//This is long. May I humbly suggest we do it up old-school, like

//in the Classic onelist years, and reply to everything until we

//agree on everything? Let's get something export-quality! Dave

//Keenan, activate your magic power ring, Voltron is needed once

//again! Monz, break out the colored chalks! Paul, I totally

//understand you wanting to take a break, and I've always been

//behind a book from you, but why not finish the paper on linear

//temperamenst first?

A paper. I think it's a great idea. And, the 569 of us who don't

have a computer set up to do calculations on linear temperaments

need a list!

Graham's catalog, "The grooviest 7-limit temperaments", Monzo's lines,

and Herman Miller's "Carl's favorite page on the internet" Warped

Canons page are huge, huge, huge. But wouldn't it be cool to really

get the goat?

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

Selection criteria

(1) Badness

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

~~~~~~~~~~~~~~~~

(1a) Gene's list

~~~~~~~~~~~~~~~~

Paul wrote...

>Gene, who's way, way ahead of any other theorist on this list (and

>possibly anywhere) has (like Dave Keenan and Graham Breed before

>him) completed a comprehensive search for linear temperaments for

>7-limit music. He proposed a 'badness' measure defined as:

>

>step^3 cent

>

>where step is a measure of the typical number of notes in a scale

>for this temperament (given any desired degree of harmonic depth),

>and cent is a measure of the deviation from JI 'consonances' in

>cents.

Is this still state-of-the-art-badness? I seem to remember

something about different exponents for each prime/odd (?) identity,

taken from coefficients of Diophantine equations, or some such?

Don't need details, just want to know if we need a new top 20.

Paul wrote...

>He then ranked his 505 temperaments by 'goodness'. The familiar

>ones don't come in until later, so bear with me . . .

Gene, you initially stopped after listing 20. Did you ever list

the requested next few needed to uncover meantone? Are you still

happy with your list?

~~~~~~~~~~~~~~~~~~~~~

(1b) The slippery six

~~~~~~~~~~~~~~~~~~~~~

I wasn't reading the tuning-math very closely back then, but Gene,

your top 20 is generated by starting with some large number of

ets and then seeing what temperaments they share, sort of like a

more precise version of looking for lines on Herman Miller's / Paul's

charts, right? But you found that some temperaments only hit a

single et up to your cutoff -- those were the slippery six, right?

Do we have a general solution to this problem -- making the cutoff

really high, an entirely different method, etc.?

Speaking of this method, nobody ever answered this:

Carl wrote...

>More to the point, every line on this plane is a linear temperament,

>right? So what makes low-numbered (less than 100) equal

>temperaments cluster on some of them?

What makes some linear temperaments belong to more than one et, out

of ets as high as some given number? They would have to share a

common generator... Is sharing a common generator related to the

un-even distribution of the rationals on the number line (such as

makes harmonic entropy work)?

Carl wrote...

>Intersections (by eye)

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

>12 - 8

>31 - 5

>22, 37, 15, 34, 19 - 4

>29, 41, 75, 61, 53, 72, 23 - 3

Why are some of the 'best' ets (ones that have gotten so much

attention on these lists for so many different reasons, for

so long) here? Is it because we've often defined "best" as

"consistent", and where two lines cross the same tuning is being

reached two different ways (via two different maps), which

requires consistency?

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

Selection criteria

(2) Maps and commas

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

A map uniquely defines a linear temperament? Or do you also

need period? Looking at Graham's catalog, I'm not sure how to

use maps with non-octave periods.

Carl wrote...

>I say the most powerful maps are the ones with the smallest

>numbers in them. Sum of abs value would work.

Or, maybe the sum of the abs values of the max and min numbers in

the map, for a given limit (or divided by the card of the map, if

you want to compare across limits). Which is better?

There's definitely some overlap with badness here, but by not

considering the quality of the approximations, doesn't this tell

us more about the abstract musical-theoretic properties of a

temperament?

Carl wrote...

>Finally, re the jumping jacks / ideal comma question... what's the

>question? How are we defining "most powerful" comma?

?

Carl wrote...

>What's the relationship between a comma vanishing and a map?

?

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

The contents of the list

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

Paul wrote...

>>Generators on the table

Carl wrote...

>Yes, I completely agree. Who can furnish rms optimums?

Paul wrote...

>i bet gene can do this in a jiffy. maybe graham too.

>and oh, we need the period as well as the generator.

I completely agree.

Paul wrote...

>actually, gene already did this back in december.

I looked. My eyes! The searching did nothing!

Paul wrote...

>i'm making a graph that includes these as well as the ets.

>

>well, i tried to, but the points get too crowded near the

>center for me to label them.

F the graph. Let's have a list!

Paul wrote...

>but it's easy to see the optimum point on the graph on monz's

>page already. simply look at the line representing the

>temperament you're interested in, and the point on that line

>that comes closest to the center ('origin') of the graph is

>the optimal one. so optimal meantone is near 50-equal, and

>optimal magic (in 5-limit at least) is near 60-equal, etc.

I figured as much. But what if the nearest et below 100 is

off the optimum some? Why not do it right?

Paul wrote...

