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Message: 11175 - Contents - Hide Contents Date: Thu, 01 Jul 2004 06:14:06 Subject: Re: my paper nears completion From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> Paul Erlich wrote: >>> I'd appreciate any comments or corrections . . . note that it's >> incomplete, and the 46 horagrams are not included -- >> >> Yahoo groups: /tuning/files/perlich/coyotepape... * [with cont.] >> Is the "Sagittal" font available for download? (Be sure to send a copy > of it with the paper.) > > Augmented should be [7 0 -3> in the table at the end. > > The mathematical parts looked easy enough to understand -- some of it > seemed a little too obvious to me, but for the average reader who hasn't > been following the tuning-math list it would be more of a challenge. A > whole paper could be written just on the musical applications of > wedgies, so it's probably just as well that they're not mentioned. But > you should at least have some description of what a "bivector" is if > you're going to include them in the tables.Yes -- as I mentioned on the tuning list, the part of the paper about combining commas (as well as temperament comnplexity) hasn't been written yet. It would be nice to have a real clear exposition on the hows and whys of 2x2 determinants . . . ;)> A couple of brief notated musical examples would be nice, like a typical > octatonic chord progression you might find in 12-ET music to illustrate > the 648;625 comma (the A minor - C minor - Eb minor - F# minor cycle on > my diminished temperament page for instance).If you could provide such notated examples for me to include, I'd be extremely grateful. Dave Keenan provided the lattices that are in there and one more that will be soon. I'll be sure to thank you both, and Gene too.

Message: 11176 - Contents - Hide Contents Date: Thu, 01 Jul 2004 19:54:35 Subject: Re: my paper nears completion From: Carl Lumma>But here's a notated version of my octatonic chord progression in 12-ET: > >ftp://ftp.io.com/pub/usr/hmiller/music/octatonic.gif >(MIDI at http://www.io.com/~hmiller/music/ex/dim12.mid - Type Ok * [with cont.] (Wayb.)) Nice! >and the porcupine chord progression in 12-ET, which illustrates the >250;243 comma (which of course doesn't vanish in 12-ET, but does in >porcupine temperament): > >ftp://ftp.io.com/pub/usr/hmiller/music/porcupine.gif >(MIDI at http://www.io.com/~hmiller/midi/porcupine-12.mid - Type Ok * [with cont.] (Wayb.))An old fav. I don't remember hearing it in 12, though. What a great comma. You should cross-post this to MMM. It might fit with the discush. Gene and Jon are having. -Carl

Message: 11177 - Contents - Hide Contents Date: Thu, 01 Jul 2004 14:16:36 Subject: Subject heading list From: Gene Ward Smith Does anyone know how to make a list, in chronological order, of the subject headings on this (or any other) Yahoo group? Such a list could be added to the Files section and periodically updated; having it would in many cases make searching the archives much faster.

Message: 11178 - Contents - Hide Contents Date: Thu, 01 Jul 2004 22:18:47 Subject: Re: Extending temperaments using TOP tuning From: Herman Miller Gene Ward Smith wrote:> I havn't made use of the "brute force" approach Dave and Herman have > had success with as a temperament finding algorithm, being an > algebraist by training. However, it occurs to me that if I want to > find higher limit tunings which accord with a lower-limit tuning and > hence deserve the same name, I could as an alternative to my algebraic > approach simply use the TOP tuned generators and search for good > approximations to the higher-limit primes. If I use the 5-limit TOP > values for 2 and 3, and if I search for exponents for each in the > range -100 to 100, I get only one value within 2 cents for 7, namely > 3^10/2^13 (leading to 7-limit meantone), and only one value for 11 > within 2 cents, 2^24/3^13, leading to 11-limit meantone, or "meanpop". > I find nothing for 13 under 2 cents, and it hardly makes sense to > increase the range of the exponents past a complexity of 100. Pushing > the error limit up to 9 cents gives 3^46/2^69, which isn't going to > break any records for wonderfulness.Hmm... if you apply this to 5-limit TOP mavila, you get a couple of potential mappings for 7-limit mavila variants: [[1, 2, 1, -5], [0, -1, 3, 18]] <<1, -3, -18, -7, -31, -33]] [[1, 2, 1, 11], [0, -1, 3, -19]] <<1, -3, 19, -7, 27, 52]] I can see this isn't going to be anyone's favorite 7-limit temperament. But if you skip 7, you have a pretty good 11 (10.15 cents flat). Of course, by the time you get it down into the range of 11/8, it ends up being exactly the same as a fourth. :-) It's only a good 11 because the octaves are wide. Actually, because of the wide octaves, probably a better mapping is [[1, 2, 1, 8], [0, -1, 3, -12]] <<1, -3, 12, -7, 16, 36]] This ends up with a 7:4 that's 12.2 cents sharp, and you'd have a nice 16-note MOS that could use this mapping.

