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Message: 3600 - Contents - Hide Contents Date: Wed, 30 Jan 2002 20:39:06 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 30, 2002 8:17 PM > Subject: [tuning-math] Re: new cylindrical meantone lattice > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > >>> Not really. >> >>>> Huh? Take a look at the "ratios" I post below. >> 2^(2/10) * 3^(-2/10) * 5^(3/10) is *mathematically* >> totally different from 2^(2pi+1)/4pi, even tho they're >> acoustically identical. >> Again, Monz, you're looking too closely at the way the math looks, > and not thinking about what it means. > > I could write pi in fifteen different ways that look totally > mathematically different to you, but they're all be the same.OK, I've admitted many times that I'm math-challenged. I'll ask more about this below ... (I've rewritten the following quote for clearer presentation)>>>> This generator is audibly indistinguishable from that >>>> of 3/10-comma quasi-meantone: >>>> >>>> >>>> 3/10-comma quasi-meantone "5th" >>>> "-" Lucytuning "5th" >>>> --------------------------------- >>>> ~0.010148131 cent = ~1/99 cent >>>> >>>> = >>>> >>>> 2^(2/10) * 3^(-2/10) * 5^(3/10) >>>> - 2^(2pi+1)/4pi >>>> ------------------------------------------ >>>> 2^(-12pi-10)/40pi * 3^(-2/10) * 5^(3/10) >> >>>> That last number is the "ratio" of the tiny xenharmonic bridge >> I'm talking about. >> That's no kind of unison vector, Monz. All that is is a tiny > difference between two functionally identical (as meantones in 5- > limit) tuning systems. A real unison vector expresses a relationship > within _one_ tuning system that allows it to take a pitch in more > than one sense (and thus lead you on the road to finity). This > interval you're talking about does not lead you in that direction at > all (as you claimed above, in case you're wondering what I'm rambling > on about).OK, in case there are other subtle distinctions between xenharmonic bridges and unison-vectors, I'm going to stick with my terminology. I'm saying that these two tunings *are* different (altho the auditory system can't hear the difference), so the xenharmonic bridge here *is* allowing the listener to accept 2^(2/10) * 3^(-2/10) * 5^(3/10) to be the same as 2^(2pi+1)/4pi . Am I missing your point simply because pi is a number that cannot be finitely quantized?>> If I could find some way to represent LucyTuning on the flat >> lattice (which means finding some way to represent pi in a >> universe where everything is factored by 3 and 5), then bend >> the lattice into a meantone cylinder, then warp the cylinder >> so that the LucyTuning "5th" occupies the same point as the >> 3/10-comma meantone "5th", I'd have it. >> >> The fact that pi is transcendental, irrational, whatever, >> makes it hard for me to figure out how to do this. >> If you want people to help you figure this out, you'll have to > determine what the above construction is all about. What does it > mean? Why do we want to see it on our perfectly good lattices-wrapped- > onto-cylinders?Sheesh, I don't know! I was hoping *you* could help me figure that out! (OK, good questions from you *do* do that.)> P.S. There are so many wonderful varieties of cylindrical tunings > your JustMusic software could be helping people visualize -- they'll > all look different --Wow, Paul, I'm amazed that you wrote this just now. Another project I've been working on for a couple of months is a MIDI-file of Beethoven's "Moonlight" Sonata, tuned in Kirnberger III well-temperament. I chose Kirnberger simply because it's quite likely to have been a tuning that Beethoven's piano tuner might have used, and because it has a simplicity and elegance (in terms of portraying it on my lattice) that make it easy to lattice, where other WTs are more complicated. Anyway, one of the things that I found fascinating about Kirnberger III (and this probably applies to many other WTs as well ... haven't looked yet) is that the ends of the tuning chain invoke the skhisma, which means in effect (assuming the ~2-cent difference falls outside the capability of a human tuner in Beethoven's day), that it's a closed tuning. Thus, a nice cylinder, perpendicular to the skhisma, which is *extremely* different from the meantone cylinder.> ... -- why not let the meantones look pretty much the > same -- compositionally, they pretty much are, wouldn't you say?Yes, I think I can agree with that. But then again, why would a composer choose one particular meantone over any other? There must be *some* reason/s ... so what's wrong with making a visual model of those choices too? Perhaps there are deeper things about the differences between different meantones that we haven't noticed yet. Having nice pictures of them would make it easier to find those as-yet undiscovered aspects, I think. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3601 - Contents - Hide Contents Date: Wed, 30 Jan 2002 20:46:35 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 30, 2002 8:24 PM > Subject: [tuning-math] Re: new cylindrical meantone lattice > >>>> Why not show mistuning as a tiny "break" in the consonant >>> connections? >> >>>> Hmm ... sounds interesting. You mean like on the recent Blackjack >> lattice you posted? >> Well, sort of. Except that all the consonant intervals would be > broken. > >> Please elaborate. >> Well, for example, you could do it like in Hall's hexagonal lattices, > where he puts a little number representing the mistuning of each > interval, but you could do it visually, like representing each > consonant interval as a twig and putting a physical "break", the size > of the mistuning, into it.Hmmm ... this sounds *really* interesting! I'd like to see more on this from you and the others who understand how to do it.>> [me, monz] >> The idea is that the user could: >> >> 1) Choose meantone as the type of tuning desired: JustMusic >> then draws the syntonic-comma based cylinder on the lattice. >> Right . . . >>> 2) Define which meantone by fraction-of-a-comma (or Lucy or Golden, >> if I ever figure out how): JustMusic draws the spiral around >> the cylinder. >> And the spiral helps me, as a musician, do . . . ?Sheesh ... I don't know. I'm just groping in the dark with this right now. If I was able to actually *implement* it into my software, I could play around with it and maybe offer some ideas. But the project has been stalled for a while now.>>> 3) Use the mouse to roll the cylinder around on the lattice >> to get whichever key-center is desired, or simply input the >> key and let JustMusic do the rolling. >> Well this I was expecting anyway.Really? Hmmm ... if you were expecting it, then why'd you ask me the bit about choosing "a 1/1, such as C, which flies in the face of the true nature of meantone as a transposible system" ? If the user can transpose/roll the cylinder at will, what difference does it make which note is chosen as 1/1 ?>> That's just a brief outline. >> I'm afraid you haven't told me anything new :) :)OK, sorry ... but I'd still like to have you on the justmusic list. Things have been extremely slow there lately, so there's no concern over dealing with the volume. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3602 - Contents - Hide Contents Date: Wed, 30 Jan 2002 05:24:02 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Tuesday, January 29, 2002 4:24 PM >> Subject: [tuning-math] Re: new cylindrical meantone lattice >> >> >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> I simply meant that I don't include 2 as part of the calculation >>> on these lattices, thus I can't graph EDOs (= 2^x). >>>> It sounds to me like you're just plugging things into the numbers >> without really understanding what they mean. > > > Huh? >> Most of my lattices don't use 2 as a factor, but it is certainly > possible to include 2, and I have done so on occasion. For example, > to produce lattices which portray ancient Greek systems, I've > occassionaly included 2, because the Greeks really weren't > thinking in terms of "8ve"-equivalence, but rather more in > terms of tetrachordal equivalence (or similarity, anyway).Right . . . but when you assume octave equivalence, you ignore 2. But what are you really plotting on the lattice, and why are you representing it mathematically the way you are? That's the kind of thinking that I'd encourage you to do more of . . . though that should carry no more weight than anyone else's opinion . . .> By including 2 in my formula, and perhaps reversing the > prime-lengths so that 2 is the longest (as I wrote in another > post), I can show EDOs as well as all the usual JI ratios, and > fraction-of-a-comma meantones too.But what happens to the cylinder? It seems that, without necessarily ignoring all the instances of the number 2, you would want to make these diagrams octave-invariant, wouldn't you?> > Well, now that I finally have a firm understanding of the > cylindrical meantone lattice, I can see at least a few of > the objections you've been leveling at me: > > > - Something I found most interesting: the distance of all > meantone pitches as represented by the circumferential > lines around the cylinder (the lines which represent > the syntonic comma), perpendicular to the cylinder itself, > is *also* identical for all meantones. > > This was a surprise when I first realized it, but upon > further reflection, it's an obvious result of the above.I could imagine a formula where these distances would be *very slightly* different from meantone to meantone. The idea is simply that you'll allow mistuning of consonant intervals to be reflected as small changes in length. But feel free to ignore that.> *But* ... those agreements noted, I don't understand why > you still object to my representation of the various meantone > spirals around the cylinder.What do those spirals represent? And what does it say about LucyTuning and ETs that you can't construct such spirals from _these_ meantones?> The different meantone systems are tuned in different ways, > and if the difference between any two systems is large enough, > it's audible. So what's wrong with showing that visually, > by having the meantones slice the cylinder in their own > particular way according to the math involved?(I mentioned another way of showing that visually above. A slight acoustical difference merits at most a slight visual difference, IMO).