 9-limit examples working perfectly, and my

code tests against those examples.

>> If not, it's a problem of geometry, and I need a

>> new lattice (of course I'm already doing this numerically

>> with no apparent problems).

>

>If not what?

If the 4-D triangular lattice failed to map consonances to

rungs.

-Carl

>> If I construct a 4-D triangular lattice

>

>This is what Gene calls the "symmetrical" lattice, right?

By "symmetrical" I think Gene simply means "unweighted".

I just meant a 4-D thing with 60 deg. angles, which I

assume exists but have no knowledge of.

I think Gene's Euclidean norm gives crow's distance

on an unweighted rectangular lattice, but I could be

wrong.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >And although you can make a lattice where expressibility corresponds

> >to a distance measure, you can't do that for odd limit //

> >Can you do it for symmetric Euclidean? Makes no sense.

>

> Why not? It's only a difference of a log.

One reason I have been assuming you mean something different by

"lattice" are the discussions like this.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I just meant a 4-D thing with 60 deg. angles, which I

> assume exists but have no knowledge of.

Well, you do now, since you have this general formula which in the

4-D (11-limit) case becomes

|| |* a b c d> || = sqrt((a^2+b^2+c^2+d^2)+(a+b+c+d)^2)/2)

> I think Gene's Euclidean norm gives crow's distance

> on an unweighted rectangular lattice, but I could be

> wrong.

No, on an unweighted "triangular" (An) lattice. Of course in the

11-limit it probably makes more sense to make 9, not 3, the same

length as 5, 7, and 11.

>> I think Gene's Euclidean norm gives crow's distance

>> on an unweighted rectangular lattice, but I could be

>> wrong.

>

>No, on an unweighted "triangular" (An) lattice. Of course in the

>11-limit it probably makes more sense to make 9, not 3, the same

>length as 5, 7, and 11.

>

>> I just meant a 4-D thing with 60 deg. angles, which I

>> assume exists but have no knowledge of.

>

>Well, you do now, since you have this general formula which in the

>4-D (11-limit) case becomes

>

>|| |* a b c d> || = sqrt((a^2+b^2+c^2+d^2)+(a+b+c+d)^2)/2)

Aha! I should have known that, since ||7/5||=1. What's the formula

for distance on a rectangular lattice?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Aha! I should have known that, since ||7/5||=1. What's the formula

> for distance on a rectangular lattice?

|| (a1, ..., an) || = sqrt(a1^2+...+an^2)

>> Aha! I should have known that, since ||7/5||=1. What's the formula

>> for distance on a rectangular lattice?

>

>|| (a1, ..., an) || = sqrt(a1^2+...+an^2)

Rad! I didn't know the pythagorean theorem generalized like this,

but it's perfectly obvious after spending 15 seconds with a pen

and scrap of paper (here at the cafe). The ^2 on the previously-

derived side will remove the sqrt there, and the new side will

get a new ^2.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Rad! I didn't know the pythagorean theorem generalized like this,

> but it's perfectly obvious after spending 15 seconds with a pen

> and scrap of paper (here at the cafe). The ^2 on the previously-

> derived side will remove the sqrt there, and the new side will

> get a new ^2.

The Pythagorean theorem can be generalized in a number of different

ways. For the purposes of this list, another good one to keep in mind

is the Law of Cosines:

c^2 = a^2 + b^2 - 2ab cos(C)

where C is the angle between sides of length a and b of a triangle,

and c is the length of the opposte side. When C=90 degrees, you get

the Pythagorean theorem. When C = 60 degrees, you get

c^2 = a^2 + b^2 - ab

When C = 120 degrees, you get

c^2 = a^2 + b^2 + ab

which is the 5-limit symmetrical lattice distance formula. It tells us

the 10/9 interval from 3/2 to 5/3, the 16/15 interval from 3/2 to 8/5,

and the 25/24 interval from 8/5 to 5/3 all have a length of sqrt(3)

and are on the vertices of an equilateral triangle with 1 as the

centroid. The same is true of 6/5, 5/4 and 4/3, and all six form an

equilateral hexagon surrounding 1.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> lattice type................Triangu. Rect.

> >> >> dimensions..................1 2 3... 1 2 3...

> >> >>

> >> >> taxicab

> >> >> odd limit

> >> >> Tenney height

> >> >> symmetric Euclidean

> >> >>

> >> >> The weighted versions of these would be

> >> >>

> >> >> "isosceles"

> >> >> expressibility

> >> >> Tenney HD

> >> >> ?

> >> >

> >> >Makes no sense to me, so I'll ignore it :)

> >>

> >> The entire chart? The entries would be things like

> >> "rhombic dodecahedron", under each of the six (in this case)

> >> columns (lattice types) for each measure.

> >

> >I'd rather use only one lattice type for each measure; the one

that

> >makes the balls of that measure most symmetrical.

>

> One way to find that out is to read the ball shapes off a chart

> like this!

But then you would need even more columns, for the different kinds of

geometry that are most symmetrical for these different measures.

> >And although you can make a lattice where expressibility

corresponds

> >to a distance measure, you can't do that for odd limit //

> >Can you do it for symmetric Euclidean? Makes no sense.

>

> Why not? It's only a difference of a log.

Distance measures must satisfy the triangle inequality: the distance

from A to B plus the distance from B to C must be less than or equal

to the distance from A to C.

> What do you mean by "distance measure"? Crow flies? Surely

> Euclidean distance as Gene's defined it measures crow's distance

> on some lattice.

Euclidean distance equals crow's distance on any lattice. But any

reference to actual ratios or musical intervals is missing here.

> >> >> >> Why do you like it better than your isosceles lattice?

> >> >> >

> >> >> >It's better for a prime limit of 5 and the new paradigm where

> >> >> >consonance/dissonance is shades-of-gray rather than black-

and-

> >> >> >white, because all the (hexagonally-determined) distances

> >> >> >correspond to complexity (expressibility), no matter how

> >> >> >complex the ratios. This is why I believe the rungs should

> >> >> >be deleted. If ratios of 9, in particular, are to be

considered

> >> >> >something other than mere dissonances in a 5-prime-limit

tuning,

> >> >> >then you want to represent them as shorter than ratios of 15.

