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Message: 9125 - Contents - Hide Contents Date: Tue, 13 Jan 2004 08:47:09 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> TOP is a single-comma technique last I heard. >>>> Where have you been? It applies to any number of commas. >> So where are the TOP 7-limit linear temperaments?They are awaiting my code. I've analyzed ets and codimension one on my web site, and the next to tackle is linear.>>>> ...did Gene or Graham say there's a version of TOP equivalent >>>> to weighted rms? And Paul, have you looked at the non-weighted >>>> Tenney lattice? >>>> I don't recall saying it, but you could do something along those >> lines if you wished. >

Message: 9126 - Contents - Hide Contents Date: Tue, 13 Jan 2004 01:14:55 Subject: Re: The Exotemperaments in the dual representation From: Carl Lumma Yahoo groups: /tuning_files/files/Erlich/dualx... * [with cont.] Beautiful! -C.

Message: 9127 - Contents - Hide Contents Date: Tue, 13 Jan 2004 09:27:40 Subject: Two questions for Gene From: Paul Erlich 1. When you're done enumerating 12-tone blocks, how about looking at 5-tone? There are some nice japanese scales that are pelograndpa :) 2. Can you explain, in as non-technical a manner as possible, the proof that there are 21.5 :) commas in the 5-limit with epimericity < 1/2? .5:) (1/1) - undefined 1.5:) 2/1 - exo 2.5:) 3/2 - exo 3.5:) 4/3 - exo 4.5:) 5/4 - exo 5.5:) 6/5 - exo 6.5:) 9/8 - exo 7.5:) 10/9 - exo 8.5:) 16/15 - ? 9.5:) 25/24 - ? 10.5:) 27/25 - ? 11.5:) 32/27 - exo 12.5:) 81/80 13.5:) 128/125 14.5:) 135/128 - ? 15.5:) 250/243 16.5:) 256/243 17.5:) 648/625 18.5:) 2048/2025 19.5:) 3125/3072 20.5:) 15625/15552 21.5:) 32768/32805 I think for 5-limit linear we can just cover these in a few pages, and separately make brief mention of anything else of "historical importance" (531441/524288 and probably that's it?) . . .

Message: 9128 - Contents - Hide Contents Date: Tue, 13 Jan 2004 10:34:22 Subject: Question for Dave Keenan From: Paul Erlich If a timbre has 2nd partial off by < 10.4 cents 3rd partial off by < 16.5 cents 4th partial off by < 20.8 cents 5th partial off by < 24.1 cents 6th partial off by < 26.9 cents does it 'hold together' as a single pitch, or does it fall apart into multiple pitches? (I'll try to prepare some examples, playing random scales . . .) If yes: If I take any inharmonic timbre with one loud partial and some quiet, unimportant ones (very many fall into this category), and use a tuning system where 2:1 off by < 10.4 cents 3:1 off by < 16.5 cents 4:1 off by < 20.8 cents 5:1 off by < 24.1 cents 6:1 off by < 26.9 cents and play a piece with full triadic harmony, doesn't it follow that the harmony should 'hold together' the way 5-limit triads should?

Message: 9129 - Contents - Hide Contents Date: Tue, 13 Jan 2004 00:24:25 Subject: Re: TOP on the web From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> "Because of the transcendence and linear independence of the logs of > odd primes, the coordinates of TOP(S) can never be the same as any of > the coordinates of JIP. TOP(S) therefore retunes every rational > number by some amount, and this includes octaves; hence the > alternative acronym of Tempered Octaves, Please for TOP." > > Something must be wrong with either the premise or the inference, > since, for example, Top Beep has pure octaves.I'm getting this for TOP(27/25): [1214.176 1879.486 2819.230] What are you getting? Here are some retunings: 2-->2.016 3/2-->1.469 4/3-->1.373 5/4-->1.253 5/3-->1.721 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9130 - Contents - Hide Contents Date: Tue, 13 Jan 2004 11:05:16 Subject: 7-limit, epi From: Paul Erlich I find sixty with n*d<10^18, but maybe there are some more complex ones? How do we know how many? 16875 16807 2430 2401 1728 1715 360 343 3125 3087 703125 702464 6144 6125 19683 19600 256 245 54 49 405 392 50 49 4000 3969 78125000 78121827 36 35 5120 5103 15 14 200 189 25 21 225 224 648 625 128 125 2048 2025 27 25 16 15 6 5 81 80 256 243 32 27 4 3 ( 1 1 ) 9 8 10 9 5 4 135 128 32805 32768 25 24 250 243 3125 3072 15625 15552 126 125 7 5 21 20 28 27 7 6 35 32 525 512 875 864 4375 4374 65625 65536 3136 3125 392 375 49 45 49 48 245 243 1029 1000 686 675 10976 10935 1029 1024 2401 2400