>Topping off Gene's list are some very funky simple temperaments,

/.../

>For these, I quote the simplest pair of unison vectors:

>

> (1) <21/20,27/25>

> (2) <8/7,15/14>

> (3) <9/8,15/14>

> (4) <25/24,49/48>

> (5) <15/14,25/24>

> (6) <21/20,25/24>

> (7) <15/14,35/32>

> (8) <7/6,16/15>

> (9) <16/15,21/20>

Can we get a list with optimum generator, et series, commas, maps,

periods for these (and the rest of the top 20)? Are any of the

"Monzo's lines" temperaments in here?

-Carl

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

> >step^3 cent

> Is this still state-of-the-art-badness? I seem to remember

> something about different exponents for each prime/odd (?) identity,

> taken from coefficients of Diophantine equations, or some such?

> Don't need details, just want to know if we need a new top 20.

No, but no consensus was reached on what we should be looking for. I like a step^2 measure of some kind, because of the log-flat business.

> Gene, you initially stopped after listing 20. Did you ever list

> the requested next few needed to uncover meantone? Are you still

> happy with your list?

No, it's not been done. I could simply go ahead and do things my way.

> ~~~~~~~~~~~~~~~~~~~~~

> (1b) The slippery six

> ~~~~~~~~~~~~~~~~~~~~~

>

> I wasn't reading the tuning-math very closely back then, but Gene,

> your top 20 is generated by starting with some large number of

> ets and then seeing what temperaments they share, sort of like a

> more precise version of looking for lines on Herman Miller's / Paul's

> charts, right? But you found that some temperaments only hit a

> single et up to your cutoff -- those were the slippery six, right?

> Do we have a general solution to this problem -- making the cutoff

> really high, an entirely different method, etc.?

I took a look at this problem, and it seems to me that starting from single ets and using generators from them ought to turn up everything of interest if we check everything which makes some sort of sense in

non-consistent cases (which are the only hard ones.) Making that precise and putting it into effect would still make for a big search; I would like something like Paul's heuristic to speed that up. Right now I'm writing, so I'm thinking through the whole thing anew, and hope that will help.

> Speaking of this method, nobody ever answered this:

>

> Carl wrote...

> >More to the point, every line on this plane is a linear temperament,

> >right? So what makes low-numbered (less than 100) equal

> >temperaments cluster on some of them?

Those are the better low-rent commas. Not all temperaments were born equal.

> What makes some linear temperaments belong to more than one et, out

> of ets as high as some given number?

Having common kernel elements.

> >12 - 8

> >31 - 5

> >22, 37, 15, 34, 19 - 4

> >29, 41, 75, 61, 53, 72, 23 - 3

>

> Why are some of the 'best' ets (ones that have gotten so much

> attention on these lists for so many different reasons, for

> so long) here?

The best ets have the best temperaments, generally speaking.

> A map uniquely defines a linear temperament? Or do you also

> need period?

I thought a map *had* a period. What do you mean by a map? A map with

two vectors (period and octave) can be wedged to get the corresponding wedgie, so it clearly defines the temperament.

> >Finally, re the jumping jacks / ideal comma question... what's the

> >question? How are we defining "most powerful" comma?

At this point, we aren't. Paul's discovery I suppose could be used to do that, if you want to define it in terms of Fourier analysis!

> >What's the relationship between a comma vanishing and a map?

If the map maps the comma to zero, the comma vanishs for the map--it's in the kernel of the map.

> Can we get a list with optimum generator, et series, commas, maps,

> periods for these (and the rest of the top 20)? Are any of the

> "Monzo's lines" temperaments in here?

These are 7-limit temperaments; Monzo would need to graph a plane.

>I took a look at this problem, and it seems to me that starting from

>single ets and using generators from them ought to turn up everything

>of interest if we check everything which makes some sort of sense in

>non-consistent cases (which are the only hard ones.) Making that

>precise and putting it into effect would still make for a big search;

>I would like something like Paul's heuristic to speed that up.

What's the status of your heuristic, Paul?

>Right now I'm writing, so I'm thinking through the whole thing anew,

>and hope that will help.

Cool!

>> Speaking of this method, nobody ever answered this:

>>

>> Carl wrote...

>> >More to the point, every line on this plane is a linear temperament,

>> >right? So what makes low-numbered (less than 100) equal

>> >temperaments cluster on some of them?

>

>Those are the better low-rent commas. Not all temperaments were born

>equal.

What makes a comma "better"?

>>A map uniquely defines a linear temperament? Or do you also

>>need period?

>

>I thought a map *had* a period. What do you mean by a map?

A list of identities given in terms of generators, ie [1 4]

for meantone. I suppose the rms optimum generator can be

constructed from this alone, and a generator plus a period

specifies a linear temerament. I really don't know where the

period comes from.

>A map with two vectors (period and octave) can be wedged to

>get the corresponding wedgie, so it clearly defines the

>temperament.

Hmmm... not sure I follow.

What do octaves have to do with anything? Isn't it just that

some temperaments have a period that is a fraction of an octave,

and so two or more periods of the temperament are used in

practice to get octaves for musical reasons?

>>>Finally, re the jumping jacks / ideal comma question... what's the

>>>question? How are we defining "most powerful" comma?

>

>At this point, we aren't. Paul's discovery I suppose could be used

>to do that, if you want to define it in terms of Fourier analysis!

Well, I'd certainly like to hear more about this. Are you referring

to the periodicity in the badness curve? You want to run a Fourier

analysis on the wave and see the commas with the biggest peaks?

>>>What's the relationship between a comma vanishing and a map?

>

>If the map maps the comma to zero, the comma vanishs for the

>map--it's in the kernel of the map.