Message: 11180 - Contents - Hide Contents Date: Thu, 01 Jul 2004 20:21:55 Subject: Wedgies and generators From: Gene Ward Smith Suppose that W is a bival wedgie for a linear temperament. Then we can take a product with a monzo u which maps to a val via W$u = ~(~W ^ u). If u is a generator for the temperament, the val will be a corresponding mapping to primes, up to sign. If u and v are a pair of generators generating the temperament, then [+-W$v, -+W$u] will be the mapping to primes for the pair u,v; we can choose signs in order to normalize but we should make one a + and the other a minus if u and v are both positive. For instance, suppose u = 15/14 and v = 7/5, and W is the wedgie for pajara. Then W$(15/14) = <2 3 5 6| in the 7-limit, <2 3 5 6 8| in the 11-limit. -W$(7/5) = <0 1 -2 -2| in the 7-limit, <0 1 -2 -2 -6| in the 11-limit. Putting them together gives a mapping for pajara; we can choose other values for u (16/15, 21/20, etc ) or v (10/7, 99/70 etc) and get the same result so long as they are equivalent under mapping to pajara. In particular, v=2 can be a period, in which case if we take the mapping [W$g, -W$2] for the generator g and 2, we get a mapping for generator g and period 2. For 7-limit meantone, for instance, we get g=3; [W$3, -W$2] = [<1 0 -4 -13|, <0 1 4 10|] g=3/2; [W$(3/2), -W$2] = [<1 1 0 -3|, <0 1 4 10|] g=4/3; [W$(4/3), -W$2] = [<-1 -2 -4 -7|, <0 1 4 10|] The mapping for 4/3 would normally be normalized by taking minus of it, [<1 2 4 7|, <0 -1 -4 -10|]. If 2 is a generator, the second part of the mapping, -W$2, can be read off directly from the wedgie by taking the first pi(p)-1 values of the wedgie, where p is the prime limit. If 2 is not a generator, it will be n times a generator, where n is the number of periods to the octave. Hence the second part of the mapping can again be read easily from the wedgie; n is the gcd of the first pi(p)-1 elements, and the mapping val is those elements divided through by n. Since we are normally concered with period-generator types of generators, the bival wedgie has a clear advantage over the multimonzo alternative, in giving the information we most want immediately and up front. The other elements of the wedgie can be thought of in various ways; for instance the first pi(p)-1 are from all the elements past the zero in -W$2, and that pattern continues, the next pi(p)-2 are the elements past the zero in -W$3, and so forth. Here's meantone: -W$2: <0 1 4 10| -W$3: <-1 0 4 13| -W$5: <-4 -4 0 12| -W$7: <-10 -13 -12 0| Put it all together and you get an antisymmetric matrix, the upper right corner of which is the wedgie. I think Herman was the one who suggested writing it in the form <<1 4 10 4 13 12|| which has a certain logic. The elements are also related to commas for the temperament with three prime factors; pi(p) choose 3 of these can be read off the wedgie; since these are commas rather than vals we can do this a little easier with the multimonzo form of the wedgie (not worrying about the signs of the exponents), but since we don't normally want or need to do this I don't see it as an important consideration. The val side and its mappings is more immediate. Of course, since only two vals are involved it is also easier to compute the wedgie, beyond the 7-limit, from vals in the first place.

Message: 11181 - Contents - Hide Contents Date: Thu, 01 Jul 2004 20:32:54 Subject: Re: my paper nears completion From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> wrote:> 4. p.4 Bras and kets? You could just say row vector and column > vector and stick to the formalism linear algebra rather than > quantum mechanics. Maybe it is hard to typeset column vectors > in MS Word?Bras and kets, or column vectors and row vectors, are just two ways of depicting the same mathematical idea. The light really began to dawn around here when we started using bras and kets; it gave us the advantage of being able to distinguish a bibra from a biket from a triket and once we did that, people suddenly seemed able to understand what was going on with the multilinear algebra. Bras and kets are easier to typeset, but the real payoff is that it makes the whole thing a lot more clear here in ascii land. Mathematicians don't normally use bras and kets, which were thought up by a physicist and which isn't the way they were taught in grad school, but there's no reason to let that worry us; our backgrounds here vary.