Message: 9131 - Contents - Hide Contents Date: Tue, 13 Jan 2004 03:15:26 Subject: Re: summary -- are these right? From: Carl Lumma Gene, I don't know how many people are beating down the xenharmony door with interest, but in any case I can only thank you if you find the time to humor me, one interested party... Thanks!>>>>> ...did Gene or Graham say there's a version of TOP equivalent >>>>> to weighted rms? And Paul, have you looked at the non-weighted >>>>> Tenney lattice? >>>>>> I don't recall saying it, but you could do something along those >>> lines if you wished. >>>> RMS lines, or unweighted lines? > >Both.Well if you can do unweighted TOP that gives min. rms over all intervals, hats off to you sir. By the way, I'm almost able to construct outlines of the stuff you write now, which can't be a bad thing. /root/tentop.htm * [with cont.] (Wayb.) Pasting in the mathworld definition of norm is a big help (or did I read in one of your e-mails...). Anyway, at "3." is "|c|" absolute value of scalar c? 4. looks like the triangle inequality. Should the first "these" on the page be changed to "as"? "A linear functional on a real vector space is a linear mapping from the space to the real numbers. It is like a val, but its coordinates can be any real number." Amazing; I'm still 100% with you! "JIP = <1 log2(3) ... log2(p)|" What's the "1" doing in there? Oh, it's log2(2) but since you've simplified it we have to guess what's going on in this series. Multiplication? Let's see, so far we've got a... () real (un-normed) vector space, call it "JI" () normed vector space, call it Tenney [which is log-weighted JI] () linear functional from JI to the reals, JIP [which sounds like one of Paul's 'stretched' vals] ...and now [drum roll]. . . "For any finite-dimensional normed vector space ..." ...such as Tenney?... "... __the space of linear functionals__ is called the dual space. __It__ ..." What's "It"? Tenney? Did you mean "__a__ space of linear functionals is __a__ dual space"? " ... has a norm induced on it defined by ||f|| = sup |f(u)|/||u||, u not zero" So now I've lost track of what f is. I'm **guessing** it's a space defined by a basis given in JIPs, perhaps something that looks like... < 1194.3343134713434 1910.9349015541493 2786.7800647664676 ] < 1202.2814046729093 1898.3390600098567 2784.2306213477896 ] What's "sup"? "We may change basis in the Tenney space by resizing the elements, so that the norm is now || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|" What, so now I'm guessing |v2| is the Euclidean distance along v2? But with |c| above, I didn't think Euc. distance would be defined for a scalar. I pray you're not mixing meanings for |x|. "Each of the basis elements now represents [big honkin' space] something of the same size as 2, but that should not worry us." Uh, ok. . . . "It is a standard fact that the dual space to L1 is the L infinity norm, and vice-versa." Oh, a standard fact. I feel much better. "We may call the dual space to the Tenney space, with this norm, the val space. Just as monzos form a lattice in Tenney space, vals form a lattice in val space." I'm lost. But wait... "A regular tuning map T is a linear functional. If c is a comma which T tempers out, then T(c) = 0. If we have a set of commas C which are tempered out, then this defines a subspace Null(C) of the val space, such that for any T in Null(C), T(c)=0 for each comma tempered out." ...I understand this! "If we have a set of vals V, this defines a subspace Span(V) of the val space consisting of the linear combinations of the vals in V." Well I don't know what "linear combinations" are (do you simply mean pairwise combinations?) but I get the gyst. "Either way we define this subspace, it corresponds to a regular temperament." ...Either of two ways, neither of which you've mentioned. "We may find this minimal distance, and the corresponding point, by finding the radius where a ball around the JIP first intersects it. In the val space, the unit ball looks like a measure polytope--which is to say a rectangle or rectangular solid of whatever the dimension of the space. It consists of all points v in the val space such that || v || <= 1 The corners of this measure polytope, one of which is the JIP, are <+-1, +-log2(3), ..., +-log2(p)| If n is the dimension of the Tenney space, and so of the val space, then there are 2^n such corners." Wow this is awesome. I understand the glaze. "The codimension of a regular temperament is the dimension of its kernel, or the number of linearly independent commas needed to define it. A temperament of codimension one is defined by a single comma c. It is a linear temperament in the 5-limit, planar in the 7-limit, spacial in the 11-limit, and so forth." Now here's something for monz's encyclopedia. This should have been written years ago. Thanks, Gene! ..... It doesn't look like you've gotten around to codimension > 1 TOPs, unweighted TOPs, or RMS-equivalent-TOPs. Oh, and try searching your page for "this this". -Carl

