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Message: 9400 - Contents - Hide Contents Date: Wed, 21 Jan 2004 18:54:56 Subject: Re: Maple code for ?(x) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Aren't 'most' continued fractions infinite? >> Obviously, Maple is not going to compute an infinte number of > convergents. What it does depends on the setting of Digits.So we're getting some *approximation* to the ? function, yes? Matlab does continued fractions, but they're strings not vectors, they use an error tolerance to determine when to stop, and they allow negative entries. I've written code to get around the last limitation; the other two shouldn't be that hard . . .

Message: 9401 - Contents - Hide Contents Date: Wed, 21 Jan 2004 18:04:11 Subject: Re: TM-reduced basis From: Carl Lumma>Could someone point me to a good definition of a TM-reduced basis? >I can see that it is arrived at by manipulating rows of a matrix, >(adding and subtracting). Is it always two or more fractions that >have the smallest numerator/denominator combo? Hi Paul,Here are some bits from the aether...>> Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of >> intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q >> and r/s independent is Minkowski reduced iff the only ratios t/u >> in the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers >> of p/q. >>So IOW, if you have a pair of unison vectors for a PB, you shouldn't >be able to stack them both in some way to get an interval that's >simpler than the more complex of the pair is by itself. // >First we need to define Tenney height: if p/q is a positive rational >number in reduced form, then the Tenney height is TH(p/q) = p q. > >Now suppose {q1, ..., qn} are n multiplicatively linearly independent >positive rational numbers. Linear independence can be equated, for >instance, with the condition that rank of the matrix whose rows are >the monzos for qi is n. Then {q1, ..., qn} is a basis for a lattice >L, consisting of every positive rational number of the form q1^e1 ... >qn^en where the ei are integers and where the log of the Tenney >height defines a norm. Let t1>1 be the shortest (in terms of Tenney >height) rational number in L greater than 1. Define ti>1 inductively >as the shortest number in L independent of {t1, ... t_{i-1}} and such >that {t1, ..., ti} can be extended to be a basis for L. In this way >we obtain {t1, ..., tn}, the TM reduced basis of L. -Carl

Message: 9402 - Contents - Hide Contents Date: Wed, 21 Jan 2004 19:07:21 Subject: Re: TOP take on 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> I appreciate this work, Gene. >> >> How about a worked-out, hand-holding example for one of these error >> and complexity calculations? >> >> P.S. Instead of using log-flat badness, why don't we use the same >> function of error and complexity that yielded epimericity in the >> codimension-1 case? >> What's your proposal specifically?Hrm. Well, I'm trying to use your webpages as a substitute for hand- holding, but why is the codimension one case here: /root/tentop.htm * [with cont.] (Wayb.) so complicated? I thought we agreed that if there's only one comma n/d, we simply temper each prime p from cents(p) to cents(p) - log(p)*cents(n/d)/log(n*d) if p is a factor of n, and to cents(p) + log(p)*cents(n/d)/log(n*d) if p is a factor of d. If p is not a factor of n or d, we probably wouldn't want to temper it at all, but it *is* tempered on all the vertices of your ball, isn't it?

Message: 9403 - Contents - Hide Contents Date: Wed, 21 Jan 2004 19:28:36 Subject: Re: TOP take on 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> I appreciate this work, Gene. >> >> How about a worked-out, hand-holding example for one of these error >> and complexity calculations? >> >> P.S. Instead of using log-flat badness, why don't we use the same >> function of error and complexity that yielded epimericity in the >> codimension-1 case? >> What's your proposal specifically?In the P.S., what I was referring to was this: Yahoo groups: /tuning/message/51221 * [with cont.] This is a function of complexity and error, so can be applied even if the actual number of commas is greater than 1. So why not use it as a badness function? Maybe it's screwy, but I think it would be informative to see what results.