Message: 11182 - Contents - Hide Contents Date: Thu, 01 Jul 2004 20:55:58 Subject: Re: my paper nears completion From: Paul Erlich Thanks for your comments; everything helps. --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> wrote:> 1. Maybe you could make it clearer what exactly the goal of the > paper is; specifically if a lay person makes a big effort and > blasts thru all the math and lattice diagrams, they will be > rewarded with knowing ...?The end of the introduction says, "The purpose of this paper is to bring to light a host of alternative temperaments alongside the familiar ones. These should not be understood merely as lists of pitches to be employed when tuning an acoustical or electronic instrument. More importantly, they should be seen as models for the conception and notation of new music, regardless of the instruments or precise tuning strategies employed in its implementation." The lists of pitches are in the horagrams, which as I said are not contained in this .doc file. How can I make this seem more "rewarding"?> 2. I would change "musical ideas" to something like " a pattern > of notes" on p.1I changed it to "patterns of notes" -- any objections?> 3. p.2 and footnote vii Why is enharmonic equivalence now important?Beethoven, Schubert, etc. would rely on such equivalence in their compositions. It's necessary in order to circumnavigate commas like 128:125 and 32805:32768. Mathieu's book does some explicit analyses showing this . . . I guess I should refer the reader to it?> 4. p.4 Bras and kets? You could just say row vector and column > vector and stick to the formalism linear algebraI don't think that would work -- if you've been following this list, we appear to need Grassmann algebra.> rather than > quantum mechanics.No, this has nothing to do with quantum mechanics. Read the mathworld links I provided. Also, I plan to include a 3-limit lattice earlier with level pitch lines and thus motivate the "ket vector" definition as a linear operator that mathworld alludes to.> 5. I think the inner product would be more acceptable to people > is you said "a convenient shorthand" rather than "a fancier way". Right. > 6. "****Importance made clear below" Make it clear up front, > otherwise people won't make the effort to read it.Umm . . . thanks, I'll try . . .> 7. Right before the Temperment section: ************* that looks > like somebody's phone number, rather than anything related to > the previous calculations (which I didn't do, by the way. sorry)OOPS!!!! Yikes. Don't call it.> 8. Middle of Temperment section: "The relevant possibilities here > include..." Why would someone want to temper out these things? > What musical goodies does it buy them?Basically all kinds of Romantic-period harmonic effects. For example, being able to re-intepret the diminished seventh chord as the dominant-function chord in four different keys, spaced 1/4 octave apart from one another. This is one of the things that made Romantic harmony able to play with a wider array of expectation/surprise effects and to coherently explore much wider harmonic terrain. Let me know what you think I should add in the paper and where. Or maybe a reference to Mathieu will do? Thanks a lot, Paul

Message: 11183 - Contents - Hide Contents Date: Thu, 01 Jul 2004 21:18:29 Subject: Re: Wedgies and generators From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Put it all together and you get an antisymmetric matrix, the upper > right corner of which is the wedgie. I think Herman was the one who > suggested writing it in the form > > <<1 4 10 > 4 13 > 12||When I asked you, some time ago, about these triangles of numbers and their great similarity to the upper triangle of certain matrix of vanishing commas, you had no response. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 11184 - Contents - Hide Contents Date: Fri, 02 Jul 2004 20:42:15 Subject: Re: Gene's mail server From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> BTW, quantum mechanics texts do refer to > > <x|y> > > as an "inner product". Are you telling me that I need to be *more* > mathematical than a quantum mechanics text?<x|y> is indeed sometimes used to notate a inner product; this is where x and y are vectors in the *same* vector space. It is *also* used to notate the effect of applying a linear fuctional <x| to a vector |y>. Students often find this confusing, I fear, and I would not mention it in your paper, but I would avoid "inner product" because it could be very confusing to those knowing enough math to be confused. If you want a "product" name for it, I suggest "bracket product", which is used for both the inner product and the linear functional and vector thing so long as the bra-ket notation, or the bracket notation for inner product, is being used.

