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Message: 9350 - Contents - Hide Contents Date: Tue, 20 Jan 2004 20:27:06 Subject: Re: Octacot? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> As usual, you need only look here: > > Yahoo groups: /tuning/database? * [with cont.] > method=reportRows&tbl=10&sortBy=6 > > or here: > > Tonalsoft Encyclopaedia of Tuning - equal-temp... * [with cont.] (Wayb.)And I was supposed to know we were talking 5-limit exactly how? I don't recall seeing the diagram of 5-limit commas on the 5-limit equilateral lattice before. It's one way of seeing what might make good Fokker blocks without doing any math.

Message: 9351 - Contents - Hide Contents Date: Tue, 20 Jan 2004 01:22:34 Subject: Re: Question for Dave Keenan From: Carl Lumma>> >he dominant seventh chord is defined >> as a local minimum of entropy. >>Who defined it as such, when, and why? > >As far as I know, the only widely accepted definition of the dominant >seventh chord is a chord containing the diatonic scale degrees V, VII, >II, IV. It may or may not be a local harmonic entropy minimum >depending how the scale is tuned.Ah; I thought you were referring to '4:5:6:7', when in fact you want bins to restrict the pitches of the chord. Finding the highest- entropy chord that fits in the bins makes sense.>It seems to me that the more >dissonant it is, the more relief is likely to be felt when it >"resolves" to a consonance.I don't personally agree with this, but it is a rather popular assertion.>>> There's also the posssibility that the dominant seventh chord >>> functions best when its harmonic entropy (or maybe only the HE of one >>> of its dyads) is locally _maximised_. >>>> What would this look like? >>Pretty much like a dominant seventh chord in 12-tET. > >Using noble-mediants to estimate them, a max entropy minor seventh >should be around 1002 cents, a max entropy diminished fifth should be >around 607 cents, and a max entropy minor third should be around 284 >cents.If you're trying to maximize pairwise entropy you'll need to consider the P-5th and M-3rd too. But tetradic entropy seems more to the point. -Carl

Message: 9352 - Contents - Hide Contents Date: Tue, 20 Jan 2004 20:32:15 Subject: Re: Octacot? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> As usual, you need only look here: >> >> Yahoo groups: /tuning/database? * [with cont.] >> method=reportRows&tbl=10&sortBy=6 >> >> or here: >> >> Tonalsoft Encyclopaedia of Tuning - equal-temp... * [with cont.] (Wayb.) >> And I was supposed to know we were talking 5-limit exactly how?I didn't say that, Gene. But dicot and tetracot are there on those tables, too, and since you brought them up, it didn't seem wholly irrelevant to bring up tricot.

Message: 9353 - Contents - Hide Contents Date: Tue, 20 Jan 2004 09:59:44 Subject: Re: A potentially informative property of tunings From: Dave Keenan Herman, I quite agree that it would be very useful to know, for any n-limit temperament, the max number of generators before we obtain a better approximation for some n-limit consonance, than that given by the temperament's mapping. I'd love to see this figure for all our old favourites. The only thing that bothers me is that I assume it will vary according to which particular optimum generator we use, and if so, then it isn't entirely a property of the temperament (i.e. the map). But otherwise, it works fine for me to say that temperament X has a _consistency_limit_ of Y generators.

Message: 9354 - Contents - Hide Contents Date: Tue, 20 Jan 2004 21:11:31 Subject: Re: Question for Dave Keenan From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> There's also the posssibility that the dominant seventh chord >>>> functions best when its harmonic entropy (or maybe only the HE > of one>>>> of its dyads) is locally _maximised_. >>>>>> What would this look like? >>>> Pretty much like a dominant seventh chord in 12-tET. >> >> Using noble-mediants to estimate them, a max entropy minor seventh >> should be around 1002 cents, a max entropy diminished fifth should > be>> around 607 cents, and a max entropy minor third should be around 284 >> cents. >> Why look at these intervals and not the major third, etc.?Well you _could_ try to locally maximise their entropy too, but I assumed the dominant triad (without the seventh) normally functions as a consonance.

Message: 9355 - Contents - Hide Contents Date: Tue, 20 Jan 2004 18:08:17 Subject: Micro-altered dominant 7th (Was: Question for Dave Keenan) 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 "Paul Erlich" <perlich@a...> wrote: >>>>> After all, Gene and others would have us >>> believe that meantone dominant seventh chords were "experienced as" >>> 4:5:6:7 chords, even though the 6:7 interval would typically be tuned >>> far closer to 5:6. >>>> Would harmonic entropy suggest something different? >> I assume you mean here the _minimisation_ of harmonic entropy. > > There's also the posssibility that the dominant seventh chord > functions best when its harmonic entropy (or maybe only the HE of one > of its dyads) is locally _maximised_. > > George Secor alluded to this recently (on the tuning list I think).The most recent thread related to this is "Just diminished 7th chords (Was: Chord names in Scala)", which began here: Yahoo groups: /tuning/message/51743 * [with cont.] but it concerns diminished rather than dominant 7th chords, and I emphasize the *melodic* properties of the intervals more than the harmonic. But I did attempt to relate the two in the last sentence of this message, which is probably what you had in mind: Yahoo groups: /tuning/message/51769 * [with cont.] There is an interesting problem I ran into when trying to incorporate all of my ideas simultaneously in attempting to resolve a dissonant dominant 7th chord to a tonic. Starting with 4:5:6 on the dominant with some sort of minor seventh added, it makes sense to lower the seventh to a 7:4 with the (dominant) root, so that the interval of resolution to the 3rd of the tonic triad is smaller. However, when you raise the leading tone a little bit (to make a smaller interval of resolution to the tonic root, and also to make the 7th chord more dissonant), you risk making the tritone so small that it ceases to sound like a tritone. The interval between a pythagorean major 3rd and harmonic 7th is ~561c -- only 10 cents larger than 8:11, and very close to 13:18. So what you end up with is something that functions very much like a dominant 7th chord, yet sounds somewhat different. An extreme example of this occurs in 19-ET: Alter both the 3rd and 7th of G B D F by 1deg, to G B# D Fb, to resolve to a C major triad. The interval between B# and Fb is equivalent to a perfect fifth -- not a tritone by anyone's stretch of the imagination! Still, I think it's musically useful in that it provides a harmonization: G-D-F-B to G-D-Fb-B# to C-C-E C that uses all of the tones of the Greek enharmonic genus as occurs in 19 (with the small intervals at the top of the tetrachord): C E Fb F G B B# C --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: 9356 - Contents - Hide Contents Date: Tue, 20 Jan 2004 21:13:06 Subject: Re: Compton/Erlich temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Gene W Smith <genewardsmith@j...> wrote:> > > On Thu, 18 Jul 2002 01:22:19 -0700 Carl Lumma <carl@l...> writes: >>> How do we classify the Compton/Erlich scheme of tuning multiple >> 12-et keyboards 15 cents apart? Some sort of planar temperament >> with the following commas? >> >> 531441/524288 (pythagorean comma) >> 5120/5103 (difference between syntonic comma and 64/63) >> >> Is this right? >> I think it's another system, discussed below. The wedgie you find from > the pyth comma and 5120/5103 gives what we are calling a linear > temperament. It is [0,12,12,-6,-19,19], and has a TM reduced basis > <50/49, 3645/3584>. The mapping is > > [[12, 19, 28, 34], [0, 0, -1, -1]] > > However, the rms optimum is 23.4 cents apart, not 15. > > I think what you want is the linear temperament with wedgie > [0,12,12,-6,-19,19], TM reduced basis > <225/224, 250047/250000> and mapping [[12,19,28,34],[0,0,-1,-1]].This is the same mapping as above. Did you mean for the last term to be -2, not -1? I know Waage proposed this system; who's Compton?