Message: 9404 - Contents - Hide Contents Date: Wed, 21 Jan 2004 19:50:37 Subject: Re: TOP take on 7-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> This is a function of complexity and error, so can be applied even if > the actual number of commas is greater than 1. So why not use it as a > badness function? Maybe it's screwy, but I think it would be > informative to see what results.It can be applied to commas individually, but how do you make it independent of the comma basis if you apply it to wedgies?

Message: 9405 - Contents - Hide Contents Date: Wed, 21 Jan 2004 20:06:11 Subject: Re: TOP take on 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> This is a function of complexity and error, so can be applied even if >> the actual number of commas is greater than 1. So why not use it as a >> badness function? Maybe it's screwy, but I think it would be >> informative to see what results. >> It can be applied to commas individually, but how do you make it > independent of the comma basis if you apply it to wedgies?*You yourself* already posted the complexity and error for various codimension-2 wedgies!!!! Those results weren't independent of the comma basis?????????

Message: 9406 - Contents - Hide Contents Date: Wed, 21 Jan 2004 21:12:53 Subject: Re: TOP take on 7-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> *You yourself* already posted the complexity and error for various > codimension-2 wedgies!!!! Those results weren't independent of the > comma basis?????????Yes they were, but I thought you didn't like what I did and were suggesting we try something else.

Message: 9407 - Contents - Hide Contents Date: Wed, 21 Jan 2004 21:22:31 Subject: Re: TOP take on 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> *You yourself* already posted the complexity and error for various >> codimension-2 wedgies!!!! Those results weren't independent of the >> comma basis????????? >> Yes they were, but I thought you didn't like what I did and were > suggesting we try something else.It was the *badness* function you used that I didn't like, which is why I was suggesting this *other* function of complexity and error which, as well, *you yourself* posted.

Message: 9408 - Contents - Hide Contents Date: Wed, 21 Jan 2004 13:29:14 Subject: Re: 114 7-limit temperaments From: Carl Lumma>>> >umber 8 Schismic >>> >>> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] >>> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750] >>> TOP generators [1200.760624, 498.1193303] >>> bad: 28.818558 comp: 5.618543 err: .912904 >>> >>> >>> Number 9 Miracle >>> >>> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] >>> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756] >>> TOP generators [1200.631014, 116.7206423] >>> bad: 29.119472 comp: 6.793166 err: .631014 //>> And I don't see how you figure schismic is less complex than >> miracle in light of the maps given. >>Probably the shortness of the fifths in the lattice wins it for >schismic . . .After I wrote that I reflected a bit on comma complexity vs. map complexity. Comma complexity gives you the number of notes you'd have to search to find the comma, on average (Kees points out that the symmetry of the lattice allows you to search 1/4 this numeber in the 5-limit, or something, but anyway...). Map complexity is the number of notes you need to complete the map *with contiguous chains of generators*. It's this contiguous-chain restriction that makes me wonder -- what good is it? I suppose it helps keep the number of step sizes (mean variety) low in the resulting scales. But it implies a variable stacking process that actually produces linear temperaments (in the DE cases) and I'm guessing planar temperaments otherwise (when there are 3 step sizes)... if so what kind of planar temperament?... in the case of Marvel, Gene says 384:whatever always goes with 225:224, and this notion of natural planar extensions seems highly interesting... Aside from the Hypothesis, it seems the link between these two ways of approaching temperament (chains vs. commas) seems little-explored. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9409 - Contents - Hide Contents Date: Thu, 22 Jan 2004 06:00:35 Subject: Re: 114 7-limit temperaments From: Carl Lumma>>> >hat kind of complexity is this? >>>> It's the complexity which arises naturally out of the Tenney space >> and dual val space point of view, as the norm on a bival. It >> therefore gives more weight to lower primes such as 2 and 3 as >> opposed to higher ones such as 5 and 7. >>>>> Do you always use the same kind? >>>> I wanted to do things from a TOP point of view, so I used >> something consistent with that. >>Carl, did you read the message that you got 3 copies of? This is >what Gene was addressing with his list.I was in some sort of trance when I read it, sorry.>In the 3-limit, there's only one kind of regular TOP temperament: >equal TOP temperament. For any instance of it, the complexity can >be assessed by either > >() Measuring the Tenney harmonic distance of the commatic unison >vector > >5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047 >12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988 > >() Calculating the number of notes per pure octave or 'tritave': > >5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave; >.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave. >12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave; >.........TOP tritave = 1901 -> 19.01 notes per pure tritave. > >The latter results are precisely the former divided by 2: in >particular, the base-2 Tenney harmonic distance gives 2 times the >number of notes per tritave, and the base-3 Tenney harmonic distance >gives 2 times the number of notes per octave. A funny 'switch' but >agreement (up to a factor of exactly 2) nonetheless. In some way, >both of these methods of course have to correspond to the same >mathematical formula . . . Ok, great! >In the 5-limit, there are both 'linear' and equal TOP temperaments. >For the 'linear' case, we can use the first method above (Tenney >harmonic distance) to calculate complexity.Did you repeat the above comparison for the two methods in the 5-limit?>For the equal case, two >commas are involved; if we delete the entries for prime p in the >monzos for each of the commatic unison vectors and calculate the >determinant of the remaining 2-by-2 matrix, we get the number of >notes per tempered p; then we can use the usual TOP formula to get >tempered p in terms of pure p and thus finally, the number of notes >per pure p.So this is a way to get map-complexity with two commas; what about comma complexity with two commas?>Note that there was no need to calculate the angle >or 'straightness' of the commas; change the angles in your lattice >and the number of notes the commas define remains the same, so >angles can't really be relevant here.Funny; when looking up TM stuff for Paul H. I ran across this same reasoning... ////>yes, a reduced basis will have good straightness, because the set of >basis vectors is, in some sense, as short as possible. and, as we >discussed before, shortness implies straightness. the "block" always >has the same "area", so if the vectors are close to parallel, >they'll have to be long to compensate. remember that whole confusing >discussion? /// -Carl