Message: 11185 - Contents - Hide Contents Date: Fri, 02 Jul 2004 20:52:02 Subject: Re: Gene's mail server From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> I care; I find the bival form very much more convenient. If you > don't care,>> why do you insist on not using it? >> Because it would add significantly to the length and complexity of > the paper.Bivals and bimonzos are equally complex.>>>> Of course >>>> switiching from one to the other is not difficult, but why saddle > us with>>>> a wholly unnecessary headache? >>>>> What is this headache, exactly? I think introducing vals would be a >>> headache.But you go ahead and use bimonzos!>> But there they are, in your table, in bimonzo form. >> The 'bimonzos' are there, yes. Geometrically they represent the > periodic unit of the lattice, when the 'commas' are applied as > equivalence relations. Relating this to vals is outside the scope of > the paper, as beautiful as they may be mathematically.Are you expecting people to read the comma values off of the bimonzo?>>> How do you know which is 5-limit and which is 7-limit? >>> If they have the same tuning, >> They don't. Different tunings, different horagrams.Ah! The light dawns--you are proposing, in effect, a naming system where two temperaments at different limits carry the same name if, and only if, they have the same TOP generators. This would indeed be simple and logical, and the only problem would arise when they have *almost* the same TOP generators, where you will be drawing a fine distinction which may not mean much in practice. I we adopt this plan, I suggest that we do link the names of related systems, as in augmented-august-augene and orson-orwell. I'd be interested in what other people think of a sweeping revision of temperament nomenclature along these lines; I've been trying to work it in this direction, but not systematically.> Different tuning, different horagrams. "Mork to Orson!" Hello?7-limit orson, according to the TOP approach, would be the +53 adjustment to orwell. It would have the orson/semicomma comma together with 4375/4374. It would also be only marginally interesting, but it would get the name anyway.

Message: 11186 - Contents - Hide Contents Date: Fri, 02 Jul 2004 20:54:24 Subject: Re: Wedgies and generators From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> If you were teaching a class, I would have failed out a long time ago.If I were teaching a class, I would be obligated to remember someone asked me a question, and not forget about it in the press of other considerations, or because I'd need to think about it to give an answer. Anyway, I don't recall the question.

Message: 11187 - Contents - Hide Contents Date: Fri, 02 Jul 2004 21:10:21 Subject: Re: Gene's mail server 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 care; I find the bival form very much more convenient. If you >> don't care,>>> why do you insist on not using it? >>>> Because it would add significantly to the length and complexity of >> the paper. >> Bivals and bimonzos are equally complex.Not when you've already introduced the "monzos" concept but haven't introduced the "vals" concept.>>>>> Of course >>>>> switiching from one to the other is not difficult, but why > saddle >> us with>>>>> a wholly unnecessary headache? >>>>>>> What is this headache, exactly? I think introducing vals would > be a >>>> headache. >> But you go ahead and use bimonzos!Yup -- and I just said why, too.>>> But there they are, in your table, in bimonzo form. >>>> The 'bimonzos' are there, yes. Geometrically they represent the >> periodic unit of the lattice, when the 'commas' are applied as >> equivalence relations. Relating this to vals is outside the scope > of>> the paper, as beautiful as they may be mathematically. >> Are you expecting people to read the comma values off of the bimonzo?No. But as long as we're on the subject here, it might be worth reviewing here for list memmbers how you do that. Not in the paper. Anyway, since these names are so ugly, does *anyone* have suggestions for renaming them (Dimipent, Dimisept, Negripent, Negrisept, Sensipent, Sensisept) that preserves their approximate alphabetical location?

