This is an
**
Opt In Archive
.
**
We would like to hear from you if you want your posts included. For the contact address see
About this archive. All posts are copyright (c).

First Previous Next

**10000**
10050
10100
10150
10200
10250
10300
10350
10400
10450
10500
10550
10600
10650
10700
10750
10800
10850
10900
10950

**10000 -**
10025 -

Message: 10000 - Contents - Hide Contents Date: Tue, 10 Feb 2004 07:51:47 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> We're trying to come up with some reasonable way to decide on which >> temperaments of each type to include in a paper on temperaments, given >> that space is always limited. We want to include those few (maybe only >> about 20 of each type) >> Our starting point (but _only_ a starting point) is the knowledge >> we've built up, over many years spent on the tuning list, regarding >> what people find musically useful, with 5-limit ETs having had the >> greatest coverage. >> You're gravely mistaken about the pertinence of this 'data source'. > Even worse than culling intervals from the Scala archive.How do you know this?>> So we add complexity and error cutoffs which >> utterly violate log-flat badness in their region of application (so >> why violate log-flat badness elsewhere and make the transition to >> non-violatedness as smooth as possible.That was meant to be "(so why not violate log-flat badness elsewhere ..."> Okay, now I have a definition of moat. How do they compare to Gene's > "acceptance regions"?As I understand it, a moat is intended to surround an acceptance region and quarantine it to some small degree from the kind of objections I mentioned.

Message: 10001 - Contents - Hide Contents Date: Tue, 10 Feb 2004 16:56:32 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >ould you do me a favor and attempt to speak to me as a human being, >> and not deal with me like a chess opponent, trying to look several >> moves ahead so that you can defeat me? >>I washed out of the first round of the US correspondence championship. >It's my brother who is the grandmaster.Is he really a grandmaster? -Carl

Message: 10002 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:02:31 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> We're trying to come up with some reasonable way to decide on which >> temperaments of each type to include in a paper on temperaments, given >> that space is always limited. We want to include those few (maybe only >> about 20 of each type) >> For musicians, I'd make the list 5 for each limit; 10 tops.If that's the case, and if we're also going to use log-flat, then my name is probably off the paper. It'll offer far too little of use for ordinary musicians.>> which we feel are most likely to actually be >> found useful by musicians, and we want to be able to answer questions >> of the kind: "since you included this and this, then why didn't you >> included this". So Gene may have a point when he talks about cluster >> analysis, I just don't find his applications of it so far to be >> producing useful results. >> I haven't seen any cluster analysis yet!It was principal components analysis, but the reasoning behind the implementation was obscure.>> Our starting point (but _only_ a starting point) is the knowledge >> we've built up, over many years spent on the tuning list, regarding >> what people find musically useful, with 5-limit ETs having had the >> greatest coverage. >> You're gravely mistaken about the pertinence of this 'data source'. > Even worse than culling intervals from the Scala archive.OK, Carl, so everyone's been sorely underestimating the true usefulness of 665-equal and 612-equal, yes?>> It may be an objective mathematical fact that log-flat badness gives >> uniform distribution, but you don't need a multiple-choice survey to >> know it is a psychological fact that musicians aren't terribly >> interested in availing themselves of the full resources of 4276-ET >> ()or whatever it was. >> So far this can be addressed with a complexity bound.Which contradicts the notion of 'badness'.>> So we add complexity and error cutoffs which >> utterly violate log-flat badness in their region of application (so >> why violate log-flat badness elsewhere and make the transition to >> non-violatedness as smooth as possible. > > ?Dave's exactly right. If we're violating it suddenly at the cutoffs but nowhere else, we're clearly not conforming to any kind of psychological badness criterion.