Message: 9410 - Contents - Hide Contents Date: Thu, 22 Jan 2004 13:01:38 Subject: Re: 114 7-limit temperaments From: Carl Lumma>> >or linear temperaments can't you use both the map-based and >> comma based approach, and see if the factor of 2 holds? >>What's the map-based approach, explicitly?The minimum number of notes that completes the map. -Carl

Message: 9411 - Contents - Hide Contents Date: Thu, 22 Jan 2004 14:44:08 Subject: Re: Maple code for ?(x) From: Graham Breed Paul Erlich wrote:> I don't have the symbolic math toolbox. I have the optimization > toolbox and the statistics toolbox, that's it. Each toolbox is like > $1000!It looks like I have it at college. Does anybody know an easy way to test it? Graham

Message: 9412 - Contents - Hide Contents Date: Thu, 22 Jan 2004 21:03:13 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> For linear temperaments can't you use both the map-based and >>> comma based approach, and see if the factor of 2 holds? >>>> What's the map-based approach, explicitly? >> The minimum number of notes that completes the map. > > -CarlHow do you define 'completes the map'? And can you show how what I did in the 3-limit satisfies this definition?

Message: 9413 - Contents - Hide Contents Date: Thu, 22 Jan 2004 13:12:09 Subject: Re: 114 7-limit temperaments From: Carl Lumma>>>> >or linear temperaments can't you use both the map-based and >>>> comma based approach, and see if the factor of 2 holds? >>>>>> What's the map-based approach, explicitly? >>>> The minimum number of notes that completes the map. >> >> -Carl >>How do you define 'completes the map'? And can you show how what I >did in the 3-limit satisfies this definition?For an et, the number of notes in an octave is guaranteed to complete the map, but it may not be the minimum number of notes to do so. So maybe the correct analog of what you did is the number of tones in the Fokker block corresponding to the TM-reduced basis of the temperament. But I'd be interested in the map thing. It's just what we used to call Graham complexity. -Carl