> >> >> >This is exactly what this lattice, with a hexagonal-norm

metric,

> >> >> >accomplishes perfectly. I should make a diagram with lots of

> >> >> >ratios on it . . .

> >> >>

> >> >> Hmm...

> >> >

> >> >As in you understand, or you don't?

> >>

> >> Don't.

> >

> >Think of a set of nested, concentric regular hexagons. These

> >represent increasing "distances". This is the notion of "distance"

we

> >need here.

>

> I have the picture, it's an example of 'grey-area dissonance and

> ratios of 9' that I'm missing.

Hmm . . . what kind of example are you looking for?

> >> I'm not sure what paradigm you're referring to.

> >> The weighted paradigm?

> >

> >No, just the paradigm where we drop the assumption of

a "consonance

> >limit",

>

> How long has this been going on?

For quite a while. Remember my posts on the tuning list about

consistency going out the window in such a paradigm?

> >> I do take it you're saying allowed motions to 9 are of length

> >> log(9)?

> >

> >Yes; if you can't see that on the diagram as it is, I'll make a

new

> >one for you.

>

> It's hard to visualize regular hexagons over that lattices skewed

> ones.

True. As I said, the rungs shouldn't even be there. I'll have to draw

a version with no rungs, with ratios written in, and with the regular

hexagons shown.

> >> If so, the "Hahn approach" does not break down above 3-D.

> >

> >Yes it does, if you ever read anything Hahn wrote.

>

> Hahn's code showed 9-limit examples working perfectly, and my

> code tests against those examples.

But what does the lattice look like?

> >> If not, it's a problem of geometry, and I need a

> >> new lattice (of course I'm already doing this numerically

> >> with no apparent problems).

> >

> >If not what?

>

> If the 4-D triangular lattice failed to map consonances to

> rungs.

>> >> >> lattice type................Triangu. Rect.

>> >> >> dimensions..................1 2 3... 1 2 3...

>> >> >>

>> >> >> taxicab

>> >> >> odd limit

>> >> >> Tenney height

>> >> >> symmetric Euclidean

>> >> >>

>> >> >> The weighted versions of these would be

>> >> >>

>> >> >> "isosceles"

>> >> >> expressibility

>> >> >> Tenney HD

>> >> >> ?

>> >> >

>> >> >Makes no sense to me, so I'll ignore it :)

>> >>

>> >> The entire chart? The entries would be things like

>> >> "rhombic dodecahedron", under each of the six (in this case)

>> >> columns (lattice types) for each measure.

>> >

>> >I'd rather use only one lattice type for each measure; the one

>that

>> >makes the balls of that measure most symmetrical.

>>

>> One way to find that out is to read the ball shapes off a chart

>> like this!

>

>But then you would need even more columns, for the different kinds of

>geometry that are most symmetrical for these different measures.

There are 6 columns in this chart.

>> >And although you can make a lattice where expressibility

>> >corresponds to a distance measure, you can't do that for

>> >odd limit //

>>

>> Why not? It's only a difference of a log.

>

>Distance measures must satisfy the triangle inequality: the distance

>from A to B plus the distance from B to C must be less than or equal

>to the distance from A to C.

That's true.

>> >Can you do it for symmetric Euclidean? Makes no sense.

//

>Euclidean distance equals crow's distance on any lattice.

>But any reference to actual ratios or musical intervals

>is missing here.

Can you give an example?

>> >> I'm not sure what paradigm you're referring to.

>> >> The weighted paradigm?

>> >

>> >No, just the paradigm where we drop the assumption of

>> >a "consonance limit",

>>

>> How long has this been going on?

>

>For quite a while. Remember my posts on the tuning list about

>consistency going out the window in such a paradigm?

No...

>> It's hard to visualize regular hexagons over that lattices skewed

>> ones.

>

>True. As I said, the rungs shouldn't even be there. I'll have to draw

>a version with no rungs, with ratios written in, and with the regular

>hexagons shown.

Sweet.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> >> lattice type................Triangu. Rect.

> >> >> >> dimensions..................1 2 3... 1 2 3...

> >> >> >>

> >> >> >> taxicab

> >> >> >> odd limit

> >> >> >> Tenney height

> >> >> >> symmetric Euclidean

> >> >> >>

> >> >> >> The weighted versions of these would be

> >> >> >>

> >> >> >> "isosceles"

> >> >> >> expressibility

> >> >> >> Tenney HD

> >> >> >> ?

> >> >> >

> >> >> >Makes no sense to me, so I'll ignore it :)

> >> >>

> >> >> The entire chart? The entries would be things like

> >> >> "rhombic dodecahedron", under each of the six (in this case)

> >> >> columns (lattice types) for each measure.

> >> >

> >> >I'd rather use only one lattice type for each measure; the one

> >that

> >> >makes the balls of that measure most symmetrical.

> >>

> >> One way to find that out is to read the ball shapes off a chart

> >> like this!

> >

> >But then you would need even more columns, for the different kinds

of

> >geometry that are most symmetrical for these different measures.

>

> There are 6 columns in this chart.

OK, OK.

> >> >And although you can make a lattice where expressibility

> >> >corresponds to a distance measure, you can't do that for

> >> >odd limit //

> >>

> >> Why not? It's only a difference of a log.

> >

> >Distance measures must satisfy the triangle inequality: the

distance

> >from A to B plus the distance from B to C must be less than or

equal

> >to the distance from A to C.

>

> That's true.

>

> >> >Can you do it for symmetric Euclidean? Makes no sense.

> //

> >Euclidean distance equals crow's distance on any lattice.

> >But any reference to actual ratios or musical intervals

> >is missing here.

>

> Can you give an example?

An example of what?

> >> >> I'm not sure what paradigm you're referring to.

> >> >> The weighted paradigm?

> >> >

> >> >No, just the paradigm where we drop the assumption of

> >> >a "consonance limit",

> >>

> >> How long has this been going on?

> >

> >For quite a while. Remember my posts on the tuning list about

> >consistency going out the window in such a paradigm?

>

> No...

Really?? It was less than two years ago . . .

> >> It's hard to visualize regular hexagons over that lattices skewed

> >> ones.