> > And I *still* don't understand how a note that I factor as, > for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the > 1/6-comma meantone "whole tone", can be represented as > anything else. There is no other combination of exponents > for 3 and 5 which will plot that point in exactly that spot.Sure there are -- they'd just be irrational exponents. But when some meantones, like LucyTuning, will require irrational exponents anyway in order to get a "spiral" happening for them, that implies to me that irrational exponents are just as meaningful as rational ones. . . . If you insist on having the spirals, then you _need_ to find a way to get them to work for LucyTuning and Golden meantone and ET meantones, etc. Otherwise I can't imagine how they could _possibly_ be meaningful. (P.S. Congratulations on discovering them -- they may be an original Monzo contribution!)> And remember ... the whole purpose in making these lattices > is to eventually include this capability in my JustMusic software.Well . . . meantones aren't "Just" by any of the definitions proposed. So maybe a new name is in order?

Message: 3604 - Contents - Hide Contents Date: Wed, 30 Jan 2002 08:02:20 Subject: Re: twintone, paultone From: clumma>> >o then what's the point of saying that inconsistency won't cause >> any problems in a regular temperament? >>There's a point if 'inconsistency' is defined in the TTTTTT, >footnote 8 sense.I don't get it. -Carl

Message: 3605 - Contents - Hide Contents Date: Wed, 30 Jan 2002 19:10:58 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Tuesday, January 29, 2002 9:24 PM >> Subject: [tuning-math] Re: new cylindrical meantone lattice >> >> >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> By including 2 in my formula, and perhaps reversing the >>> prime-lengths so that 2 is the longest (as I wrote in another >>> post), I can show EDOs as well as all the usual JI ratios, and >>> fraction-of-a-comma meantones too. >>>> But what happens to the cylinder? It seems that, without necessarily >> ignoring all the instances of the number 2, you would want to make >> these diagrams octave-invariant, wouldn't you? > >> I'll have to try to answer this question and the previous > (which I snipped) more fully another time. But I have one here: > what the heck is the difference between "octave equivalent" and > "octave invariant"? Is there a difference?Not that I can think of right now.> >>>> *But* ... those agreements noted, I don't understand why >>> you still object to my representation of the various meantone >>> spirals around the cylinder. >>>> What do those spirals represent? >> The actual mathematical tuning of the fraction-of-a-comma meantones.Hmm . . .>> And what does it say about LucyTuning and ETs that you can't >> construct such spirals from _these_ meantones? > >> I says that while LucyTuning and meantone-like ETs are audibly > indistinguishable from certain fraction-of-a-comma meantones, > they are mathematically entirely different. Not really. > Again, I refer you to my (very vague but seemingly always > getting clearer) ideas on finity. Xenharmonic Bridges in > effect here.Can you elaborate, please?>>> The different meantone systems are tuned in different ways, >>> and if the difference between any two systems is large enough, >>> it's audible. So what's wrong with showing that visually, >>> by having the meantones slice the cylinder in their own >>> particular way according to the math involved? >>>> (I mentioned another way of showing that visually above. A slight >> acoustical difference merits at most a slight visual difference, IMO). >> OK, Paul, I can buy that! As I've said before many times, I'd > love to enlist your help and for the two of us to work together > to create some really killer lattice formulae. > > You know that I'm very fond of my particular formula, but > I'm open-minded and willing to revise it, or better, to create > new kinds of lattices from scratch.Somewhere a long time ago, perhaps in the Mills times, I posted a proposed formula for these lengths. But I'm not too picky about it. Why not show mistuning as a tiny "break" in the consonant connections?>>>> And I *still* don't understand how a note that I factor as, >>> for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the >>> 1/6-comma meantone "whole tone", can be represented as >>> anything else. There is no other combination of exponents >>> for 3 and 5 which will plot that point in exactly that spot. >>>> Sure there are -- they'd just be irrational exponents. But when some >> meantones, like LucyTuning, will require irrational exponents anyway >> in order to get a "spiral" happening for them, that implies to me >> that irrational exponents are just as meaningful as rational >> ones. . . . >> >> If you insist on having the spirals, then you _need_ to find a way to >> get them to work for LucyTuning and Golden meantone and ET meantones, >> etc. Otherwise I can't imagine how they could _possibly_ be >> meaningful. (P.S. Congratulations on discovering them -- they may be >> an original Monzo contribution!) > >> Hmmm ... yes, I'm thinking that the meantone-spiral thing really > might be a useful new contribution to tuning theory. Thanks for > the acknowledgement!You got it, though I'm not banking on the "useful" part :) Particularly irksome is that you must choose a 1/1, such as C, which flies in the face of the true nature of meantone as a transposible system.

