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Message: 9425 - Contents - Hide Contents Date: Thu, 22 Jan 2004 20:39:13 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> 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. >> 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? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9427 - Contents - Hide Contents Date: Thu, 22 Jan 2004 03:24:59 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:>> I've put these in the photos section. >> >> Recall that "standard" ? HE is a Gaussian smoothing of ?(2^ > (x/1200))',>> whereas the poor man simply contents himself with a central > difference>> operator on ?(2^(x/1200)). I've put up graphs of poor man for Del_s, >> with s 5, 10, 25 and 50 cents. These are very fast and easy to >> compute, and might be useful for that reason. >> Unfortunately, their features bear little resemblance to those of > harmonic entropy curves. Yet they are interesting in their own right. > 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. If this isn't HE, what is it, I wonder? Of course, we haven't yet done what I first suggested, and did Gaussian smoothing. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9428 - Contents - Hide Contents Date: Sat, 24 Jan 2004 02:47:18 Subject: Re: 114 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: > > 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.The cross-checking that I showed in the 3-limit case (except I was off by a factor of 2). In each limit and dimension, the complexity measure should arise from a single general formula -- ||Wedgie||, I suppose, but with a full elaboration for the grassmann-unaware -- and our paper should show how this reduces, in the dimension-1 case, to the number of notes per log(frequency) unit (assume we will also explain fokker determinants), and in the codimension-1 case, to the length (scaled with a factor of 1/2 or however it works out) of the comma = the width of the 'periodicity slice'. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9429 - Contents - Hide Contents Date: Sun, 25 Jan 2004 05:04:45 Subject: top complexity From: Paul Erlich orthogonalization. since it doesn't matter which comma basis you choose, you can always choose a basis where the commas each involve n-1 primes (this is probably one of the matrix reduction or decomposition methods matlab is happy to do). then it's trivial, up to torsion, to express the complexity in terms of the complexities of the commas, since the relevant {length, area, volume . . .} measure will just be that of a rectangular solid . . . or do you have to iteratively cascade down the dimensions (brain foggy . . .)?

Message: 9430 - Contents - Hide Contents Date: Sun, 25 Jan 2004 19:13:34 Subject: Re: 114 7-limit temperaments From: Herman Miller On Wed, 21 Jan 2004 09:08:14 -0000, "Gene Ward Smith" <gwsmith@xxxxx.xxx> wrote:>Number 82 > >[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]] >TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030] >TOP generators [601.7004928, 230.8749260] >bad: 79.825592 comp: 4.619353 err: 3.740932There are a number of interesting and potentially useful new tunings in this list, but I'd like to draw attention to this one. I've been making charts of the ET's produced by equal divisions of slightly stretched octaves, from 1201 to 1205 cents, and noticed that the appearance of these is quite different from the usual ones centered around the 12-19-22 triangle. Stretched octave ET's tend to cluster around 19, and the region of meantone between 19 and 26 starts to get filled with new ET's. There's a couple of new temperaments on the 1205 cent octave chart that don't show up on the old familiar chart; one of these goes through 26 from 10-ET to 16-ET, and also includes 42, 68, 94, and 36-ET (plus some inconsistent ones). When I plug 10 and 16 into the temperament finder, this is what I end up with. 5/13, 229.4 cent generator basis: (0.5, 0.191135896755) mapping by period and generator: [(2, 0), (2, 3), (5, -1), (6, -1)] mapping by steps: [(16, 10), (25, 16), (37, 23), (45, 28)] highest interval width: 4 complexity measure: 8 (10 for smallest MOS) highest error: 0.014573 (17.488 cents) The decatonic version of this scale seems to have some possibilities. I've been playing around with it, originally in the 5-limit version with TOP tuning [1203.571465, 1896.294363, 2778.021029, 3368.825906]. -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Message: 9431 - Contents - Hide Contents Date: Sun, 25 Jan 2004 05:38:15 Subject: Re: top complexity From: Paul Erlich The orthogonalized basis for pajara would be {64:63, 50:49}. The complexity should be the product of the complexities of these 3, with the suitable "dimension boost" factors = log of the remaining prime. yup? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9432 - Contents - Hide Contents Date: Sun, 25 Jan 2004 22:43:03 Subject: rank complexity explanation updated From: Carl Lumma When a listener hears a melody in a fixed scal... * [with cont.] (Wayb.) -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: 9433 - Contents - Hide Contents Date: Tue, 27 Jan 2004 02:33:30 Subject: Re: 114 7-limit temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> This list is attractive, but Meantone, Magic, Pajara, maybe >>> Injera to name a few are too low for my taste, if I'm reading >>> these errors right (they're weighted here, I take it). >>>> I think log-flat badness has outlived its popularity :) >> Not with me. However, an alternative which isn't simply ad-hoc > randomness would be nice.Thanks for the list Gene. Gene or Paul, Can one of you easily plot these 7-limit temperaments on an error vs. complexity graph (log log or whatever seemed best with 5-limit) so we can all think about what our subjective badness contours might look like. Please label the points with the numbers-plus-names Gene gave them in his list.

