Tuning-Math Digests messages 4125 - 4149

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Message: 4125

Date: Sun, 10 Mar 2002 20:59:50

Subject: Re: 32 best 5-limit linear temperaments redux

From: Carl Lumma

>> i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7}, 
>> {3,5,7}, and {2,3,5,7}. isn't that right?
>
>That's ok by me. But if we're listing these separately (which seems 
>the right thing to me), then the maximum interval width (which is also 
>the width of the complete chord - Carl's measure) seems more relevant 
>to me than the rms interval width.

That is a good one, but mine is that divided by the number of elements
in your chord (I said "card map", which is the same if you define map
like me, and remember to mulitply by periods in your interval of
equivalence at the end), which would be 3, 3, 3, 3, and 4 above,
respectively.

-Carl


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Message: 4126

Date: Sun, 10 Mar 2002 12:17:17

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Dave wrote...
> >Another reason it isn't ad hoc. The perceptual "pain" caused by 
> >mistuning is not directly proportional to the error in cents. Even 
the 
> >best microtonal ear on the planet apparently experiences 
essentially 
> >zero pain with a 0.5 cent mistuning. Most people aren't 
significantly 
> >bothered by a 3c mistuning (depending on the interval and how long 
it 
> >is sustained). But a 30 cent mistuning is so bad that a 40 cent one 
> >could hardly be much worse. 
> 
> Gene's "cents" are already rms, which we've long ago decided is the
> best single error measure.

Whether it is rms or max-absolute is irrelevant to my argument. I'm 
happy to use minimum rms error as the input to badness (though I think 
it's good to know the minimum max-absolute as well). If you're 
suggesting that using rms somehow removes the need to apply a 
nonlinear "pain" function, then you're mistaken. You've put "cents" in 
scare quotes above as if you think the units aren't really cents when 
it's rms. They are.

>  Chopping off anything less than .5 is a
> hack, and hopefully an un-necessary one.

It is definitely unnecessary and I do not propose to chop anything 
off. Gene's use of straight cents gives far too much credit to 
temperaments that have very small sub-half-cent errors. It is allowed 
to compensate so much for large gens that a temperament with a 
complexity of 35.5 gens (rms) can be considered the best 5-limit 
temperament! If you were to take a 0.5 cent threshold into account in 
a discontinuous manner, you would treat any temperament whose rms 
error was less than 0.5 c as if its error _was_ 0.5 c. But I'm not 
proposing we do that. My pain function is non-linear but smooth. A 
temperament still gets some credit for being sub-half-cent, just not 
so much.

> Gene wrote...
> >>For example, I think meantone must be in the top 3, or the badness
> >>measure is nonsense.
> >
> >This is like saying 12-et must be in the top three.

I don't have a problem with that, of course 12-ET is in the top 3 ETs 
for 5-limit.

> It's great in its 
> >size range-- if that happens to be the range you are interested in. 
If 
> >it doesn't suit your requirements then it isn't great, whatever 
> >number you come up with for it. What's top or not top depends on 
what 
> >tone group you are looking at (5-limit, 7-limit?) and what sort of 
> >accuracy you want.
> 
> Dave's just saying that you're not weighting the span of the map
> enough, since he considers musical history to be a worthy badness
> measure in its own right -- one that selected meantone, diminished,
> augmented, over the infinity of temperaments bigger than schismic.

That's a good way of putting it, except I wouldn't say that Gene's not 
weighting the complexity enough, I'd say he's failing to level off 
(asymptote) with sub-cent errors and not weighting super-20-cent 
errors enough.

> You want something that exposes the pattern in the series of
> best temperaments.
> 
> It would be nice to see a list with a much stronger penalty
> for size, but I can live with a flat measure with a sharp cutoff.

You don't _have_ to live with it.


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Message: 4127

Date: Mon, 11 Mar 2002 03:18:47

Subject: Re: Dave's 18 best 5-limit temperaments

From: dkeenanuqnetau

I've updated my spreadsheet to include the new discoveries.

http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Ok *

The names of my top 26 5-limit temperaments are (in order of keenan 
badness):
meantone
kleismic
augmented
diaschismic
schismic
small diesic
wuerschmidt
tiny diesic
porcupine
orwell
neutral thirds
diminished
minimal diesic
amt
semisuper
1990656/1953125 ?
16875/16384 ?
twin meantone
half meantone-fifth
half meantone-fourth
twin kleismic
half kleismic-minor-third
half kleismic-major-sixth
parakleismic
pelogic
hemithird


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Message: 4128

Date: Mon, 11 Mar 2002 16:38:08

Subject: Re: 32 best 5-limit linear temperaments redux

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Dave wrote:

> orwell

From George Orwell?