Message: 9357 - Contents - Hide Contents Date: Tue, 20 Jan 2004 21:19:14 Subject: the case of the disappearing 250047/250000 From: Paul Erlich From tuning . . . I was eliminating commas where the greatest common divisor of the entries in the monzo was greater than 1, since these are merely powers of other commas. But when I moved to 7-limit, I was unfortunately still checking only the first three components of the monzo. Hence I erroneously eliminated 250047/250000 and, I think, 1077 other commas under 600 cents where -20<e3<20, -14<e5<14, and - 12<e7<12. Both this comma and the most complex one on our recent lists were mentioned here: Yahoo groups: /tuning-math/message/6390 * [with cont.] and 250047/250000 is mentioned in many posts, for example: Yahoo groups: /tuning-math/message/4542 * [with cont.] but even earlier here: Yahoo groups: /tuning-math/message/1213 * [with cont.] and its earliest mention on this list was here: Yahoo groups: /tuning-math/message/200 * [with cont.]

Message: 9358 - Contents - Hide Contents Date: Tue, 20 Jan 2004 21:19:56 Subject: Attn: Gene 2 From: Paul Erlich All right, folks . . . I'm not sure if I missed anything important since I last posted, but before I catch up . . . 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 . . . 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. 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. 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. As I understand it, the determinant measures *area* not only in Euclidean geometry, but also in 'affine' geometry, where angles are left undefined . . . Anyhow, since both of these methods could be used to address a 3-limit TOP temperament, in 5-limit could they be still both be expressible in a single form in a general enough framework, say exterior algebra? In the 7-limit, the two methods give us, respectively, the complexity of a 'planar' temperament as a distance, and the cardinality of a 7- limit equal temperament as a volume. But 7-limit 'linear' temperaments get left out in the cold. The appropriate measure would seem to have to be an *area* of some sort -- from what I understand from exterior algebra, this is the area of the *bivector* formed by taking the *wedge product* of any two linearly independent commatic unison vectors (barring torsion). If the generalization I referred to above is attainable, all three of the 7-limit cases could be expressed in a single way. Anyhow, if this is all correct, I want details, details, details. The goal, of course, is to produce complexity vs. TOP error graphs for 7-limit linear temperaments, something I currently don't know how to do. If someone can fill in the missing links on the above, preferably showing the rigorous collapse to a single formula in the 3-limit and with some intuitive guidance on how to visualize the bivector area in affine geometry (or whatever), I'd be extremely grateful.

Message: 9359 - Contents - Hide Contents Date: Tue, 20 Jan 2004 13:48:18 Subject: Re: Attn: Gene 2 From: Carl Lumma This is the 3rd version of this post I've recieved, this time with a different subject. Which version should I refer to? -Carl>All right, folks . . . I'm not sure if I missed anything important >since I last posted, but before I catch up . . . > >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 . . . > >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. 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. 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. As I understand it, the determinant >measures *area* not only in Euclidean geometry, but also in 'affine' >geometry, where angles are left undefined . . . Anyhow, since both of >these methods could be used to address a 3-limit TOP temperament, in >5-limit could they be still both be expressible in a single form in a >general enough framework, say exterior algebra? > >In the 7-limit, the two methods give us, respectively, the complexity >of a 'planar' temperament as a distance, and the cardinality of a 7- >limit equal temperament as a volume. But 7-limit 'linear' >temperaments get left out in the cold. The appropriate measure would >seem to have to be an *area* of some sort -- from what I understand >from exterior algebra, this is the area of the *bivector* formed by >taking the *wedge product* of any two linearly independent commatic >unison vectors (barring torsion). If the generalization I referred to >above is attainable, all three of the 7-limit cases could be >expressed in a single way. Anyhow, if this is all correct, I want >details, details, details. The goal, of course, is to produce >complexity vs. TOP error graphs for 7-limit linear temperaments, >something I currently don't know how to do. If someone can fill in >the missing links on the above, preferably showing the rigorous >collapse to a single formula in the 3-limit and with some intuitive >guidance on how to visualize the bivector area in affine geometry (or >whatever), I'd be extremely grateful. > > > > > >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: 9360 - Contents - Hide Contents Date: Tue, 20 Jan 2004 22:03:43 Subject: Re: the case of the disappearing 250047/250000 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> From tuning . . . > > I was eliminating commas where the greatest common divisor of the > entries in the monzo was greater than 1, since these are merely > powers of other commas. But when I moved to 7-limit, I was > unfortunately still checking only the first three components of the > monzo. Hence I erroneously eliminated 250047/250000 and, I think, > 1077 other commas under 600 cents where -20<e3<20, -14<e5<14, and - > 12<e7<12.And apparently 391 others under 600 with log(n*d)/log(10)<18, which is what the graphs were showing :( I've uploaded the 3 corrected graphs.

Message: 9361 - Contents - Hide Contents Date: Tue, 20 Jan 2004 22:06:15 Subject: Grassmann Algebra question From: Paul Erlich Can someone explain equation (7) here: File Not Found: /People/dfs/Papers/GrassmannLi... * [with cont.] Search for http://www.maths.utas.edu.au/People/dfs/Papers/GrassmannLinAlgpaper.pd in Wayback Machine f ? Why do we 'assume' it in accordance with what is written above it?

Message: 9362 - Contents - Hide Contents Date: Tue, 20 Jan 2004 22:35:46 Subject: Maple code for ?(x) From: Gene Ward Smith This shows how easy it is to compute, and may inspire anyone who wants to code it in something other than Maple. Paul was using a log-based system for defining harmonic entropy, and that makes more sense, so the actual function to do Stieltjes integration with would be ?(2^x) or ?(2^(x/1200)), but either way you want a ? function to start out with. quest := proc(x) # Minkowsky question mark function local i, j, d, l, s, t; l := convert(x, confrac); d := nops(l); s := l[1]; for i from 2 to d do t := 1; for j from 2 to i do t := t - l[j] od; s := s + (-1)^i * 2^t od; s end:

Message: 9363 - Contents - Hide Contents Date: Tue, 20 Jan 2004 22:53:08 Subject: Re: Maple code for ?(x) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> This shows how easy it is to compute, and may inspire anyone who wants > to code it in something other than Maple. Paul was using a log-based > system for defining harmonic entropy, and that makes more sense, so > the actual function to do Stieltjes integration with would be ?(2^x) > or ?(2^(x/1200)), but either way you want a ? function to start out with. > > quest := proc(x) > # Minkowsky question mark function > local i, j, d, l, s, t; > l := convert(x, confrac); > d := nops(l); > s := l[1]; > for i from 2 to d do > t := 1; > for j from 2 to i do > t := t - l[j] od; > s := s + (-1)^i * 2^t od; > s end:what is nops?