Message: 9132 - Contents - Hide Contents Date: Tue, 13 Jan 2004 12:02:11 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Gene, > > I don't know how many people are beating down the xenharmony door > with interest, but in any case I can only thank you if you find > the time to humor me, one interested party... Thanks!I suspect there isn't much interest but I don't track it. Thanks for the positive comment.> Well if you can do unweighted TOP that gives min. rms over all > intervals, hats off to you sir.Over *all* intervals? I didn't know you were asking for that! First thing would be to construct your all-intervals norm.> By the way, I'm almost able to construct outlines of the stuff you > write now, which can't be a bad thing. > > /root/tentop.htm * [with cont.] (Wayb.) > > Pasting in the mathworld definition of norm is a big help (or did > I read in one of your e-mails...). Anyway, at "3." is "|c|" absolute > value of scalar c? It is. > 4. looks like the triangle inequality.No normed vector space would be complete without it.> Should the first "these" on the page be changed to "as"?If you mean where I defined the Tenney lattice I can see that needs fixing.> "A linear functional on a real vector space is a linear mapping from > the space to the real numbers. It is like a val, but its coordinates > can be any real number." > > Amazing; I'm still 100% with you! > > "JIP = <1 log2(3) ... log2(p)|" > > What's the "1" doing in there? Oh, it's log2(2) but since you've > simplified it we have to guess what's going on in this series. > Multiplication? > > Let's see, so far we've got a... > > () real (un-normed) vector space, call it "JI"I didn't introduce that.> () normed vector space, call it Tenney [which is log-weighted JI] > () linear functional from JI to the reals, JIP [which sounds > like one of Paul's 'stretched' vals] > > ...and now [drum roll]. . . > > "For any finite-dimensional normed vector space ..." > > ...such as Tenney?... > > "... __the space of linear functionals__ is called the dual space. > __It__ ..." > > What's "It"? Tenney?No, "it" is the dual space to our original normed vector space.> Did you mean "__a__ space of linear functionals is __a__ dual space"?No. There is only one dual to the given space.> " ... has a norm induced on it defined by > > ||f|| = sup |f(u)|/||u||, u not zero" > > So now I've lost track of what f is. I'm **guessing** it's a space > defined by a basis given in JIPs, perhaps something that looks like..."f" is a linear functional.> < 1194.3343134713434 1910.9349015541493 2786.7800647664676 ] > < 1202.2814046729093 1898.3390600098567 2784.2306213477896 ] > > What's "sup"?"Max" will work.> "We may change basis in the Tenney space by resizing the elements, > so that the norm is now > > || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|" > > What, so now I'm guessing |v2| is the Euclidean distance along v2?I've simply changed the basis so that instead of the second coordinate being a log2(3), it corresponds to something the same size as 2, but in the 3 direction; so it;s of size log2(2)=1 now.> But with |c| above, I didn't think Euc. distance would be defined > for a scalar. I pray you're not mixing meanings for |x|.I'm not sure what you mean by Euclidean distance, but I suspect it isn't what I would mean.> "Each of the basis elements now represents > [big honkin' space] > something of the same size as 2, but that should not worry us." > > Uh, ok. . . . > > "It is a standard fact that the dual space to L1 is the L infinity > norm, and vice-versa." > > Oh, a standard fact. I feel much better. > > "We may call the dual space to the Tenney space, with this norm, the > val space. Just as monzos form a lattice in Tenney space, vals form a > lattice in val space." > > I'm lost. But wait...Where did you get lost? We have vals sitting at lattice points, and a means of measuring distance.> "A regular tuning map T is a linear functional. If c is a comma which > T tempers out, then T(c) = 0. If we have a set of commas C which are > tempered out, then this defines a subspace Null(C) of the val space, > such that for any T in Null(C), T(c)=0 for each comma tempered out." > > ...I understand this! > > "If we have a set of vals V, this defines a subspace Span(V) of the val > space consisting of the linear combinations of the vals in V." > > Well I don't know what "linear combinations" are (do you simply mean > pairwise combinations?) but I get the gyst.If v1, v2, ..., vn are n vals, c1 v1 + c2 v2 + ... + cn vn, where the c's are any real number (they don't need to be integers now) is an element in the span of {v1, ..., vn}.> "Either way we define this subspace, it corresponds to a regular > temperament." > > ...Either of two ways, neither of which you've mentioned.You quoted it--Span(V) or Null(C).> "We may find this minimal distance, and the corresponding point, by > finding the radius where a ball around the JIP first intersects it. In > the val space, the unit ball looks like a measure polytope--which is > to say a rectangle or rectangular solid of whatever the dimension of > the space. It consists of all points v in the val space such that > > || v || <= 1 > > The corners of this measure polytope, one of which is the JIP, are > > <+-1, +-log2(3), ..., +-log2(p)| > > If n is the dimension of the Tenney space, and so of the val space, > then there are 2^n such corners." > > Wow this is awesome. I understand the glaze. > > "The codimension of a regular temperament is the dimension of its > kernel, or the number of linearly independent commas needed to define > it. A temperament of codimension one is defined by a single comma c. > It is a linear temperament in the 5-limit, planar in the 7-limit, > spacial in the 11-limit, and so forth." > > Now here's something for monz's encyclopedia. This should have been > written years ago. Thanks, Gene!Yer welcome. Your reaction is a distinct improvement on sullen resentment.

