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Message: 7750

Date: Sun, 26 Oct 2003 14:26:57

Subject: Re: comma search (was Re: Polyphonic notation)

From: Carl Lumma

>> > where badness is defined as log_2(ratio)^2 * prime-limit(ratio)...
>> 
>> Any comments on this badness measure? 
>
>You haven't defined it yet.

What's lacking in the above definition?

-Carl


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Message: 7751

Date: Sun, 26 Oct 2003 15:02:16

Subject: Re: [tuning] Re: Polyphonic notation

From: Carl Lumma

>> Since every prime limit contains an infinite number of ratios, and
>> neither size nor complexity behave smoothly as one searches farther
>> out, it seems we'll never know the top 10 lowest-badness ratios at
>> any prime limit....
> 
>For any limit, zero will be an accumulation point of log2(q)^2, since 
>p-limit commas are arbitrarily small; but this hardly matters, since 
>whatever it is you are calculating, it clearly isn't log2(q)^2 
>primelimit(q). Can we start over?

Whoops, I wasn't actually taking the log2 of q.  The formula used
was...

q^2 * primelimit(q)

Probably I should use (log2(q) + 1)^2 * primelimit(q).  In this case
the 10 lowest-scoring ratios <= 600 cents with denominator <= 500
are...

(badness, primelimit, ratio)
((3.468084457207407 3 256/243)
 (4.106173526999384 3 9/8)
 (4.6509153968061785 3 32/27)
 (5.180825053903934 5 81/80)
 (5.348010729259556 5 128/125)
 (5.385594090689787 3 81/64)
 (5.418111244777691 5 250/243)
 (5.6062792235873795 5 25/24)
 (5.797659150259687 5 135/128)
 (5.974440849845497 5 16/15))

-Carl


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Message: 7752

Date: Mon, 27 Oct 2003 17:03:57

Subject: comma search (was Re: Polyphonic notation)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Paul, any thoughts on a badness heuristic
> 
>  log(d) * |n-d|/log(d) = |n-d|
> 
> ?
> 
> Thanks,
> 
> -Carl

it's a good one, but how is it derived? it almost looks like the term 
between the '*' and the '=' is the error heuristic, but it's missing 
a factor of d in the denominator.


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Message: 7753

Date: Mon, 27 Oct 2003 17:49:45

Subject: Re: heuristic and straightness

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "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?


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Message: 7754

Date: Mon, 27 Oct 2003 10:44:47

Subject: Re: comma search (was Re: Polyphonic notation)

From: Carl Lumma

>> Paul, any thoughts on a badness heuristic
>> 
>>  log(d) * |n-d|/log(d) = |n-d|
>> 
>> ?
>> 
>> Thanks,
>> 
>> -Carl
>
>it's a good one, but how is it derived? it almost looks like the term 
>between the '*' and the '=' is the error heuristic, but it's missing 
>a factor of d in the denominator.

Drat!  Ok, howabout this...

log(d) * |n-d|/d*log(d) = |n-d|/d

-Carl


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Message: 7755

Date: Mon, 27 Oct 2003 19:02:51

Subject: comma search (was Re: Polyphonic notation)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> Paul, any thoughts on a badness heuristic
> >> 
> >>  log(d) * |n-d|/log(d) = |n-d|
> >> 
> >> ?
> >> 
> >> Thanks,
> >> 
> >> -Carl
> >
> >it's a good one, but how is it derived? it almost looks like the 
term 
> >between the '*' and the '=' is the error heuristic, but it's 
missing 
> >a factor of d in the denominator.
> 
> Drat!  Ok, howabout this...
> 
> log(d) * |n-d|/d*log(d) = |n-d|/d
> 
> -Carl

another decent badness measure -- complexity times error.


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Message: 7756

Date: Mon, 27 Oct 2003 11:04:37

Subject: Re: comma search (was Re: Polyphonic notation)

From: Carl Lumma

>Drat!  Ok, howabout this...
>
>log(d) * |n-d|/d*log(d) = |n-d|/d

Difficult to see how we could get some of our favorites like
135/128 to come out of this, without some kind of restriction
on prime limit.

-Carl


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Message: 7757

Date: Mon, 27 Oct 2003 19:48:11

Subject: Re: heuristic and straightness

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "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?


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Message: 7758

Date: Mon, 27 Oct 2003 01:06:26

Subject: Re: [tuning] Re: Polyphonic notation

From: Carl Lumma

I wrote...

>...each triple is (badness, prime-limit, ratio).  The search took
>less than 10 minutes on a P3 600 laptop (code available).  Performance
>would be drastically better by using anything other than the slowest
>conceivable factoring algorithm, which I chose for expediency.