Message: 9435 - Contents - Hide Contents Date: Tue, 27 Jan 2004 04:31:54 Subject: Characteristic polynomials and inverse matriies of interval matricies From: Gene Ward Smith It seems these might be worth looking at, especially for a normalization of intervals to the octave. For 1--9/8--5/4--4/3--3/2--5/3--15/8 I find that the characteristic polynomial is 2*x^7 - (x+1)^7 = x^7-7*x^6-21*x^5-35*x^4-35*x^3-21*x^2-7*x-1 The inverse matrix has -1 along the main diagonal, with the steps of the scale in a circulating diagonal below it--9/8, 10/9, 16/15, 9/8, 10/9, 9/8 plus a 16/15 in the upper right-hand corner to complete the circulating diagonal.

Message: 9436 - Contents - Hide Contents Date: Tue, 27 Jan 2004 13:09:49 Subject: Re: rank complexity explanation updated From: Carl Lumma>> >ttp://lumma.org/tuning/rank-complexity.txt >>Of course, you are talking about "external" intervals here, right? >If you take ALL the intervals in a septachord, you get an interval >vector that totals up to 21. (Hexachords, 15, Pentachords, 10) It >would be cool if someone could tie "interval-vectors" (The kind >Jon Wild has compiled- hey that rhymes!) into the main discussion. >I'm still trying to find a good use for them! -And trying to find >a rule for Z-relations... Hi Paul,I'm afraid I don't know what an "external" interval is. Here's the interval matrix of the diatonic scale in 12-equal, as given by Scala... 100.0 : 2 4 5 7 9 11 12 200.0 : 2 3 5 7 9 10 12 400.0 : 1 3 5 7 8 10 12 500.0 : 2 4 6 7 9 11 12 700.0 : 2 4 5 7 9 10 12 900.0 : 2 3 5 7 8 10 12 1100.0: 1 3 5 6 8 10 12 The ruler is... 0...1...2...3...4...5...6...7...8...9...10..11..12 The list of adjacent marks on the ruler: 1 The rank complexity: 0 As for the things that Jon Wild compiled, I don't have a clue what they are... which is akin to saying I don't know of any use for them. -Carl

Message: 9437 - Contents - Hide Contents Date: Tue, 27 Jan 2004 04:48:37 Subject: Re: 114 7-limit temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> On Wed, 21 Jan 2004 09:08:14 -0000, "Gene Ward Smith" <gwsmith@s...> > wrote: > >> Number 82 >>>> [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]] >> TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030] >> TOP generators [601.7004928, 230.8749260] >> bad: 79.825592 comp: 4.619353 err: 3.740932 ...> When I plug 10 and 16 into the temperament finder, this is what I > end up with. > > 5/13, 229.4 cent generator > > basis: > (0.5, 0.191135896755) > > mapping by period and generator: > [(2, 0), (2, 3), (5, -1), (6, -1)] > > mapping by steps: > [(16, 10), (25, 16), (37, 23), (45, 28)] > > highest interval width: 4 > complexity measure: 8 (10 for smallest MOS) > highest error: 0.014573 (17.488 cents)This comparison of different outputs for the same temperament shows up the need to correctly normalise the new weighted error and complexity figures so they actually have units we can relate to. i.e. cents for the error and gens per interval for the complexity. This should be simple to do. I think the correct normalisation of a weighted norm is the one where, if every individual value happened to be X then the, the norm would also be X, irrespective of the weights. e.g. if the individual errors are E1, E2, ... En, and the respective weights are W1, W2, ... Wn (all positive), I think the p-norm should not be [(|W1E1|**p + |W2E2|**p + ... |WnEn|**p)/n]**(1/p) but instead [(|W1E1|**p + |W2E2|**p + ... |WnEn|**p)/(W1**p + W2**p + ... Wn**p)]**(1/p) i.e. n is replaced by (W1**p + W2**p + ... Wn**p) However it bothers me slightly that for minimax (p -> oo), this is equivalent to Max(|W1E1|, |W2E2|, ... |WnEn|)/Max(W1, W2, ... Wn) It seems like I'd rather have Max(|W1E1|, |W2E2|, ... |WnEn|)/Mean(W1, W2, ... Wn) but I guess that would be inconsistent.