Message: 11188 - Contents - Hide Contents Date: Fri, 02 Jul 2004 22:44:12 Subject: Re: my paper nears completion From: Herman Miller Paul Erlich wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> > wrote: >>> Paul Erlich wrote: >> >>>>> If you could provide such notated examples for me to include, I'd > > be >>>> extremely grateful. Dave Keenan provided the lattices that are in >>> there and one more that will be soon. I'll be sure to thank you > > both, >>>> and Gene too. >>>> Well, I don't have any good notation software, but I managed to put > > a >>> couple of examples together with Voyetra Digital Orchestrator and > > some >>> cutting and pasting in Paint Shop Pro. Unfortunately I couldn't > > figure >>> out how to tell it to use sharps instead of flats, if it can even > > do >>> that (probably not, since it's a MIDI editor). >> >> But here's a notated version of my octatonic chord progression in > > 12-ET: > >> ftp://ftp.io.com/pub/usr/hmiller/music/octatonic.gif >> (MIDI at http://www.io.com/~hmiller/music/ex/dim12.mid - Type Ok * [with cont.] (Wayb.)) >> >> and the porcupine chord progression in 12-ET, which illustrates the >> 250;243 comma (which of course doesn't vanish in 12-ET, but does in >> porcupine temperament): >> >> ftp://ftp.io.com/pub/usr/hmiller/music/porcupine.gif >> (MIDI at http://www.io.com/~hmiller/midi/porcupine-12.mid - Type Ok * [with cont.] (Wayb.)) > >> Thanks Herman. I can't view these in IE, for some reason. Do I have > to do something special?No, I don't know why it wouldn't work, unless you've got problems with displaying GIF files in general, or have some problem connecting to FTP sites. Try these: http://www.io.com/~hmiller/music/octatonic.gif - Type Ok * [with cont.] (Wayb.) http://www.io.com/~hmiller/music/porcupine.gif - Type Ok * [with cont.] (Wayb.) Or these: http://www.io.com/~hmiller/music/octatonic.png - Type Ok * [with cont.] (Wayb.) http://www.io.com/~hmiller/music/porcupine.png - Type Ok * [with cont.] (Wayb.)

Message: 11189 - Contents - Hide Contents Date: Fri, 02 Jul 2004 01:22:10 Subject: Re: Extending temperaments using TOP tuning From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The TOP generators are only slightly different. 112 equal is one > tuning choice, but the temperament really makes no sense, since you > may as well simply use 31.The lack of logic could be the basis for a naming system; since nothing makes sense as a 13-limit meantone, don't call any of them "meantone". In the same way, nothing seems to make sense as 13-limit miracle, or 11-limit ennealimmal, etc, since the complexities are so high you may as well be in an equal temperament.

Message: 11191 - Contents - Hide Contents Date: Fri, 02 Jul 2004 17:36:38 Subject: The Sagittal website is officially open From: Dave Keenan I expect there probably isn't anyone who reads this list and doesn't also read the main tuning list, but since most of Sagittal was developed here, I figure we deserve our own announcement so it is there in the archives. Hey Paul, you'd better not go here until you've finish your Xenharmonikon paper. ;-) See Sagittal * [with cont.] (Wayb.) A few items will not be up for another few days. But there's definitely enough there to make a visit worthwhile, and no reason to delay the announcement further. Even if you aren't particularly interested in a universal microtonal notation system, I think you will enjoy the mythology of its creation. :-) Gift of the Gods * [with cont.] (Wayb.) -- Dave Keenan

Message: 11192 - Contents - Hide Contents Date: Fri, 02 Jul 2004 17:42:02 Subject: Re: The Sagittal website is officially open From: Dave Keenan P.S. Please post any non-mathematical responses on this topic to the main tuning list. Yahoo! - * [with cont.] (Wayb.) Thanks.

