Tuning-Math Archive Section 11: 10775

Message: 10775 - Contents - Hide Contents 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.

Message: 10776 - Contents - Hide Contents 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

Message: 10777 - Contents - Hide Contents Date: Wed, 07 Apr 2004 16:29:45 Subject: Re: Comma names From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- 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,

Please don't. There is already the Miracle "sec(ond, min)or". Having two intervals named after anyone would be too much (as well as potentially confusing), unless one interval is clearly related to the other.

> 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,

"Sagittal schismina" will do nicely, then, because disregarding it results in an economical use of symbol-elements in the notation. (We don't refer to 4095:4096 as a schisma, because there are a couple of schismas, 32768:32805 and 512:513, that _are_ symbolized in the notation. A schismina is small enough that it may be disregarded when notating JI.)

> and makes it likely > that you'll only have to rename them again later when an earlier > mention comes to light.

Even then I would not be too hasty to name an interval after somebody simply because they happened to mention it in passing (without drawing attention to its special significance) or apparently happened to be the first to list it (along with a bunch of other intervals). If you're in doubt as to whether it would be appropriate to attach someone's name to a comma or schisma, then don't do it unless you are reasonably sure that that person recognized something of practical or theoretical value that is associated with that interval.

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

Yes, and I'm continuing to make good progress on it. 8-} I just need a break from time to time to see what's going on in the outside world, and I responded to this because I thought my good friend Archytas hasn't been getting the recognition he deserves for his advocacy of ratios of 7 in a musical scale.

> I'm not here either.

Then I guess I'll need to copy you off-list with the above. ;-) --George ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10779 - Contents - Hide Contents 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 * [with cont.] (Wayb.) (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 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10781 - Contents - Hide Contents 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: 10782 - Contents - Hide Contents Date: Fri, 09 Apr 2004 19:27:28 Subject: Re: Fokker pentatonics, known and unknown From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

Scala checks modes automatically. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10783 - Contents - Hide Contents 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. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------

Message: 10784 - Contents - Hide Contents Date: Sun, 11 Apr 2004 19:55:44 Subject: Re: 126 7-limit linears From: Gene Ward Smith --- 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.

> 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.

Message: 10785 - Contents - Hide Contents 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 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10786 - Contents - Hide Contents 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 # Temperament finding library -- definitions * [with cont.] (Wayb.) 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

Message: 10787 - Contents - Hide Contents Date: Mon, 12 Apr 2004 18:26:51 Subject: Re: Optimal octave stretching From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> 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.

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.

> 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.

It would be interested to find the Zeta tunings for comparison; I'll need to move to the Linux side to do that.

> 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.

You could try 441 also.

Message: 10788 - Contents - Hide Contents 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

Message: 10791 - Contents - Hide Contents 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 * [with cont.] (Wayb.) gives me "This page cannot be displayed" . . . :(

Message: 10792 - Contents - Hide Contents Date: Tue, 13 Apr 2004 06:23:03 Subject: 32201 seven limit linear temperaments From: Gene Ward Smith 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.

Message: 10793 - Contents - Hide Contents 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 * [with cont.] (Wayb.) 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?

Message: 10794 - Contents - Hide Contents Date: Tue, 13 Apr 2004 06:24:19 Subject: Re: Optimal octave stretching From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> 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?

One of several things I keep mentioning related to the Zeta function.

Message: 10796 - Contents - Hide Contents 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

Message: 10797 - Contents - Hide Contents Date: Tue, 13 Apr 2004 23:02:24 Subject: Re: 32201 temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Gene, your zip file is corrupted. > > -C.

Thanks, I uploaded again.