> >

> >True. As I said, the rungs shouldn't even be there. I'll have to

draw

> >a version with no rungs, with ratios written in, and with the

regular

> >hexagons shown.

>

> Sweet.

Don't let me forget!

>> >> >Can you do it for symmetric Euclidean? Makes no sense.

I think "it" referred to:

>> >> >make a lattice where ___________

>> >> >corresponds to a distance measure

>> >Euclidean distance equals crow's distance on any lattice.

>> >But any reference to actual ratios or musical intervals

>> >is missing here.

>>

>> Can you give an example?

>

>An example of what?

How a lack reference to ratios or musical intervals causes

Euclidean distance to fail to be a distance measure.

>> >> >No, just the paradigm where we drop the assumption of

>> >> >a "consonance limit",

>> >>

>> >> How long has this been going on?

>> >

>> >For quite a while. Remember my posts on the tuning list about

>> >consistency going out the window in such a paradigm?

>>

>> No...

>

>Really?? It was less than two years ago . . .

I you sure I wasn't on hiatus at the time?

>> I'll have to draw a version with no rungs, with ratios written

>> in, and with the regular hexagons shown.

>>

>> Sweet.

>

>Don't let me forget!

Heh! I can't even remember to tie my shoes.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> >Can you do it for symmetric Euclidean? Makes no sense.

>

> I think "it" referred to:

>

> >> >> >make a lattice where ___________

> >> >> >corresponds to a distance measure

>

> >> >Euclidean distance equals crow's distance on any lattice.

> >> >But any reference to actual ratios or musical intervals

> >> >is missing here.

> >>

> >> Can you give an example?

> >

> >An example of what?

>

> How a lack reference to ratios or musical intervals causes

> Euclidean distance to fail to be a distance measure.

How can I supply an example for what isn't there? I meant it isn't a

distance measure on JI intervals, but you seemed to want to compare

it directly against distance measures on JI intervals. Meanwhile, it

seems it *is* comparable to the things you wanted to put in the

*columns* (rather than rows) of your table.

> >> >> >No, just the paradigm where we drop the assumption of

> >> >> >a "consonance limit",

> >> >>

> >> >> How long has this been going on?

> >> >

> >> >For quite a while. Remember my posts on the tuning list about

> >> >consistency going out the window in such a paradigm?

> >>

> >> No...

> >

> >Really?? It was less than two years ago . . .

>

> I you sure I wasn't on hiatus at the time?

Maybe . . .

> >> I'll have to draw a version with no rungs, with ratios written

> >> in, and with the regular hexagons shown.

> >>

> >> Sweet.

> >

> >Don't let me forget!

>

> Heh! I can't even remember to tie my shoes.

OK . . . I'll be dreaming of this lattice every night until I make it

for you.

>> >> >> >Can you do it for symmetric Euclidean? Makes no sense.

>>

>> I think "it" referred to:

>>

>> >> >> >make a lattice where ___________

>> >> >> >corresponds to a distance measure

>>

>> >> >Euclidean distance equals crow's distance on any lattice.

>> >> >But any reference to actual ratios or musical intervals

>> >> >is missing here.

>> >>

>> >> Can you give an example?

>> >

>> >An example of what?

>>

>> How a lack reference to ratios or musical intervals causes

>> Euclidean distance to fail to be a distance measure.

>

>How can I supply an example for what isn't there? I meant it isn't a

>distance measure on JI intervals,

Then can you give an example of where it breaks the triangle

inequality?

>Meanwhile, it seems it *is* comparable to the things you wanted

>to put in the *columns* (rather than rows) of your table.

We must be talking about different "it"s here.

The rows are measures, all of which should obey the triangle

inequality. The columns are dimensions of either triangular

or rectangular latti. The entries are ball shapes.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> >> >Can you do it for symmetric Euclidean? Makes no sense.

> >>

> >> I think "it" referred to:

> >>

> >> >> >> >make a lattice where ___________

> >> >> >> >corresponds to a distance measure

> >>

> >> >> >Euclidean distance equals crow's distance on any lattice.

> >> >> >But any reference to actual ratios or musical intervals

> >> >> >is missing here.

> >> >>

> >> >> Can you give an example?

> >> >

> >> >An example of what?

> >>

> >> How a lack reference to ratios or musical intervals causes

> >> Euclidean distance to fail to be a distance measure.

> >

> >How can I supply an example for what isn't there? I meant it isn't

a

> >distance measure on JI intervals,

>

> Then can you give an example of where it breaks the triangle

> inequality?

It doesn't -- Tenney height and 'odd limit' do.

> >Meanwhile, it seems it *is* comparable to the things you wanted

> >to put in the *columns* (rather than rows) of your table.

>

> We must be talking about different "it"s here.

>

> The rows are measures, all of which should obey the triangle

> inequality. The columns are dimensions of either triangular

> or rectangular latti.

Dimensions? I thought they were different varieties. If you don't

have different varieties of triangular lattice, you don't have enough

columns to get at the thing you're interested in, which is to find

which lattice gives most "sphericity" to each measure. So you'd need

more columns.

Anyway, "symmetrical Euclidean" to me refers to a particular variety

of triangular lattice. If there's a ratio-complexity measure that

agrees well with this lattice, that's great, but using the former as

a name for it is asking for confusion, particularly when you're

planning to look at the ball shapes produced by every possible ratio-

complexity measure in every possible lattice!

> The entries are ball shapes.

>> Then can you give an example of where it breaks the triangle

>> inequality?

>

>It doesn't -- Tenney height and 'odd limit' do.

Ah, right, they should be taken off the list (rows). Or

better yet, their entries should be "N/A".

>> The rows are measures, all of which should obey the triangle

>> inequality. The columns are dimensions of either triangular

>> or rectangular latti.

>

>Dimensions?

That's why it says "dimensions"

>> >> lattice type................Triangu. Rect.

>> >> dimensions..................1 2 3... 1 2 3...

>> >>

>> >> taxicab

>> >> odd limit

>> >> Tenney height

>> >> symmetric Euclidean

>> >>

>> >> The weighted versions of these would be

>> >>

>> >> "isosceles"

>> >> expressibility

>> >> Tenney HD

>> >> ?

