Tuning-Math Digests messages 10775 - 10799

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Message: 10775

Date: Wed, 07 Apr 2004 07:58:26

Subject: Re: Comma names

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
> wrote:
> > Conspicuous by its absence is:
> > 
> > 4096/4095 tridecimal schisma or schismina
> 
> That's because it is 13-limit, of course, but Manuel does have it 
> listed as "tridecimal schisma". There are no names for a lot of other 
> 13-limit supers, such as 2080/2079, 4225/4224, 6656/6655, and 
> 123201/123200, but he has 10648/10647 down as the "harmonisma".
> 
> > Perhaps you will want to suggest another name for 4095:4096 that 
> > would acknowledge it as the linchpin of the Sagittal symbol-flag 
> > economy.
> 
> I think you should do that, if you wish. Of course "sagittal schisma" 
> suggests itself.

Please don't rename 225/224. There's nothing wrong with "septimal
kleisma" and it's been in use for a long time.

If you want to name a comma after George then I suggest that 4096/4095
could be Secor's schismina, although I am generally no longer in
favour of naming things after people, since that gives you no clue as
to what the things are or what they are good for, and makes it likely
that you'll only have to rename them again later when an earlier
mention comes to light.

Haven't you got something better to work on George. :-)

I'm not here either.


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Message: 10776

Date: Wed, 7 Apr 2004 10:12:07

Subject: Re: Comma names

From: Manuel Op de Coul

I'll add the name Sagittal schismina to 4096/4095.

>Which is why I was objecting--isn't diatonic semitone the
>historically established name for 16/15?

Yes but 15/14 also has been named major diatonic semitone.

>Because "undecimal 1/4-tone" is kind of ugly, and Schoenberg seems to
>have considered this comma according to Monzo's analysis.

I'll make it al-Farabi's 1/4-tone.

Manuel


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Message: 10779

Date: Thu, 08 Apr 2004 09:36:33

Subject: Re: Comma names

From: Dave Keenan

I'm still not here. But it seems a good time to post this.

The page cannot be found *  (181 KB)

It's an Excel spreadsheet that automatically generates a unique
systematic name for any 31-limit comma (almost). It includes over 200
commas. It has the commas from Scala's intnam.par with their common
names (although some may be out of date) and it has all the commas
smaller than an apotome which can be represented exactly in Sagittal
notation, with their symbols (in the ASCII longhand representation).

The "almost" above, is because I have not properly implemented the
"complexity level" calculation, but instead used a quick and dirty
heuristic that works for all the commas listed (and probably any that
anyone is likely to want to add in the near future).

I did most of this spreadsheet months ago, but it was waiting for me
to code the "complexity level" algorithm. So I added the heuristic
today, just so I could release it.

Regards,
-- Dave Keenan



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Message: 10781

Date: Thu, 08 Apr 2004 22:54:30

Subject: Re: Fokker pentatonics, known and unknown

From: Paul Erlich

Since you aren't being specific as to mode, are you checking is Scala 
lists *modes* of these scales?

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> Below I list what the Scala scl database knows about the Fokker 
> pentatonics I computed. Two scales appeared as all three kinds of 
> blocks; tranh-prime_5 and its inverse. It surprises me that such an 
> evidently important pentatonic as tranh-inverse was not listed.
> 
> 
> 16/15 27/25
> [1, 6/5, 4/3, 3/2, 5/3] harrison_min.scl
> From Lou Harrison, a symmetrical pentatonic with minor thirds   
> 
> [1, 6/5, 4/3, 3/2, 9/5] tranh.scl inverse equal key 0
> [1, 6/5, 5/4, 3/2, 5/3] unknown
> [1, 6/5, 4/3, 8/5, 5/3] unknown
> 
> [1, 6/5, 4/3, 8/5, 9/5] tranh.scl key 4, prime_5.scl key 2
> Bac Dan Tranh scale, Vietnam
> What Lou Harrison calls "the Prime Pentatonic", a widely used 
scale   
> 
> 16/15 81/80
> 
> [1, 9/8, 4/3, 3/2, 16/9] hexany16.scl, chin_5.scl key 3
> 1.3.9.27 Hexany, a degenerate pentatonic form
> Chinese pentatonic from Zhou period 
> 
> [1, 9/8, 4/3, 3/2, 9/5] korea_5.scl
> According to Lou Harrison, called "the Delightful" in Korea 
> 
> [1, 6/5, 4/3, 3/2, 9/5] same as first 16/15 27/25 scale, inverse 
tranh
> [1, 6/5, 4/3, 8/5, 9/5] same as fifth 16/15 27/25 scale, tranh and 
> prime_5
> [1, 9/8, 4/3, 3/2, 5/3] inverse korea_5, key 3
> 
> 27/25 81/80
> 
> [1, 10/9, 4/3, 3/2, 9/5] unknown
> [1, 10/9, 27/20, 3/2, 9/5] unknown, inverse of fifth scale below
> [1, 6/5, 4/3, 3/2, 9/5] same as first 16/15 27/25 and second 16/15 
> 81/80, inverse tranh
> [1, 6/5, 4/3, 8/5, 9/5] same as fifth 16/15 27/25 and fourth 16/15 
> 81/80, tranh&prime_5
> [1, 10/9, 27/20, 3/2, 5/3] unknown, inverse of second scale above