Message: 9438 - Contents - Hide Contents Date: Tue, 27 Jan 2004 13:13:25 Subject: Re: Graef article on rationalization of scales From: Carl Lumma>Does anyone know why Barlow changed Euler's (p-1) weighting to >2(p-1+1/(2p))? > >Graef gives examples of "rationaizing" 12-equal and 1/4-comma >meantone in the 5-limit. Rationalizing means moving to a nearby value >so that a certain badness figure is minimized, where we look at the >entire matrix of intervals and work locally (interval pair by >interval pair) not globally. Adding the last condition makes the >problem much more complicated, and I don't see the point in it. He >got the duodene by rationalizing equal temperament, and >syndie2=fogliano1 by rationalizing 1/4-comma.This raises an interesting question. What is our approved method for finding Fokker blocks for an arbitrary irrational scale? Such a method would surely make Graf's look silly. -Carl

Message: 9439 - Contents - Hide Contents Date: Tue, 27 Jan 2004 04:54:30 Subject: Re: Characteristic polynomials and inverse matriies of interval matricies From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> It seems these might be worth looking at, especially for a > normalization of intervals to the octave. For > > 1--9/8--5/4--4/3--3/2--5/3--15/8 > > I find that the characteristic polynomial is > > 2*x^7 - (x+1)^7 = x^7-7*x^6-21*x^5-35*x^4-35*x^3-21*x^2-7*x-1Unfortunately, this is merely a consequence of the fact that it is an interval matrix; it tells us nothing about the scale.

Message: 9441 - Contents - Hide Contents Date: Tue, 27 Jan 2004 19:15:47 Subject: Graef article on rationalization of scales From: Gene Ward Smith Does anyone know why Barlow changed Euler's (p-1) weighting to 2(p-1+1/(2p))? Graef gives examples of "rationaizing" 12-equal and 1/4-comma meantone in the 5-limit. Rationalizing means moving to a nearby value so that a certain badness figure is minimized, where we look at the entire matrix of intervals and work locally (interval pair by interval pair) not globally. Adding the last condition makes the problem much more complicated, and I don't see the point in it. He got the duodene by rationalizing equal temperament, and syndie2=fogliano1 by rationalizing 1/4-comma. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9442 - Contents - Hide Contents Date: Tue, 27 Jan 2004 22:55:52 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- 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 "Paul Erlich" <perlich@a...> wrote:>>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> This list is attractive, but Meantone, Magic, Pajara, maybe >>>> Injera to name a few are too low for my taste, if I'm reading >>>> these errors right (they're weighted here, I take it). >>>>>> I think log-flat badness has outlived its popularity :) >>>> Not with me. However, an alternative which isn't simply ad-hoc >> randomness would be nice. >> Thanks for the list Gene. > > Gene or Paul, > > Can one of you easily plot these 7-limit temperaments on an error vs. > complexity graph (log log or whatever seemed best with 5-limit) so we > can all think about what our subjective badness contours might look like.I could do that, but as this was done with a log-flat badness cutoff, there will be a huge gaping hole in the graph. That's why I'm trying to figure out the whole deal for myself, but no one's helping.

Message: 9443 - Contents - Hide Contents Date: Tue, 27 Jan 2004 23:09:00 Subject: Re: Graef article on rationalization of scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Does anyone know why Barlow changed Euler's (p-1) weighting to >> 2(p-1+1/(2p))? >> >> Graef gives examples of "rationaizing" 12-equal and 1/4-comma >> meantone in the 5-limit. Rationalizing means moving to a nearby value >> so that a certain badness figure is minimized, where we look at the >> entire matrix of intervals and work locally (interval pair by >> interval pair) not globally. Adding the last condition makes the >> problem much more complicated, and I don't see the point in it. He >> got the duodene by rationalizing equal temperament, and >> syndie2=fogliano1 by rationalizing 1/4-comma. >> This raises an interesting question. What is our approved method > for finding Fokker blocks for an arbitrary irrational scale? > Such a method would surely make Graf's look silly.All such methods are silly, but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees blocks that result from the min-"odd- limit" criterion. But the whole idea of rationalizing a tempered scale is completely backwards and misses the point in a big way.