You mean: if I factor a comma in terms of the identities in a map,

sum the corresponding number of generators on each side of the

fraction, subtract the numerator's sum from the denominator's, and

get zero? So, 81/80 -> (3^4)/(5* 2^4), so if the map is [1 4],

I get 4 - 4 = 0.

>>Can we get a list with optimum generator, et series, commas, maps,

>>periods for these (and the rest of the top 20)? Are any of the

>>"Monzo's lines" temperaments in here?

>

>These are 7-limit temperaments; Monzo would need to graph a plane.

You mean a volume? Anyway, yes, they are 7-limit -- I've been

thinking about the relationship between good linear temperaments

at limit n and good ones at higher limits. You just append the

extra identities onto the map. Porcupine, for example, is on

Monzo's chart but extends to the 7-limit. I'm interested in how

the commas change... 64:63 becomes a porcupine comma in the 7-limit,

for example. IOW, will we need separate top-20 lists for each

limit?

-Carl

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

> What makes a comma "better"?

For 5-limit commas, simply the badness score of the associated temperament quantifies it.

> >>A map uniquely defines a linear temperament? Or do you also

> >>need period?

> >

> >I thought a map *had* a period. What do you mean by a map?

>

> A list of identities given in terms of generators, ie [1 4]

> for meantone.

That maps one generator, except it should really be [0 1 4]. You need another map for the octave, which is the other generator.

> What do octaves have to do with anything? Isn't it just that

> some temperaments have a period that is a fraction of an octave,

> and so two or more periods of the temperament are used in

> practice to get octaves for musical reasons?

The period is one of the generators, if you decide to have one generator proportional to octaves.

> >If the map maps the comma to zero, the comma vanishs for the

> >map--it's in the kernel of the map.

>

> You mean: if I factor a comma in terms of the identities in a map,

> sum the corresponding number of generators on each side of the

> fraction, subtract the numerator's sum from the denominator's, and

> get zero? So, 81/80 -> (3^4)/(5* 2^4), so if the map is [1 4],

> I get 4 - 4 = 0.

Pretty much; however you should really take the whole map: you have the period part as well, [1,1,0]; so 81/80 -> -4*1 + 4*0 -1*0 = 0.

IOW, will we need separate top-20 lists for each

> limit?

For each finitely generated group of intervals we are interested in, which covers a lot more ground, I fear.

>> What makes a comma "better"?

>

>For 5-limit commas, simply the badness score of the associated

>temperament quantifies it.

We've defined good commas as those that result in good temperaments.

What is it about a comma that results in a better temperaments than

another? All the lines on Monz's chart could have just two dots.

The badness of the associated temperaments would still quantify the

situation. But instead, some lines have many dots.

>>A list of identities given in terms of generators, ie [1 4]

>>for meantone.

>

>That maps one generator, except it should really be [0 1 4]. You

>need another map for the octave, which is the other generator.

Okay, thanks. So is the map for the octave in meantone [1 0 -2]?

>> >If the map maps the comma to zero, the comma vanishs for the

>> >map--it's in the kernel of the map.

>>

>> You mean: if I factor a comma in terms of the identities in a map,

>> sum the corresponding number of generators on each side of the

>> fraction, subtract the numerator's sum from the denominator's, and

>> get zero? So, 81/80 -> (3^4)/(5* 2^4), so if the map is [1 4],

>> I get 4 - 4 = 0.

>

>Pretty much; however you should really take the whole map: you have

>the period part as well, [1,1,0];

What's this!? Two 1's?

>so 81/80 -> -4*1 + 4*0 -1*0 = 0.

?

>>IOW, will we need separate top-20 lists for each limit?

>

>For each finitely generated group of intervals we are interested in,

>which covers a lot more ground, I fear.

If we're potentially interested in all the subsets, we make software

which accepts queries. For being interested in on paper, odd limits

through 19 are quite enough.

-Carl

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

> >Pretty much; however you should really take the whole map: you have

> >the period part as well, [1,1,0];

>

> What's this!? Two 1's?

>

> >so 81/80 -> -4*1 + 4*0 -1*0 = 0.

2 ~ (3/2)^0 2^1

3 ~ (3/2)^1 2^1

5 ~ (3/2)^4 2^0

So the map is

[0 1]

[1 1]

[4 0]

>>>Pretty much; however you should really take the whole map: you have

>>>the period part as well, [1,1,0];

>>

>> What's this!? Two 1's?

>>

>> >so 81/80 -> -4*1 + 4*0 -1*0 = 0.

>

>2 ~ (3/2)^0 2^1

>3 ~ (3/2)^1 2^1

>5 ~ (3/2)^4 2^0

>

>So the map is

>

>[0 1]

>[1 1]

>[4 0]

Aha! Thanks.

-Carl

In-Reply-To: <4.0.1.20020223185452.019836c0@lumma.org>

Carl Lumma wrote:

> A paper. I think it's a great idea. And, the 569 of us who don't

> have a computer set up to do calculations on linear temperaments

> need a list!

You can get "set up" with only free software, and all 569 of you could do

that if you really wanted. I've also got my ISP's computer set up to do

the calculations at <http://microtonal.co.uk/temper/>. That is limited,

mainly because you can't choose your own badness measures, and it can only

handle the odd limits.

As my free time's limited, what would people prefer I do this evening?

Improving the CGI, writing up the algorithms and playing with some drum

loops are all options.