Probably those column labels should have been above

the columns.... :)

>If you don't

>have different varieties of triangular lattice, you don't have enough

>columns to get at the thing you're interested in, which is to find

>which lattice gives most "sphericity" to each measure. So you'd need

>more columns.

I'm cool with more columns (and rows).

>Anyway, "symmetrical Euclidean" to me refers to a particular variety

>of triangular lattice. If there's a ratio-complexity measure that

>agrees well with this lattice, that's great, but using the former as

>a name for it is asking for confusion, particularly when you're

>planning to look at the ball shapes produced by every possible ratio-

>complexity measure in every possible lattice!

They don't have to be "ratio" complexity measures (as in a simple

rule for ratios?). They just have to be complexity measures.

By "Symmetric Euclidean" I mean the actual thing, not a simplified

version. It's true that "Symmetric Euclidean" and "taxicab",

unlike "Tenney HD" and "expressibility" require different formulas

for their tri. and rect. varieties.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >Anyway, "symmetrical Euclidean" to me refers to a particular

variety

> >of triangular lattice. If there's a ratio-complexity measure that

> >agrees well with this lattice, that's great, but using the former

as

> >a name for it is asking for confusion, particularly when you're

> >planning to look at the ball shapes produced by every possible

ratio-

> >complexity measure in every possible lattice!

>

> They don't have to be "ratio" complexity measures (as in a simple

> rule for ratios?). They just have to be complexity measures.

But what are they measuring the complexity of if not ratios?

> By "Symmetric Euclidean" I mean the actual thing, not a simplified

> version.

Why would anyone think otherwise?

> It's true that "Symmetric Euclidean" and "taxicab",

> unlike "Tenney HD" and "expressibility" require different formulas

> for their tri. and rect. varieties.

>> >Anyway, "symmetrical Euclidean" to me refers to a particular

>> >variety of triangular lattice. If there's a ratio-complexity

>> >measure that agrees well with this lattice, that's great, but

>> >using the former as a name for it is asking for confusion,

>> >particularly when you're planning to look at the ball shapes

>> >produced by every possible ratio-complexity measure in every

>> >possible lattice!

>>

>> They don't have to be "ratio" complexity measures (as in a simple

>> rule for ratios?). They just have to be complexity measures.

>

>But what are they measuring the complexity of if not ratios?

You dropped the term "ratio-complexity measure". What does it

mean? Clearly the formula Gene gave for symmetrical Euclidean

distance takes ratios as input.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >Anyway, "symmetrical Euclidean" to me refers to a particular

> >> >variety of triangular lattice. If there's a ratio-complexity

> >> >measure that agrees well with this lattice, that's great, but

> >> >using the former as a name for it is asking for confusion,

> >> >particularly when you're planning to look at the ball shapes

> >> >produced by every possible ratio-complexity measure in every

> >> >possible lattice!

> >>

> >> They don't have to be "ratio" complexity measures (as in a simple

> >> rule for ratios?). They just have to be complexity measures.

> >

> >But what are they measuring the complexity of if not ratios?

>

> You dropped the term "ratio-complexity measure".

Dropped it?

> What does it

> mean?

It means something that measures the complexity of ratios.

> Clearly the formula Gene gave for symmetrical Euclidean

> distance takes ratios as input.

Then it needs a name to distinguish it from things that are supposed

to go in *columns* of your table. That's what I was trying to say in

the bit at the top.

>> >> >Anyway, "symmetrical Euclidean" to me refers to a particular

>> >> >variety of triangular lattice. If there's a ratio-complexity

>> >> >measure that agrees well with this lattice, that's great, but

>> >> >using the former as a name for it is asking for confusion,

>> >> >particularly when you're planning to look at the ball shapes

>> >> >produced by every possible ratio-complexity measure in every

>> >> >possible lattice!

>> >>

>> >> They don't have to be "ratio" complexity measures (as in a simple

>> >> rule for ratios?). They just have to be complexity measures.

>> >

>> >But what are they measuring the complexity of if not ratios?

>>

>> You dropped the term "ratio-complexity measure".

>

>Dropped it?

>

>> What does it

>> mean?

>

>It means something that measures the complexity of ratios.

>

>> Clearly the formula Gene gave for symmetrical Euclidean

>> distance takes ratios as input.

>

>Then it needs a name to distinguish it from things that are supposed

>to go in *columns* of your table. That's what I was trying to say in

>the bit at the top.

The columns of the table are lattice types... 2-D triangular, 3-D rect.,

etc. Symmetric Euclidean is on a row in the table.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> >Anyway, "symmetrical Euclidean" to me refers to a particular

> >> >> >variety of triangular lattice. If there's a ratio-complexity

> >> >> >measure that agrees well with this lattice, that's great, but

> >> >> >using the former as a name for it is asking for confusion,

> >> >> >particularly when you're planning to look at the ball shapes

> >> >> >produced by every possible ratio-complexity measure in every

> >> >> >possible lattice!

> >> >>

> >> >> They don't have to be "ratio" complexity measures (as in a

simple

> >> >> rule for ratios?). They just have to be complexity measures.

> >> >

> >> >But what are they measuring the complexity of if not ratios?

> >>

> >> You dropped the term "ratio-complexity measure".

> >

> >Dropped it?

> >

> >> What does it

> >> mean?

> >

> >It means something that measures the complexity of ratios.

> >

> >> Clearly the formula Gene gave for symmetrical Euclidean

> >> distance takes ratios as input.

> >

> >Then it needs a name to distinguish it from things that are

supposed

> >to go in *columns* of your table. That's what I was trying to say

in

> >the bit at the top.

>

> The columns of the table are lattice types... 2-D triangular, 3-D

rect.,

> etc. Symmetric Euclidean is on a row in the table.

Yes, I know this is what you said, and the above was a response to

that. As I stated before, you'll need different kinds of triangular

lattice types if you want the balls of each measure to be as

spherical as possible in one of the lattice types. One of these

triangular lattice types would be the symmetric type. So, to repeat

myself, I'm trying to encourage you to avoid confusion by not

referring to the *rows* by names that sound more appropriate for

*columns*. At least, you confused *me* by doing that. So why are you

merely repeating yourself here? Did you misunderstand what I was

saying?