Message: 9133 - Contents - Hide Contents Date: Tue, 13 Jan 2004 05:25:30 Subject: Re: summary -- are these right? From: Carl Lumma>> >ell if you can do unweighted TOP that gives min. rms over all >> intervals, hats off to you sir. >>Over *all* intervals? I didn't know you were asking for that!What did you think I wanted? (Seriously; it might be interesting.)>First thing would be to construct your all-intervals norm. hrm...>> By the way, I'm almost able to construct outlines of the stuff you >> write now, which can't be a bad thing. >> >> /root/tentop.htm * [with cont.] (Wayb.) >> >> Pasting in the mathworld definition of norm is a big help (or did >> I read in one of your e-mails...). Anyway, at "3." is "|c|" absolute >> value of scalar c? > >It is. >>> 4. looks like the triangle inequality. >>No normed vector space would be complete without it.Great, some of this is finally sinking in.>> Should the first "these" on the page be changed to "as"? >>If you mean where I defined the Tenney lattice I can see that needs >fixing.Yep. Ctrl+F, by the way, is man's best friend. Don't forget "this this".>> Let's see, so far we've got a... >> >> () real (un-normed) vector space, call it "JI" >>I didn't introduce that.Hrm, looks like I was wrong then and JIP is a linear functional from Tenney to the reals (not JI to the reals).>> "For any finite-dimensional normed vector space ..." >> >> ...such as Tenney?... >> >> "... __the space of linear functionals__ is called the dual space. >> __It__ ..." >> >> What's "It"? Tenney? >>No, "it" is the dual space to our original normed vector space.What does a basis for this dual space look like? Does it have the same dimension as Tenney?>> Did you mean "__a__ space of linear functionals is __a__ dual space"? >>No. There is only one dual to the given space.Only one way to map coordinates in a space to the reals? I find that counterintuitive but I'll take your word for it.>> " ... has a norm induced on it defined by >> >> ||f|| = sup |f(u)|/||u||, u not zero" >> >> So now I've lost track of what f is. I'm **guessing** it's a space >> defined by a basis given in JIPs, perhaps something that looks like... >>"f" is a linear functional.You're defining a norm on f and I'm unable to imagine at the moment what a norm on a mapping would be like. A norm on a space seems much more intuitive.>> What's "sup"? >>"Max" will work. Check.>> "We may change basis in the Tenney space by resizing the elements, >> so that the norm is now >> >> || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|" >> >> What, so now I'm guessing |v2| is the Euclidean distance along v2? >>I've simply changed the basis so that instead of the second coordinate >being a log2(3), it corresponds to something the same size as 2, but >in the 3 direction; so it;s of size log2(2)=1 now. >>> But with |c| above, I didn't think Euc. distance would be defined >> for a scalar. I pray you're not mixing meanings for |x|. >>I'm not sure what you mean by Euclidean distance, but I suspect it >isn't what I would mean.If v2 is a vector, what is |v2|? What's the absolute value of a vector?