Message: 3606 - Contents - Hide Contents Date: Wed, 30 Jan 2002 19:16:26 Subject: Re: kleismic (say, is it good for klezmer?) From: paulerlich --- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:>>> Now, can you figure out how "kleismic" is defined? Hint: the kleisma >> = 15625:15552 >> >> It is the ratio of octave_normalized( 5^6/3^5 ), that I would probably > think of more as... difference between "B" found by cycling fifths and > Ax found by cycling major thirds. > > Now, is there a special term for when > > best( 5/4 ) = 9 * best( 3/2 ) > > (in other words, D# ~= 5/4, which I believe is the case in 22 but is not > a general rule for diaschismic).I don't think there's a special term for that -- but there are plenty of simpler ways of characterizing 22 -- for example, 22 is the only tuning where all three of the commas 50:49 64:63 245:243 vanish. This is the 7-limit Minkowski-reduced basis for 22 -- meaning, the simplest set of three 7-limit commas that define 22. Perhaps Gene can provide the 5-limit Minkowski reduced basis, which would have only two commas. I bet the diaschisma is one of them . . .

Message: 3607 - Contents - Hide Contents Date: Wed, 30 Jan 2002 21:17:41 Subject: Re: ET that does adaptive-JI? From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 30, 2002 9:13 PM > Subject: [tuning-math] Re: ET that does adaptive-JI? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> From: paulerlich <paul@s...> >>> To: <tuning-math@y...> >>> Sent: Wednesday, January 30, 2002 8:08 PM >>> Subject: [tuning-math] Re: ET that does adaptive-JI? >>> >>> >>> I have offered 152-tET as a Universal Tuning -- one >>> reason for this is that it supports the wonderful >>> adaptive JI system of two (or three, or more, if necessary) >>> 1/3-comma meantone chains, tuned 1/3-comma apart. This >>> gives you 5-limit adaptive JI with no drift problems, >>> and the pitch shifts reduced to normally imperceptible >>> levels. 1/152 oct. ~= 1/150 oct. = 8 cents. >> >>>> Would those be equidistant chains of 1/3-comma MT? >> Yes, equidistant at intervals of about 1/3-comma -- > 1/152 oct. ~= 1/150 oct. = 8 cents. >>> Or is there some special interval between chains? >> Yes, about 1/3-comma -- 1/152 oct. ~= 1/150 oct. = 8 cents.Oh, sorry Paul ... I only noticed just now that you already said "tuned 1/3-comma apart" in your original description. My bad.>> Does it also work equivalently as chains of 19-EDO, >> since that's so close to 1/3-comma MT? >> Right. Just like Vicentino's second tuning, where two 31-tET chains > 1/4-comma apart could do all its tricks really well, here two (or > three, if you need certain chords besides major and minor triads) 19- > tET chains 1/3-comma apart do those same kinds of tricks.Hmmm... and I see that the number 8 pops up again, because 152 = 19 * 8. So 152-EDO is like 8 bicycle chains of 19-EDO, just like 72-EDO is like 6 bicycle chains of 12-EDO. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3608 - Contents - Hide Contents Date: Thu, 31 Jan 2002 05:52:31 Subject: Fwd: Re: interval of equivalence, unison-vector, periodd From: paulerlich --- In metatuning@y..., "paulerlich" <paul@s...> wrote: --- In metatuning@y..., "monz" <joemonz@y...> wrote:> Hi guys, > > > I've been diligently studying the tuning-math archives, and > am really confused about one thing. > > (OK, many things ... but let's start here...) > >>> tuning-math message 823 >> From: graham@m... >> Date: Thu Aug 23, 2001 7:22 am >> Subject: Re: Interpreting Graham's matrix > Yahoo groups: /tuning-math/message/823?expand=1 * [with cont.] >>>> The things that make this system different to the one >> before is that it isn't unitary, and only one column of >> the inverse depends on the first generator. It's the second >> criterion that allows us to draw the non-arbitrary >> distinction between "interval of equivalence" and >> "unison vector", and so throw away the former. > >> I'm having a really hard time understanding the differences > between "interval of equivalence", "period", and "unison-vector". > > Why aren't they *all* unison-vectors?The period is often 1/2-octave, 1/3-octave, 1/4-octave, 1/9- octave, . . . so that's clearly not a "unison-vector". The "interval of equivalence" is a unison vector in Graham's system, but Graham's system seems more limited than Gene's. Gene treats it as only one of the "constructing" consonant intervals, and then somehow "sticks it back in at the end" with some LLL reduction of something. --- End forwarded message ---