Message: 9364 - Contents - Hide Contents Date: Tue, 20 Jan 2004 23:54:03 Subject: Re: Maple code for ?(x) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> quest := proc(x) >> # Minkowsky question mark function >> local i, j, d, l, s, t; >> l := convert(x, confrac); >> d := nops(l); >> s := l[1]; >> for i from 2 to d do >> t := 1; >> for j from 2 to i do >> t := t - l[j] od; >> s := s + (-1)^i * 2^t od; >> s end: >> what is nops?Nops gives the number of operands for a variety of expressions. Convert(x, confrac) returns a list of integers, and for a list, nops gives the number of elements of the list. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9365 - Contents - Hide Contents Date: Wed, 21 Jan 2004 01:18:24 Subject: Re: Attn: Gene 2 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: '> geometry, where angles are left undefined . . . Anyhow, since both of > these methods could be used to address a 3-limit TOP temperament, in > 5-limit could they be still both be expressible in a single form in a > general enough framework, say exterior algebra?That was my suggestion. You normalize by dividing each coordinate by log2(p) and take the wedge product up to the (normalized) multival, and then measure complexity by taking the max of the absolute values of the coefficients. If you start from the monzo side, you get the same normalized coefficients up to a constant factor, but now you might rather take the sum of the absolute values (L1 vs L infinity.) The goal, of course, is to produce> complexity vs. TOP error graphs for 7-limit linear temperaments, > something I currently don't know how to do.Why not use the formula I gave before: || <<w1 w2 w3 w4 w5 w6|| || = max(|w1|/p3, |w2|/p5, |w3|/p7, |w4|/p3p5, |w5|/p3p7, |w6|/p5p7) to measure complexity? Here p3=log2(3), etc. The corresponding log-flat badness would be BAD = ||TOP - JIP|| * ||Wedgie||^2 Here ||Wedgie|| is as above, and ||TOP-JIP|| is the maximum weighted error, or distance to the JIP. If <t2 t3 t5 t7| is the top tuning for the temperament given by Wedgie, then this is ||TOP - JIP|| = Max(|t2-1|, |t3/p3-1|, |t5/p5-1|, |t7/p7-1|) Your choice whether to do everything in log2 or cents terms, of course.

Message: 9366 - Contents - Hide Contents Date: Wed, 21 Jan 2004 13:38:47 Subject: Re: 114 7-limit temperaments From: Carl Lumma>> >his 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). >> >> If you could make this list finite with badness bounds only, >> I'd be more impressed by claims that log-flat badness is >> desirable (allows the comparison of ennealimmal with all >> temperaments in a sense, not just the others on the list, or >> whatever). >>Log flat badness is deliberately designed not to be finite. and it >seems to me your objection is strange--do you think epimericity >allows comparison of one comma with another, while a log flat badness >does not?What's the rub again? Within equally-sized complexity bins, log-flat badness returns roughly the same number of temperaments? I guess that makes sense.>As for meantone, magic and pajara being too low, they are all near tht >top of the list. It would seem the list is doing exactly what you want >it to do.I did say it was attractive...>You can make the list finite by bounding complexity, which is what >I've done. >>> And I don't see how you figure schismic is less complex than >> miracle in light of the maps given. >>Schismic gets to 3/2 in one generator step, and miracle takes six.What kind of complexity is this? Do you always use the same kind? It seems you happily switch between geometric, weighted map-based, and 3 other flavors when giving these lists. Providing a template at the top of the lists showing units or something for each key might help your readers. -Carl

Message: 9367 - Contents - Hide Contents Date: Wed, 21 Jan 2004 01:51:45 Subject: Re: Compton/Erlich temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Gene W Smith <genewardsmith@j...> > wrote: >> >>>> On Thu, 18 Jul 2002 01:22:19 -0700 Carl Lumma <carl@l...> writes: >>>>> How do we classify the Compton/Erlich scheme of tuning multiple >>> 12-et keyboards 15 cents apart? Some sort of planar temperament >>> with the following commas? >>> >>> 531441/524288 (pythagorean comma) >>> 5120/5103 (difference between syntonic comma and 64/63) >>> >>> Is this right? >>>> I think it's another system, discussed below. The wedgie you find > from>> the pyth comma and 5120/5103 gives what we are calling a linear >> temperament. It is [0,12,12,-6,-19,19], and has a TM reduced basis >> <50/49, 3645/3584>. The mapping is >> >> [[12, 19, 28, 34], [0, 0, -1, -1]] >> >> However, the rms optimum is 23.4 cents apart, not 15. >> >> I think what you want is the linear temperament with wedgie >> [0,12,12,-6,-19,19], TM reduced basis >> <225/224, 250047/250000> and mapping [[12,19,28,34],[0,0,-1,-1]]. >> This is the same mapping as above. Did you mean for the last term to > be -2, not -1? I know Waage proposed this system; who's Compton?That's it. Maybe this should be the Waage or Compton temperament? Wedgie: <<0 12 24 19 38 22|| TM basis: {225/224, 250047/250000} TOP tuning: <1200.617051 1900.976998 2784.880964 3368.630668| TOP error: 0.617051 cents TOP complexity: 8.548972490 TOP badness: 45.097 Not that the last two of these figures mean much without something to compare them to.

Message: 9368 - Contents - Hide Contents Date: Wed, 21 Jan 2004 21:42:14 Subject: Re: TOP take on 7-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> 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.That was in a discussion of commas, not wedgies. How do you apply it to wedgies?