Message: 10003 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:10:18 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> It was principal components analysis, but the reasoning behind the >> implementation was obscure. >> The reasoning was to draw an elliptical moat.OK, I'd be happy to revisit that, then.>> OK, Carl, so everyone's been sorely underestimating the true >> usefulness of 665-equal and 612-equal, yes? >> Sounds like you are. Not everyone plays live music and has that as > their focus, like you.But are you using these to approximate JI or truly for their inherent properties?>> Dave's exactly right. If we're violating it suddenly at the cutoffs >> but nowhere else, we're clearly not conforming to any kind of >> psychological badness criterion. >> And if you are simply drawing squiggly lines on a graph, you are?What squiggly lines? Your lines, with their two "corners", are a lot squigglier than ours.

Message: 10004 - Contents - Hide Contents Date: Tue, 10 Feb 2004 22:16:50 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Years ago, when you first made be aware of this fact, I was seduced >> by it, to Dave's dismay. Did you forget? Now, I'm thinking about it >> from a musician's point of view. Simply put, music based on >> constructs requiring large numbers of pitches doesn't seem to be able >> to cohere in the way almost all the world's music does. >> You've gotten all the way up to 22 notes to the octave.False, and I don't appreciate the sarcastic tone of this either.

Message: 10005 - Contents - Hide Contents Date: Tue, 10 Feb 2004 01:16:32 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Carl Lumma>> >elated how? Via log-flat badness? Meanwhile, error and complexity >> are related? >>If C is the complexity and E is the error for a 7-limit linear >temperament which belongs to an infinite list of best examples, >then E ~ k C^2.Whoa dude, howdoyoufigure? -Carl

Message: 10006 - Contents - Hide Contents Date: Tue, 10 Feb 2004 08:09:47 Subject: Re: loglog! From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Yahoo groups: /tuning-math/files/Paul/et5loglo... * [with cont.] > > Ok, easy! No moat needed, at least for ETs. Just draw a > circle around the origin and grow the radius until it would > include something that exceeds a single bound -- a "TOP > notes per 1200 cents" bound. For ETs at least. Choose a > bound according to sensibilities in the 5-limit, round it > to the nearest ten, and use it for all limits. > > -CarlSomething like that may be worth looking at, except for the fact that the origin (as in the point 0,0) does not appear anywhere on a log log graph, as Paul has been at pains to point out several times over the past few days. But I agree that one could use a circle to take the bite out of the side that I talked about elsewhere. At least we agree on that general shape! One can freely choose not only the radius, but also the center point. And it would still be wise to try to fit it into a moat, for reasons already given. Whether the same circle can be generalised in such a simple way to other limits remains to be seen, but seems unlikely. And it still seems simpler to me to draw a straight line on a linear plot than a circle on a log plot. And the linear axes themselves are closer to representing "pain" than log ones.

Message: 10007 - Contents - Hide Contents Date: Tue, 10 Feb 2004 17:13:55 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>>>>> >..still trying to understand why the rectangle doesn't enclose >>>>>> a finite number of temperaments... >>>>> >>>>> Which rectangle? >>>>>>>> The rectangle enclosed by error and complexity bounds. >>>>>> Yes, that would enclose a finite number of temperaments. >>>> Then why the hell do we need a badness bound? >>To keep the utter crap at bay, and allow us not to try to publish a >list of 1000000 "temperaments". Did you see the vast clouds of >darkness on Paul's plots?I'm looking at... Yahoo groups: /tuning-math/files/Paul/et5loglo... * [with cont.] ...is a badness bound here shown by a line, say, through 12 and 171, as on Dave's mock-up ASCI plot? Oh, so you want to keep the likes of 3 and 2513 off the list? If so, a circle would do this, or an error-comp. rectangle with a diagonal at about 45deg. to either axis. -Carl

Message: 10008 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:02:42 Subject: Re: ! 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: >> Your plots>> make it clear that loglog is the right approach. Look at them! >> Geez, you must really be thinking like a mathematician and not a > musician.A musician is going to look at these plots, see that they show a slantwise arrangement of ets, and conclude circles are the way to analyze them, and select out the best ones? I don't think musicians are brain-damaged, sorry. Can you take me seriously enough not to blow me off this this bilge, and give a real argument for a clearly stated position?

