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Message: 9100 - Contents - Hide Contents Date: Mon, 12 Jan 2004 04:21:29 Subject: Re: Temperament agreement From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I don't like these two-curve boundaries when it's clear one simple > curve could do. I personally could do without 78732/78125 and > 20000/19683, but not without 531441/524288.Is this subjective, or can you quantify it?

Message: 9101 - Contents - Hide Contents Date: Mon, 12 Jan 2004 04:24:04 Subject: Re: The Two Diadiaschisma Scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I've been seeing to on all the latest charts. Note its appearance as > the "misty" comma here, connecting 12, (51,) 63, 75, and the > excellent 87 and 99:"Misty" is certainly less clumbersome than diaschisma-schisma. I think I'll call these the Diamisty scales.

Message: 9102 - Contents - Hide Contents Date: Mon, 12 Jan 2004 04:26:56 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> You should hang it on your refrigerator. Once you do, you may be able > to understand this: for the kernel of a temperament, it will be a > list of linearly independent commas that don't lead to torsion; for a > temperament, it will be a list of linearly independent intervals that > generate the whole temperament.Do you mean "vals" and not "intervals"?>> ---- > >> Standard val -> canonical val >> >> ...the standard val is just the best approximation of each >> identity in the ET, right? Are there any other contenders >> for canonical val? > > Yes.Now that we've got TOP, we might take the n-division val closest to JIP, or such that its TOP tuning is closest to the JIP.

Message: 9103 - Contents - Hide Contents Date: Mon, 12 Jan 2004 04:28:38 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Now that we've got TOP, we might take the n-division val closest to > JIPBy which I meant, such that V/V(2) is closest to the JIP.

Message: 9104 - Contents - Hide Contents Date: Mon, 12 Jan 2004 18:13:59 Subject: TOP on the web From: Gene Ward Smith 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.) Paul, could you tell me what you want attributed to you?

Message: 9105 - Contents - Hide Contents Date: Mon, 12 Jan 2004 19:03:58 Subject: Re: summary -- are these right? From: Carl Lumma>> >OP 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?>>> ...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? -Carl

Message: 9106 - Contents - Hide Contents Date: Mon, 12 Jan 2004 19:42:23 Subject: Re: Temperament agreement From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> I don't like these two-curve boundaries when it's clear one simple >> curve could do. I personally could do without 78732/78125 and >> 20000/19683, but not without 531441/524288. >> Is this subjective, or can you quantify it?Actually, it's absurd. I misremembered the four curves that I drew, which was easy since no one has referred to them yet. What I really meant (:)) was that probably, all three of these should be in, or all three should be out.

Message: 9107 - Contents - Hide Contents Date: Mon, 12 Jan 2004 19:21:47 Subject: Re: summary -- are these right? From: Carl Lumma>>> >here have you been? It applies to any number of commas. >>>> So where are the TOP 7-limit linear temperaments? > >Yahoo groups: /tuning-math/message/8504 * [with cont.]Oh dear. Does anybody besides Gene understand this yet? Last I heard Paul was trying to use heron's formula to get around straightness. -Carl

Message: 9108 - Contents - Hide Contents Date: Mon, 12 Jan 2004 19:43:50 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> You should hang it on your refrigerator. Once you do, you may be > able>> to understand this: for the kernel of a temperament, it will be a >> list of linearly independent commas that don't lead to torsion; for > a>> temperament, it will be a list of linearly independent intervals > that>> generate the whole temperament. >> Do you mean "vals" and not "intervals"?No, I mean intervals -- for example, for meantone temperament, the list could consist of the meantone fifth and octave.

Message: 9109 - Contents - Hide Contents Date: Mon, 12 Jan 2004 20:29:53 Subject: Re: TOP on the web From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 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.) > > Paul, could you tell me what you want attributed to you?My eyes glazed over. I'll have to look at this again some other time. I came up with the idea of tempering uniformly by length, and then with the observation that this minimizes maximum weighted error over all intervals. Strictly codimension one. I also came up with a way to minimize maximum weighted error over all intervals, for dimension one (ETs).

Message: 9110 - Contents - Hide Contents Date: Mon, 12 Jan 2004 20:34:43 Subject: Re: TOP on the web From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I came up with the idea of tempering uniformly by length, and then > with the observation that this minimizes maximum weighted error over > all intervals. Strictly codimension one. OK.I also came up with a way to> minimize maximum weighted error over all intervals, for dimension one > (ETs).I think several people were doing ets; I know I was.

Message: 9111 - Contents - Hide Contents Date: Mon, 12 Jan 2004 21:32:19 Subject: Re: TOP on the web From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> I came up with the idea of tempering uniformly by length, and then >> with the observation that this minimizes maximum weighted error > over>> all intervals. Strictly codimension one. > > OK. >> I also came up with a way to>> minimize maximum weighted error over all intervals, for dimension > one >> (ETs). >> I think several people were doing ets; I know I was.Yes, you and Graham both, but from what I could tell, you were both using more complicated methods. Though mine was fairly obvious and I'm sure either of you could have come up with it anyway.