Message: 9414 - Contents - Hide Contents Date: Thu, 22 Jan 2004 19:50:21 Subject: Re: Maple code for ?(x) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> The computational engine underlying the [symbolic math] toolboxes >>>> is the kernel of Maple, a system developed primarily at the >>>> University of Waterloo, Canada, and, more recently, at the >>>> Eidgenössiche Technische Hochschule, Zürich, Switzerland. Maple >>>> is marketed and supported by Waterloo Maple, Inc. >>>>>> Once again I learn not to suspect Dan of being wrong. >>>> I don't have the symbolic math toolbox. I have the optimization >> toolbox and the statistics toolbox, that's it. Each toolbox is like >> $1000! >> How can one tell what toolboxes are installed?Go to your "toolbox" subdirectory. Within that, you'll have "matlab" and "local" subdirectories, plus ones for whatever toolboxes you have.> I have 27 toolboxes > in my "launch pad", whatever that means. > > -CarlBut you don't know which ones?

Message: 9415 - Contents - Hide Contents Date: Thu, 22 Jan 2004 19:52:26 Subject: Re: TM-reduced basis From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Could someone point me to a good definition of a TM-reduced basis? >>> I can see that it is arrived at by manipulating rows of a matrix, >>> (adding and subtracting). Is it always two or more fractions that >>> have the smallest numerator/denominator combo? >> >> Hi Paul, >>>> Here are some bits from the aether... >>>>>> Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of >>>> intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q >>>> and r/s independent is Minkowski reduced iff the only ratios t/u >>>> in the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers >>>> of p/q. >>>>>> So IOW, if you have a pair of unison vectors for a PB, you > shouldn't>>> be able to stack them both in some way to get an interval that's >>> simpler than the more complex of the pair is by itself. >> Really. Not even in between the two unison vectors?Don't know what you mean by "between" but, a simple example is 81:80 and 128:125 (defining 5-limit 12-equal). You can't get a simpler comma by multiplying and dividing powers of these.

Message: 9416 - Contents - Hide Contents Date: Thu, 22 Jan 2004 21:23:41 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> For linear temperaments can't you use both the map-based and >>>>> comma based approach, and see if the factor of 2 holds? >>>>>>>> What's the map-based approach, explicitly? >>>>>> The minimum number of notes that completes the map. >>> >>> -Carl >>>> How do you define 'completes the map'? And can you show how what I >> did in the 3-limit satisfies this definition? >> For an et, the number of notes in an octave is guaranteed to > complete the map, but it may not be the minimum number of notes > to do so. So maybe the correct analog of what you did is the > number of tones in the Fokker block corresponding to the TM-reduced > basis of the temperament.. . . of the kernel of the temperament. Right, except this block only exists in the case of equal temperament. Otherwise you have a "Fokker strip", a "Fokker sheet", or what have you. In the co-dimension 1 case, comma complexity gives us the thickness of the Fokker sheet. In the co-dimension 2 case, we need an affine-geometrical measure of the cross-sectional area of the Fokker strip. Gene seems to be implying that the norm of the wedgie gives this, but I'd love to see him (when he has time) show how the cross-checking for the 3- and 5-limit cases works out.

Message: 9417 - Contents - Hide Contents Date: Thu, 22 Jan 2004 19:53:53 Subject: Re: Poor man's harmonic entropy graphs uploaded From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:>> Are those global maxima at the golden ratio or something? >> There ought to be a global maximum there, so I assume that's what >it is.The global maximum should really be at 65-70 cents or so, but your function is pretty low there.