>> The columns of the table are lattice types... 2-D triangular, 3-D

>> rect., etc. Symmetric Euclidean is on a row in the table.

>

>Yes, I know this is what you said, and the above was a response to

>that. As I stated before, you'll need different kinds of triangular

>lattice types if you want the balls of each measure to be as

>spherical as possible in one of the lattice types.

These should be added to the columns, I thought I said. The

entries only report the ball shapes. The 'most spherical' ones

are to be determined by the reader.

>I'm trying to encourage you to avoid confusion by not

>referring to the *rows* by names that sound more appropriate for

>*columns*. At least, you confused *me* by doing that. So why are you

>merely repeating yourself here? Did you misunderstand what I was

>saying?

The rows are distance measures. "Symmetric Euclidean" is a

distance measure. What's confusing about that?

You correctly pointed out that odd limit and Tenney height are

not distance measures, and I said they should be deleted from

the rows.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> The columns of the table are lattice types... 2-D triangular, 3-

D

> >> rect., etc. Symmetric Euclidean is on a row in the table.

> >

> >Yes, I know this is what you said, and the above was a response to

> >that. As I stated before, you'll need different kinds of

triangular

> >lattice types if you want the balls of each measure to be as

> >spherical as possible in one of the lattice types.

>

> These should be added to the columns, I thought I said.

Yes, you did.

> The

> entries only report the ball shapes. The 'most spherical' ones

> are to be determined by the reader.

Well right now it seems you and I would agree on 'most spherical' --

I just meant you need enough columns so that each of the rows gets

one colums where the corresponding measure gives a ball that is as

spherical as possible.

> >I'm trying to encourage you to avoid confusion by not

> >referring to the *rows* by names that sound more appropriate for

> >*columns*. At least, you confused *me* by doing that. So why are

you

> >merely repeating yourself here? Did you misunderstand what I was

> >saying?

>

> The rows are distance measures. "Symmetric Euclidean" is a

> distance measure. What's confusing about that?

I thought the rows were supposed to be measures of ratio-complexity.

Each measure of ratio complexity corresponds to a different distance

measure on each lattice, since each lattice is a different way of

arranging the ratios as points in space.

Anyway, what's confusing (to repeat myself once again) is that this

one sounds more like a lattice type than a ratio-complexity measure.

>I just meant you need enough columns so that each of the rows gets

>one colums where the corresponding measure gives a ball that is as

>spherical as possible.

Agreed.

>> >I'm trying to encourage you to avoid confusion by not

>> >referring to the *rows* by names that sound more appropriate for

>> >*columns*. At least, you confused *me* by doing that. So why are

>> >you merely repeating yourself here? Did you misunderstand what I

>> >was saying?

>>

>> The rows are distance measures. "Symmetric Euclidean" is a

>> distance measure. What's confusing about that?

>

>I thought the rows were supposed to be measures of ratio-complexity.

Distance measures. That's why odd limit and Tenney height need

to be deleted.

>Each measure of ratio complexity corresponds to a different distance

>measure on each lattice, since each lattice is a different way of

>arranging the ratios as points in space.

Ok, I see how you're thinking of this. I was thinking, start with

the distance measure you like and then find the best lattice for

them (by plotting the points in each ball and comparing them to a

sphere). You're thinking, start with a ratio complexity measure,

then find the lattice on which there is some corresponding distance

measure... yes? But I'm not clear on what difference there is

between these two approaches, since a ratio complexity measure

like odd limit could never correspond to a distance measure, could

it?

>Anyway, what's confusing (to repeat myself once again) is that this

>one sounds more like a lattice type than a ratio-complexity measure.

Ah, I see! I was just trying to follow Gene's usage.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> You dropped the term "ratio-complexity measure". What does it

> mean? Clearly the formula Gene gave for symmetrical Euclidean

> distance takes ratios as input.

Equivalence classes, actually.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >I just meant you need enough columns so that each of the rows gets

> >one colums where the corresponding measure gives a ball that is as

> >spherical as possible.

>

> Agreed.

>

> >> >I'm trying to encourage you to avoid confusion by not

> >> >referring to the *rows* by names that sound more appropriate

for

> >> >*columns*. At least, you confused *me* by doing that. So why are

> >> >you merely repeating yourself here? Did you misunderstand what I

> >> >was saying?

> >>

> >> The rows are distance measures. "Symmetric Euclidean" is a

> >> distance measure. What's confusing about that?

> >

> >I thought the rows were supposed to be measures of ratio-

complexity.

>

> Distance measures. That's why odd limit and Tenney height need

> to be deleted.

>

> >Each measure of ratio complexity corresponds to a different

distance

> >measure on each lattice, since each lattice is a different way of

> >arranging the ratios as points in space.

>

> Ok, I see how you're thinking of this. I was thinking, start with

> the distance measure you like and then find the best lattice for

> them (by plotting the points in each ball

You mean plotting the ball in each lattice?

> and comparing them to a

> sphere).

If you agree with the substitution above, that's what I'm thinking.

> You're thinking, start with a ratio complexity measure,

> then find the lattice on which there is some corresponding distance

> measure... yes?

No.

> But I'm not clear on what difference there is

> between these two approaches, since a ratio complexity measure

> like odd limit could never correspond to a distance measure, could

> it?

It can if you allow a series of concentric regular hexagons, instead

of circles, to define a distance measure -- then the required lattice

is the lattice that we've been talking about so much lately (the

second-to-last one on that http://www.kees.cc/tuning/lat_perbl.html

page, preferably with the rungs deleted). Technically, any convex

figure can be used to define a distance measure or _metric_.

> >Anyway, what's confusing (to repeat myself once again) is that

this

> >one sounds more like a lattice type than a ratio-complexity

measure.

>

> Ah, I see!

Whew!

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> But I'm not clear on what difference there is

> between these two approaches, since a ratio complexity measure

> like odd limit could never correspond to a distance measure, could

> it?

Eh? What about the Kees norm?

> >Anyway, what's confusing (to repeat myself once again) is that this

> >one sounds more like a lattice type than a ratio-complexity measure.