Message: 11193 - Contents - Hide Contents Date: Fri, 02 Jul 2004 19:44:22 Subject: Re: Gene's mail server From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >> Hi Gene, >>>> Your mail server rejected my e-mail response to you as "probable >> SPAM". >> >> Assuming you disagree with your server's assessment, let me know how >> I should proceed. >> Considering the huge quantity of spam which makes it through, this is > kind of depressing. Maybe upload it, or post it if it is postable. I > suppose I should get an alternative email address.Anyway, here we go:>>> "Temperament, when implemented in a regular manner, reduces the often >>> bewildering variety of interval sizes in a scale or tuning system to a >>> manageable few." >>> >>> It only produces a finite number in the case equal temperaments; >>> otherwise, the intervals are dense in the real continuum of pitches. Why >>> not say it makes intervals more managable? >>>> I should indeed say that in a few places. But here, I'm not talking about >> an *infinite* tuning system, let alone an infinite scale. So isn't my >> assertion true?>If you have an equal temperament, it is theoretically infinite, but finite in >practice since only a finite number of tones are audible. If you have >something with more than one generator, whether JI or temperament, you get >density. Meantone is finite only in the same way and for the same reason that >5-limit JI is finite; you can't distinguish an infinite number of tones.Are you completely ignoring what I'm saying here, or what? I changed "tuning system" to "finite tuning system". Is my assertion now true, or not?>>> "A fancier way of writing this _expression is >>> <1200 1901.96 2786.31|-4 4 -1> = 21.5; >>> this is an example of an inner product operation between two vectors." >>> >>> Technically, it would be better to call this an angle bracket.Most of the>>> time, an inner product means between two vectors in the same space. >>> Is it an interior product?>No, though there is a relationship. It's the linear mapping of an element of >the dual space V* acting on an element of the original vector space V. >Here's what Math World has on dual vector spaces: >Dual Vector Space -- from MathWorld * [with cont.] >Linear Functional -- from MathWorld * [with cont.]Can't I just refer to it as some kind of product? I really don't want to burden the reader with abstruse mathematics, and I think 99% of them would thank me for this.>>> and >>> the mapping to primes can be read off the bival more readily. >>> Maybe the generator part of the mapping is a tiny bit more direct. But the >> period part of the mapping, without which the other part does little good, >> doesn't.follow directly from either. The table tells you both, so who >> cares?>I care; I find the bival form very much more convenient. If you don't care, >why do you insist on not using it?Because it would add significantly to the length and complexity of the paper.>>> Of course >>> switiching from one to the other is not difficult, but why saddle us with >>> a wholly unnecessary headache? >>> What is this headache, exactly? I think introducing vals would be a >> headache.>But there they are, in your table, in bimonzo form.The 'bimonzos' are there, yes. Geometrically they represent the periodic unit of the lattice, when the 'commas' are applied as equivalence relations. Relating this to vals is outside the scope of the paper, as beautiful as they may be mathematically. However, I will be explaining the meaning and role of the "ket vector" a little more fully in the 3-limit case, so an opportunity might open up for you to provide a very informative footnote.>> There's a footnote about Graham Breed's "melodic" approach being >> the algebraic dual to this one.>Graham, can you explain (maybe on tuning-math) what the melodic approach is?Take a look at Graham's webpages on meantone, schismic, diaschismic . . . It seems to be closely wedded to the val approach, which is why I suggested we use the term "breeds" to refer to vals (as long as we're using "monzos" to refer to lattice vectors).>> How do you know which is 5-limit and which is 7-limit?>If they have the same tuning,They don't. Different tunings, different horagrams. Take another look. These systems do *not* have identical tunings in the 5-limit and 7-limit (at least according to the 7-limit results you furnished). However, for meantone, porcupine, superpyth, and magic, I'm already doing this, because the tunings are indeed the same. Did you really not see this?>I guess I *could* > redo these as long as whatever you propose stays in the same place > alphabetically . . .Please think about this again. Otherwise you'll be "saddled" with the ugly names>>> Why orson? >>> Son of Orwell, or ORson WELLs, or whatever . . .>Why not just orwell?Different tuning, different horagrams. "Mork to Orson!" Hello? :)

Message: 11194 - Contents - Hide Contents Date: Fri, 02 Jul 2004 19:44:59 Subject: Re: Wedgies and generators From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote: >>>>> Put it all together and you get an antisymmetric matrix, the upper >>> right corner of which is the wedgie. I think Herman was the one who >>> suggested writing it in the form >>> >>> <<1 4 10 >>> 4 13 >>> 12|| >>>> When I asked you, some time ago, about these triangles of numbers and >> their great similarity to the upper triangle of certain matrix of >> vanishing commas, you had no response. >> Sorry. I did discuss it fairly extensively when I first introduced the > wedge product, if I recall correctly, in connection with the commutator.If you were teaching a class, I would have failed out a long time ago.

Message: 11195 - Contents - Hide Contents Date: Fri, 02 Jul 2004 19:48:39 Subject: Re: my paper nears completion From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> Paul Erlich wrote: >>> If you could provide such notated examples for me to include, I'd be >> extremely grateful. Dave Keenan provided the lattices that are in >> there and one more that will be soon. I'll be sure to thank you both, >> and Gene too. >> Well, I don't have any good notation software, but I managed to put a > couple of examples together with Voyetra Digital Orchestrator and some > cutting and pasting in Paint Shop Pro. Unfortunately I couldn't figure > out how to tell it to use sharps instead of flats, if it can even do > that (probably not, since it's a MIDI editor). > > But here's a notated version of my octatonic chord progression in 12-ET: > > ftp://ftp.io.com/pub/usr/hmiller/music/octatonic.gif > (MIDI at http://www.io.com/~hmiller/music/ex/dim12.mid - Type Ok * [with cont.] (Wayb.)) > > and the porcupine chord progression in 12-ET, which illustrates the > 250;243 comma (which of course doesn't vanish in 12-ET, but does in > porcupine temperament): > > ftp://ftp.io.com/pub/usr/hmiller/music/porcupine.gif > (MIDI at http://www.io.com/~hmiller/midi/porcupine-12.mid - Type Ok * [with cont.] (Wayb.))Thanks Herman. I can't view these in IE, for some reason. Do I have to do something special?