Message: 9418 - Contents - Hide Contents Date: Thu, 22 Jan 2004 21:25:05 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> For linear temperaments can't you use both the map-based and >>>>> comma based approach, and see if the factor of 2 holds? >>>>>>>> What's the map-based approach, explicitly? >>>>>> The minimum number of notes that completes the map. >>> >>> -Carl >>>> How do you define 'completes the map'? And can you show how what I >> did in the 3-limit satisfies this definition? >> For an et, the number of notes in an octave is guaranteed to > complete the map, but it may not be the minimum number of notes > to do so. So maybe the correct analog of what you did is the > number of tones in the Fokker block corresponding to the TM-reduced > basis of the temperament.TM reduction is irrelevant; any other kernel basis will give the same result.

Message: 9419 - Contents - Hide Contents Date: Thu, 22 Jan 2004 19:59:37 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:>>>> What kind of complexity is this? >>>>>> It's the complexity which arises naturally out of the Tenney space >>> and dual val space point of view, as the norm on a bival. It >>> therefore gives more weight to lower primes such as 2 and 3 as >>> opposed to higher ones such as 5 and 7. >>>>>>> Do you always use the same kind? >>>>>> I wanted to do things from a TOP point of view, so I used >>> something consistent with that. >>>> Carl, did you read the message that you got 3 copies of? This is >> what Gene was addressing with his list. >> I was in some sort of trance when I read it, sorry. >>> In the 3-limit, there's only one kind of regular TOP temperament: >> equal TOP temperament. For any instance of it, the complexity can >> be assessed by either >> >> () Measuring the Tenney harmonic distance of the commatic unison >> vector >> >> 5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047 >> 12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988 >> >> () Calculating the number of notes per pure octave or 'tritave': >> >> 5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave; >> .........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave. >> 12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave; >> .........TOP tritave = 1901 -> 19.01 notes per pure tritave. >> >> The latter results are precisely the former divided by 2: in >> particular, the base-2 Tenney harmonic distance gives 2 times the >> number of notes per tritave, and the base-3 Tenney harmonic distance >> gives 2 times the number of notes per octave. A funny 'switch' but >> agreement (up to a factor of exactly 2) nonetheless. In some way, >> both of these methods of course have to correspond to the same >> mathematical formula . . . > > Ok, great! >>> In the 5-limit, there are both 'linear' and equal TOP temperaments. >> For the 'linear' case, we can use the first method above (Tenney >> harmonic distance) to calculate complexity. >> Did you repeat the above comparison for the two methods in the > 5-limit?How can you? Linear temperaments and equal temperaments are different entities in the 5-limit. However, I'd like to see Gene's general formula and how it handles all these cases.>> For the equal case, two >> commas are involved; if we delete the entries for prime p in the >> monzos for each of the commatic unison vectors and calculate the >> determinant of the remaining 2-by-2 matrix, we get the number of >> notes per tempered p; then we can use the usual TOP formula to get >> tempered p in terms of pure p and thus finally, the number of notes >> per pure p. >> So this is a way to get map-complexity with two commas; what about > comma complexity with two commas?The determinant is an affine-geometric measure of "area" which of course is invariant under change of comma basis, so it certainly seems to represent the latter at least as clearly as it represents the former.>> Note that there was no need to calculate the angle >> or 'straightness' of the commas; change the angles in your lattice >> and the number of notes the commas define remains the same, so >> angles can't really be relevant here. >> Funny; when looking up TM stuff for Paul H. I ran across this same > reasoning... > > ////>> yes, a reduced basis will have good straightness, because the set of >> basis vectors is, in some sense, as short as possible. and, as we >> discussed before, shortness implies straightness. the "block" always >> has the same "area", so if the vectors are close to parallel, >> they'll have to be long to compensate. remember that whole confusing >> discussion? /// > > -Carl