Message: 9369 - Contents - Hide Contents Date: Wed, 21 Jan 2004 05:44:14 Subject: TOP take on 7-limit temperaments From: Gene Ward Smith Here is my old list of 45 7-limit linear temperaments, this time using top to sort it all out. The goodness winner is ennealimmal by a huge margin, with meantone in second and magic coming up third. Ones with badness under 30 are dominant seventh, augmented, pajara, meantone, magic, miracle, schismic, and ennealimmal. Decimal <4 2 2 -6 -8 -1] [1207.657798 1914.092323 2768.532858 3372.361757] [[2 4 5 6] [0 -2 -1 -1]] [600.0000000 249.0224992] err: 7.657798 comp: 2.523719 bad: 48.773723 Dominant seventh <1 4 -2 4 -6 -16] [1195.228951 1894.576888 2797.391744 3382.219933] [[1 2 4 2] [0 -1 -4 2]] [1200. 497.7740225] err: 4.771049 comp: 2.454561 bad: 28.744957 Diminished <4 4 4 -3 -5 -2] [1194.128460 1892.648830 2788.245174 3385.309404] [[4 6 9 11] [0 1 1 1]] [300.0000000 85.69820677] err: 5.871540 comp: 2.523719 bad: 37.396767 Blackwood <0 5 0 8 0 -14] [1195.893464 1913.429542 2786.313713 3348.501698] [[5 8 12 14] [0 0 -1 0]] [240.0000000 90.61325640] err: 7.239629 comp: 2.173813 bad: 34.210608 Augmented <3 0 6 -7 1 14] [1199.976630 1892.649878 2799.945472 3385.307546] [[3 5 7 9] [0 -1 0 -2]] [400.0000000 110.2596913] err: 5.870879 comp: 2.147741 bad: 27.081145 Pajara <2 -4 -4 -11 -12 2] [1196.893422 1901.906680 2779.100462 3377.547174] [[2 3 5 6] [0 1 -2 -2]] [600.0000000 108.8143299] err: 3.106578 comp: 2.988993 bad: 27.754421 Hexadecimal <1 -3 5 -7 5 20] [1208.959294 1887.754858 2799.450479 3393.977822] [[1 2 1 5] [0 -1 3 -5]] [1200. 526.8909182] err: 8.959294 comp: 3.068202 bad: 84.341555 Negri <4 -3 2 -14 -8 13] [1203.187308 1907.006766 2780.900506 3359.878000] [[1 2 2 3] [0 -4 3 -2]] [1200. 125.4687958] err: 3.187309 comp: 3.804173 bad: 46.125884 Kleismic <6 5 3 -6 -12 -7] [1203.187308 1907.006766 2792.359613 3359.878000] [[1 0 1 2] [0 6 5 3]] [1200. 316.6640534] err: 3.187309 comp: 3.785579 bad: 45.676063 Tripletone <3 0 -6 -7 -18 -14] [1197.060039 1902.640406 2793.140092 3377.079420] [[3 5 7 8] [0 -1 0 2]] [400.0000000 88.72066409] err: 2.939961 comp: 4.045351 bad: 48.112067 Hemifourth <2 8 1 8 -4 -20] [1203.668842 1902.376967 2794.832500 3358.526166] [[1 2 4 3] [0 -2 -8 -1]] [1200. 252.7423121] err: 3.668842 comp: 3.445412 bad: 43.552336 Meantone <1 4 10 4 13 12] [1201.698521 1899.262909 2790.257556 3370.548328] [[1 2 4 7] [0 -1 -4 -10]] [1200. 503.3520320] err: 1.698521 comp: 3.562072 bad: 21.551439 Injera <2 8 8 8 7 -4] [1201.777814 1896.276546 2777.994928 3378.883835] [[2 3 4 5] [0 1 4 4]] [600.0000000 93.65102578] err: 3.582707 comp: 3.445412 bad: 42.529834 Double wide <8 6 6 -9 -13 -3] [1198.553882 1907.135354 2778.724633 3378.001574] [[2 5 6 7] [0 -4 -3 -3]] [600.0000000 274.3886321] err: 3.268439 comp: 5.047438 bad: 83.268810 Porcupine <3 5 -6 1 -18 -28] [1196.905961 1906.858938 2779.129576 3367.717888] [[1 2 3 2] [0 -3 -5 6]] [1200. 162.3778142] err: 3.094040 comp: 4.295482 bad: 57.088650 Superpythagorean <1 9 -2 12 -6 -30] [1197.596121 1905.765059 2780.732078 3374.046608] [[1 2 6 2] [0 -1 -9 2]] [1200. 489.6151808] err: 2.403879 comp: 4.602303 bad: 50.917015 Muggles <5 1 -7 -10 -25 -19] [1203.148010 1896.965522 2785.689126 3359.988323] [[1 0 2 5] [0 5 1 -7]] [1200. 377.6398800] err: 3.148011 comp: 5.618543 bad: 99.376477 Beatles <2 -9 -4 -19 -12 16] [1197.104145 1906.544822 2793.037680 3369.535226] [[1 1 5 4] [0 2 -9 -4]] [1200. 356.3080304] err: 2.895855 comp: 5.162806 bad: 77.187771 Flattone <1 4 -9 4 -17 -32] [1202.536420 1897.934872 2781.593812 3361.705278] [[1 2 4 -1] [0 -1 -4 9]] [1200. 506.5439220] err: 2.536420 comp: 4.909123 bad: 61.126418 Magic <5 1 12 -10 5 25] [1201.276744 1903.978592 2783.349206 3368.271877] [[1 0 2 -1] [0 5 1 12]] [1200. 380.5064473] err: 1.276744 comp: 4.274486 bad: 23.327687 Nonkleismic <10 9 7 -9 -17 -9] [1198.828458 1900.098151 2789.033948 3368.077085] [[1 -1 0 1] [0 10 9 7]] [1200. 309.9514712] err: 1.171542 comp: 6.309298 bad: 46.635848 Semisixths <7 9 13 -2 1 5] [1198.389531 1903.732520 2790.053106 3364.304748] [[1 -1 -1 -2] [0 7 9 13]] [1200. 443.6203855] err: 1.610469 comp: 4.630693 bad: 34.533812 Orwell <7 -3 8 -21 -7 27] [1199.532657 1900.455530 2784.117029 3371.481834] [[1 0 3 1] [0 7 -3 8]] [1200. 271.3263635] err: .946061 comp: 5.706260 bad: 30.805067 Miracle <6 -7 -2 -25 -20 15] [1200.631014 1900.954868 2784.848544 3368.451756] [[1 1 3 3] [0 6 -7 -2]] [1200. 116.5729472] err: .631014 comp: 6.793166 bad: 29.119472 Quartaminorthirds <9 5 -3 -13 -30 -21] [1199.792743 1900.291122 2788.751252 3365.878770] [[1 1 2 3] [0 9 5 -3]] [1200. 77.70708732] err: 1.049791 comp: 6.742251 bad: 47.721346 Supermajor seconds <3 12 -1 12 -10 -36] [1201.698521 1899.262909 2790.257556 3372.574099] [[1 1 0 3] [0 3 12 -1]] [1200. 232.1235474] err: 1.698521 comp: 5.522763 bad: 51.806440 Schismic <1 -8 -14 -15 -25 -10] [1200.760625 1903.401919 2784.194017 3371.388750] [[1 2 -1 -3] [0 -1 8 14]] [1200. 497.8598384] err: .912904 comp: 5.618543 bad: 28.818563 Superkleismic <9 10 -3 -5 -30 -35] [1201.371918 1904.129438 2783.128219 3369.863245] [[1 4 5 2] [0 -9 -10 3]] [1200. 321.8581276] err: 1.371918 comp: 6.742251 bad: 62.364566 Squares <4 16 9 16 3 -24] [1201.698521 1899.262909 2790.257556 3372.067656] [[1 3 8 6] [0 -4 -16 -9]] [1200. 425.9591136] err: 1.698521 comp: 6.890825 bad: 80.651668 Semififth <2 8 -11 8 -23 -48] [1201.698521 1899.262909 2790.257556 3373.586984] [[1 1 0 6] [0 2 8 -11]] [1200. 348.3528922] err: 1.698521 comp: 7.363684 bad: 92.100337 Diaschismic <2 -4 -16 -11 -31 -26] [1198.732403 1901.885616 2789.256983 3365.267311] [[2 3 5 7] [0 1 -2 -8]] [600.0000000 103.7370914] err: 1.267597 comp: 6.966993 bad: 61.527901 Octacot <8 18 11 10 -5 -25] [1199.031259 1903.490418 2784.064367 3366.693863] [[1 1 1 2] [0 8 18 11]] [1200. 88.14540671] err: .968741 comp: 7.752178 bad: 58.217715 Tritonic <5 -11 -12 -29 -33 3] [1201.023211 1900.333250 2785.201472 3365.953391] [[1 4 -3 -3] [0 -5 11 12]] [1200. 580.4242150] err: 1.023211 comp: 7.880073 bad: 63.536850 Supersupermajor <3 17 -1 20 -10 -50] [1200.231588 1903.372996 2784.236389 3366.314293] [[1 1 -1 3] [0 3 17 -1]] [1200. 234.4104084] err: .894655 comp: 7.670504 bad: 52.638504 Shrutar <4 -8 14 -22 11 55] [1198.920873 1903.665377 2786.734051 3365.796415] [[2 3 5 5] [0 2 -4 7]] [600.0000000 52.89351739] err: 1.079127 comp: 8.437555 bad: 76.825572 Catakleismic <6 5 22 -6 18 37] [1200.536356 1901.438376 2785.068335 3370.331646] [[1 0 1 -3] [0 6 5 22]] [1200. 316.7238784] err: .536356 comp: 7.836558 bad: 32.938503 Hemiwuerschmidt <16 2 5 -34 -37 6] [1199.692003 1901.466838 2787.028860 3368.496143] [[1 -1 2 2] [0 16 2 5]] [1200. 193.9099372] err: .307997 comp: 10.094876 bad: 31.386987 Hemikleismic <12 10 -9 -12 -48 -49] [1199.411231 1902.888178 2785.151380 3370.478790] [[1 0 1 4] [0 12 10 -9]] [1200. 158.7324720] err: .588769 comp: 10.787602 bad: 68.516458 Hemithird <15 -2 -5 -38 -50 -6] [1200.363229 1901.194685 2787.427555 3367.479202] [[1 4 2 2] [0 -15 2 5]] [1200. 193.2841225] err: .479706 comp: 11.237086 bad: 60.573479 Wizard <12 -2 20 -31 -2 52] [1200.639571 1900.941305 2784.828674 3368.342104] [[2 1 5 2] [0 6 -1 10]] [600.0000000 216.7129477] err: .639571 comp: 8.423526 bad: 45.381303 Duodecimal <0 12 24 19 38 22] [1200.617051 1900.976998 2785.844725 3370.558188] [[12 19 28 34] [0 0 -1 -2]] [100.0000000 15.94743281] err: .617051 comp: 8.548972 bad: 45.097159 Slender <13 -10 6 -46 -27 42] [1200.337238 1901.055858 2784.996493 3370.418508] [[1 2 2 3] [0 -13 10 -6]] [1200. 38.46612667] err: .567296 comp: 12.499426 bad: 88.631905 Amity <5 13 -17 9 -41 -76] [1199.723894 1902.392618 2786.717797 3369.601033] [[1 3 6 -2] [0 -5 -13 17]] [1200. 339.4147297] err: .276106 comp: 11.659166 bad: 37.532790 Hemififth <2 25 13 35 15 -40] [1199.700353 1902.429930 2785.617954 3368.041901] [[1 1 -5 -1] [0 2 25 13]] [1200. 351.4712147] err: .299647 comp: 10.766914 bad: 34.737019 Ennealimmal <18 27 18 1 -22 -34] [1200.036377 1902.012658 2786.350298 3368.723784] [[9 15 22 26] [0 -2 -3 -2]] [133.3333333 48.99915090] err: .036377 comp: 11.628267 bad: 4.918774