Message: 3610 - Contents - Hide Contents Date: Thu, 31 Jan 2002 05:52:45 Subject: Fwd: Re: interval of equivalence, unison-vector, periodd From: paulerlich --- In metatuning@y..., "paulerlich" <paul@s...> wrote: And recall that the interval of equivalence is usually a _large_ interval, usually an octave, so not really much like a _unison_ at all! --- End forwarded message ---

Message: 3611 - Contents - Hide Contents Date: Thu, 31 Jan 2002 09:22:25 Subject: Re: archives From: Robert Walker Hi there, I've done the tuning-math archive. Tuning-Math messages - Contents * [with cont.] (Wayb.) Excluded from search engines for now using meta tags on each page, (<meta name="robots" content="noindex,nofollow">) to give an opportunity to delete posts. To delete posts from it you need to send me a list of the dates - see the About this archive link for more details of how to do it. The zip is 2.4 Mb. I've done it so that the posts in the digests start at number 4000, with no posts between the program works, and because I wanted each one to be a 1000 and 4000 - it was an easy way to do it, because of the way single section to make it easier to show the list of dates for ones posts. I'll upload the main tuning digests next, later today (already made, but quite large, 3 Mb zip and 21 Mb as html). Then have a go at harmonic entropy. I've uploaded the zip to the group files area: Yahoo groups: /tuning-math/files/tuning-math_h... * [with cont.] 2.3 Mb Robert

Message: 3613 - Contents - Hide Contents Date: Thu, 31 Jan 2002 01:23:23 Subject: Re: new cylindrical meantone lattice From: monz ----- Original Message ----- From: paulerlich <paul@xxxxxxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Wednesday, January 30, 2002 9:33 PM Subject: [tuning-math] Re: new cylindrical meantone lattice> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> OK, in case there are other subtle distinctions between xenharmonic >> bridges and unison-vectors, I'm going to stick with my terminology. >> You're depriving it of much-needed meaning. You'll have to revise > your definition, then.How so? You were the one who pointed out to me that my xenharmonic bridge concept, while very similar, was not identical to Fokker's unison-vector concept. So since I know in my mind -- even if I'm not expressing it entirely clearly to others, or in the Dictionary -- what a xenharmonic bridge is, then I'll use that term when I mean that concept. Keep the criticism and questions coming ... it should help me to hone the definition.>> I'm saying that these two tunings *are* different (altho >> the auditory system can't hear the difference), so the >> xenharmonic bridge here *is* allowing the listener to accept >> 2^(2/10) * 3^(-2/10) * 5^(3/10) to be the same as 2^(2pi+1)/4pi . >> Now you're claiming that our auditory system cares about these > irrational ratios???Huh? No, I'm not claiming that. As always, I believe that our auditory system cares about fairly-low-integer/prime ratios. I'm stating that LucyTuning is not the same as 3/10-comma meantone. Sure, it sounds the same. But then why didn't Harrison and Lucy simply write about 3/10-comma instead?, which I think would be a lot easier to understand mathematically. There is a difference, and there's a little tiny xenharmonic bridge in effect which blurs that difference and allows us to accept them as being exactly the same. *This* kind of fudging and blurring is what I originally was trying to express with the xenharmonic bridge concept.>> Am I missing your point simply because pi is a number that cannot >> be finitely quantized? > > No.So then please try to elaborate more ... I still don't get what you're saying.>>> ... -- why not let the meantones look pretty much the >>> same -- compositionally, they pretty much are, wouldn't you say? >> >>>> Yes, I think I can agree with that. >> >> But then again, why would a composer choose one particular >> meantone over any other? >> It would depend on the criteria they chose to determine their > meantone. Are they concentrating on the fifths? Do they care about > the thirds only? Do they minimize maximum error, or total error? See > my table in your meantone definition page for examples of what > meantones different desiderata can lead to.Well there you go: there's the answer as to why the spirals are useful! By seeing how a particular meantone spirals around the cylindrical lattice -- which still has the JI points and distances marked on it, albeit warped a bit so as to fit around the circle -- one