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Message: 8826 - Contents - Hide Contents Date: Fri, 19 Dec 2003 21:03:42 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > George wrote:>> Another question is: why 15/14 and 15/8 (when 16/15 would have been >> the inversion of 15/8)? >> Then it wouldn't be epimorphic anymore, nor a constant structure.Manuel, that can't be right.> The alternatives are limited to changes by the unison vectors of > the PB.Correct, and 225:224 is indeed one of the unison vectors!

Message: 8828 - Contents - Hide Contents Date: Fri, 19 Dec 2003 21:05:05 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> After fixing my program, here is what I am getting for Prooijen and > geometric 11-limit reductions: > > ! red72_11pro.scl > Prooijen 11-limit reduced scale > 72 > ! > 81/80 > 64/63Gene -- why isn't this 45/44?

Message: 8829 - Contents - Hide Contents Date: Fri, 19 Dec 2003 21:15:44 Subject: Re: Chromatic Unison Vector From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > It's just a unison vector in the sense that it defines the > periodicity block in the same way, but it's a different one in > the strict sense because the periodicity block will have > intervals smaller than this chromatic unison vector, which is > normally avoided.That's not accurate. For example, the JI diatonic scale has no intervals smaller than the chromatic unison vector, whether you consider the latter to be 25:24, 135:128, or 250:243.> So it's always the largest unison vector of > the set,although its JI pre-image may not the largest.

Message: 8830 - Contents - Hide Contents Date: Fri, 19 Dec 2003 21:23:51 Subject: Re: epimorphism From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:>>> Also it seems implied that non-torsion = epimorphic. Is >> that true? >> It's not true because I found a counterexample. The > [225/224, 1029/1024, 25/24] block is not a Constant Structure > and it has no torsion. > > Manuelugh! Is this, Gene, one of the cases where the notes are "in the wrong order"?

Message: 8831 - Contents - Hide Contents Date: Fri, 19 Dec 2003 21:25:14 Subject: Re: Chromatic Unison Vector From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx paul.hjelmstad@u... wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" > <manuel.op.de.coul@e...> wrote: >>>> It's just a unison vector in the sense that it defines the >> periodicity block in the same way, but it's a different one in >> the strict sense because the periodicity block will have >> intervals smaller than this chromatic unison vector, which is >> normally avoided. So it's always the largest unison vector of >> the set, and called "chromatic" because it's not supposed to >> "vanish". >> >> Manuel >> Thanks. Can you point me to an example? I am trying to use Graham's > method of calculating generators from unison vector and commas. I'm > pretty close, but don't always know what unison vector to use...Graham says "chromatic unison vector" in that method, but doesn't really mean it. Any non-unison works as well. Try 3/2, since that's rarely a chromatic unison!

Message: 8832 - Contents - Hide Contents Date: Fri, 19 Dec 2003 21:47:07 Subject: Re: A higher dimensional continued fraction From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx d.keenan@b... wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx gwsmith@s... wrote:>> Below I list successively better standard vals for cap limits from >> 5 to 13, and n from 1 to 1000; as usual it makes no real difference to >> restrict to standard vals because we are looking at best of breed >> anyway. I list the equal division n, and next to it the log base 2 of >> the L-cap badness measure (which is an ugly looking rational number I >> don't want to mess with.) > ... >> I'd appreciate it if you could give these sequences for integer- limits > to 31 and ETs to 5000. > > The reason I'm interested is because it may help us decide the best > places to site the various precision-levels of Sagittal JI notation.But why is this particular ET badness measure of so much interest to you, Dave? I mean, I'm *very* interested in learning more about it, and I may end up advocating it myself (who knows?), but there are so many others we've used . . .

Message: 8833 - Contents - Hide Contents Date: Fri, 19 Dec 2003 21:48:47 Subject: Re: Chromatic Unison Vector From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx paul.hjelmstad@u... wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" >>>> >>> It's just a unison vector in the sense that it defines the >>> periodicity block in the same way, but it's a different one in >>> the strict sense because the periodicity block will have >>> intervals smaller than this chromatic unison vector, which is >>> normally avoided. So it's always the largest unison vector of >>> the set, and called "chromatic" because it's not supposed to >>> "vanish". >>> >>> Manuel >>>> Thanks. Can you point me to an example? I am trying to use Graham's >> method of calculating generators from unison vector and commas. I'm >> pretty close, but don't always know what unison vector to use... >> Graham says "chromatic unison vector" in that method, but doesn't > really mean it. Any non-unison works as well. Try 3/2, since that's > rarely a chromatic unison!I meant rarely a *commatic* unison -- the thing you have to avoid for Graham's method!