> Graham's catalog, "The grooviest 7-limit temperaments", Monzo's lines,

> and Herman Miller's "Carl's favorite page on the internet" Warped

> Canons page are huge, huge, huge. But wouldn't it be cool to really

> get the goat?

What else do you want?

> Is this still state-of-the-art-badness? I seem to remember

> something about different exponents for each prime/odd (?) identity,

> taken from coefficients of Diophantine equations, or some such?

> Don't need details, just want to know if we need a new top 20.

I don't think any badness will ever be good enough for everybody. My

solutions are

1) Put the calculations online, so people can specify what they want in

terms of accuracy and complexity, instead of generating as many as 20

options.

2) Allow custom badness measures. Hopefully this can be done by entering

Python expressions in text boxes. I will need to check the security on

this. The badness as a function of error (RMS or minimax) and complexity

should be the most useful thing to customize.

> ~~~~~~~~~~~~~~~~~~~~~

> (1b) The slippery six

> ~~~~~~~~~~~~~~~~~~~~~

>

> I wasn't reading the tuning-math very closely back then, but Gene,

> your top 20 is generated by starting with some large number of

> ets and then seeing what temperaments they share, sort of like a

> more precise version of looking for lines on Herman Miller's / Paul's

> charts, right? But you found that some temperaments only hit a

> single et up to your cutoff -- those were the slippery six, right?

> Do we have a general solution to this problem -- making the cutoff

> really high, an entirely different method, etc.?

I'm not sure about Gene, but that's how I do it. Raising the cutoff --

not insisting on consistency -- does get you more temperaments, and this

is an option on the CGI. You'll still miss some because they aren't

represented by a pair of nearest-prime ETs. One way round this is to

consider all sane options of inconsistent temperaments. Another is to use

a different method.

> Speaking of this method, nobody ever answered this:

>

> Carl wrote...

> >More to the point, every line on this plane is a linear temperament,

> >right? So what makes low-numbered (less than 100) equal

> >temperaments cluster on some of them?

>

> What makes some linear temperaments belong to more than one et, out

> of ets as high as some given number? They would have to share a

> common generator... Is sharing a common generator related to the

> un-even distribution of the rationals on the number line (such as

> makes harmonic entropy work)?

I'd say ETs belong to linear temperaments rather than the other way round.

And ET is a linear temperament where the large and small steps happen to

make a rational number. How many you get depends on your badness cutoff.

> Carl wrote...

> >Intersections (by eye)

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

> >12 - 8

> >31 - 5

> >22, 37, 15, 34, 19 - 4

> >29, 41, 75, 61, 53, 72, 23 - 3

>

> Why are some of the 'best' ets (ones that have gotten so much

> attention on these lists for so many different reasons, for

> so long) here? Is it because we've often defined "best" as

> "consistent", and where two lines cross the same tuning is being

> reached two different ways (via two different maps), which

> requires consistency?

No, because they don't have to be consistent. But the approximations for

a linear temperament must be at least as good as its best constituent ET.

The complexity must be somehow related as well. I've conjectured that the

complexity of an LT must be no worse than the larger number of notes of a

pair of consistent ETs consistent with it.

> A map uniquely defines a linear temperament? Or do you also

> need period? Looking at Graham's catalog, I'm not sure how to

> use maps with non-octave periods.

The simple, one-dimensional maps assume an equivalence interval. All you

need to do is set this other period to be that interval. Otherwise you

have to give the octave specific map, as for the Bohlen-Pierce entry. Or

describe it in words, as the golden non-meantone entry.

> Carl wrote...

> >I say the most powerful maps are the ones with the smallest

> >numbers in them. Sum of abs value would work.

>

> Or, maybe the sum of the abs values of the max and min numbers in

> the map, for a given limit (or divided by the card of the map, if

> you want to compare across limits). Which is better?

You have to consider 0 as well, for the unison. Then, the max-min is my

complexity. I think Gene's using the RMS.

> There's definitely some overlap with badness here, but by not

> considering the quality of the approximations, doesn't this tell

> us more about the abstract musical-theoretic properties of a

> temperament?

My keyboard mapping lists (.key suffix) do this.

> Carl wrote...

> >Finally, re the jumping jacks / ideal comma question... what's the

> >question? How are we defining "most powerful" comma?

One rule of thumb, which relates to Paul's heuristic, is to consider the

number of consonances that make up the comma, and the size of the comma.

The simpler the interval the better, and the smaller it is the more

accurate a temperament can be. The optimal accuracy is the size of the

comma in cents divided by its complexity in consonances.

Here, the "consonances" are the list of intervals you want to optimize.

> Carl wrote...

> >What's the relationship between a comma vanishing and a map?

>

> ?

The map will approximate all vanishing commas to a unison.

> Paul wrote...

> >i bet gene can do this in a jiffy. maybe graham too.

> >and oh, we need the period as well as the generator.

>

> I completely agree.

I can't tell from the context what you want here. I probably thought the

CGI could do it originally. I can't be wiping your noses forever.

> > (1) <21/20,27/25>

> > (2) <8/7,15/14>

> > (3) <9/8,15/14>

> > (4) <25/24,49/48>

> > (5) <15/14,25/24>

> > (6) <21/20,25/24>

> > (7) <15/14,35/32>

> > (8) <7/6,16/15>

> > (9) <16/15,21/20>

>

> Can we get a list with optimum generator, et series, commas, maps,

> periods for these (and the rest of the top 20)?