Message: 11196 - Contents - Hide Contents Date: Fri, 02 Jul 2004 19:57:12 Subject: Re: my paper nears completion From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>>> The end of the introduction says, >>>> "The purpose of this paper is to bring to light a host of > alternative>> temperaments alongside the familiar ones. These should not be >> understood merely as lists of pitches to be employed when tuning an >> acoustical or electronic instrument. More importantly, they should > be>> seen as models for the conception and notation of new music, >> regardless of the instruments or precise tuning strategies employed >> in its implementation." >> >> The lists of pitches are in the horagrams, which as I said are not >> contained in this .doc file. >> >> How can I make this seem more "rewarding"? >> >> "You are going to see some really erotic horagrams if you make it > to the end of this paper. No one under 21 admitted" > > I'm kidding :-) > > I'm just of the opinion that you should maybe put more description > of the results in the opening paragraph (or abstract) so the general > reader will get fired up to read the whole thing.Oh, yes. There's still going to be an abstract added. Thanks.> That raises a question though: I assumed this paper was for a journal > like Xenharmonikon...Yes, it's for Xenharmonikon.>I think you said something like "the editor made > me cut out a lot of complicated math".Well, he really didn't want the paper to be mathematical or mainly concerned with math.> If this is not the case, then > a lot of my comments were not really relevant! (and probably sound > needlessly nitpicky)I'm trying to use the minimum of math needed to show where my results are coming from, and why.>>> 3. p.2 and footnote vii Why is enharmonic equivalence now > important? >>>> Beethoven, Schubert, etc. would rely on such equivalence in their >> compositions. It's necessary in order to circumnavigate commas like >> 128:125 and 32805:32768. Mathieu's book does some explicit analyses >> showing this . . . I guess I should refer the reader to it? >> >> This is really interesting, to me. If possible, I would throw in > more details about this, although maybe that is a different paper...I'll try to expand on that footnote, thanks. Yes, I'd recommend Mathieu's book, though he really ignores the whole "middle path" idea (including meantone!) and jumps straight to 12-equal. Speaking of "jumps", his theory in that regard is goofy, if I may say so. But he does a great job showing how certain great pieces of Western music require certain commas to vanish.

Message: 11197 - Contents - Hide Contents Date: Fri, 02 Jul 2004 19:58:49 Subject: Re: Gene's mail server From: Paul Erlich BTW, quantum mechanics texts do refer to <x|y> as an "inner product". Are you telling me that I need to be *more* mathematical than a quantum mechanics text? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 11198 - Contents - Hide Contents Date: Sat, 03 Jul 2004 17:14:42 Subject: Re: bimonzos, and naming tunings (was: Gene's mail server)) From: monz hi Paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:>> now that we have so much broader a view of large numbers >> of tunings, we should subject the whole "tuning universe" >> to deep review, and come up with a really good and logical >> system of classification and naming. >> OK, but this was asked in the context of my paper, which > has to be submitted very soon. Did you see the draft?yes, but i only had time to skim it quickly. actually, i'm printing it out right now so that i can give it a proper reading, which i'll do later this morning. but it does seem to me that the time may be right for a drastic reconsideration of existing nomenclature ... considering especially the advances made in tuning theory since Gene joined the tuning cyber-community, and the fact that sagittal notation and my own little contribution (Tonalsoft software) are about to be born to the world. -monz

Message: 11199 - Contents - Hide Contents Date: Sat, 03 Jul 2004 18:40:47 Subject: Re: temperament names (was: Gene's mail server) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi Gene and Paul > > can you explain why TOP generators are so important?They have an intrinsic definition in terms of bounding the error relative to Tenney height, which in turn seems the best way to define interval complexity. Hence, they define a tuning standard which is less arbitary than the alternatives, and so consequently a seemingly better choice on which to base a naming system.

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