I measured the slowdown due to factoring by comparing the prime-limit complexity mode with the n*d complexity mode.  It's there, but the
main cause of slowness was sorting all the results just to get the
top r of them, using insertsort which is generally O(n^2).  So I cooked
up a procedure that just gets the top r results and leaves the rest
unsorted in O(n*r).  The above search only takes a few seconds now.

Anyway, the point of posting this here is to find out how you guys
(Gene, Paul, Graham) cook up commas.  Gene, is there a particular
maple function I should look at?  I see lists of commas doped into
the code in various places...

-Carl


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Message: 7759

Date: Mon, 27 Oct 2003 19:51:12

Subject: comma search (was Re: Polyphonic notation)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Drat!  Ok, howabout this...
> >
> >log(d) * |n-d|/d*log(d) = |n-d|/d
> 
> Difficult to see how we could get some of our favorites like
> 135/128 to come out of this, without some kind of restriction
> on prime limit.
> 
> -Carl

you just need to penalize complexity more.


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Message: 7760

Date: Mon, 27 Oct 2003 09:31:53

Subject: [tuning] Re: Polyphonic notation

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Anyway, the point of posting this here is to find out how you guys
> (Gene, Paul, Graham) cook up commas.  Gene, is there a particular
> maple function I should look at?  I see lists of commas doped into
> the code in various places...

Extremely small commas are easy to find using integer relation 
algorithms. The trick comes if you want a complete list of them 
satisfying certain conditions. One way to do that is via what I call 
notations, where you pass to notations using progressively smaller 
commas (or equivalently, larger equal temperaments) taking care while 
doing so to ensure you don't miss anything fulfilling your conditions.


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Message: 7761

Date: Mon, 27 Oct 2003 20:15:02

Subject: 1076168025:1073741824

From: Paul Erlich

a web search on 1076168025 took me to this rameau article:!!!!!!!!!

RAMNOU TEXT * [with cont.]  (Wayb.)

1073741824 is just 2^30, so maybe rameau actually considered this 
interval?


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Message: 7762

Date: Mon, 27 Oct 2003 09:55:14

Subject: Re: Polyphonic notation

From: Graham Breed

Carl Lumma wrote:

> Anyway, the point of posting this here is to find out how you guys
> (Gene, Paul, Graham) cook up commas.  Gene, is there a particular
> maple function I should look at?  I see lists of commas doped into
> the code in various places...

I don't cook up commas, because it looks like a difficult problem that 
I'll leave for those who care about it.  If you have commas, I can find 
temperaments from them, but it may take a very long time.  This is 
because the number of commas per temperament increases the more prime 
numbers you consider.

I cook up linear temperaments by combining linear temperaments.  This is 
roughly O(n**2) in the number of equal temperaments, and although it can 
be slow, is never intolerably so, given the investment required to 
actually make music in a linear temperament.

For 5-limit linear temperaments it doesn't make any difference, as there 
is only one comma.  But then the 5-limit case is easy however you go 
about it.


                  Graham


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Message: 7763

Date: Mon, 27 Oct 2003 21:12:36

Subject: Re: heuristic and straightness

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:


>                 1076168025                1073741824 UNNAMED!!!!!!!!

Unnamed since it is a schisma squared.


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Message: 7764

Date: Mon, 27 Oct 2003 13:50:20

Subject: Re: heuristic and straightness

From: Carl Lumma

>> > 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


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Message: 7765

Date: Mon, 27 Oct 2003 22:00:02

Subject: Re: heuristic and straightness

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> 
> >                 1076168025                1073741824 
UNNAMED!!!!!!!!
> 
> Unnamed since it is a schisma squared.

OOPS!!!!!!!! (i can feel that torsion in my gut)


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Message: 7766

Date: Mon, 27 Oct 2003 22:00:25

Subject: Re: heuristic and straightness

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx 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.


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Message: 7767

Date: Mon, 27 Oct 2003 14:08:36

Subject: Re: heuristic and straightness

From: Carl Lumma

>> >> > 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


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Message: 7768

Date: Mon, 27 Oct 2003 22:20:58

Subject: Re: heuristic and straightness

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx 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?


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Message: 7769

Date: Mon, 27 Oct 2003 14:37:25

Subject: Re: heuristic and straightness

From: Carl Lumma

>> >> 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


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Message: 7770

Date: Mon, 27 Oct 2003 22:42:55

Subject: Re: heuristic and straightness

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx 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.