I've got a CGI that can do some of that.

Graham

>You can get "set up" with only free software, and all 569 of you could do

>that if you really wanted. I've also got my ISP's computer set up to do

>the calculations at <http://microtonal.co.uk/temper/>. That is limited,

>mainly because you can't choose your own badness measures, and it can

>only handle the odd limits.

Rock! I hadn't looked at these for a while. Some nice-to-haves:

() document it

() allow input of identities, not just odd limit

() return the name of the temperament, if known

() return all fields for each temperament (ie "not unique" or "unique")

>As my free time's limited, what would people prefer I do this evening?

>Improving the CGI, writing up the algorithms and playing with some drum

>loops are all options.

Hope you enjoyed what you picked.

My biggest complaint is too many worthy options. But it's better than

not enough! Really shows how spoiled I am.

>>Graham's catalog, "The grooviest 7-limit temperaments", Monzo's lines,

>>and Herman Miller's "Carl's favorite page on the internet" Warped

>>Canons page are huge, huge, huge. But wouldn't it be cool to really

>>get the goat?

>

>What else do you want?

Slightly improved cgi stuff (see above) and/or a list.

My ideal list would have:

() Top 20 temperaments, by Gene's favorite badness measure, in each

odd limit from 5 to 17.

() Show a name, map, rms optimum generator, rms error, simplest commas,

and complexity for each.

() Make sure the names hold for a given LT if it makes it into the

top 20 of higher and higher limits.

() Uniqueness level.

>>Is this still state-of-the-art-badness? I seem to remember

>>something about different exponents for each prime/odd (?) identity,

>>taken from coefficients of Diophantine equations, or some such?

>>Don't need details, just want to know if we need a new top 20.

>

>I don't think any badness will ever be good enough for everybody. My

>solutions are

>

>1) Put the calculations online, so people can specify what they want in

>terms of accuracy and complexity, instead of generating as many as 20

>options.

It would still be nice to report the top 20 to people who aren't going

to learn to use the script.

>2) Allow custom badness measures. Hopefully this can be done by entering

>Python expressions in text boxes. I will need to check the security on

>this. The badness as a function of error (RMS or minimax) and complexity

>should be the most useful thing to customize.

This is a good idea, but it would be the last thing I would spend time

implementing. IMO rms is always better than minimax, and complexity

should be kept separate from badness, and for badness I'll happy to trust

Gene!

>I'm not sure about Gene, but that's how I do it. Raising the cutoff --

>not insisting on consistency -- does get you more temperaments, and this

>is an option on the CGI. You'll still miss some because they aren't

>represented by a pair of nearest-prime ETs.

Nearest-prime? Anyway, all you have to do is show why a temperament

with low badness is bound to appear in multiple ets, as you claim

above, and I'll be happy.

>>What makes some linear temperaments belong to more than one et, out

>>of ets as high as some given number? They would have to share a

>>common generator... Is sharing a common generator related to the

>>un-even distribution of the rationals on the number line (such as

>>makes harmonic entropy work)?

>

>I'd say ETs belong to linear temperaments rather than the other way

>round.

Okay, fine. But I don't see how this answers the question.

>> Carl wrote...

>> >I say the most powerful maps are the ones with the smallest

>> >numbers in them. Sum of abs value would work.

>>

>> Or, maybe the sum of the abs values of the max and min numbers in

>> the map, for a given limit (or divided by the card of the map, if

>> you want to compare across limits). Which is better?

>

>You have to consider 0 as well, for the unison. Then, the max-min

>is my complexity.

Okay, great! We agree.

>I think Gene's using the RMS.

Wow. z'that true, Gene?

>>>Finally, re the jumping jacks / ideal comma question... what's the

>>>question? How are we defining "most powerful" comma?

>

>One rule of thumb, which relates to Paul's heuristic, is to consider the

>number of consonances that make up the comma, and the size of the comma.

>The simpler the interval the better, and the smaller it is the more

>accurate a temperament can be. The optimal accuracy is the size of the

>comma in cents divided by its complexity in consonances.

Makes sense.

>The map will approximate all vanishing commas to a unison.

Gene showed me this. Cool.

>>>i bet gene can do this in a jiffy. maybe graham too.

>>>and oh, we need the period as well as the generator.

>>

>> I completely agree.

>

>I can't tell from the context what you want here.

More details on Monzo's chart. But Monzo's chart is just a list

of LTs that we eyeballed from Paul's graph. But if what you say

is right about LTs sharing ETs and badness, the eyeballing might

not be so bad afterall. But still, we shouldn't bother with it

any more. We should create a badness-ranked list!

>I probably thought the CGI could do it originally. I can't be

>wiping your noses forever.

I think your cgi can do most of it.

-Carl

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

> Paul wrote...

> >actually, gene already did this back in december.

>

> I looked. My eyes! The searching did nothing!

>

> Paul wrote...

> >i'm making a graph that includes these as well as the ets.

> >

> >well, i tried to, but the points get too crowded near the

> >center for me to label them.

>

> F the graph. Let's have a list!

tuning-math messages 1985, 1997, 2009, 2064, 2121, and one more i

can't find right now.

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

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

> > I wasn't reading the tuning-math very closely back then, but Gene,

> > your top 20 is generated by starting with some large number of

> > ets

no, i don't think gene approached it this way. he didn't answer this

part of your question, though, carl, so you should really get it from

the horse's mouth.