>

> Ah, I see! I was just trying to follow Gene's usage.

Even though I have no idea what all this is about.

>> Ok, I see how you're thinking of this. I was thinking, start with

>> the distance measure you like and then find the best lattice for

>> them (by plotting the points in each ball

>

>You mean plotting the ball in each lattice?

Yes!

>> and comparing them to a sphere).

>

>If you agree with the substitution above, that's what I'm thinking.

Oh.

>> You're thinking, start with a ratio complexity measure,

>> then find the lattice on which there is some corresponding distance

>> measure... yes?

>

>No.

Oh.

>> But I'm not clear on what difference there is

>> between these two approaches, since a ratio complexity measure

>> like odd limit could never correspond to a distance measure, could

>> it?

>

>It can if you allow a series of concentric regular hexagons, instead

>of circles, to define a distance measure -- then the required lattice

>is the lattice that we've been talking about so much lately (the

>second-to-last one on that http://www.kees.cc/tuning/lat_perbl.html

>page, preferably with the rungs deleted). Technically, any convex

>figure can be used to define a distance measure or _metric_.

How could this be, since odd limit doesn't obey the triangle

inequality?

>> >Anyway, what's confusing (to repeat myself once again) is that

>> >this one sounds more like a lattice type than a ratio-complexity

>> >measure.

>>

>> Ah, I see!

>

>Whew!

What should we call it?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> Ok, I see how you're thinking of this. I was thinking, start

with

> >> the distance measure you like and then find the best lattice for

> >> them (by plotting the points in each ball

> >

> >You mean plotting the ball in each lattice?

>

> Yes!

>

> >> and comparing them to a sphere).

> >

> >If you agree with the substitution above, that's what I'm thinking.

>

> Oh.

>

> >> You're thinking, start with a ratio complexity measure,

> >> then find the lattice on which there is some corresponding

distance

> >> measure... yes?

> >

> >No.

>

> Oh.

>

> >> But I'm not clear on what difference there is

> >> between these two approaches, since a ratio complexity measure

> >> like odd limit could never correspond to a distance measure,

could

> >> it?

> >

> >It can if you allow a series of concentric regular hexagons,

instead

> >of circles, to define a distance measure -- then the required

lattice

> >is the lattice that we've been talking about so much lately (the

> >second-to-last one on that

http://www.kees.cc/tuning/lat_perbl.html

> >page, preferably with the rungs deleted). Technically, any convex

> >figure can be used to define a distance measure or _metric_.

>

> How could this be, since odd limit doesn't obey the triangle

> inequality?

Sorry, I was thinking log of odd limit, or expressibility.

> >> >Anyway, what's confusing (to repeat myself once again) is that

> >> >this one sounds more like a lattice type than a ratio-complexity

> >> >measure.

> >>

> >> Ah, I see!

> >

> >Whew!

>

> What should we call it?

I don't know but for the purposes of the table, we could just label

it Gene1 or whatever. As long as it doesn't sound like it could be

the label for a column, we're good.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> How could this be, since odd limit doesn't obey the triangle

> inequality?

As applied to pitch classes, it does.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> I don't know but for the purposes of the table, we could just label

> it Gene1 or whatever. As long as it doesn't sound like it could be

> the label for a column, we're good.

Could one of you explain what these rows and columns are?

>> I don't know but for the purposes of the table, we could just label

>> it Gene1 or whatever. As long as it doesn't sound like it could be

>> the label for a column, we're good.

>

>Could one of you explain what these rows and columns are?

The rows are different distance measures and the columns

are types of things you don't like calling lattices, and

the entries are ball shapes. Ex. (hit reply to view on

the web):

3-D triangular

Gene1 (aka symmetric Euclidean distance) cuboctahedron

-Carl

But I wouldn't be too bothered by this incredibly convoluted

discussion Paul and I are having. I think the key points

are, we're curious about the lattice on Kees' website with

the angles you recently supplied...

"Then put 3/2 and 5/4 at 60 degrees from each other, and

scale then so that the length of 3/2 to 5/4 is in the

proportion 1:log3(5)."

And I'm curious about anything you have on the size of the

smallest interval we should expect to find as we search n

notes of whatever-limit JI.

-Carl

At 06:15 PM 11/21/2005, you wrote:

>>> I don't know but for the purposes of the table, we could

>>> just label it Gene1 or whatever. As long as it doesn't

>>> sound like it could be the label for a column, we're good.

>>

>>Could one of you explain what these rows and columns are?

>

>The rows are different distance measures and the columns

>are types of things you don't like calling lattices, and

>the entries are ball shapes. Ex. (hit reply to view on

>the web):

>

> 3-D triangular

>Gene1 (aka symmetric Euclidean distance) cuboctahedron

>

>-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> And I'm curious about anything you have on the size of the

> smallest interval we should expect to find as we search n

> notes of whatever-limit JI.

A partial answer to that is the n-diamond comma, which is the smallest

comma in the n-limit tonality diamond. It's either n^2/(n^2-1) or

(n^2+1)/n^2.

>> And I'm curious about anything you have on the size of the

>> smallest interval we should expect to find as we search n

>> notes of whatever-limit JI.

>

>A partial answer to that is the n-diamond comma, which is the smallest

>comma in the n-limit tonality diamond. It's either n^2/(n^2-1) or

>(n^2+1)/n^2.

That's a pretty restricted case that I don't think would be

generally applicable to lattice searches.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> That's a pretty restricted case that I don't think would be

> generally applicable to lattice searches.

The answer is going to depend on what norm you use, however. For Hahn

in the 7-limit case, the smallest interval greater than one in the

various shells from 1 to 10 goes 8/7, 50/49, 126/125, 2401/2400,

2401/2400, 2401/2400, 4375/4374, 4375/4374, 250047/250000,

250047/250000. It's possible log log regression would come up with a

rate of growth.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

> The answer is going to depend on what norm you use, however. For Hahn

> in the 7-limit case, the smallest interval greater than one in the

> various shells from 1 to 10 goes 8/7, 50/49, 126/125, 2401/2400,

> 2401/2400, 2401/2400, 2401/2400, 4375/4374, 4375/4374, 250047/250000,

> 250047/250000. It's possible log log regression would come up with a

> rate of growth.