Message: 9112 - Contents - Hide Contents Date: Mon, 12 Jan 2004 13:57:42 Subject: Re: summary -- are these right? From: Carl Lumma>> >nd is there >> a definition of "basis" somewhere? >>You should hang it on your refrigerator. Once you do, you may be able >to understand this: for the kernel of a temperament, it will be a >list of linearly independent commas that don't lead to torsion; for a >temperament, it will be a list of linearly independent intervals that >generate the whole temperament.? I can't hang it on my refrigerator if I don't have it!>> Standard val -> canonical val >> >> ...the standard val is just the best approximation of each >> identity in the ET, right? Are there any other contenders >> for canonical val? > >Yes. >>(I'm in a hurry, my apologies)Paul, please take your time. At your convenience, I'd love to have your full comments on my message. And did you see the posts where I compare zeta, gram, and TOP-et tunings? Thanks, -Carl

Message: 9113 - Contents - Hide Contents Date: Mon, 12 Jan 2004 22:07:40 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> and is there >>> a definition of "basis" somewhere? >>>> You should hang it on your refrigerator. Once you do, you may be able >> to understand this: for the kernel of a temperament, it will be a >> list of linearly independent commas that don't lead to torsion; for a >> temperament, it will be a list of linearly independent intervals that >> generate the whole temperament. >> ? I can't hang it on my refrigerator if I don't have it! Vector Space Basis -- from MathWorld * [with cont.]>>> Standard val -> canonical val >>> >>> ...the standard val is just the best approximation of each >>> identity in the ET, right? Are there any other contenders >>> for canonical val? >> >> Yes. >>>> (I'm in a hurry, my apologies) >> Paul, please take your time. At your convenience, I'd love to have > your full comments on my message. > > And did you see the posts where I compare zeta, gram, and TOP-et > tunings?Yup . . .

Message: 9114 - Contents - Hide Contents Date: Mon, 12 Jan 2004 23:10:14 Subject: Re: TOP on the web From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 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.)"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.

Message: 9115 - Contents - Hide Contents Date: Tue, 13 Jan 2004 00:59:03 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 was implicity assuming all primes were in the facorization of the comma. This only works for primes which appear in that factorization, and 2 does not appear in 27/25.

Message: 9116 - Contents - Hide Contents Date: Tue, 13 Jan 2004 01:20:25 Subject: Re: TOP on the web From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- 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?See Yahoo groups: /tuning/message/51193 * [with cont.] -- the same 3 and 5, but not the same 2. I guess the tuning of 2 actually has a range of possibilities while maintaining the same maximum weighted error for the tuning as a whole. Hopefully you've seen my formula for TOP tempering of a single comma . . .

Message: 9117 - Contents - Hide Contents Date: Tue, 13 Jan 2004 02:06:31 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> is there a definition of "basis" somewhere? > // >> Vector Space Basis -- from MathWorld * [with cont.] >> Ah, good. That's what I thought. >>>>> You should hang it on your refrigerator. Once you do, you may be >>>> able to understand this: for the kernel of a temperament, it will >>>> be a list of linearly independent commas that don't lead to >>>> torsion; >> This is the only sense I've ever noticed it used around here, and > it's what I meant by "TM reduction -> canonical basis". >>>>> for a temperament, it will be a list of linearly independent >>>> intervals that generate the whole temperament. >> Generate the pitches in the temperament. One also needs the map.If it's a regular tuning, rather than a regular temperament, one doesn't need a map.>>> And did you see the posts where I compare zeta, gram, and TOP-et >>> tunings? >>>> Yup . . . >> I've been wondering about working backwards from the technique > to TOP for codimension > 1 temperaments. How would it apply to > a pair of vals?A pair of vals -> dimension = 2. How would what apply?> Which commas is it tempering in the single-val > case?Nothing new to TOP here.> etc. > > -CarlNot sure what these questions mean, but working forwards from my technique for ETs to dimension>1 seems possible, a linearly- constrained minimax problem, good for linear programming . . .

Message: 9118 - Contents - Hide Contents Date: Tue, 13 Jan 2004 02:54:44 Subject: Re: TOP on the web From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> See Yahoo groups: /tuning/message/51193 * [with cont.] -- the same 3 > and 5, but not the same 2. I guess the tuning of 2 actually has a > range of possibilities while maintaining the same maximum weighted > error for the tuning as a whole.Right, but this is exactly what I wanted to avoid.> Hopefully you've seen my formula for TOP tempering of a single > comma . . .I remember looking at it, and then deciding it would be easier to start from scratch. I'm glad I did, since formulating things in terms of normed vector spaces and their duals makes it all crystal clear, in case you are a mathematician.

Message: 9119 - Contents - Hide Contents Date: Tue, 13 Jan 2004 02:56:27 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> I've been wondering about working backwards from the technique > to TOP for codimension > 1 temperaments. How would it apply to > a pair of vals? Which commas is it tempering in the single-val > case? etc.A pair of vals can be used to define the space for the temperament with two parameters, which would be a good starting point.

Message: 9120 - Contents - Hide Contents Date: Tue, 13 Jan 2004 02:57:56 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Not sure what these questions mean, but working forwards from my > technique for ETs to dimension>1 seems possible, a linearly- > constrained minimax problem, good for linear programming . . .All this TOP stuff can be set up as a linear programming problem.

Message: 9121 - Contents - Hide Contents Date: Tue, 13 Jan 2004 03:01:54 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.>> ...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: 9122 - Contents - Hide Contents Date: Tue, 13 Jan 2004 03:07:40 Subject: Re: summary -- are these right? From: Paul Erlich --- 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? Yahoo groups: /tuning-math/message/8504 * [with cont.]

Message: 9123 - Contents - Hide Contents Date: Tue, 13 Jan 2004 03:53:54 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Where have you been? It applies to any number of commas. >>>>>> So where are the TOP 7-limit linear temperaments? >> >> Yahoo groups: /tuning-math/message/8504 * [with cont.] >> Oh dear. Does anybody besides Gene understand this yet? > Last I heard Paul was trying to use heron's formula to get > around straightness.Right, but that was pre-TOP. Now I can at least understand how to do these as a linear programming exercise, if no other way . . .

Message: 9124 - Contents - Hide Contents Date: Tue, 13 Jan 2004 06:15:15 Subject: The Exotemperaments in the dual representation From: Paul Erlich Yahoo groups: /tuning_files/files/Erlich/dualx... * [with cont.]

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