>> "Each of the basis elements now represents >> [big honkin' space] >> something of the same size as 2, but that should not worry us." >> >> Uh, ok. . . . >> >> "It is a standard fact that the dual space to L1 is the L infinity >> norm, and vice-versa." >> >> Oh, a standard fact. I feel much better. >> >> "We may call the dual space to the Tenney space, with this norm, the >> val space. Just as monzos form a lattice in Tenney space, vals form a >> lattice in val space." >> >> I'm lost. But wait... >>Where did you get lost? We have vals sitting at lattice points, and a >means of measuring distance.More is clicking now. You can measure the distance between vals. Wild. I think I got lost with "something the same size as 2".>> "If we have a set of vals V, this defines a subspace Span(V) of the val >> space consisting of the linear combinations of the vals in V." >> >> Well I don't know what "linear combinations" are (do you simply mean >> pairwise combinations?) but I get the gyst. >>If v1, v2, ..., vn are n vals, c1 v1 + c2 v2 + ... + cn vn, where the >c's are any real number (they don't need to be integers now) is an >element in the span of {v1, ..., vn}.Oh, v1 is a val now? At the top of the page v is a vector and this seems confusing.>> "Either way we define this subspace, it corresponds to a regular >> temperament." >> >> ...Either of two ways, neither of which you've mentioned. >>You quoted it--Span(V) or Null(C). Aha!>> "We may find this minimal distance, and the corresponding point, by >> finding the radius where a ball around the JIP first intersects it. In >> the val space, the unit ball looks like a measure polytope--which is >> to say a rectangle or rectangular solid of whatever the dimension of >> the space. It consists of all points v in the val space such that >> >> || v || <= 1 >> >> The corners of this measure polytope, one of which is the JIP, are >> >> <+-1, +-log2(3), ..., +-log2(p)| >> >> If n is the dimension of the Tenney space, and so of the val space, >> then there are 2^n such corners." >> >> Wow this is awesome. I understand the glaze. >> >> "The codimension of a regular temperament is the dimension of its >> kernel, or the number of linearly independent commas needed to define >> it. A temperament of codimension one is defined by a single comma c. >> It is a linear temperament in the 5-limit, planar in the 7-limit, >> spacial in the 11-limit, and so forth." >> >> Now here's something for monz's encyclopedia. This should have been >> written years ago. Thanks, Gene! >>Yer welcome. Your reaction is a distinct improvement on sullen >resentment.Am I prone to sullen resentment? I don't think I've ever resented anything of yours. I often wish more hand-holdy treatments were available, and I think the availability of such treatments would generally help the cause (if there is a cause). But of course the availability of mathematical treatments is a positive thing. -Carl