Message: 9420 - Contents - Hide Contents Date: Thu, 22 Jan 2004 23:37:46 Subject: Play Misty for me From: Gene Ward Smith I'm gearing up to compose some two-part music in misty, and thought I'd share some of the mistyfactitions I've calculated in the process. TOP data for Misty Mapping: [<3 5 6 6 7 7|, <0 -1 4 10 14 17|] 5-limit {67108864/66430125} tuning: <1199.661465 1902.491567 2787.099766| generators: [399.8871550 96.94420930] 7-limit <<3 -12 -30 -26 -56 -36|| {3136/3125, 5120/5103} tuning: <1199.661465 1902.491566 2787.099767 3368.765021| generators: [399.8871550 96.94420930] 11-limit <<3 -12 -30 -42 -26 -56 -77 -36 -56 -14|| {441/440, 896/891, 3136/3125} tuning: <1199.859836 1903.049660 2786.586602 3366.886999, 4153.707209| generators: [399.9532785, 96.71673279] 13-limit <<3 -12 -30 -42 -51 -26 -56 -77 -92 -36 -56 -74 -14 -32 -21|| {196/195, 364/363, 352/351, 625/624} tuning: <1199.918503 1903.182745 2786.562713 3366.651274 4153.349815 4443.394096| generators: [399.9728342 96.68142684]

Message: 9421 - Contents - Hide Contents Date: Thu, 22 Jan 2004 20:10:05 Subject: Re: Poor man's harmonic entropy graphs uploaded From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>>> Are those global maxima at the golden ratio or something? >>>> There ought to be a global maximum there, so I assume that's what >> it is. >> The global maximum should really be at 65-70 cents or so, but your > function is pretty low there.I'll look more carefully at what you did and see if I can get ? or Conway's box function (inverse ?) to replicate it.

Message: 9422 - Contents - Hide Contents Date: Thu, 22 Jan 2004 23:53:04 Subject: Re: 114 7-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: Gene seems to be implying> that the norm of the wedgie gives this, but I'd love to see him (when > he has time) show how the cross-checking for the 3- and 5-limit cases > works out.I'd need to know what you mean by cross-checking first. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9423 - Contents - Hide Contents Date: Thu, 22 Jan 2004 12:13:35 Subject: Re: Maple code for ?(x) From: Carl Lumma>> >ow can one tell what toolboxes are installed? >>Go to your "toolbox" subdirectory. Within that, you'll have "matlab" >and "local" subdirectories, plus ones for whatever toolboxes you have.Yep, looks like I have the works. No wonder the bugger was 2 full cds.>> I have 27 toolboxes >> in my "launch pad", whatever that means. >>But you don't know which ones?It says which ones. Symbolic math and Optimization among them. But since I already have Maple it's kinda moot in this case. -Carl

Message: 9424 - Contents - Hide Contents Date: Thu, 22 Jan 2004 12:16:41 Subject: Re: 114 7-limit temperaments From: Carl Lumma>>> >n the 3-limit, there's only one kind of regular TOP temperament: >>> equal TOP temperament. For any instance of it, the complexity can >>> be assessed by either >>> >>> () Measuring the Tenney harmonic distance of the commatic unison >>> vector >>> >>> 5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047 >>> 12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988 >>> >>> () Calculating the number of notes per pure octave or 'tritave': >>> >>> 5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave; >>> .........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave. >>> 12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave; >>> .........TOP tritave = 1901 -> 19.01 notes per pure tritave. >>> >>> The latter results are precisely the former divided by 2: in >>> particular, the base-2 Tenney harmonic distance gives 2 times the >>> number of notes per tritave, and the base-3 Tenney harmonic >distance>>> gives 2 times the number of notes per octave. A funny 'switch' but >>> agreement (up to a factor of exactly 2) nonetheless. In some way, >>> both of these methods of course have to correspond to the same >>> mathematical formula . . . >> >> Ok, great! >>>>> In the 5-limit, there are both 'linear' and equal TOP >>> temperaments. >>> For the 'linear' case, we can use the first method above (Tenney >>> harmonic distance) to calculate complexity. >>>> Did you repeat the above comparison for the two methods in the >> 5-limit? >>How can you? Linear temperaments and equal temperaments are different >entities in the 5-limit.For linear temperaments can't you use both the map-based and comma based approach, and see if the factor of 2 holds? -Carl

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