I left off a 2401/2400, there are three of them.

>> That's a pretty restricted case that I don't think would be

>> generally applicable to lattice searches.

>

>The answer is going to depend on what norm you use, however. For

>Hahn in the 7-limit case, the smallest interval greater than one

>in the various shells from 1 to 10 goes 8/7, 50/49, 126/125,

>2401/2400, 2401/2400, 2401/2400, 4375/4374, 4375/4374,

>250047/250000, 250047/250000. It's possible log log regression

>would come up with a rate of growth.

The only reason I asked is that Paul said you had already posted

formulas for this.

I wouldn't expect it to depend much on the norm as the number of

notes gets large, but I'm prepared to be surprised. But maybe a

better first question is: for your favorite norm, how does this

change in the 3-, 5-, 7-limit?

I don't know how to do log log regression, but maybe it's time

I learned. Neither wikipedia or mathworld have entries for it.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> I don't know but for the purposes of the table, we could just

label

> >> it Gene1 or whatever. As long as it doesn't sound like it could be

> >> the label for a column, we're good.

> >

> >Could one of you explain what these rows and columns are?

>

> The rows are different distance measures

!!!

They are??

I thought we had this clear, that the rows were ratio-complexity

measures, not distance measures. What happened?

> and the columns

> are types of things you don't like calling lattices,

Are you sure about that?

> and

> the entries are ball shapes. Ex. (hit reply to view on

> the web):

>

> 3-D triangular

> Gene1 (aka symmetric Euclidean distance) cuboctahedron

I know you don't mean for this to be nitpicked, but if it were, I'd

change the above to "equilateral triangular", since other forms of

triangular lattice are going to be needed as columns in this table if

some of the other ratio-complexity measures are going to show up as

having fairly symmetrical balls in some lattice -- for example odd

limit or expressibility has "regular" cuboctahedron balls when the

lattice is based on scalene triangles (with only one 60-degree angle)

of the kind we've been discussing lately.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> That's a pretty restricted case that I don't think would be

> >> generally applicable to lattice searches.

> >

> >The answer is going to depend on what norm you use, however. For

> >Hahn in the 7-limit case, the smallest interval greater than one

> >in the various shells from 1 to 10 goes 8/7, 50/49, 126/125,

> >2401/2400, 2401/2400, 2401/2400, 4375/4374, 4375/4374,

> >250047/250000, 250047/250000. It's possible log log regression

> >would come up with a rate of growth.

>

> The only reason I asked is that Paul said you had already posted

> formulas for this.

I thought the idea was to compare different prime limits. Comma size

as a function of comma complexity (the latter is directly related to

the "number of notes" you need to search) is related to temperament

error as a function of comma complexity. For the latter, Gene found

some formulas that depend on the number of independent factors, so I

figured we could therefore derive some formulas for the former from

that.

> I wouldn't expect it to depend much on the norm as the number of

> notes gets large, but I'm prepared to be surprised. But maybe a

> better first question is: for your favorite norm, how does this

> change in the 3-, 5-, 7-limit?

>

> I don't know how to do log log regression, but maybe it's time

> I learned. Neither wikipedia or mathworld have entries for it.

You take the log of one quantity and plot it against the log of the

other quantity. If you find a straight line, then you know the

relationship between the original quantities was an exponential one,

where the slope of the line gives you the exponent:

y=x^a

implies that

log(y) = a*log(x)

>> >> I don't know but for the purposes of the table, we could just

>> >> label it Gene1 or whatever. As long as it doesn't sound like

>> >> it could be the label for a column, we're good.

>> >

>> >Could one of you explain what these rows and columns are?

>>

>> The rows are different distance measures

>

>!!!

>

>They are??

>

>I thought we had this clear, that the rows were ratio-complexity

>measures, not distance measures. What happened?

No, you never told me what you mean (and I still don't know)

by "ratio-complexity measure". You were the one who mentioned

that odd limit and Tenney height should be removed since

they didn't qualify as distance measures.

>> and the columns are types of things you don't like calling

>> lattices,

>

>Are you sure about that?

I guess you can put anything there you want. But I expect to

all of the major "lattices" discussed on these lists in the last

10 years.

>> and the entries are ball shapes. Ex. (hit reply to view on

>> the web):

>>

>> 3-D triangular

>> Gene1 (aka symmetric Euclidean distance) cuboctahedron

>

>I know you don't mean for this to be nitpicked, but if it were, I'd

>change the above to "equilateral triangular", since other forms of

>triangular lattice are going to be needed as columns in this table

Yes, very good.

-Carl

>> I don't know how to do log log regression, but maybe it's time

>> I learned. Neither wikipedia or mathworld have entries for it.

>

>You take the log of one quantity and plot it against the log of the

>other quantity. If you find a straight line, then you know the

>relationship between the original quantities was an exponential one,

>where the slope of the line gives you the exponent:

>

>y=x^a

>implies that

>log(y) = a*log(x)

Thanks!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> >> >> I don't know but for the purposes of the table, we could just

> >> >> label it Gene1 or whatever. As long as it doesn't sound like

> >> >> it could be the label for a column, we're good.

> >> >

> >> >Could one of you explain what these rows and columns are?

> >>

> >> The rows are different distance measures

> >

> >!!!

> >

> >They are??

> >

> >I thought we had this clear, that the rows were ratio-complexity

> >measures, not distance measures. What happened?

>

> No, you never told me what you mean (and I still don't know)

> by "ratio-complexity measure".

I thought we had already come to agreement on this -- did you forget

so soon? The input is a JI interval (two pitch-ratios or one interval-

ratio, octave-equivalence may or may not be assumed), and the output

is some reckoning of the "complexity" of that interval.

> You were the one who mentioned

> that odd limit and Tenney height should be removed since

> they didn't qualify as distance measures.

I said that no distance measure could agree with them. And you seemed

to understand me perfectly when you responded that the relevant

entries in these rows would therefore be "N/A". That seemed fine --

what happened to the mutual understanding we had come to?

> >> and the columns are types of things you don't like calling

> >> lattices,

> >

> >Are you sure about that?