Message: 9134 - Contents - Hide Contents Date: Tue, 13 Jan 2004 19:41:29 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Am I prone to sullen resentment?No, but I get tired of comlaints that I discuss tuning on the tuning list and tuning math on the tuning-math list.

Message: 9136 - Contents - Hide Contents Date: Tue, 13 Jan 2004 23:45:24 Subject: Re: TOP on the web From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> Gene, > > I've read through this web page, very interesting. Just a few quick > and albeit too-obvious questions: Is TOP(c) the solutions shown on > the right side of the tables?Top(c) is the top tuning for the temperament defined by the comma c. Is this "Tenney complexity"? Is it> anything like Graham complexity?"Tenney complexity" would be for example the norm on linear wedgies I gave, and it is a sort of weighted Graham complexity from a point of view which gives no special role to octaves.> Thanks. I've printed this out and plan to read it until I get it, > hopefully...I'll be adding to it. Carl pointed out one place where it needed translation into some known language, for that matter. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9137 - Contents - Hide Contents Date: Wed, 14 Jan 2004 03:16:55 Subject: Re: summary -- are these right? From: Carl Lumma>>> >ell if you can do unweighted TOP that gives min. rms over all >>> intervals, hats off to you sir. >>>> Over *all* intervals? I didn't know you were asking for that! >>What did you think I wanted? (Seriously; it might be interesting.) >>> First thing would be to construct your all-intervals norm. > >hrm...By "unweighted" I probably mean a norm without coefficents for an interval's coordinates. This ruins the correspondence with taxicab distance on the odd-limit lattice given by Paul's/Tenney's norm, which Paul thinks has pyschoacoustic import, as on that lattice intervals do not have unique factorizations and thus a metric based on unit lengths is likely to fail the triangle inequality. In a glossy way I can see how TOP, where intervals relatively prime to the comma(s) being tempered out unaffected, might lead to minimax. This suggests that all intervals will be affected by ROP (RMS-OPtimal). Maybe something akin to drawing a radius from the origin to the interval in Euclidean space and uniformly shrinking the sphere so defined, whereas TOP would warp the space by shrinking it more in some dimensions than others. Does any of this make any sense? -Carl

Message: 9138 - Contents - Hide Contents Date: Wed, 14 Jan 2004 12:26:35 Subject: Re: summary -- are these right? From: Carl Lumma>> >ell I don't know what "linear combinations" are > >Linear Combination -- from MathWorld * [with cont.]Thanks! It's sometimes hard to tell when a piece of language is specialized. -Carl

Message: 9139 - Contents - Hide Contents Date: Wed, 14 Jan 2004 12:55:08 Subject: Re: summary -- are these right? From: Carl Lumma>> >y "unweighted" I probably mean a norm without coefficents for >> an interval's coordinates. > >?The norm on Tenney space... || |u2 u3 u5 ... up> || = log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up| The 'coefficients on the intervals coordinates' here are log2(2), log2(3) etc.>> This ruins the correspondence with >> taxicab distance on the odd-limit lattice given by Paul's/Tenney's >> norm, >>Huh? Which odd-limit lattice and which norm?It's the same norm on a triangular lattice with a dimension for each odd number. The taxicab distance on this lattice is log(odd-limit). It's also the same distance as on the Tenney lattice, except perhaps for the action of 2s in the latter (I forget the reasoning there).>> as on that >> lattice intervals do not have unique factorizations and thus a >> metric based on unit lengths is likely to fail the triangle >> inequality. > >Not following.Hmm, maybe I was wrong. I was thinking stuff like ||9|| = ||3|| = 1 and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems bad though, since the 3s are pointed in the same direction. -Carl

Message: 9140 - Contents - Hide Contents Date: Wed, 14 Jan 2004 16:51:53 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Well I don't know what "linear combinations" are Linear Combination -- from MathWorld * [with cont.]