Message: 10009 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:11:01 Subject: Re: ! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Did either of you guys look at the loglog version of the moat-of- 23 7- >> limit linear temperaments? >> I have a plot with unlabled axes and a curved red line on it. > Obviously, since I don't know what is being plotted, I draw no >conclusion.I've already clarified this for you!!

Message: 10010 - Contents - Hide Contents Date: Tue, 10 Feb 2004 22:20:03 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>>> OK, Carl, so everyone's been sorely underestimating the true >>>> usefulness of 665-equal and 612-equal, yes? >>>>>> Sounds like you are. Not everyone plays live music and has that as >>> their focus, like you. >>>> But are you using these to approximate JI or truly for their inherent >> properties? >> I'm in the middle of working on an ennealimmal piece now. Inherent > properties are a major aspect for this kind of thing.You're using a full basis for the kernel? And it's audible? (Real questions, not rhetorical or riddles.)> 612 is a fine > way to tune ennealimmal, though I plan on using TOP for this one. This > stuff really is practical if you care to practice it. > > In terms of commas, we have a sort of complexity of the harmonic > relationships they imply--distance measured in terms of the > symmetrical lattice norm possibly being more relevant here than > Tenney.How so? You really think a progression by perfect fifths is as complex as a progression by ratios of 7?> Past a certain point the equivalencies aren't going to make > any differences to you, and there is another sort of complexity bound > to think about.I thought this was the only kind. Can you elaborate?

Message: 10011 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:16:34 Subject: Re: loglog! From: Carl Lumma>One can freely choose not only the radius, but also the center point. >And it would still be wise to try to fit it into a moat, for reasons >already given. > >Whether the same circle can be generalised in such a simple way to >other limits remains to be seen, but seems unlikely.Why do you say that?>And it still seems simpler to me to draw a straight line on a linear >plot than a circle on a log plot. And the linear axes themselves are >closer to representing "pain" than log ones.Didn't Gen ask for a linear plot? -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: 10012 - Contents - Hide Contents Date: Tue, 10 Feb 2004 01:23:26 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:>>> Related how? Via log-flat badness? Meanwhile, error and complexity >>> are related? >>>> If C is the complexity and E is the error for a 7-limit linear >> temperament which belongs to an infinite list of best examples, >> then E ~ k C^2. >> Whoa dude, howdoyoufigure?Should be E~k/C^2, sorry. The convex hull thing I was talking about is relevant here.

Message: 10013 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:06:23 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> No way, dude! The decision is virtually made for us.Prove it. Give log-log plots for your proposed moats, and let us see what you've got. It's possible we could come to some kind of consensus if you would attempt to treat people with something better than the contempt you have shown lately. Work with me, work with Carl.

Message: 10014 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:12:34 Subject: Re: The same page From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>> Forgot 'em, but you seem to have them figured out. Modulo some >> slight>>> fiddling if you must fiddle, >>>> I'd like to understand this slight fiddling, and apply this >> understanding to the 7-limit linear case (and elsewhere). >> You can take the val and simply choose the first number in it, the > number of steps in an octave. Or, you can normalize it by > 1/log2(prime), and take the maximum. Or, you can TOP tune it, > normalize that, and take the maximum.Or you can take the sum. What I'm trying to get at is what these *mean*, beginning with the 3-limit cases in the "Attn: Gene 2" post.