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

>

> >I think Gene's using the RMS.

>

> Wow. z'that true, Gene?

gene's complexity measures involve "step" and "cent", and "step" is

the RMS number of generators in the consonant intervals.

>

> More details on Monzo's chart. But Monzo's chart is just a list

> of LTs that we eyeballed from Paul's graph. But if what you say

> is right about LTs sharing ETs and badness, the eyeballing might

> not be so bad afterall. But still, we shouldn't bother with it

> any more. We should create a badness-ranked list!

gene gave the top 5 -- search for 'top 5'.

>>>I think Gene's using the RMS.

>>

>> Wow. z'that true, Gene?

>

>gene's complexity measures involve "step" and "cent", and "step" is

>the RMS number of generators in the consonant intervals.

Wait- is this complexity or badness? I wasn't talking about anything

with cents there.

>> More details on Monzo's chart. But Monzo's chart is just a list

>> of LTs that we eyeballed from Paul's graph. But if what you say

>> is right about LTs sharing ETs and badness, the eyeballing might

>> not be so bad afterall. But still, we shouldn't bother with it

>> any more. We should create a badness-ranked list!

>

>gene gave the top 5 -- search for 'top 5'.

Well, top 5 isn't enough. Also, Gene has given many lists of things,

and it isn't clear to me what is supposed to be "the" list. Best I

can tell, there isn't one. Which is what this thread is supposed to

be about.

-Carl

In-Reply-To: <200203050710.g257Ae819262@satyr.host4u.net>

Carl Lumma wrote:

> () document it

Is this the program, the method or the CGI? The first two need doing.

I'd rather make the CGI easy enough to use that it doesn't need any other

documentation.

> () allow input of identities, not just odd limit

I'm not sure what this means, but I don't think the CGI does it. You can

do anything with the original script.

> () return the name of the temperament, if known

Yes, I suppose that should be done. It'll mean I need to keep a list of

them somewhere. I wonder if it can be integrated with the catalog.

> () return all fields for each temperament (ie "not unique" or "unique")

You think that's important? It won't be difficult to change.

> My biggest complaint is too many worthy options. But it's better than

> not enough! Really shows how spoiled I am.

The odd limit, worst complexity and worst error have to be there. Other

things would ideally be guessed by the script, but I don't know how to

guess them yet. The defaults should do fine most of the time.

> >What else do you want?

>

> Slightly improved cgi stuff (see above) and/or a list.

There are already lists on the site, but I'm not planning to update them

now the CGI's there.

> My ideal list would have:

>

> () Top 20 temperaments, by Gene's favorite badness measure, in each

> odd limit from 5 to 17.

You can get close to Gene's badness measure with the CGI, and I am

allowing 20 results now, provided the server doesn't kill the script.

> () Show a name, map, rms optimum generator, rms error, simplest commas,

> and complexity for each.

Those are all reasonable, and you can expect them by the end of the year.

Certainly before I publish anything on dead trees.

> () Make sure the names hold for a given LT if it makes it into the

> top 20 of higher and higher limits.

Yes, I'll try and sort that out.

> () Uniqueness level.

Ooh! I can only do this up to 2nd order so far. Would that be okay?

> It would still be nice to report the top 20 to people who aren't going

> to learn to use the script.

If people can understand the top 20 but not use the script, there must be

something wrong with the script. I'd rather fix it than circumvent it.

Still, you're welcome to generate your own lists and put them on your own

site.

How about if a particular calculation could be referenced by a URL? That

might already work, but it could be simplified by allowing the script to

substitute default values.

> >2) Allow custom badness measures. Hopefully this can be done by

> entering >Python expressions in text boxes. I will need to check the

> security on >this. The badness as a function of error (RMS or minimax)

> and complexity >should be the most useful thing to customize.

>

> This is a good idea, but it would be the last thing I would spend time

> implementing. IMO rms is always better than minimax, and complexity

> should be kept separate from badness, and for badness I'll happy to

> trust

> Gene!

Well, it's implemented now. As there's an ongoing discussion about

badness, at least people have something to do test runs with. There isn't

a choice of minimax, because it's slower to calculate and isn't always

correct anyway.

It can't enforce uniqueness either, or know about the simplest MOS, so it

doesn't duplicate my static lists.

> >I'm not sure about Gene, but that's how I do it. Raising the cutoff

> -- >not insisting on consistency -- does get you more temperaments, and

> this >is an option on the CGI. You'll still miss some because they

> aren't >represented by a pair of nearest-prime ETs.

>

> Nearest-prime? Anyway, all you have to do is show why a temperament

> with low badness is bound to appear in multiple ets, as you claim

> above, and I'll be happy.

All temperaments will contain multiple ETs, it's only a question of the

algorithm being general enough to find them. They aren't always

consistent. A lot of the high-ranking temperaments for 17 and higher

limits don't include a consistent ET. I don't know if they really are

good temperaments or not because I don't have experience of 17-limit, and

I don't think anybody does in this way.

> >>What makes some linear temperaments belong to more than one et, out

> >>of ets as high as some given number? They would have to share a

> >>common generator... Is sharing a common generator related to the

> >>un-even distribution of the rationals on the number line (such as

> >>makes harmonic entropy work)?

> >

> >I'd say ETs belong to linear temperaments rather than the other way

> >round.