can see which JI intervals it favors. Thus, it can be seen from my 1/4-comma lattice at the bottom of Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) that 1/4-comma meantone gives all the "major 3rds" and "minor 6ths" exactly. If you refer to the dotted lines on these graphs, Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) which show the fraction-of-a-comma tempering of each meantone note, and imagine that these lattices are wrapped around a cylinder, then for the "tonic" chord: - it could be seen from the spiral of 1/3-comma meantone that it gives the "minor 3rd" and "major 6th" exactly; - it could be seen from the 2/7-comma spiral that both the "major" and "minor" "3rd" and "6th" all have an equal amount of error. - 1/5-comma favors the "5ths/4ths" and "major 3rd / minor 6th" (which have the same error) over the "minor 3rd / major 6th"; Etc. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3614 - Contents - Hide Contents Date: Thu, 31 Jan 2002 01:30:28 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 30, 2002 9:35 PM > Subject: [tuning-math] Re: new cylindrical meantone lattice > >>>> Well, for example, you could do it like in Hall's hexagonal >>> lattices, where he puts a little number representing the >>> mistuning of each interval, but you could do it visually, >>> like representing each consonant interval as a twig and >>> putting a physical "break", the size of the mistuning, into it. >> >>>> Hmmm ... this sounds *really* interesting! I'd like to see >> more on this from you and the others who understand how to do it. >> Seems simple enough. What more do you want? Did Hall do the hexagonal > ones in the article I sent you? Do I need to send you another Hall > article?Yes, he did the hexagonal ones. Does he have more articles that I'd find of interest? What I meant was: perhaps you could make a quick-and-dirty graphic illustrating that "twig" idea. That was what I found interesting.>>>> 3) Use the mouse to roll the cylinder around on the lattice >>>> to get whichever key-center is desired, or simply input the >>>> key and let JustMusic do the rolling. >>>>>> Well this I was expecting anyway. >> >>>> Really? Hmmm ... if you were expecting it, then why'd you ask >> me the bit about choosing "a 1/1, such as C, which flies in the >> face of the true nature of meantone as a transposible system" ? >> >> If the user can transpose/roll the cylinder at will, what >> difference does it make which note is chosen as 1/1 ? >> Because the *spiral* will be *pinned* to 1/1 no matter how we roll > the cylinder, correct?Doesn't have to be. It could be shifted along either axis by any fraction of 3 or 5 or both that you'd like. It doesn't affect the *relative* relationships between the meantone and the JI, only the specific relationships between specific pairs of notes. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3615 - Contents - Hide Contents Date: Thu, 31 Jan 2002 01:33:24 Subject: Re: IM conversation with Monz From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 30, 2002 9:45 PM > Subject: [tuning-math] IM conversation with Monz > > > paulerlich: It can only work its magic as adaptive JI if the > separation between the two chains is either 1/4-comma for 1/4-comma > meantone, or 1/3-comma for 1/3-comma meantone. No other combination > of system and separation would work. > Monz: Hmmm ... can you quote some of this IM to tuning-math with a > nice elaboration on how this works. I don't quite see it. > paulerlich: Simple: > paulerlich: in 1/4-comma meantone, the major thirds are just, and the > fifth and minor third are 1/4-comma flat -- so you simply go to the > 1/4-comma meantone chain 1/4-comma away to get your just minor third > and just fifth. > paulerlich: in 1/3-comma meantone, the minor thirds are just, and the > fifth and major third are 1/3-comma flat -- so you simply go to the > 1/3-comma meantone chain 1/3-comma away to get your just major third > and just fifth.That's cool! So in other words, a 38-tone subset of 152-EDO would give you a nice adaptive-JI based on 1/3-comma meantone? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3616 - Contents - Hide Contents Date: Thu, 31 Jan 2002 01:16:53 Subject: Re: Some Minkowsi reduced bases From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> I noticed that. BTW, have you had a chance to think about > my "question for Gene"?I was going to do some calculations when I had a chance--I'm going to see A Beautiful Mind and prepping for tomorrow's classs tonight.