Message: 10015 - Contents - Hide Contents Date: Tue, 10 Feb 2004 22:24:39 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>>> It's possible we could come to some kind of consensus >>> if you would attempt to treat people with something better than the >>> contempt you have shown lately. >>>> I can take your attitude in no other way, unless you either ignored >> completely or have an abominably low level of respect for the >> discussions Dave and I posted on the topic. >> Dave has been treating the rest of us with respect and discussing > things. I'd like the same from you.Then let's start over, and out with the sarcasm!>> Let's start over. If I'm willing to tolerate a certain level of >> error, and a certain level of complexity, why wouldn't I be willing >> to tolerate both together? >> From some points of view complexity doesn't even matter, ? > so the whole > premise we've all been operating under can be questioned by someone > who is interested in the character of the commas in the kernel, not > what complexity they give.Please elaborate on this point of view -- I'm not seeing it.> As for your question, are you arguing *for* > straight line error and complexity bounds, because I don't see where > else you can possibly go with it?Could you do me a favor and attempt to speak to me as a human being, and not deal with me like a chess opponent, trying to look several moves ahead so that you can defeat me?> Tolerating both together is exactly > what Dave doesn't tolerate, and I thought you agreed with that.I'll let Dave speak for himself, but I was hoping we would be starting over, rather than arguing about what was said before.

Message: 10017 - Contents - Hide Contents Date: Tue, 10 Feb 2004 06:28:09 Subject: Re: Loglog From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I actually meant the right, not the left -- but this isn't so much of > a problem for the loglog graph I made for you before and for the > current batch, is it?No, the good stuff lies along lines, making the whole moat business both much easier and far more logical--in case that matters to anyone.

Message: 10018 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:05:57 Subject: Re: ! 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:>>> Why should we want to multiply instead of add? >>>> Oh, for God's sake Paul-have you looked at your own plots? Did you >> notice how straight the thing looks in loglog coordinates? Your plots >> make it clear that loglog is the right approach. Look at them! >> I don't much care how it's plotted, so long as we zoom in on the > interesting bit. So, on these plots, what shape would you make a > smooth curve that encloses only (or mostly) those ETs that musicians > have actually found useful (or that you think are likely to be found > useful) for approximating JI to the relevant limit? Having regard for > the difficulty caused by complexity as well as error.Did either of you guys look at the loglog version of the moat-of-23 7- limit linear temperaments?> I wonder if, when you say that there is no particular problem with > complexity you are thinking of cases where you may use a subset of an > ET, in the way that Joseph Pehrson is using a 21 note subset of 72- ET. > In that case you are really using a linear temperament, not the ET > itself. I think the complexity of an ET should be considered as if you > planned to use _all_ its notes.I'd say that if you planned to use any set of commas that generate the ET's kernel (for chord-pump progressions, say), we're justified to consider that you're planning to use the ET itself.

Message: 10019 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:13:56 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Thus it's great for a paper for mathematicians. Not for musicians. >>>> The *contents* of the list is what's great for musicians, not >> how it was generated. >> No; I agree with Graham that we should "teach a man to fish".I disagree. It's just too hard for non-mathematicians. Unless by "fish" you mean "go to Graham's web site and use the temperament finder there" in which case I'm all for it! And this would let us not worry too much that we may have left some temperament out of the paper that someone someday may find useful.