Message: 9141 - Contents - Hide Contents Date: Wed, 14 Jan 2004 16:55:22 Subject: Re: TOP on the web From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> "Tenney complexity"? Is it > anything like Graham complexity?For a commatic unison vector n/d, the Tenney Complexity is log(n*d). I think Tenney used base-2 logs.

Message: 9142 - Contents - Hide Contents Date: Wed, 14 Jan 2004 21:29:50 Subject: Re: Two questions for Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Hmm . . . that severely reduces the appeal of using epimericity to > define our badness contour. In this case, we should probably use > something that crosses zero error at some finite complexity.A curious remark for a physics major to make. We know the list is compelte, we just can't prove it, and if there were one more comma we missed it would be of no concievable musical use anyway, as Dave would be quick to point out.

Message: 9143 - Contents - Hide Contents Date: Wed, 14 Jan 2004 16:59:42 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Well if you can do unweighted TOP that gives min. rms over all >>>> intervals, hats off to you sir. >>>>>> Over *all* intervals? I didn't know you were asking for that! >>>> What did you think I wanted? (Seriously; it might be interesting.) >>>>> First thing would be to construct your all-intervals norm. >> >> hrm... >> By "unweighted" I probably mean a norm without coefficents for > an interval's coordinates. ? > This ruins the correspondence with > taxicab distance on the odd-limit lattice given by Paul's/Tenney's > norm,Huh? Which odd-limit lattice and which norm?> which Paul thinks has pyschoacoustic import,I think odd-limit has import if a composer wishes to treat all octave- related interval classes as a single entity.

Message: 9144 - Contents - Hide Contents Date: Wed, 14 Jan 2004 21:31:15 Subject: Re: Two questions for Gene 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: >>> The proof is a proof that the number of p-limit commas >> with epimericity e < 1 is finite. >> There are an infinite number of commas with e < 1, but for any > positive d, there are a finite number of commas with e < 1-d -- > correct?That's what I was trying to say. There are a finite number of commas (in some prime limit) with comma < e < 1.

Message: 9145 - Contents - Hide Contents Date: Wed, 14 Jan 2004 13:59:27 Subject: Dave's Way (was Re: Re: summary -- are these right?) From: Carl Lumma>In a glossy way I can see how TOP, where intervals relatively >prime to the comma(s) being tempered out unaffected, might >lead to minimax. This suggests that all intervals will be affected >by ROP (RMS-OPtimal). Maybe something akin to drawing a radius >from the origin to the interval in Euclidean space and uniformly >shrinking the sphere so defined, whereas TOP would warp the space >by shrinking it more in some dimensions than others. > >Does any of this make any sense?Maybe not. It's hard to see how tempering an interval with no factors in common with any of the commas being tempered out could help matters. Let's visit Dave's Method for Optimally Distributing Any Comma... A method for optimally distributing any comma * [with cont.] (Wayb.) "The particular kind of optimisation I'm referring to here is the one where we want to minimise the maximum of the absolute values of the errors of all the intervals that we care about; assuming they relate to the comma under consideration" ...this sounds an awful lot like TOP, as I think Graham mentioned. "The RMS (or sum of squares) error cannot be minimised by this method but since it is a continuous function its minima may be found by equating its partial derivatives to zero and solving." ...this sounds exactly like what Paul was telling me to do to get unweighted RMS the 'old way'. Here's one thing that puzzles me, though... "The optimum distribution will occur when the comma is equally distributed over all those factors in the longest line, with zero errors for those in the shortest line." That's not like TOP. "the real proviso is, it works for ratios where the counts, of tempered prime factors on each side of the ratio, differ by at most one. So what do we do to make it work when they differ by more than one? We "weight" the count for the offending prime so they don't differ by more than 1 for any of the intervals under consideration. In this case, if we weight the count of 2's by 1/2 then 3:4 = 3:(2 * 2) will have one tempered prime on top and none on the bottom, 5:8 (= 5:(2 * 2 * 2)) will have 1.5 tempered primes on the top and 1 on the bottom." What kind of weighting is this? -Carl