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Message: 10783

Date: Sat, 10 Apr 2004 22:08:14

Subject: Re: 126 7-limit linears

From: Paul Erlich

Hi Gene,

I hope you're making progress on un-culling the list.

Would it be rude of me to request a similar list for 11-
limit 'linears'? Dave told me I should include these in my paper, and 
I agree.

Thanks,
Paul


--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> Hi Gene,
> 
> Would you be so kind as to produce a file like the one below, but 
> instead of culling to 126 lines, leave all 32201 in there? That 
would 
> be great. If that's too much, you could cut off the error and 
> complexity wherever you see fit. The idea, though, is to produce a 
> graph, and as most pieces of paper are rectangular, the data should 
> fill a rectangular region. I'm *not* arguing for a rectangular 
> badness function.
> 
> Also could you provide the TM-reduced kernel bases -- at least for 
> the 126 below?
> 
> Thanks so much,
> Paul
> 
> 
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> > I first made a candidate list by the kitchen sink method:
> > 
> > (1) All pairs n,m<=200 of standard vals
> > 
> > (2) All pairs n,m<=200 of TOP vals
> > 
> > (3) All pairs 100<=n,m<400 of standard vals
> > 
> > (4) All pairs 100<=n,m<=400 of TOP vals
> > 
> > (5) Generators of standard vals up to 100
> > 
> > (6) Generators of certain nonstandard vals up to 100
> > 
> > (7) Pairs of commas from Paul's list of relative error < 0.06,
> > epimericity < 0.5
> > 
> > (8) Pairs of vals with consistent badness figure < 1.5 up to 5000
> > 
> > This lead to a list of 32201 candidate wedgies, most of which of
> > course were incredible garbage. 



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Message: 10785

Date: Sun, 11 Apr 2004 21:26:38

Subject: Re: 126 7-limit linears

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > Hi Gene,
> > 
> > I hope you're making progress on un-culling the list.
> 
> I think I'll finish by today.

Awesome!

> > Would it be rude of me to request a similar list for 11-
> > limit 'linears'? Dave told me I should include these in my paper, 
> and 
> > I agree.
> 
> I think getting a big list of 11-limits would be nice. I hope I 
don't 
> need to get 32000.

As long as we're sure we're getting a complete list up to some fairly 
modest error and complexity bounds, I'm happy. I hope to 
perform/verify all these calculations myself eventually, but don't 
yet have code to calculate TOP tuning in the general case, and time 
is running out.

Much appreciated,
Paul



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Message: 10786

Date: Mon, 12 Apr 2004 18:12:48

Subject: Optimal octave stretching

From: Graham Breed

I've been looking at the RMS weighted error of equal temperaments.  A 
weighted RMS for an unbound set of intervals is more appropriate than 
the worst weighted error used by TOP.  The worst error is used to ensure 
that none of the intervals you plan to use fall outside an acceptable 
range of mistuning.  The more complex and less frequently used an 
interval is, the more important that it lie within this range if it is 
to be heard as a consonance.  Dissonance curves of whatever kind also 
show larger basins for more simple consonances, so that the range of 
mistuning is likely to be smaller for more complex intervals.  These 
factors together conspire to make the most complex intervals the ones 
that dominate the result, so that if there's no cutoff the result will 
not converge.  Hence you decide in advance where the cutoff should be.

The RMS error, on the other hand, gives the average pain associated with 
a mistuning.  In this case, it is appropriate to give simple or common 
intervals (generally the two will be the same) a higher weighting 
because their mistuning will lead to greater pain.  It doesn't matter if 
an interval is classed as a consonance or not, because its presence can 
still take pain away from a chord.  Ideally, we'd only consider 
intervals that pass a worst-error threshold, but for simplicity's sake 
an unbounded set can be considered.