Message: 10020 - Contents - Hide Contents Date: Tue, 10 Feb 2004 23:34:29 Subject: Re: ! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Again: The horizontal axis, as always, is *complexity*. The vertical > axis, as always, is *error*.I don't want complexity and error, I want log(complexity) and log(error), and labeling the axes if possible is always a good plan. We've already established we're on the> same page on those. It's easy to see, by the tick marks, if either or > both of the axes is scaled logarithmically.I prefer knowing to guessing. The red line is our> proposed moat. And again, the 7-limit 'linear' temperaments are > indexed as follows (I show the first three numbers in the val-wedgie, > since you feel they are the most important):Putting in the names would have made it a hell of a lot clearer than numbers I had no clue about. I'm looking at a Windows file in Linux, so the name comes out as "7lin23~1.gif"; I hope the actual name means something I can understand. The convex hull of your selected temperaments looks by eyeballing to be 20, 17, 23, 21, 22, 12 which corresponds to blackwood, augmented, porcupine, supermajor seconds, nonkleismic, and miracle. This ought to correspond to what I found by doing the same thing using Maple, but I'm not sure it does. I'm not impressed with the idea of arbitarily cooking the books to keep miralce on but leave ennealimmal off, incidentally, if this is part of the plan. The slope from blackwood to augmented is not much different than the slope from augmented to porcupine, and it seems you could start off simply with the slope of -1, and then drop it off gradually. That would already began to make more sense out of this; I don't think imagining you see a moat curve when there are in fact lots of ways to draw the curve and they look quite different is a very good start. Here is TOP complexity times TOP error for the above: Blackwood: 46.9 Augmented: 48.8 Porcupine: 45.8 Supermajor seconds: 31.3 Nonkleismic: 23.8 Miracle: 13.3 I checked my big list of 32201 temperaments, and found exactly four where the error times the complexity was less than 50, and the complexity was less than 10: blackwood, augmented, diminished, and dominant seventh. You could simply start out like this, and roll down gradually, and you'd already be making a hell of a lot more sense.

Message: 10021 - Contents - Hide Contents Date: Tue, 10 Feb 2004 02:03:19 Subject: The same page From: Gene Ward Smith Just to be sure we are on it, in terms of defintions of compexity and error, here is my page. 5-limit, comma = n/d Complexity is log2(n*d), so log(complexity) is loglog(n*d). Error is distance from the TOP tuning to the JIP, or in other words the max of the absolute values of the errors for 2, 3 and 5 in TOP tuning, divided by log2(2), log2(3) and log2(5) respectively. Log(error) is the log of this. Loglog plots compare loglog(n*d) with log(error). 7-limit planar, comma = n/d Complexity is log2(n*d) (this is always the TOP complexity in codimension one) and error is again the distance from the JIP for the 7-limit planar TOP tuning for n/d. Log(error) is log of this error. 7-limit linear Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5 a6|| is the wedgie, then complexity is |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|. Log complexity is log of this. Error is the distance from the JIP of the 7-limit TOP tuning for the temperament; log(complexity) and log(error) are logs of complexity and error, so defined.

Message: 10022 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:07:38 Subject: Re: loglog! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Yahoo groups: /tuning-math/files/Paul/et5loglo... * [with cont.] > > Ok, easy! No moat needed, at least for ETs. Just draw a > circle around the originWhere's the origin, Carl? I don't see it.> and grow the radius until it would > include something that exceeds a single bound -- a "TOP > notes per 1200 cents" bound.I'm not following.> For ETs at least. Choose a > bound according to sensibilities in the 5-limit, round it > to the nearest ten, and use it for all limits.The complexity measures cannot be compared across different dimensionalities, any more than lengths can be compared with areas can be compared with volumes.

Message: 10023 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:18:49 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> It's not either one when I'm doing it, to me log(n/d)/log(n*d) is >> just a variant on epimericity. >> I'm not following you, and I'm at a loss to understand why the above > definition of TOP error is suddenly a problem for you.Sorry, brain short-circuit.

Message: 10024 - Contents - Hide Contents Date: Tue, 10 Feb 2004 23:37:42 Subject: Re: 23 "pro-moated" 7-limit linear temps 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:>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >>>>> The rectangle enclosed by error and complexity bounds. You > answered>>> that the axes were infinitely far away, but the badness line AB >>> doesn't seem to be helping that. >>>> If you simply bound complexity alone, you get a finite number of >> temperaments. >> That doesn't seem to be true. There are lots of low-complexity > temperaments with arbitrarily high error."Lots" is not the same as "infintely many". If you bound complexity, you bound the size of the numbers in a wedgie, and hence bound the number of wedgies.

First Previous Next

**10000**
10050
10100
10150
10200
10250
10300
10350
10400
10450
10500
10550
10600
10650
10700
10750
10800
10850
10900
10950

**10000 -**
10025 -