Message: 9444 - Contents - Hide Contents Date: Tue, 27 Jan 2004 22:59:31 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:>> On Wed, 21 Jan 2004 09:08:14 -0000, "Gene Ward Smith" <gwsmith@s...> >> wrote: >> >>> Number 82 >>>>>> [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]] >>> TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030] >>> TOP generators [601.7004928, 230.8749260] >>> bad: 79.825592 comp: 4.619353 err: 3.740932 > > ... >>> When I plug 10 and 16 into the temperament finder, this is what I >> end up with. >> >> 5/13, 229.4 cent generator >> >> basis: >> (0.5, 0.191135896755) >> >> mapping by period and generator: >> [(2, 0), (2, 3), (5, -1), (6, -1)] >> >> mapping by steps: >> [(16, 10), (25, 16), (37, 23), (45, 28)] >> >> highest interval width: 4 >> complexity measure: 8 (10 for smallest MOS) >> highest error: 0.014573 (17.488 cents) >> This comparison of different outputs for the same temperament shows up > the need to correctly normalise the new weighted error and complexity > figures so they actually have units we can relate to. i.e. cents for > the error and gens per interval for the complexity. > > This should be simple to do. > > I think the correct normalisation of a weighted norm is the one where, > if every individual value happened to be X then the, the norm would > also be X, irrespective of the weights. > > e.g. if the individual errors are E1, E2, ... En,You realize that there are an infinite number of errors in the TOP case.

Message: 9445 - Contents - Hide Contents Date: Tue, 27 Jan 2004 15:53:46 Subject: Re: rank complexity explanation updated From: Carl Lumma>> >ere's the interval matrix of the diatonic scale in >> 12-equal, as given by Scala... >> >> 100.0 : 2 4 5 7 9 11 12 >> 200.0 : 2 3 5 7 9 10 12 >> 400.0 : 1 3 5 7 8 10 12 >> 500.0 : 2 4 6 7 9 11 12 >> 700.0 : 2 4 5 7 9 10 12 >> 900.0 : 2 3 5 7 8 10 12 >> 1100.0: 1 3 5 6 8 10 12 //>Interesting. What I meant was really "adjacent, outer" intervals: >This row: > >100.0 : 2 4 5 7 9 11 12 Has a vector count of >(2,5,0,0,0,0) "outer" intervals.I'm lost. There are 2, 5 and 0 of what? -Carl

Message: 9447 - Contents - Hide Contents Date: Tue, 27 Jan 2004 16:11:22 Subject: Re: 114 7-limit temperaments From: Carl Lumma>> >an one of you easily plot these 7-limit temperaments on an error >> vs. complexity graph (log log or whatever seemed best with 5-limit) >> so we can all think about what our subjective badness contours might >> look like. >>I could do that, but as this was done with a log-flat badness cutoff, >there will be a huge gaping hole in the graph. That's why I'm trying >to figure out the whole deal for myself, but no one's helping.Since I usually like the way you figure things out, I'll do whatever I can to help, which may not be much. If there are any particular msg. #s associated with this, I'll reread them. I didn't follow your orthogonalization posts at all. :( Part of the problem is these contours represent musical values, so they're ultimately a matter of opinion. Log-flat badness has some nice things going for it, I suppose. Back when I was coding it I was asking things like, 'every time I double the number of commas I search, how much of my top-10 list ought to change?'... without much closure, I might add. -Carl

Message: 9448 - Contents - Hide Contents Date: Tue, 27 Jan 2004 18:27:12 Subject: Re: rank complexity explanation updated From: Carl Lumma>>>> >ttp://lumma.org/tuning/rank-complexity.txt >>> >>>> Hi Paul, >>>> I'm afraid I don't know what an "external" interval is. Here's >> the interval matrix of the diatonic scale in 12-equal, as given >> by Scala... >> >> 100.0 : 2 4 5 7 9 11 12 >> 200.0 : 2 3 5 7 9 10 12 >> 400.0 : 1 3 5 7 8 10 12 >> 500.0 : 2 4 6 7 9 11 12 >> 700.0 : 2 4 5 7 9 10 12 >> 900.0 : 2 3 5 7 8 10 12 >> 1100.0: 1 3 5 6 8 10 12 >>Q: Shouldn't the first row be 000.0?It is quite safe to ignore the values before the colon, as they are merely an artifact of scala's output.>Another point, > >One can obtain "my" interval vector from "your" interval matrix >by tallying all the intervals from 1 to 6 and ignoring 7 to 12. >You subsequently obtain (2,5,4,3,6,1)Sorry, but how does tallying numbers in the above matrix lead to (2,5,4,3,6,1)? -Carl

Message: 9449 - Contents - Hide Contents Date: Tue, 27 Jan 2004 18:29:50 Subject: Re: Graef article on rationalization of scales From: Carl Lumma>> >his raises an interesting question. What is our approved method >> for finding Fokker blocks for an arbitrary irrational scale? >> Such a method would surely make Graf's look silly. >>All such methods are silly,Perhaps you mean Graf's idea of people wanting "just" versions of arbitrary scales is silly. That's for sure. A method which could show when there is (and isn't) a reasonable Fokker-block interp. of, say scales taken from field measurements silly? I think not.>but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees >blocks that result from the min-"odd-limit" criterion. But the >whole idea of rationalizing a tempered scale is completely >backwards and misses the point in a big way.I think you missed my point. -Carl

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