It would be nice if a particular error converged for some set of 
intervals as its size approached infinity -- say an integer limit 
containing only the primes we're interested in.  Unfortunately, I can't 
find one so I'll use the simplest error for a set of primes -- the 
weighted RMS of the prime intervals.  Adding or averaging the errors of 
the primes is usually the first thing people think of.  We tell them 
they're wrong because they don't take account of things like 15:8 being 
more complex than 6:5.  This objection doesn't apply when you consider 2 
on a par with other prime numbers, because then 15:8 becomes naturally 
more complex than 6:5.  A temperament, such as 19-equal in the 5-limit, 
where the errors of 5:4 and 3:2 cancel out in 6:5 will be a good fit for 
octave stretching.  It will then have a naturally reduced prime error.

The obvious weighting to use is the size of the prime interval in 
octaves.  That should give an indication of the average Tenny-weighted 
error for an arbitrary set of intervals.  This weighting essentially 
ensures that prime and composite numbers are treated equally.  If you 
like, you can set a weighting such that high primes have a much smaller 
weight than low ones, so that you don't need to specify the prime limit.

This is all a bit arbitrary, but so is any algorithm in the absence of 
sound, empirical data on the strength and tolerance of mistuning for 
each interval.  This is a particular problem for octave-specific 
measures, because interval size and perceptual octave stretching come 
into play.  So we may as well stick with the simplest method if we're 
going to bother at all.

The weighted, square error for an equally tempered interval is given by

[(km - p)w]**2

where

   k is the size of a scale step
   m is the number of scale steps to this tempered interval
   p is the untempered pitch difference of this interval
   w is the weight given to this interval
   **2 is "squared"

For weighting by interval size, w=1/p, so

[(km - p)/p]**2 = (km/p - 1)**2

The mean squared error is then

Avg[(km/p - 1)**2]

over m an p for all primes.  Setting x=m/p to be the ideal number of 
steps to a just octave for each prime, that becomes

Avg[(kx - 1)**2] = Avg[(kx)**2 - 2kx + 1]

The optimum value for k is found by setting the derivative with respect 
to k equal to zero, so

Avg[2k(x**2) - 2x] = 0
k = Avg(x)/Avg(x**2)

Then, rearranging the formula for the mean squared error, and plugging 
in this optimum step size

Avg[(kx)**2 - 2kx + 1]
= k**2 Avg(x**2) - 2k Avg(x) + 1
= [Avg(x)/Avg(x**2)]**2 Avg(x**2) - 2[Avg(x)/Avg(x**2)] Avg(x) + 1
= Avg(x)**2 / Avg(x**2) - 2[Avg(x)**2]/Avg(x**2) + 1
= 1 - Avg(x)**2/Avg(x**2)

So the RMS error is

Sqrt[1 - Avg(x)**2/Avg(x**2)]

This is quite similar to the sample standard deviation of {x} (that is, 
the standard deviation you shouldn't use in error estimation):

STD(x) = Sqrt[Avg(x**2) - Avg(x)**2]

So you could write the RMS error as STD(x)/Sqrt(Avg(x**2)) if you happen 
to have a convenient way of calculating the standard deviation.  As each 
x will be close to n, the number of steps to a tempered octave, you can 
simplify the RMS as STD(x)/n.

Anyway, I've adapted my python module at

############################################################################### *

to do these calculations.  Here are some examples:

 >>> temper.PrimeET(12, temper.primes[:2]).getPORMSWE()
0.0025886343681387008
 >>> (temper.PrimeET(12, temper.primes[:2]).getPORMSWEStretch()-1)*1200
-1.5596534250319039

That means 5-limit 12-equal has a prime, optimum, RMS, weighted error of 
around 0.003.  This is a dimensionless value hopefully comparable to the 
TOP error.  The optimum octave is flattened by around 1.6 cents.

 >>> temper.PrimeET(19, temper.primes[:2]).getPORMSWE()
0.0015921986407487665
 >>> (temper.PrimeET(19, temper.primes[:2]).getPORMSWEStretch()-1)*1200
2.5780456079649738
 >>> temper.PrimeET(22, temper.primes[:2]).getPORMSWE()
0.0022460185834616815
 >>> (temper.PrimeET(22, temper.primes[:2]).getPORMSWEStretch()-1)*1200
-0.86081876412746894
 >>> temper.PrimeET(29, temper.primes[:2]).getPORMSWE()
0.0025604733781234741
 >>> (temper.PrimeET(29, temper.primes[:2]).getPORMSWEStretch()-1)*1200
1.6758871121345997
 >>> temper.PrimeET(31, temper.primes[:2]).getPORMSWE()
0.0013562866803350085
 >>> (temper.PrimeET(31, temper.primes[:2]).getPORMSWEStretch()-1)*1200
0.9757470533824808
 >>> temper.PrimeET(50, temper.primes[:2]).getPORMSWE()
0.0013261119467051412
 >>> (temper.PrimeET(50, temper.primes[:2]).getPORMSWEStretch()-1)*1200
1.5845318713727963