Message: 9370 - Contents - Hide Contents Date: Wed, 21 Jan 2004 13:43:27 Subject: Re: Maple code for ?(x) From: Carl Lumma>> >bviously, 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 . . .By the way, somebody once told me that maple code can be executed in Matlab. Dunno if that's true... -Carl

Message: 9371 - Contents - Hide Contents Date: Wed, 21 Jan 2004 09:08:14 Subject: 114 7-limit temperaments From: Gene Ward Smith This is a list of linear temperaments with top complexity < 15, top error < 15, and top badness < 100. I searched extensively without adding to the list, which is probably complete. Most of the names are old ones. In some cases I extended a 5-limit name to what seemed like the appropriate 7-limit temperament, and in the case of The Temperament Formerly Known as Duodecimal, am suggesting Waage or Compton if one of these gentlemen invented it. There are a few new names being suggested, none of which are yet etched in stone--not even when the name is Bond, James Bond. Number 1 Ennealimmal [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784] TOP generators [133.3373752, 49.02398564] bad: 4.918774 comp: 11.628267 err: .036377 Number 2 Meantone [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328] TOP generators [1201.698520, 504.1341314] bad: 21.551439 comp: 3.562072 err: 1.698521 Number 3 Magic [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877] TOP generators [1201.276744, 380.7957184] bad: 23.327687 comp: 4.274486 err: 1.276744 Number 4 Beep [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548] TOP generators [1194.642673, 254.8994697] bad: 23.664749 comp: 1.292030 err: 14.176105 Number 5 Augmented [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546] TOP generators [399.9922103, 107.3111730] bad: 27.081145 comp: 2.147741 err: 5.870879 Number 6 Pajara [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174] TOP generators [598.4467109, 106.5665459] bad: 27.754421 comp: 2.988993 err: 3.106578 Number 7 Dominant Seventh [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933] TOP generators [1195.228951, 495.8810151] bad: 28.744957 comp: 2.454561 err: 4.771049 Number 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 Number 10 Orwell [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834] TOP generators [1199.532657, 271.4936472] bad: 30.805067 comp: 5.706260 err: .946061 Number 11 Hemiwuerschmidt [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]] TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143] TOP generators [1199.692003, 193.8224275] bad: 31.386908 comp: 10.094876 err: .307997 Number 12 Catakleismic [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]] TOP tuning [1200.536356, 1901.438376, 2785.068335, 3370.331646] TOP generators [1200.536355, 316.9063960] bad: 32.938503 comp: 7.836558 err: .536356 Number 13 Father [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]] TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477] TOP generators [1185.869125, 447.3863410] bad: 33.256527 comp: 1.534101 err: 14.130876 Number 14 Blackwood [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698] TOP generators [239.1786927, 83.83059859] bad: 34.210608 comp: 2.173813 err: 7.239629 Number 15 Semisixths [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]] TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748] TOP generators [1198.389531, 443.1602931] bad: 34.533812 comp: 4.630693 err: 1.610469 Number 16 Hemififths [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]] TOP tuning [1199.700353, 1902.429930, 2785.617954, 3368.041901] TOP generators [1199.700353, 351.3647888] bad: 34.737019 comp: 10.766914 err: .299647 Number 17 Diminished [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]] TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404] TOP generators [298.5321149, 101.4561401] bad: 37.396767 comp: 2.523719 err: 5.871540 Number 18 Amity [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]] TOP tuning [1199.723894, 1902.392618, 2786.717797, 3369.601033] TOP generators [1199.723894, 339.3558130] bad: 37.532790 comp: 11.659166 err: .276106 Number 19 Pelogic [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]] TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957] TOP generators [1209.734056, 532.9412251] bad: 39.824125 comp: 2.022675 err: 9.734056 Number 20 Parakleismic [13, 14, 35, -8, 19, 42] [[1, 5, 6, 12], [0, -13, -14, -35]] TOP tuning [1199.738066, 1902.291445, 2786.921905, 3368.090564] TOP generators [1199.738066, 315.1076065] bad: 40.713036 comp: 12.467252 err: .261934 Number 21 {21/20, 28/27} [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]] TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876] TOP generators [1214.253642, 509.4012304] bad: 42.300772 comp: 1.722706 err: 14.253642 Number 22 Injera [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]] TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835] TOP generators [600.8889070, 93.60982493] bad: 42.529834 comp: 3.445412 err: 3.582707 Number 23 Dicot [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]] TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113] TOP generators [1204.048159, 356.3998255] bad: 42.920570 comp: 2.137243 err: 9.396316 Number 24 Hemifourths [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]] TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166] TOP generators [1203.668841, 252.4803582] bad: 43.552336 comp: 3.445412 err: 3.668842 Number 25 Waage? Compton? Duodecimal? [0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]] TOP tuning [1200.617051, 1900.976998, 2785.844725, 3370.558188] TOP generators [100.0514209, 16.55882096] bad: 45.097159 comp: 8.548972 err: .617051 Number 26 Wizard [12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]] TOP tuning [1200.639571, 1900.941305, 2784.828674, 3368.342104] TOP generators [600.3197857, 216.7702531] bad: 45.381303 comp: 8.423526 err: .639571 Number 27 Kleismic [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]] TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000] TOP generators [1203.187309, 317.8344609] bad: 45.676063 comp: 3.785579 err: 3.187309 Number 28 Negri [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]] TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000] TOP generators [1203.187309, 124.8419629] bad: 46.125886 comp: 3.804173 err: 3.187309 Number 29 Nonkleismic [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085] TOP generators [1198.828458, 309.8926610] bad: 46.635848 comp: 6.309298 err: 1.171542 Number 30 Quartaminorthirds [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]] TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770] TOP generators [1199.792743, 77.83315314] bad: 47.721352 comp: 6.742251 err: 1.049791 Number 31 Tripletone [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420] TOP generators [399.0200131, 92.45965769] bad: 48.112067 comp: 4.045351 err: 2.939961 Number 32 Decimal [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]] TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757] TOP generators [603.8288989, 250.6116362] bad: 48.773723 comp: 2.523719 err: 7.657798 Number 33 {1029/1024, 4375/4374} [12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]] TOP tuning [1200.421488, 1901.286959, 2785.446889, 3367.642640] TOP generators [600.2107440, 183.2944602] bad: 50.004574 comp: 10.892116 err: .421488 Number 34 Superpythagorean [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]] TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608] TOP generators [1197.596121, 489.4271829] bad: 50.917015 comp: 4.602303 err: 2.403879 Number 35 Supermajor seconds [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099] TOP generators [1201.698520, 232.5214630] bad: 51.806440 comp: 5.522763 err: 1.698521 Number 36 Supersupermajor [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]] TOP tuning [1200.231588, 1903.372996, 2784.236389, 3366.314293] TOP generators [1200.231587, 234.3804692] bad: 52.638504 comp: 7.670504 err: .894655 Number 37 {6144/6125, 10976/10935} Hendecatonic? [11, -11, 22, -43, 4, 82] [[11, 17, 26, 30], [0, 1, -1, 2]] TOP tuning [1199.662182, 1902.490429, 2787.098101, 3368.740066] TOP generators [109.0601984, 48.46705632] bad: 53.458690 comp: 12.579627 err: .337818 Number 38 {3136/3125, 5120/5103} Misty [3, -12, -30, -26, -56, -36] [[3, 5, 6, 6], [0, -1, 4, 10]] TOP tuning [1199.661465, 1902.491566, 2787.099767, 3368.765021] TOP generators [399.8871550, 96.94420930] bad: 53.622498 comp: 12.585536 err: .338535 Number 39 {1728/1715, 4000/3993} [11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]] TOP tuning [1199.083445, 1901.293958, 2784.185538, 3371.399002] TOP generators [1199.083445, 45.17026643] bad: 55.081549 comp: 7.752178 err: .916555 Number 40 {36/35, 160/147} Hystrix? [3, 5, 1, 1, -7, -12] [[1, 2, 3, 3], [0, -3, -5, -1]] TOP tuning [1187.933715, 1892.564743, 2758.296667, 3402.700250] TOP generators [1187.933715, 161.1008955] bad: 55.952057 comp: 2.153383 err: 12.066285 Number 41 {28/27, 50/49} [2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]] TOP tuning [1191.599639, 1915.269258, 2766.808679, 3362.608498] TOP generators [595.7998193, 127.8698005] bad: 56.092257 comp: 2.584059 err: 8.400361 Number 42 Porcupine [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]] TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888] TOP generators [1196.905960, 162.3176609] bad: 57.088650 comp: 4.295482 err: 3.094040 Number 43 [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]] TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174] TOP generators [598.4467109, 162.3159606] bad: 57.621529 comp: 4.306766 err: 3.106578 Number 44 Octacot [8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]] TOP tuning [1199.031259, 1903.490418, 2784.064367, 3366.693863] TOP generators [1199.031259, 88.05739491] bad: 58.217715 comp: 7.752178 err: .968741 Number 45 {25/24, 81/80} Jamesbond? [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]] TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906] TOP generators [172.7759159, 86.69241190] bad: 58.637859 comp: 2.493450 err: 9.431411 Number 46 Hemithirds [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]] TOP tuning [1200.363229, 1901.194685, 2787.427555, 3367.479202] TOP generators [1200.363229, 193.3505488] bad: 60.573479 comp: 11.237086 err: .479706 Number 47 [12, 34, 20, 26, -2, -49] [[2, 4, 7, 7], [0, -6, -17, -10]] TOP tuning [1200.284965, 1901.503343, 2786.975381, 3369.219732] TOP generators [600.1424823, 83.17776441] bad: 61.101493 comp: 14.643003 err: .284965 Number 48 Flattone [1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]] TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278] TOP generators [1202.536419, 507.1379663] bad: 61.126418 comp: 4.909123 err: 2.536420 Number 49 Diaschismic [2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]] TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311] TOP generators [599.3662015, 103.7870123] bad: 61.527901 comp: 6.966993 err: 1.267597 Number 50 Superkleismic [9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]] TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245] TOP generators [1201.371918, 322.3731369] bad: 62.364585 comp: 6.742251 err: 1.371918 Number 51 [8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]] TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814] TOP generators [1201.135545, 387.5841360] bad: 62.703297 comp: 6.411729 err: 1.525246 Number 52 Tritonic [5, -11, -12, -29, -33, 3] [[1, 4, -3, -3], [0, -5, 11, 12]] TOP tuning [1201.023211, 1900.333250, 2785.201472, 3365.953391] TOP generators [1201.023211, 580.7519186] bad: 63.536850 comp: 7.880073 err: 1.023211 Number 53 [1, 33, 27, 50, 40, -30] [[1, 2, 16, 14], [0, -1, -33, -27]] TOP tuning [1199.680495, 1902.108988, 2785.571846, 3369.722869] TOP generators [1199.680495, 497.2520023] bad: 64.536886 comp: 14.212326 err: .319505 Number 54 [6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]] TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076] TOP generators [1202.659696, 82.97467050] bad: 64.556006 comp: 4.306766 err: 3.480440 Number 55 [0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]] TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906] TOP generators [99.80617249, 24.58395811] bad: 65.630949 comp: 4.295482 err: 3.557008 Number 56 [2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]] TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460] TOP generators [1204.567524, 355.9419091] bad: 66.522610 comp: 2.696901 err: 9.146173 Number 57 [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]] TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203] TOP generators [1185.869125, 223.6931705] bad: 66.774944 comp: 2.173813 err: 14.130876 Number 58 [5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]] TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528] TOP generators [1194.335372, 99.13879319] bad: 67.244049 comp: 3.445412 err: 5.664628 Number 59 [3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]] TOP tuning [1193.415676, 1912.390908, 2789.512955, 3350.341372] TOP generators [1193.415676, 158.1468146] bad: 67.670842 comp: 3.205865 err: 6.584324 Number 60 [3, 0, 9, -7, 6, 21] [[3, 5, 7, 9], [0, -1, 0, -3]] TOP tuning [1193.415676, 1912.390908, 2784.636577, 3350.341372] TOP generators [397.8052253, 76.63521863] bad: 68.337269 comp: 3.221612 err: 6.584324 Number 61 Hemikleismic [12, 10, -9, -12, -48, -49] [[1, 0, 1, 4], [0, 12, 10, -9]] TOP tuning [1199.411231, 1902.888178, 2785.151380, 3370.478790] TOP generators [1199.411231, 158.5740148] bad: 68.516458 comp: 10.787602 err: .588769 Number 62 [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]] TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105] TOP generators [592.7342285, 146.7842660] bad: 68.668284 comp: 2.173813 err: 14.531543 Number 63 [8, 13, 23, 2, 14, 17] [[1, 2, 3, 4], [0, -8, -13, -23]] TOP tuning [1198.975478, 1900.576277, 2788.692580, 3365.949709] TOP generators [1198.975478, 62.17183489] bad: 68.767371 comp: 8.192765 err: 1.024522 Number 64 [3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]] TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070] TOP generators [1202.900537, 570.4479508] bad: 69.388565 comp: 4.891080 err: 2.900537 Number 65 [3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]] TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323] TOP generators [1202.624742, 569.0491468] bad: 70.105427 comp: 5.168119 err: 2.624742 Number 66 [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]] TOP tuning [1199.379215, 1900.971080, 2787.482526, 3370.568669] TOP generators [1199.379215, 464.5804210] bad: 71.416917 comp: 10.725806 err: .620785 Number 67 [11, 13, 17, -5, -4, 3] [[1, 3, 4, 5], [0, -11, -13, -17]] TOP tuning [1198.514750, 1899.600936, 2789.762356, 3371.570447] TOP generators [1198.514750, 154.1766650] bad: 71.539673 comp: 6.940227 err: 1.485250 Number 68 [3, -24, -1, -45, -10, 65] [[1, 1, 7, 3], [0, 3, -24, -1]] TOP tuning [1200.486331, 1902.481504, 2787.442939, 3367.460603] TOP generators [1200.486331, 233.9983907] bad: 72.714599 comp: 12.227699 err: .486331 Number 69 [23, -1, 13, -55, -44, 33] [[1, 9, 2, 7], [0, -23, 1, -13]] TOP tuning [1199.671611, 1901.434518, 2786.108874, 3369.747810] TOP generators [1199.671611, 386.7656515] bad: 73.346343 comp: 14.944966 err: .328389 Number 70 [6, 29, -2, 32, -20, -86] [[1, 4, 14, 2], [0, -6, -29, 2]] TOP tuning [1200.422358, 1901.285580, 2787.294397, 3367.645998] TOP generators [1200.422357, 483.4006416] bad: 73.516606 comp: 13.193267 err: .422358 Number 71 [7, -15, -16, -40, -45, 5] [[1, 5, -5, -5], [0, -7, 15, 16]] TOP tuning [1200.210742, 1900.961474, 2784.858222, 3370.585685] TOP generators [1200.210742, 585.7274621] bad: 74.053446 comp: 10.869066 err: .626846 Number 72 [5, 3, 7, -7, -3, 8] [[1, 1, 2, 2], [0, 5, 3, 7]] TOP tuning [1192.540126, 1890.131381, 2803.635005, 3361.708008] TOP generators [1192.540126, 139.5182509] bad: 74.239244 comp: 3.154649 err: 7.459874 Number 73 [4, 21, -3, 24, -16, -66] [[1, 0, -6, 4], [0, 4, 21, -3]] TOP tuning [1199.274449, 1901.646683, 2787.998389, 3370.862785] TOP generators [1199.274449, 475.4116708] bad: 74.381278 comp: 10.125066 err: .725551 Number 74 [3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]] TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692] TOP generators [1195.486066, 559.3589487] bad: 74.989802 comp: 4.075900 err: 4.513934 Number 75 [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]] TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936] TOP generators [399.8000105, 155.5708520] bad: 76.576420 comp: 3.804173 