Message: 8835 - Contents - Hide Contents Date: Sat, 20 Dec 2003 22:53:44 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul Paul wrote:>Aha -- looks like Manuel was making an arbitrary choice in the case >of a tie, perhaps letting Tenney complexity break the tie.Yes it's arbitrary, and that latter would be a useful addition. Manuel

Message: 8836 - Contents - Hide Contents Date: Sun, 21 Dec 2003 19:44:11 Subject: Re: A higher dimensional continued fraction From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> The badness measure wasn't of interest at all. I don't even have any >> idea what it is. I probably misunderstood, but I imagined that the >> actual sequence of ETs wasn't dependent on the badness measure, and >> was something like the sequence of convergents for a ratio. Since > Gene>> claimed he was using a higher-D generalisation of continued fraction >> approximation (which I also don't understand). >> If you want to approximate log2(3), and take as your badness measure > the size of the Pythagorean comma, you get exactly the convergents of > the continued fraction. I generalized that, but whether anyone finds > it interesting is another question. I asked Jeff Shallit, who didn't > know and suggested I ask Jeff Lagarias, who hasn't answered my email > yet. Chances are it's a new idea.J. Murray Barbour used multi-term continued fractions too, as did many other people. There's actually been a huge amount of discussion surrounding this topic here and on the tuning list, as it's connected to so many issues. The problem is that there's no clear choice for the 'best' way of defining them, and there have been some proofs of this statement (made more precise, of course), but then there's the ferguson-forcade algorithm . . .

Message: 8837 - Contents - Hide Contents Date: Sun, 21 Dec 2003 01:49:23 Subject: Re: A higher dimensional continued fraction From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx d.keenan@b... wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx gwsmith@s... wrote:>>> Below I list successively better standard vals for cap limits from >>> 5 to 13, and n from 1 to 1000; as usual it makes no real > difference to>>> restrict to standard vals because we are looking at best of breed >>> anyway. I list the equal division n, and next to it the log base > 2 of>>> the L-cap badness measure (which is an ugly looking rational > number I>>> don't want to mess with.) >> ... >>>> I'd appreciate it if you could give these sequences for integer- > limits>> to 31 and ETs to 5000. >> >> The reason I'm interested is because it may help us decide the best >> places to site the various precision-levels of Sagittal JI notation. >> But why is this particular ET badness measure of so much interest to > you, Dave? I mean, I'm *very* interested in learning more about it, > and I may end up advocating it myself (who knows?), but there are so > many others we've used . . .The badness measure wasn't of interest at all. I don't even have any idea what it is. I probably misunderstood, but I imagined that the actual sequence of ETs wasn't dependent on the badness measure, and was something like the sequence of convergents for a ratio. Since Gene claimed he was using a higher-D generalisation of continued fraction approximation (which I also don't understand).

Message: 8838 - Contents - Hide Contents Date: Sun, 21 Dec 2003 20:18:42 Subject: Re: A higher dimensional continued fraction From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> There's actually been a huge amount of discussion > surrounding this topic hereor at least the harmonic entropy list.

Message: 8839 - Contents - Hide Contents Date: Sun, 21 Dec 2003 02:17:39 Subject: Re: A higher dimensional continued fraction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> The badness measure wasn't of interest at all. I don't even have any > idea what it is. I probably misunderstood, but I imagined that the > actual sequence of ETs wasn't dependent on the badness measure, and > was something like the sequence of convergents for a ratio. Since Gene > claimed he was using a higher-D generalisation of continued fraction > approximation (which I also don't understand).If you want to approximate log2(3), and take as your badness measure the size of the Pythagorean comma, you get exactly the convergents of the continued fraction. I generalized that, but whether anyone finds it interesting is another question. I asked Jeff Shallit, who didn't know and suggested I ask Jeff Lagarias, who hasn't answered my email yet. Chances are it's a new idea.

Message: 8840 - Contents - Hide Contents Date: Sun, 21 Dec 2003 20:29:49 Subject: Re: A higher dimensional continued fraction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> J. Murray Barbour used multi-term continued fractions too, as did > many other people.I know. That was an algorithm, and not a very good one. This is a definition.