Manuel


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Message: 4129

Date: Mon, 11 Mar 2002 03:31:59

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> ok. let our paper have two sets of lists then.

OK. The mathematicians list and the musicians list. But if we're not 
including any badness measure and we're not listing the temperaments 
in order of any badness measure, then that may not be necessary. I've 
just given cutoffs for Gene, so that he is guaranteed to include my 
list in his (assuming my list doesn't grow when Gene finds out what 
went wrong with his attempt to generate my list).

> > > gene, by insisting on a log-flat measure, best 
> > > permits the musician to make this decision for him/herself. 
> > > going too far in both directions doesn't hurt anyone.
> > 
> > How can you say that!? 
> > 
> > I just showed that Gene's list failed to include at least two 
> > temperaments that are _way_ more interesting and useful 
> than 
> > "fourth-thirds" and the one with 49 gens to the fifth.
> 
> not in their respective ranges of complexity 

That's not relevant. My point is that it _does_ hurt someone. It hurts 
the person interested in middle-of-the-road temperaments.

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

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

> *don't* put the badness numbers in there. the important part of 
> the paper should be about how temperaments work and why 
> they're important, not about how to *rank* them!

OK. I can accept that.


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Message: 4130

Date: Mon, 11 Mar 2002 18:58:58

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 
wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 
> > wrote:
> > > then the maximum interval width (which is also 
> > > the width of the complete chord - Carl's measure) seems 
more 
> > relevant 
> > > to me than the rms interval width.
> > 
> > and shouldn't it be the maximum or rms *weighted* interval 
> > width?
> 
> You also asked why max abs? I'll answer that first.
> 
> One of the reasons given for using rms was that if there was 
an 
> outlier the LT still got credit for the intervals it did well. But if 
> we're looking at say {1,3,5,7} then if _any_ {1,3,5,7} interval is 
> very wide, I don't think it is a good {1,3,5,7} temperament. Let it 
be 
> found to be a good {2,5,7} temperament or whatever.

it might not -- again, what if it's really a good {2, 3, 7/5} 
temperament?

> 
> As far as the weighting by odd-limit thing goes. I'm reasonably 
in 
> favour. But it's a very musician-type "subjective" thing to do and 
not 
> at all a mathematician type thing. And I'm not prepared to argue 
over 
> it, mainly because I expect the good temperaments will still 
make it 
> on the list without it. Also, I have no easy way of finding 
optimum 
> generators (which you'd have to do all over again before you 
can find 
> the error).

huh? the optimum generator would be exactly the same as 
before, since we're not weighting the 'cents' part of it, just the 
'steps' part of it . . .


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Message: 4131

Date: Mon, 11 Mar 2002 03:58:10

Subject: Re: 32 best 5-limit linear temperaments redux

From: genewardsmith

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

> Would
you mind saying why?

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


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Message: 4132

Date: Mon, 11 Mar 2002 21:46:46

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> it might not -- again, what if it's really a good {2, 3, 7/5} 
> temperament?

Ok let's stick with rms error. We seem to have the most agreement on 
that.

> huh? the optimum generator would be exactly the same as 
> before, since we're not weighting the 'cents' part of it, just the 
> 'steps' part of it . . .

You're right. I got confused. So ask gene to do a run with weighted 
rms gens to see if anything new comes up.

Oh and Manuel, Yes that's from George Orwell. The generator for that 
temperament is essentially 19/84 oct. Gene's idea.


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Message: 4133

Date: Mon, 11 Mar 2002 04:10:37

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 
> wrote:
> > Would you mind saying why?
> 
> because the systems of similar complexity should be next to 
> each other, so a musician who's interested in a particular 
> complexity range can immediately compare and contrast the 
> systems in that range.
> 
> > I'm just as likely to be looking for the best temperament whose 
> > generator size is in a particular range
> 
> why? seems overly specific. who is likely to have that as their 
> priority?

e.g. Folks considering making a LT guitar and wanting to keep close to 
standard open string tuning would favour a generator that was a 
fourth.

Fourth/fifth generators are favoured in general.

Or someone might be specifically looking for generators that will be 
steps of the MOS.