err: 5.291448 Number 76 [13, 2, 30, -27, 11, 64] [[1, 6, 3, 13], [0, -13, -2, -30]] TOP tuning [1200.672456, 1900.889183, 2786.148822, 3370.713730] TOP generators [1200.672456, 407.9342733] bad: 76.791305 comp: 10.686216 err: .672456 Number 77 Shrutar [4, -8, 14, -22, 11, 55] [[2, 3, 5, 5], [0, 2, -4, 7]] TOP tuning [1198.920873, 1903.665377, 2786.734051, 3365.796415] TOP generators [599.4604367, 52.64203308] bad: 76.825572 comp: 8.437555 err: 1.079127 Number 78 [12, 10, 25, -12, 6, 30] [[1, 6, 6, 12], [0, -12, -10, -25]] TOP tuning [1199.028703, 1903.494472, 2785.274095, 3366.099130] TOP generators [1199.028703, 440.8898120] bad: 77.026097 comp: 8.905180 err: .971298 Number 79 Beatles [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]] TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226] TOP generators [1197.104145, 354.7203384] bad: 77.187771 comp: 5.162806 err: 2.895855 Number 80 [6, -12, 10, -33, -1, 57] [[2, 4, 3, 7], [0, -3, 6, -5]] TOP tuning [1199.025947, 1903.033657, 2788.575394, 3371.560420] TOP generators [599.5129735, 165.0060791] bad: 78.320453 comp: 8.966980 err: .974054 Number 81 [4, 4, 0, -3, -11, -11] [[4, 6, 9, 11], [0, 1, 1, 0]] TOP tuning [1212.384652, 1905.781495, 2815.069985, 3334.057793] TOP generators [303.0961630, 63.74881402] bad: 78.879803 comp: 2.523719 err: 12.384652 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 Number 83 [1, 6, 5, 7, 5, -5] [[1, 2, 5, 5], [0, -1, -6, -5]] TOP tuning [1211.970043, 1882.982932, 2814.107292, 3355.064446] TOP generators [1211.970043, 540.9571536] bad: 79.928319 comp: 2.584059 err: 11.970043 Number 84 Squares [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]] TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656] TOP generators [1201.698520, 426.4581630] bad: 80.651668 comp: 6.890825 err: 1.698521 Number 85 [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]] TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166] TOP generators [199.0788921, 88.83392059] bad: 80.672767 comp: 3.820609 err: 5.526647 Number 86 [7, 26, 25, 25, 20, -15] [[1, 5, 15, 15], [0, -7, -26, -25]] TOP tuning [1199.352846, 1902.980716, 2784.811068, 3369.637284] TOP generators [1199.352846, 584.8262161] bad: 81.144087 comp: 11.197591 err: .647154 Number 87 [18, 15, -6, -18, -60, -56] [[3, 6, 8, 8], [0, -6, -5, 2]] TOP tuning [1200.448679, 1901.787880, 2785.271912, 3367.566305] TOP generators [400.1495598, 83.18491309] bad: 81.584166 comp: 13.484503 err: .448679 Number 88 [9, -2, 14, -24, -3, 38] [[1, 3, 2, 5], [0, -9, 2, -14]] TOP tuning [1201.918556, 1904.657347, 2781.858962, 3363.439837] TOP generators [1201.918557, 189.0109248] bad: 81.594641 comp: 6.521440 err: 1.918557 Number 89 [1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]] TOP tuning [1195.155395, 1894.070902, 2774.763716, 3382.790568] TOP generators [1195.155395, 496.2398890] bad: 82.638059 comp: 4.075900 err: 4.974313 Number 90 [3, 7, -1, 4, -10, -22] [[1, 1, 1, 3], [0, 3, 7, -1]] TOP tuning [1205.820043, 1890.417958, 2803.215176, 3389.260823] TOP generators [1205.820043, 228.1993049] bad: 82.914167 comp: 3.375022 err: 7.279064 Number 91 [6, 5, -31, -6, -66, -86] [[1, 0, 1, 11], [0, 6, 5, -31]] TOP tuning [1199.976626, 1902.553087, 2785.437532, 3369.885264] TOP generators [1199.976626, 317.0921813] bad: 83.023430 comp: 14.832953 err: .377351 Number 92 [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]] TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574] TOP generators [599.2769413, 272.3123381] bad: 83.268810 comp: 5.047438 err: 3.268439 Number 93 [4, 2, 9, -6, 3, 15] [[1, 3, 3, 6], [0, -4, -2, -9]] TOP tuning [1208.170435, 1910.173796, 2767.342550, 3391.763218] TOP generators [1208.170435, 428.5843770] bad: 83.972208 comp: 3.205865 err: 8.170435 Number 94 Hexidecimal [1, -3, 5, -7, 5, 20] [[1, 2, 1, 5], [0, -1, 3, -5]] TOP tuning [1208.959294, 1887.754858, 2799.450479, 3393.977822] TOP generators [1208.959293, 530.1637287] bad: 84.341555 comp: 3.068202 err: 8.959294 Number 95 [6, 0, 15, -14, 7, 35] [[3, 5, 7, 9], [0, -2, 0, -5]] TOP tuning [1197.060039, 1902.856975, 2793.140092, 3360.572393] TOP generators [399.0200131, 46.12154491] bad: 84.758945 comp: 5.369353 err: 2.939961 Number 96 [0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]] TOP tuning [1198.015473, 1896.857833, 2778.846497, 3377.854234] TOP generators [99.83462277, 16.52294019] bad: 85.896401 comp: 5.168119 err: 3.215955 Number 97 [11, -6, 10, -35, -15, 40] [[1, 4, 1, 5], [0, -11, 6, -10]] TOP tuning [1200.950404, 1901.347958, 2784.106944, 3366.157786] TOP generators [1200.950404, 263.8594234] bad: 85.962459 comp: 9.510433 err: .950404 Number 98 Slender [13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]] TOP tuning [1200.337238, 1901.055858, 2784.996493, 3370.418508] TOP generators [1200.337239, 38.43220154] bad: 88.631905 comp: 12.499426 err: .567296 Number 99 [0, 5, 10, 8, 16, 9] [[5, 8, 12, 15], [0, 0, -1, -2]] TOP tuning [1195.598382, 1912.957411, 2770.195472, 3388.313857] TOP generators [239.1196765, 99.24064453] bad: 89.758630 comp: 3.595867 err: 6.941749 Number 100 [1, -1, -5, -4, -11, -9] [[1, 2, 2, 1], [0, -1, 1, 5]] TOP tuning [1185.210905, 1925.395162, 2815.448458, 3410.344145] TOP generators [1185.210905, 445.0266480] bad: 90.384580 comp: 2.472159 err: 14.789095 Number 101 [2, 8, -11, 8, -23, -48] [[1, 1, 0, 6], [0, 2, 8, -11]] TOP tuning [1201.698521, 1899.262909, 2790.257556, 3373.586984] TOP generators [1201.698520, 348.7821945] bad: 92.100337 comp: 7.363684 err: 1.698521 Number 102 [3, 12, 18, 12, 20, 8] [[3, 5, 8, 10], [0, -1, -4, -6]] TOP tuning [1202.260038, 1898.372926, 2784.451552, 3375.170635] TOP generators [400.7533459, 105.3938041] bad: 92.910783 comp: 6.411729 err: 2.260038 Number 103 [4, -8, -20, -22, -43, -24] [[4, 6, 10, 13], [0, 1, -2, -5]] TOP tuning [1199.003867, 1903.533834, 2787.453602, 3371.622404] TOP generators [299.7509668, 105.0280329] bad: 93.029698 comp: 9.663894 err: .996133 Number 104 [3, 0, -3, -7, -13, -7] [[3, 5, 7, 8], [0, -1, 0, 1]] TOP tuning [1205.132027, 1884.438632, 2811.974729, 3337.800149] TOP generators [401.7106756, 124.1147448] bad: 94.336372 comp: 2.921642 err: 11.051598 Number 105 [4, 7, 2, 2, -8, -15] [[1, 2, 3, 3], [0, -4, -7, -2]] TOP tuning [1190.204869, 1918.438775, 2762.165422, 3339.629125] TOP generators [1190.204869, 115.4927407] bad: 94.522719 comp: 3.014736 err: 10.400103 Number 106 [13, 19, 23, 0, 0, 0] [[1, 0, 0, 0], [0, 13, 19, 23]] TOP tuning [1200.0, 1904.187463, 2783.043215, 3368.947050] TOP generators [1200., 146.4759587] bad: 94.757554 comp: 8.202087 err: 1.408527 Number 107 [2, -6, -6, -14, -15, 3] [[2, 3, 5, 6], [0, 1, -3, -3]] TOP tuning [1206.548264, 1891.576247, 2771.109113, 3374.383246] TOP generators [603.2741324, 81.75384943] bad: 94.764743 comp: 3.804173 err: 6.548265 Number 108 [2, -6, -6, -14, -15, 3] [[2, 3, 5, 6], [0, 1, -3, -3]] TOP tuning [1206.548264, 1891.576247, 2771.109113, 3374.383246] TOP generators [603.2741324, 81.75384943] bad: 94.764743 comp: 3.804173 err: 6.548265 Number 109 [1, -13, -2, -23, -6, 32] [[1, 2, -3, 2], [0, -1, 13, 2]] TOP tuning [1197.567789, 1904.876372, 2780.666293, 3375.653987] TOP generators [1197.567789, 490.2592046] bad: 94.999539 comp: 6.249713 err: 2.432212 Number 110 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]] TOP tuning [1197.060039, 1897.499011, 2793.140092, 3360.572393] TOP generators [133.0066710, 35.40561749] bad: 95.729260 comp: 5.706260 err: 2.939961 Number 111 [5, 1, 9, -10, 0, 18] [[1, 0, 2, 0], [0, 5, 1, 9]] TOP tuning [1193.274911, 1886.640142, 2763.877849, 3395.952256] TOP generators [1193.274911, 377.3280283] bad: 99.308041 comp: 3.205865 err: 9.662601 Number 112 Muggles [5, 1, -7, -10, -25, -19] [[1, 0, 2, 5], [0, 5, 1, -7]] TOP tuning [1203.148010, 1896.965522, 2785.689126, 3359.988323] TOP generators [1203.148011, 379.3931044] bad: 99.376477 comp: 5.618543 err: 3.148011 Number 113 [11, 6, 15, -16, -7, 18] [[1, 1, 2, 2], [0, 11, 6, 15]] TOP tuning [1202.072164, 1905.239303, 2787.690040, 3363.008608] TOP generators [1202.072164, 63.92428535] bad: 99.809415 comp: 6.940227 err: 2.072164 Number 114 [1, -8, -26, -15, -44, -38] [[1, 2, -1, -8], [0, -1, 8, 26]] TOP tuning [1199.424969, 1900.336158, 2788.685275, 3365.958541] TOP generators [1199.424969, 498.5137806] bad: 99.875385 comp: 9.888635 err: 1.021376