Message: 8841 - Contents - Hide Contents Date: Sun, 21 Dec 2003 03:40:08 Subject: Re: A higher dimensional continued fraction From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> The badness measure wasn't of interest at all. I don't even have any >> idea what it is. I probably misunderstood, but I imagined that the >> actual sequence of ETs wasn't dependent on the badness measure, and >> was something like the sequence of convergents for a ratio. Since > Gene>> claimed he was using a higher-D generalisation of continued fraction >> approximation (which I also don't understand). >> If you want to approximate log2(3), and take as your badness measure > the size of the Pythagorean comma, you get exactly the convergents of > the continued fraction. I generalized that, but whether anyone finds > it interesting is another question. I asked Jeff Shallit, who didn't > know and suggested I ask Jeff Lagarias, who hasn't answered my email > yet. Chances are it's a new idea.OK. Well it still sounds pretty interesting to me. Any chance of going up to the 32-integer-limit and 5000-ET?

Message: 8842 - Contents - Hide Contents Date: Sun, 21 Dec 2003 06:08:35 Subject: Re: A higher dimensional continued fraction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Any chance of going up to the 32-integer-limit and 5000-ET?Maple is slow, and I use the computer overnight for other things. Others on this list could compute it much faster. ________________________________________________________________________ ________________________________________________________________________ To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ 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: 8844 - Contents - Hide Contents Date: Mon, 22 Dec 2003 18:03:24 Subject: Re: Chord mapping From: Carl Lumma>Here is a poor man's method for retuning to meantone. I take all 4096 >12-et chords (pitch class sets, or set classes, or whatever gibberish >you prefer) and map to the circle of fifths. I then find the reduced >version of this (smallest corresponding base two number over a cyclic >orbit) and take that to be the preferred version of the chord. I then >find the one with midpoint closest to 2.5. The results, along with the >simple and simple-minded Maple routines I wrote to calculate this I've >placed here: > >Yahoo groups: /tuning-math/files/Gene/chordlist * [with cont.] > >There are only 4096 chords to contend with, including the empty and >full chords, so we can do a first pass at remapping simply by means of >a lookup table. This file gives a possible version of such a table. We >could use this as a starting point for more sophsiticated retunings.Just a note, chord lookup table with common-tone matching is the basis of my adaptive tuning algorithm... MIDI-based adaptive tuning by common-tone matc... * [with cont.] (Wayb.) -Carl

Message: 8846 - Contents - Hide Contents Date: Mon, 22 Dec 2003 23:58:30 Subject: Re: Chord mapping From: Carl Lumma>> >ust a note, chord lookup table with common-tone matching is the >> basis of my adaptive tuning algorithm... >> >> MIDI-based adaptive tuning by common-tone matc... * [with cont.] (Wayb.) >>So when do you code it?When I get around to it. -Carl

Message: 8847 - Contents - Hide Contents Date: Tue, 23 Dec 2003 19:46:26 Subject: Re: epimorphism From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" > <manuel.op.de.coul@e...> wrote: >>>>> Also it seems implied that non-torsion = epimorphic. Is >>> that true? >>>> It's not true because I found a counterexample. The >> [225/224, 1029/1024, 25/24] block is not a Constant Structure >> and it has no torsion. >> >> Manuel >> ugh! Is this, Gene, one of the cases where the notes are "in the > wrong order"?Manuel, you are wrong. This is indeed a torsional block. The four determinants are 20, 32, 46, and 56 -- obviously these are all multiples of 2, so we have torsion! Is everyone asleep on this list? :) :)

Message: 8848 - Contents - Hide Contents Date: Tue, 23 Dec 2003 01:48:14 Subject: Chord mapping From: Gene Ward Smith Here is a poor man's method for retuning to meantone. I take all 4096 12-et chords (pitch class sets, or set classes, or whatever gibberish you prefer) and map to the circle of fifths. I then find the reduced version of this (smallest corresponding base two number over a cyclic orbit) and take that to be the preferred version of the chord. I then find the one with midpoint closest to 2.5. The results, along with the simple and simple-minded Maple routines I wrote to calculate this I've placed here: Yahoo groups: /tuning-math/files/Gene/chordlist * [with cont.] There are only 4096 chords to contend with, including the empty and full chords, so we can do a first pass at remapping simply by means of a lookup table. This file gives a possible version of such a table. We could use this as a starting point for more sophsiticated retunings.

Message: 8849 - Contents - Hide Contents Date: Tue, 23 Dec 2003 20:51:48 Subject: Attention Gene From: Paul Erlich Yahoo groups: /tuning-math/message/8269 * [with cont.] --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> After fixing my program, here is what I am getting for Prooijen and >> geometric 11-limit reductions: >> >> ! red72_11pro.scl >> Prooijen 11-limit reduced scale >> 72 >> ! >> 81/80 >> 64/63 >> Gene -- why isn't this 45/44?

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