That reminds me, we must list _all_ possible generators less than an 
octave, for each temperament. i.e gen, period-gen, period+gen, 
2*period-gen, etc.

> > or whose period is a 
> > particular fraction of an octave,
> 
> ditto.

e.g. Someone might prefer not to have a temperament with multiple 
chains, because dealing with that can be an added complexity in 
itself.
 
> temperaments are just mappings of ji. it's more important, 
> compositionally, *which* commas vanish than what the 
> generator is or what the period is. 

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

> of course a fifth generator is 
> preferable and that's why i cling to my unpopular idea of a 
> *weighted* complexity calculation.

I support that idea, but that won't help me with actually _finding_ 
the best fourth/fifth generators in the list.

> > or whose rms error is in a 
> > particular range,
> 
> you can always introduce more error by other means. really one 
> wants the tunings that *minimize* the error within the musician's 
> preferred sphere of 'complexity'.

I have done this very thing already, looking thru graham's lists for 
microtemperaments suitable for fretting a guitar for some JI scale.  I 
wanted errors in the range of 2 to 3 cents, no more no less. Keeping 
the error above 2 c served to limit the complexity, below 3c limited 
the mistuning pain.

>  
> > Why favour any one of these (and thereby 
> > make the others much more difficult to find) by sorting the list 
> on 
> > it?
> 
> hopefully this makes it clear. 

Well a little. But it doesn't matter. If the only way we can move 
forward is to agree not to list them in order of _any_ badness, 
then I'm happy for them to be listed in order of complexity.

> maybe this is why you didn't 
> understand my liking of log-flat badness -- it's because i've 
> assumed this is how you'd present the tunings.

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

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

1. No badness shown.

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

neutral thirds
augmented
meantone
pelogic
diminished
porcupine
small diesic
diaschismic
kleismic
16875/16384
twin meantone
half meantone-fifth
half meantone-fourth
wuerschmidt
minimal diesic
1990656/1953125
tiny diesic
schismic
orwell
twin kleismic
half kleismic-minor-third
half kleismic-major-sixth
amt
semisuper
parakleismic
hemithird
and maybe a few others

3. Smith badness < 861
rms complexity < 13.2 gens
rms error < 28.9 cents

You can of course round the numbers up so it doesn't look so contrived 
provided that doesn't add more than 2 or 3 extra temperaments.


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Message: 4134

Date: Mon, 11 Mar 2002 22:49:49

Subject: Re: Dave's 18 best 5-limit temperaments

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > hmm? i thought dave said 6561:6400 was half-meantone. 
> 
> No I called that twin meantone.
> 
> > someone needs to write a gentle introduction to contortion, just 
> > to list all the issues in one place.
> 
> I guess you haven't looked at my spreadsheet yet.
> http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Ok *
> 
> I named them like this
> 
> Comma     Name                 Mapping           Gen   Period
> -------------------------------------------------------------
> 6561/6400 twin meantone	       [0,-1,-4] [2,4,8] 503.8 600

isn't this just meantone itself? isn't this an instance of torsion?

> 81/80     half meantone-fifth  [0,2,8]   [1,2,4] 348.1 1200
> 81/80     half meantone-fourth [0,-2,-8] [1,2,4] 251.9 1200

these are the valid instances of contortion i was talking about. they 
are not temperaments, but they are quite interesting. since 
their 'badness' should simply be 4 or 8 times that of meantone, such 
cases can be handled with an introductory passage and never mentioned 
again -- it would be easy enough for the reader to supply them for 
any given badness cutoff.


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Message: 4135

Date: Mon, 11 Mar 2002 04:18:48

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7}, 
> {3,5,7}, and {2,3,5,7}. isn't that right?

That's ok by me. But if we're listing these separately (which seems 
the right thing to me), then the maximum interval width (which is also 
the width of the complete chord - Carl's measure) seems more relevant 
to me than the rms interval width.


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Message: 4136

Date: Mon, 11 Mar 2002 22:54:06

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >>I know I said "limit", but I meant it
> >> figuratively, not literally.  It's an abuse of terminology to
> >> call [1 5 11] the "[1 5 11]-limit", for example, but that's what
> >> I meant.
> >
> >is [2 3 7/5] a limit too?
> 
> In the sense that you can give a list for it or tell the reader
> to try Graham's script with input X, so that you don't have to
> use rms of the "highest interval width" (Graham's term for the
> length of the chain).

so do you want to propose a modification of the list of cases to be 
studied in our paper?