>

> Okay, fine. But I don't see how this answers the question.

It comes from the scale tree. A linear temperament will be a particular

branch. It's up to you where you cut it off.

Graham

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

> >I took a look at this problem, and it seems to me that starting

from

> >single ets and using generators from them ought to turn up

everything

> >of interest if we check everything which makes some sort of sense

in

> >non-consistent cases (which are the only hard ones.) Making that

> >precise and putting it into effect would still make for a big

search;

> >I would like something like Paul's heuristic to speed that up.

>

> What's the status of your heuristic, Paul?

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)).

> >>>Finally, re the jumping jacks / ideal comma question... what's

the

> >>>question? How are we defining "most powerful" comma?

> >

> >At this point, we aren't. Paul's discovery I suppose could be used

> >to do that, if you want to define it in terms of Fourier analysis!

>

> Well, I'd certainly like to hear more about this. Are you referring

> to the periodicity in the badness curve? You want to run a Fourier

> analysis on the wave and see the commas with the biggest peaks?

that seems a bit tangential at this point, since it's related to ets

and not linear temperaments. but eventually, the whole business

should be unified.

>> () document it

>

>Is this the program, the method or the CGI? The first two need doing.

>I'd rather make the CGI easy enough to use that it doesn't need any other

>documentation.

It's the answers to the questions I've been asking. So people can

use it without... having to ask the questions I'm asking. That falls

under "the program", I guess.

>> () allow input of identities, not just odd limit

>

>I'm not sure what this means, but I don't think the CGI does it. You can

>do anything with the original script.

I just want to enter identities 7 9 11. If I want 11-limit, make me

enter 1 3 5 7 9 11.

>> () return the name of the temperament, if known

>

>Yes, I suppose that should be done. It'll mean I need to keep a list of

>them somewhere. I wonder if it can be integrated with the catalog.

It could. One gripe I have with the catalog is that it gives different

names for the same temperament at different limits. IIRC Cassandra

1 and 2 are two different extensions of schismic, one of which should

be schismic, and the other Cassandra x, if its complexity (or g) is low

enough to warrant a place in the catalog (which I don't think it is,

but I could see including it for historical reasons).

>> () return all fields for each temperament (ie "not unique" or "unique")

>

>You think that's important? It won't be difficult to change.

It makes the output of the one that lists multiple temperaments easier

to parse. Ideal would be uniqueness level.

>The odd limit, worst complexity and worst error have to be there. Other

>things would ideally be guessed by the script, but I don't know how to

>guess them yet. The defaults should do fine most of the time.

Required fields should be marked with a star, or something.

>There are already lists on the site, but I'm not planning to update them

>now the CGI's there.

That's fine by me. One man can't do everything.

>> () Top 20 temperaments, by Gene's favorite badness measure, in each

>> odd limit from 5 to 17.

>

>You can get close to Gene's badness measure with the CGI, and I am

>allowing 20 results now, provided the server doesn't kill the script.

What's Gene's measure called there? Is it steps^3 or steps^2?

I was able to get 20 results.

>> () Show a name, map, rms optimum generator, rms error, simplest commas,

>> and complexity for each.

>

>Those are all reasonable, and you can expect them by the end of the year.

>Certainly before I publish anything on dead trees.

:)

>> () Make sure the names hold for a given LT if it makes it into the

>> top 20 of higher and higher limits.

>

>Yes, I'll try and sort that out.

>

>> () Uniqueness level.

>

>Ooh! I can only do this up to 2nd order so far. Would that be okay?

2nd order = triads, 1st order = dyads? That would be splendid.

Up to hexads would be nice. Don't let anybody tell you they need

more than that.

>>It would still be nice to report the top 20 to people who aren't going

>>to learn to use the script.

>

>If people can understand the top 20 but not use the script, there must be

>something wrong with the script.

If you're looking at it that way, then good!

>Still, you're welcome to generate your own lists and put them on your own

>site.

I was referring to the paper. No cgi support there yet. :)

>How about if a particular calculation could be referenced by a URL? That

>might already work, but it could be simplified by allowing the script to

>substitute default values.

Sounds like a good idea to me.

>> This is a good idea, but it would be the last thing I would spend time

>> implementing. IMO rms is always better than minimax, and complexity

>> should be kept separate from badness, and for badness I'll happy to

>> trust

>> Gene!

>

>Well, it's implemented now. As there's an ongoing discussion about

>badness, at least people have something to do test runs with. There

>isn't a choice of minimax, because it's slower to calculate and isn't

>always correct anyway.

Cool.

>It can't enforce uniqueness either, or know about the simplest MOS, so

>it doesn't duplicate my static lists.

That's fine.

-Carl

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

> Well, top 5 isn't enough. Also, Gene has given many lists of things,

> and it isn't clear to me what is supposed to be "the" list.

I gave a top 20, which might do. How many do you want for "the" list?

>> Well, top 5 isn't enough. Also, Gene has given many lists of things,

>> and it isn't clear to me what is supposed to be "the" list.

>

>I gave a top 20, which might do. How many do you want for "the" list?

I think around 20 is good... however many it is, and whatever badness

measure is used, we just have to be sure to get augmented, diminished,

schismic, and meantone. That will impress the ivory tower types.

-Carl

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

> >>>I think Gene's using the RMS.

> >>

> >> Wow. z'that true, Gene?

> >

> >gene's complexity measures involve "step" and "cent", and "step"

is

> >the RMS number of generators in the consonant intervals.