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Message: 7771

Date: Mon, 27 Oct 2003 22:54:30

Subject: comma search

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx 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:

Searching Small Intervals * [with cont.]  (Wayb.)


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Message: 7772

Date: Mon, 27 Oct 2003 17:38:49

Subject: Re: heuristic and straightness

From: Carl Lumma

>> 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


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Message: 7773

Date: Mon, 27 Oct 2003 18:23:16

Subject: Re: Linear temperament names?

From: Carl Lumma

>Graham, have you ever thought of spelling it "majic" since it's
>generated by MAJor thirds?

I thought it was an acronym.

-Carl


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Message: 7774

Date: Tue, 28 Oct 2003 16:23:40

Subject: Re: Linear temperament names?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> > Have names been proposed for any of the linear microtemperaments
below?
> > 
> > The generators and errors are for minimax. The mappings given are
> > octave equivalent:
> > [gens_per_3 gens_per_5 gens_per_7 gens_per_11 gens_per_13;
> > periods_per_3 periods_per_5 etc...]
> 
> Graham convinced me to switch to his convention. Do you have a reason
> for preferring [generator, period] over [period, generator]? I think
> we should try for some degree of standardization.

The article deals only with linear temperaments of octave-repeating
octave-equivalent scales, so the reader is only interested in how many
periods there are modulo the number in the octave. So when the period
_is_ the octave these are all zero and I prefer to omit them. It's
easier to omit them without confusion if they come _last_. I do not
want to use any vector or matrix math in the article. It's pitched at
an audience with more basic math skills.

But I agree that for the greatest generality the period should come
first, followed by the generator(s).

> Noreover, you are
> ignoring 2, and to me this is simply not acceptable. 

Oh blow it out your ear. :-)

The article deals only with octave-repeating octave-equivalent scales
so why should I bother saying that there are zero generators in the
1:2 every time. And the size of the period, given as a fraction of an
octave, is a bit of a giveaway as to how many of _them_ are in the
1:2. Also I have limited space to fit many things about each LT in
columns across the width of a page.

But I certainly agree that for greatest generality 2 should be
included in all the matrices and vectors.

The main thing is that I explain the format I'm using.

> > Limit     Period    Gen  Max gens  Max err  Prime mapping       
Rep ET
> > 7-limit   1 oct     193.87 c  16   1.4 c    [16 2 5]  
> 
> Hemiwuerschmidt. You should give all of the mapping and give it in a
> canonical reduced form, or a give a reduced comma basis, or a
> wedgie--or best of all, all three.

Commas and wedgies are utterly irrelevant to my article. My canonical
generator is the smallest one (less than half the period), what's yours?

> > 11-limit  1/2 oct   216.74 c  30   3.1 c    [-6 -1 10 -3; 1 1 0 0]

No name for this one? Is there any other LT more deserving of being
called "twin thirds"?

> > 11-limit  1/2 oct   183.21 c  30   2.4 c    [-6 -11 2 3; 1 0 1 0]    
> 
> Unidec.

Please explain. Why not call it "twin minortones" since the generator
represents 9:10 in the temperament.

> > 15-limit  1/3 oct    83.02 c  48   2.8 c [-6 -5 2 -3 -14; 0 2 2 2 2] 
> 
> Trikleismic.

That makes sense, but I would have said "triple kleismic". If you use
the prefix tri- to mean 3 equispaced chains of a generator then what
would you use to mean a single chain of 1/3 of that generator? i.e. in
the way that you use hemi- to mean a single chain of 1/2 a generator?

> > 15-lm-wo-13  1 oct  193.24 c  35   2.8 c    [-15 2 5 -22]           
> 
> For the 7-limit temperament, I have it listed as Hemithird.

Makes sense too. But I don't understand why you use hemi- when it is
already established that semi- is used to halve a musical interval, as
in semitone and semisharp. I've asked you that before, but I don't
remember a satisfactory answer.

> > If you do propose a name, please also say why you think it is
> > appropriate. 
> 
> The names I give are ones which have already been used; they are not
> new proposals.

So what? They are still only proposals as far as my article is
concerned. I don't have to use the name you give me, and so I'd still
like explanations for the less than obvious ones, like "unidec".

> If I don't think the name makes much sense I may just
> > include the temperament without a name.
> 
> My preference is for you not to sow confusion by introducing new 
> names for already named and cataloged temperaments;

That's why I'm asking.

How about this one?

Period    Gen  Max gens  Max err  Prime mapping (no 2s)        
  1 oct   351.45 c   10  1.9 c    [2 25 13]

"cata neutral thirds"?


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