> >> So if your paper's really going to cover all these (by that I
> >> mean have a list for each one), I'd suggest ranking by
> >> complexity, with a sharp cutoff.  Just show the most accurate
> >> three temperaments for each integer of complexity up to 15 or
> >> so. 
> >
> >this is flat badness.
> 
> Which badness formulas produce flat?

this is a question gene can surely answer (if only to say, 'none').

> >i think log-flat is better, and it's what
> >gene's using.
> 
> Ok.  Is steps^n*cents log-flat for any n > 1?

no. n depends on the number of independent intervals in the map, 
i.e., the number of 'terms' in the 'limit'.


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Message: 4137

Date: Mon, 11 Mar 2002 05:44:17

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >>Height functions can be thought of as distance measures, 
but 
> >>I'm not getting your point.
> >
> >a _lattice_ distance function.
> 
> Does that mean a taxicab one?  Whatever it is, if it can do
> n*d, who cares?

i care because it gives us the natural lattice-based complexity 
measure for linear temperaments in the 7-limit and higher, 
where a single comma won't do, and a sort of 'angle' between 
the two is at work.


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Message: 4138

Date: Mon, 11 Mar 2002 23:04:31

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> Dave wrote:
> 
> > orwell
> 
> From George Orwell?

gene calls it that because the generator is about 19/84 octave.


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Message: 4139

Date: Mon, 11 Mar 2002 04:25:48

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >> I agree, but then it's just a good temperament at a different
> >> limit.
> >
> >not necessarily.
> 
> Example?

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

and what if the subset of 'primes' you're thinking about is not 
really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?


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Message: 4140

Date: Mon, 11 Mar 2002 05:46:20

Subject: Microtempered guitar design

From: dkeenanuqnetau

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> (simpsons reference -- Answer: Yes, Kent.)

And people let their children watch that crap.

> Did you ever manage to automate the process at all, Dave?

Only parts of it and not in a user friendly way.


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Message: 4141

Date: Mon, 11 Mar 2002 23:09:14

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > it might not -- again, what if it's really a good {2, 3, 7/5} 
> > temperament?
> 
> Ok let's stick with rms error. We seem to have the most agreement 
on 
> that.
> 
> > huh? the optimum generator would be exactly the same as 
> > before, since we're not weighting the 'cents' part of it, just 
the 
> > 'steps' part of it . . .
> 
> You're right. I got confused. So ask gene to do a run with weighted 
> rms gens to see if anything new comes up.

ok. gene, once again, this means that in the 'gens' calculation, the 
number of generators in the 3:1 should be multiplied by log(3), the 
number of generators in the 5:3 should be multiplied by log(5), the 
number of generators in the 5:1 should be multiplied by log(5). this 
will cause temperaments generated by the fifth to look better than 
they currently do, relative to those that aren't. this is important 
since augmented (especially) and diminished (a little less so) are 
far harder for the ear to understand than meantone, even when all are 
*tuned* in 12-equal, and the badness values would no longer put 
meantone as the 'best'.


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Message: 4142

Date: Mon, 11 Mar 2002 04:29:57

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

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

> That's not relevant. My point is that it _does_ hurt someone. It 
hurts 
> the person interested in middle-of-the-road temperaments.
> 
> > -- and there were 
> > other reasons for excluding "fourth-thirds", as i recall.
> 
> OK. Just substitute "pelogic" for "fourth-thirds" in the above.

then i'd definitely disagree with you.


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Message: 4143

Date: Mon, 11 Mar 2002 05:53:21

Subject: Re: 32 best 5-limit linear temperaments redux

From: genewardsmith

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

> Hmmm.  You once told me it had to be primes.  Does this have anything
> to do with that?

What had to be primes? The generators? Nope.


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Message: 4144

Date: Mon, 11 Mar 2002 15:20:35

Subject: Re: 32 best 5-limit linear temperaments redux

From: Carl Lumma

>> In the sense that you can give a list for it or tell the reader
>> to try Graham's script with input X, so that you don't have to
>> use rms of the "highest interval width" (Graham's term for the
>> length of the chain).
>
>so do you want to propose a modification of the list of cases to be 
>studied in our paper?

I'm just saying I think rms(complexity) smooths over information
I want.  You objected that mean(complexity) might be unfair to
temperaments that only did one thing badly.  I suggested such
temperaments be listed separately, as doing everything well over
a restricted set of identities.