>

> I guess you can put anything there you want.

I mean are you sure Gene doesn't like calling them lattices?

> But I expect to

> all of the major "lattices" discussed on these lists in the last

> 10 years.

Oh, ok, so that might include some things which are really

only "graphs" and not "lattices"?

>> >I thought we had this clear, that the rows were ratio-complexity

>> >measures, not distance measures. What happened?

>>

>> No, you never told me what you mean (and I still don't know)

>> by "ratio-complexity measure".

>

>I thought we had already come to agreement on this -- did you forget

>so soon? The input is a JI interval (two pitch-ratios or one interval-

>ratio, octave-equivalence may or may not be assumed), and the output

>is some reckoning of the "complexity" of that interval.

>

>> You were the one who mentioned

>> that odd limit and Tenney height should be removed since

>> they didn't qualify as distance measures.

>

>I said that no distance measure could agree with them. And you seemed

>to understand me perfectly when you responded that the relevant

>entries in these rows would therefore be "N/A". That seemed fine --

>what happened to the mutual understanding we had come to?

Why would anyone include a row if all its entries were N/A?

It seems that if the ratio-complexity measure is not a distance

measure, it will have more than one ball shape per lattice. So

I don't know why you want to keep these on the chart.

>> >> and the columns are types of things you don't like calling

>> >> lattices,

>> >

>> >Are you sure about that?

>>

>> I guess you can put anything there you want.

>

>I mean are you sure Gene doesn't like calling them lattices?

It doesn't look like either of us fully understand Gene's position

on this.

>> But I expect to all of the major "lattices" discussed on these

>> lists in the last 10 years.

>

>Oh, ok, so that might include some things which are really

>only "graphs" and not "lattices"?

The latter term is apparently under contention, so I don't know

how to answer this.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > >> and the columns are types of things you don't like calling

> > >> lattices,

> > >

> > >Are you sure about that?

> >

> > I guess you can put anything there you want.

>

> I mean are you sure Gene doesn't like calling them lattices?

I'm fine calling them lattices so long as they live inside a space

with a distance measure on it.

> Oh, ok, so that might include some things which are really

> only "graphs" and not "lattices"?

Or things which are abelian groups and not lattices, or things which are

tessellation but not lattices, etc.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> The latter term is apparently under contention, so I don't know

> how to answer this.

I suggest that the definition this group needs is "discrete subgroup

of a finite-dimensional normed real vector space".

>> The latter term is apparently under contention, so I don't know

>> how to answer this.

>

>I suggest that the definition this group needs is "discrete subgroup

>of a finite-dimensional normed real vector space".

The only problem with this that I can see is that intuitively,

the A3 "lattice" is the A3 "lattice" whether we're using Hahn

diameter or symmetric Euclidean distance. Whereas if I understand

the above, they'd actually be different lattices.

By the way, things like "A3" are impossible to search for.

Does anybody know of a good resource listing the various

instances of this nomenclature? Or is there a name for

this nomenclature ("Whostartedit symbols" or something)?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >I suggest that the definition this group needs is "discrete subgroup

> >of a finite-dimensional normed real vector space".

>

> The only problem with this that I can see is that intuitively,

> the A3 "lattice" is the A3 "lattice" whether we're using Hahn

> diameter or symmetric Euclidean distance. Whereas if I understand

> the above, they'd actually be different lattices.

They are, but then I think they should be. But there can be two

different lattices by this definition which clearly are both "the" A3

lattice even without non-Euclidean distances.

> By the way, things like "A3" are impossible to search for.

> Does anybody know of a good resource listing the various

> instances of this nomenclature? Or is there a name for

> this nomenclature ("Whostartedit symbols" or something)?

I thought you owned a copy of Conway and Sloane?

>> >I suggest that the definition this group needs is "discrete

>> >subgroup of a finite-dimensional normed real vector space".

>>

>> The only problem with this that I can see is that intuitively,

>> the A3 "lattice" is the A3 "lattice" whether we're using Hahn

>> diameter or symmetric Euclidean distance. Whereas if I understand

>> the above, they'd actually be different lattices.

>

>They are, but then I think they should be. But there can be two

>different lattices by this definition which clearly are both "the"

>A3 lattice even without non-Euclidean distances.

Only two? What are they?

>> By the way, things like "A3" are impossible to search for.

>> Does anybody know of a good resource listing the various

>> instances of this nomenclature? Or is there a name for

>> this nomenclature ("Whostartedit symbols" or something)?

>

>I thought you owned a copy of Conway and Sloane?

This one

http://www.amazon.com/gp/product/0387985859/

?

Nah, I have Conway & Guy _The Book of Numbers_. And at $77, it's

a library trip for me...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >They are, but then I think they should be. But there can be two

> >different lattices by this definition which clearly are both "the"

> >A3 lattice even without non-Euclidean distances.

>

> Only two? What are they?

I didn't mean only two. Take an A3 lattice, and rotate, reflect or

dilate it and you get an A3 lattice.

> >I thought you owned a copy of Conway and Sloane?

>

> This one

>

> http://www.amazon.com/gp/product/0387985859/

>

> ?

That's the one. Your basic resource for Euclidean lattices.

>> >They are, but then I think they should be. But there can be two

>> >different lattices by this definition which clearly are both "the"

>> >A3 lattice even without non-Euclidean distances.

>>

>> Only two? What are they?

>

>I didn't mean only two. Take an A3 lattice, and rotate, reflect or

>dilate it and you get an A3 lattice.

What's the name of the class of things to which A3 belongs?

>> >I thought you owned a copy of Conway and Sloane?

>>

>> This one

>>

>> http://www.amazon.com/gp/product/0387985859/

>>

>> ?

>

>That's the one. Your basic resource for Euclidean lattices.

I'll pick it up.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >I didn't mean only two. Take an A3 lattice, and rotate, reflect or

> >dilate it and you get an A3 lattice.

>

> What's the name of the class of things to which A3 belongs?

It belongs to a wider class of "root lattices", but the point is that

often you will equate lattices by similarity. All equilateral

triangles are the same shape, and so forth. On the other hand for

number theoretic purposes you might require that the corresponding

quadratic form is reduced with integer coefficients, for example.