50-equal is probably close to the RMS meantone optimum, so the stretch 
of 1.6 cents is refreshingly close to the 1.7 cents Gene gave for TOP 
meantone on metatuning.  In fact, it's a fix because stretched 31-equal 
is closer to the TOP meantone, and 81 is closer to the meantone PORMSWE:

 >>> temper.PrimeET(81, temper.primes[:2]).getPORMSWE()
0.0013189616858225524
 >>> (temper.PrimeET(81, temper.primes[:2]).getPORMSWEStretch()-1)*1200
1.3515272079124507

But, anyway, the stretching is of the same order of magnitude.

I would do more comparisons, but I haven't implemented the TOP 
optimization yet.  For that matter, I can only do it for 3-limit equal 
temeperaments, 5-limit linear temperaments, 7-limit planar temperaments, 
etc.  I'm sure I could work out how to do PORMSWE for linear 
temperaments, but I haven't done so yet.  So for now, it's a case of 
finding a representative equal temperament.

Guessing the ennealimmal optimum is difficult, because it seems to lie 
close to the point where the octave goes from being sharp to flat.  But 
it looks close to 612-equal, with the stretch stable either side.

 >>> (temper.PrimeET(612, temper.primes[:3]).getPORMSWEStretch()-1)*1200
0.020981278370690859

That's the same order of magnitude as the TOP optimum Gene gave of 0.036 
cents.


                  Graham


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Message: 10788

Date: Mon, 12 Apr 2004 19:53:53

Subject: Re: Optimal octave stretching

From: Graham Breed

Gene Ward Smith wrote:

> Zeta tuning works along the lines you want here. A Python script 
> which found the Zeta tuning might be a bit of a pain to write, 
> though, and it only works for rank one (equal or "dimension zero") 
> temperaments.

That's the thing you keep mentioning related to the zeta function, is it?

> You could try 441 also.

Oh, I did, and many more.  It gives an octave flat by 0.012 cents.  I've 
shown that the optimum lies between 5679- 7173-equal.


                   Graham


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Message: 10791

Date: Tue, 13 Apr 2004 19:08:39

Subject: Re: 32201 seven limit linear temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> I put a link on the xenharmony home page to these. This project 
took 
> longer than anticipated because of computer problems; I don't know 
if 
> my computer and/or Linux install is up to the job of 11 limit any 
> more.

I'm salivating, but Xenharmony * gives me "This page 
cannot be displayed" . . . :(


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Message: 10793

Date: Tue, 13 Apr 2004 20:32:54

Subject: Re: 32201 seven limit linear temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> > I put a link on the xenharmony home page to these. This project 
> took 
> > longer than anticipated because of computer problems; I don't 
know 
> if 
> > my computer and/or Linux install is up to the job of 11 limit any 
> > more.
> 
> I'm salivating, but Xenharmony * gives me "This page 
> cannot be displayed" . . . :(

Now it works . . . I downloaded the file, but then it says "Cannot 
open file . . . does not appear to be a valid archive".

Does it work for anyone else?


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Message: 10795

Date: Tue, 13 Apr 2004 14:07:27

Subject: 32201 temps

From: Carl Lumma

Gene, your zip file is corrupted.

-C.


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Message: 10796

Date: Tue, 13 Apr 2004 11:40:55

Subject: Re: Optimal octave stretching

From: Graham Breed

I wrote:

> Oh, I did, and many more.  It gives an octave flat by 0.012 cents.  I've 
> shown that the optimum lies between 5679- 7173-equal.

I've got it working with linear temperaments now.  Here's the 
ennealimmal and 5-limit meantone results:

 >>> enne = temper.Temperament(171,612,temper.limit9)
 >>> enne.optimizePORMSWE()
 >>> enne.getPRMSWError()
2.4769849465587193e-05
 >>> (enne.mapping[0][0]*enne.basis[0] - 1)*1200
0.021691213712138335
 >>> enne.basis[1]*1200
49.021363311937186
 >>> meantone = temper.Temperament(19,31,temper.limit5)
 >>> meantone.optimizePORMSWE()
 >>> meantone.getPRMSWError()
0.001318517728382543
 >>> (meantone.basis[0] - 1)*1200
1.3968513622916845
 >>> meantone.basis[1]*1200
504.34774072728203

                     Graham


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