If I were to say some temperaments are good over some list of
identities, you'd expect I meant all the identities, wouldn't
you?  Isn't the idea behind a "limit" that you need the lower
identities to make the higher ones work (not that I think this
is correct)?

>> >i think log-flat is better, and it's what
>> >gene's using.
>> 
>> Ok.  Is steps^n*cents log-flat for any n > 1?
>
>no. n depends on the number of independent intervals in the map, 
>i.e., the number of 'terms' in the 'limit'.

Oh!!  Howso?

-Carl


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Message: 4145

Date: Mon, 11 Mar 2002 04:30:06

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > Would you mind saying why?
> 
> Supposing you saw a list which rated calculators, portable PCs, 
desktops, workstations, servers, mainframes and supercomputers with a 
numerical score, which combined price with performance. Would it make 
much sense to find cheap $15 calculaors next to supercomputers?
>

If the analogy is say

dollars <-> complexity
megaflop.gigabytes <-> 1/error

then badness would be in dollars per megaflop.gigabyte. 

Well if that's what your badness is, 1/value-for-money, then yes there 
might well be a calculator next to a supercomputer.

Anyway. I'm willing to accept a list ordered by complexity provided 
smith-badness is not shown anywhere.


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Message: 4146

Date: Mon, 11 Mar 2002 05:55:49

Subject: Re: 32 best 5-limit linear temperaments redux

From: genewardsmith

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

> i care because it gives us the natural lattice-based complexity 
> measure for linear temperaments in the 7-limit and higher, 
> where a single comma won't do, and a sort of 'angle' between 
> the two is at work.

Any metric on R^n allows you to define lattices, so I don't see what your angle is.


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Message: 4147

Date: Mon, 11 Mar 2002 23:26:28

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >> In the sense that you can give a list for it or tell the reader
> >> to try Graham's script with input X, so that you don't have to
> >> use rms of the "highest interval width" (Graham's term for the
> >> length of the chain).
> >
> >so do you want to propose a modification of the list of cases to 
be 
> >studied in our paper?
> 
> I'm just saying I think rms(complexity) smooths over information
> I want.  You objected that mean(complexity) might be unfair to
> temperaments that only did one thing badly.  I suggested such
> temperaments be listed separately, as doing everything well over
> a restricted set of identities.

by identities, you really mean all the consonant intervals, right?

well, i think if you look on a case-by-case basis, you won't be able 
to object to what rms does. because only doing 'one thing badly' 
isn't really possible in this context. find an example, if you can!

> >> >i think log-flat is better, and it's what
> >> >gene's using.
> >> 
> >> Ok.  Is steps^n*cents log-flat for any n > 1?
> >
> >no. n depends on the number of independent intervals in the map, 
> >i.e., the number of 'terms' in the 'limit'.
> 
> Oh!!  Howso?

gene derived this from diophantine approximation theory.


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Message: 4148

Date: Mon, 11 Mar 2002 04:32:04

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Well if that's what your badness is, 1/value-for-money, then yes 
there 
> might well be a calculator next to a supercomputer.

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

> Anyway. I'm willing to accept a list ordered by complexity provided 
> smith-badness is not shown anywhere.


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Message: 4149

Date: Mon, 11 Mar 2002 16:16:48

Subject: Re: 32 best 5-limit linear temperaments redux

From: Carl Lumma

>by identities, you really mean all the consonant intervals, right?

Right.

>well, i think if you look on a case-by-case basis, you won't be able 
>to object to what rms does. because only doing 'one thing badly' 
>isn't really possible in this context. find an example, if you can!

Well, you just gave Orwell, so let's look at it.  I still have no
idea about the map.  My best guess was:

[2] [0 1]
[3] [7 0]
[5] [-3 2]
[7] [8 1]
[11] [2 3]

Which disagrees with Gene:

>[ 0 1]
>[ 7 0]
>[-3 3]

Going by my map:

          max int width  card(map)  me    g
5-limit   10             2          5     7.257
11-limit  11             4          2.75  ??

I have no idea how Gene got 7.257, so  I can't fill
g in for the 11-limit.

Still don't know why we can't include 9 in the map.


>>>> Ok.  Is steps^n*cents log-flat for any n > 1?
>>>
>>>no. n depends on the number of independent intervals in the map, 
>>>i.e., the number of 'terms' in the 'limit'.
>> 
>>Oh!!  Howso?
>
>gene derived this from diophantine approximation theory.

I remember now.

-Carl


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