Message: 9372 - Contents - Hide Contents Date: Wed, 21 Jan 2004 21:47:24 Subject: Re: 114 7-limit temperaments From: Gene Ward Smith --- 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.

Message: 9373 - Contents - Hide Contents Date: Wed, 21 Jan 2004 02:15:15 Subject: Re: 114 7-limit temperaments From: Carl Lumma 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). If you could make this list finite with badness bounds only, I'd be more impressed by claims that log-flat badness is desirable (allows the comparison of ennealimmal with all temperaments in a sense, not just the others on the list, or whatever). And I don't see how you figure schismic is less complex than miracle in light of the maps given. -Carl>Number 1 Ennealimmal > >[18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] >TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784] >TOP generators [133.3373752, 49.02398564] >bad: 4.918774 comp: 11.628267 err: .036377 > > >Number 2 Meantone > >[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] >TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328] >TOP generators [1201.698520, 504.1341314] >bad: 21.551439 comp: 3.562072 err: 1.698521 > > >Number 3 Magic > >[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] >TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877] >TOP generators [1201.276744, 380.7957184] >bad: 23.327687 comp: 4.274486 err: 1.276744 > > >Number 4 Beep > >[2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] >TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548] >TOP generators [1194.642673, 254.8994697] >bad: 23.664749 comp: 1.292030 err: 14.176105 > > >Number 5 Augmented > >[3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] >TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546] >TOP generators [399.9922103, 107.3111730] >bad: 27.081145 comp: 2.147741 err: 5.870879 > > >Number 6 Pajara > >[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] >TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174] >TOP generators [598.4467109, 106.5665459] >bad: 27.754421 comp: 2.988993 err: 3.106578 > > >Number 7 Dominant Seventh > >[1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] >TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933] >TOP generators [1195.228951, 495.8810151] >bad: 28.744957 comp: 2.454561 err: 4.771049 > > >Number 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 > > >Number 10 Orwell > >[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] >TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834] >TOP generators [1199.532657, 271.4936472] >bad: 30.805067 comp: 5.706260 err: .946061

Message: 9374 - Contents - Hide Contents Date: Wed, 21 Jan 2004 21:52:22 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Number 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*.Thus it will depend on the choice of generators. For so-called linear temperaments, this is only made definite by fixing one of them to be 1/N octaves. For planar and higher-dimensional temperaments, the choice is even more arbitrary. Comma complexity, or wedgie complexity for higher codimensions, is well-defined, and is (according to Gene) the natural generalization of the complexity measures we all agree on for the simplest cases.

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