Message: 9146 - Contents - Hide Contents Date: Wed, 14 Jan 2004 19:28:52 Subject: Re: Two questions for Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> 1. When you're done enumerating 12-tone blocks, how about looking at > 5-tone? There are some nice japanese scales that are pelograndpa :)Hadn't thought about doing that; 7 and 10 was a little on my mind.> 2. Can you explain, in as non-technical a manner as possible, the > proof that there are 21.5 :) commas in the 5-limit with epimericity < > 1/2?I'm afraid I don't have a proof, and proving it would probably be difficult. The proof is a proof that the number of p-limit commas with epimericity e < 1 is finite. In practice, you get to a point where it seems obvious no futher commas are going to show up; if they did, it would be very odd from a number theoretic point of view.

Message: 9147 - Contents - Hide Contents Date: Wed, 14 Jan 2004 19:36:02 Subject: Re: Two questions for Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:>> 2. Can you explain, in as non-technical a manner as possible, the >> proof that there are 21.5 :) commas in the 5-limit with epimericity > < >> 1/2? >> I'm afraid I don't have a proof, and proving it would probably be > difficult. The proof is a proof that the number of p-limit commas > with epimericity e < 1 is finite. In practice, you get to a point > where it seems obvious no futher commas are going to show up; if they > did, it would be very odd from a number theoretic point of view.Hmm . . . that severely reduces the appeal of using epimericity to define our badness contour. In this case, we should probably use something that crosses zero error at some finite complexity.

Message: 9148 - Contents - Hide Contents Date: Wed, 14 Jan 2004 20:36:37 Subject: Re: TOP on the web From: Herman Miller On Mon, 12 Jan 2004 18:13:59 -0000, "Gene Ward Smith" <gwsmith@xxxxx.xxx> wrote:>I've put up a TOP web page. It needs to have, at least, a discussion >of equal and linear temperaments and Tenney complexity and badness >added to it, but it should be valuable as a starter. Here it is: > >/root/tentop.htm * [with cont.] (Wayb.)I can plug the formulas into a program, and they seem to produce accurate results, but I have no idea how to generalize them to higher limits. There seems to be a pattern to them, but it'd take a long time and some guesswork to figure it out. Could you be more explicit about how this works (in language a non-mathematician programmer can understand)? Also, how would this work for more than one comma? I'd like to eventually be able to figure out things like the TOP for a linear tuning with a mapping of [(1, 0), (2, -2), (1, 6), (3, -1), (3, 2)] and commas of [-7 3 1], [-4 -1 0 2], and [-5 1 0 0 1] (what I'm calling superpelog), or any other interesting and unusual things that come up. I'd like to know if the 260.76 cent generator and 1206.55 cent period is still optimal, or whether the extension to 11 limit changes anything. In fact, I'd like to see the derivation for this particular example if it isn't too nasty. (I've noticed there's a new notation for monzos that involves different symbols like angle brackets and vertical bars, but I'm continuing to use the old notation with square brackets since I don't know the appropriate usage of the new notation.) -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Message: 9149 - Contents - Hide Contents Date: Wed, 14 Jan 2004 19:38:02 Subject: Re: Two questions for Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The proof is a proof that the number of p-limit commas > with epimericity e < 1 is finite.There are an infinite number of commas with e < 1, but for any positive d, there are a finite number of commas with e < 1-d -- correct? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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.)

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