>

> Wait- is this complexity or badness?

badness. sorry.

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

> All temperaments will contain multiple ETs, it's only a question of the

> algorithm being general enough to find them. They aren't always

> consistent.

An algorithm which will find some of them, consistent or not, is to wedge with commas until you get an et.

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

> I think around 20 is good... however many it is, and whatever badness

> measure is used, we just have to be sure to get augmented, diminished,

> schismic, and meantone. That will impress the ivory tower types.

Do you mean by augmented and diminished 5-limit temperaments?

I get a badness of 142 for 128/125 and of 385 for 648/625. They both made my list of 32 (not 20) best, and are discussed on

Paul followed up and suggested "octo-diminished" for the 648/625 system, but "diminished" should be fine. The "octo" part came, I think, since I pointed out how well 64-et could do this system.

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

> Paul followed up and suggested "octo-diminished" for the 648/625

>system,

only when it occurs in 64-equal.

>but "diminished" should be fine.

i hope monz will catch on. also, on his et page, it seems he forgot

that "<" meant "less than" and he appended ">" to each entry that

referred to a mention *before* a certain date. but i should wait for

him to catch up on all the other stuff first . . .

>The "octo" part came, I think, >since I pointed out how well 64-et

>could do this system.

right -- i think i was making a rough analogy to the hepta-diminished

system of 28-equal. i guess 32-equal is actually octo-diminished . . .

>>>actually, gene already did this back in december.

>>

>>I looked. My eyes! The searching did nothing!

>

>tuning-math messages 1985, 1997, 2009, 2064, 2121, and one more i

>can't find right now.

Thanks, Paul. I've nabbed all these.

-C.

In-Reply-To: <200203061801.g26I1Mt30684@satyr.host4u.net>

Carl Lumma wrote:

> I just want to enter identities 7 9 11. If I want 11-limit, make me

> enter 1 3 5 7 9 11.

Well, you're in luck! That's the other thing I added on Monday night.

> >Yes, I suppose that should be done. It'll mean I need to keep a list

> of >them somewhere. I wonder if it can be integrated with the catalog.

>

> It could. One gripe I have with the catalog is that it gives different

> names for the same temperament at different limits. IIRC Cassandra

> 1 and 2 are two different extensions of schismic, one of which should

> be schismic, and the other Cassandra x, if its complexity (or g) is low

> enough to warrant a place in the catalog (which I don't think it is,

> but I could see including it for historical reasons).

The problem of integration is that the catalog's in static HTML, whereas

the script would have to get the names from a database or a global

dictionary somewhere. The catalog could be made dynamic, but it'd mean

the comments would also have to go in a database and some provision made

for the inharmonic one. It's easier not to bother, and have two different

lists, that are bound to get out of sync...

The intention of the catalog is to record temperaments that have been

singled out. There's no value judgement on my part as to what goes in

there. The Cassandras could go under schismic, and Shrutar could go under

diaschismic. Currently Paultone/Twintone/Pajara is under diaschismic.

> >> () return all fields for each temperament (ie "not unique" or

> > "unique")

> >

> >You think that's important? It won't be difficult to change.

>

> It makes the output of the one that lists multiple temperaments easier

> to parse. Ideal would be uniqueness level.

If some of the temperaments don't have "unique" at the bottom, you can get

more on the screen at once to compare them.

> >The odd limit, worst complexity and worst error have to be there.

> Other >things would ideally be guessed by the script, but I don't know

> how to >guess them yet. The defaults should do fine most of the time.

>

> Required fields should be marked with a star, or something.

But as I supply defaults for most (which I think makes it easier to see

what they're intended for) it wouldn't make any difference to normal

operation if they were option. A separation between "important" and

"geeky" might be better.

> >> () Uniqueness level.

> >

> >Ooh! I can only do this up to 2nd order so far. Would that be okay?

>

> 2nd order = triads, 1st order = dyads? That would be splendid.

> Up to hexads would be nice. Don't let anybody tell you they need

> more than that.

It'd be "all second order intervals are unique" because I've got a routine

for generating the second order intervals. That could easily be adapted

for fourth order but not, as it stands, third order.

> >Still, you're welcome to generate your own lists and put them on your

> own >site.

>

> I was referring to the paper. No cgi support there yet. :)

Oh. Well, my paper will include examples I found promising after trying

them on my ZTar (which I'll hopefully be getting soon). Also, a link to

the CGIs and the source code for anybody who wants to duplicate it all.

That's going to take a while to finish, so until then I'm trying to make

what's on the web as accessible as possible.

Graham

In-Reply-To: <a663hl+fj4c@eGroups.com>

Me:

> > All temperaments will contain multiple ETs, it's only a question of

> > the algorithm being general enough to find them. They aren't always

> > consistent.

Gene:

> An algorithm which will find some of them, consistent or not, is to

> wedge with commas until you get an et.

It's not finding the ETs from the LT that's difficult. You can walk the

scale tree, and one day I will. No, the difficult bit is finding the LT

in the first place, and a good pair or ETs is an efficient way of doing

that. Checking all mappings of each individual ET may be better in some

cases.

Another way I forgot to mention is to take all combinations of a set of

unison vectors. I've implemented this now, and I find it to be very slow

when it goes beyond the 11-limit. It may be improvable, but I think

checking all versions of inconsistent ETs will be much more productive.

Graham