Message: 3617 - Contents - Hide Contents Date: Thu, 31 Jan 2002 01:38:46 Subject: Re: Fwd: Re: interval of equivalence, unison-vector, periodd From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 30, 2002 9:52 PM > Subject: [tuning-math] Fwd: Re: interval of equivalence, unison-vector, period > > >>>> I'm having a really hard time understanding the differences >> between "interval of equivalence", "period", and "unison-vector". >> >> Why aren't they *all* unison-vectors? >> The period is often 1/2-octave, 1/3-octave, 1/4-octave, 1/9- > octave, . . . so that's clearly not a "unison-vector".But what *is* the period? I mean, not what interval or size, but what is it? What significance does it have?> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 30, 2002 9:52 PM > Subject: [tuning-math] Fwd: Re: interval of equivalence, unison-vector, period > > > And recall that the interval of equivalence is usually a _large_ > interval, usually an octave, so not really much like a _unison_ at > all!But ... but ... Say the syntonic comma is a unison-vector. So pick a reference note; the note a comma away is considered equivalent. But on an "8ve"-equivalent lattice, the note *an "8ve" and a comma away (~1222 cents) is also considered equivalent*!! So then why not the "8ve" itself? I don't get it. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3618 - Contents - Hide Contents Date: Thu, 31 Jan 2002 01:56:08 Subject: Re: new cylindrical meantone lattice From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Cc: <justmusic@xxxxxxxxxxx.xxx>; Ken Fasano <kenfasano@xxxxxxx.xxx> > Sent: Thursday, January 31, 2002 1:30 AM > Subject: Re: [tuning-math] Re: new cylindrical meantone lattice > > >>> [me, monz]>>> If the user can transpose/roll the cylinder at will, what >>> difference does it make which note is chosen as 1/1 ? >> >> [Paul]>> Because the *spiral* will be *pinned* to 1/1 no matter how we roll >> the cylinder, correct? > > [monz]> Doesn't have to be. It could be shifted along either axis by any > fraction of 3 or 5 or both that you'd like. It doesn't affect the > *relative* relationships between the meantone and the JI, only the > specific relationships between specific pairs of notes.There's a flat-lattice example of this at the bottom of Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) The last two pictures on the page shows a Duodene JI periodicity-block with 2/9-comma meantone latticed within it. The first picture has the bounding unison-vectors and the meantone both centered on 1/1. But the Duodene ends up having three pitches lying on the right (i.e., positive-3) boundary, one of which is at a corner. So all three of them may be exchanged for pitches a comma away, which would fall on the other boundary, and either of the two corner pitches may be exchanged for pitches a diesis away, which would fall on the opposite corners. So I slid things around a bit. About the last picture, I quote from the page:>> If one keeps the JI pitches in place, and moves the bounding >> unison-vectors and the meantone 1/2-step to the right along >> the 3-axis, so that the boundaries enclose only 12 >> pitches (the minimal set for this pair of unison-vectors) >> and the meantone is exactly centered and symmetrical to those >> 12 pitches, the entire system is centered and symmetrical >> around the ratio 3^(1/2), or in terms of "8ve"-equivalent >> ratios, the square-root of 3/2. >> >> In fact this structure more accurately portrays what the >> meantone really represents: because of the disappearance >> of the syntonic comma unison-vector, this flat lattice >> should be imagined to wrap around as a cylinder, so that >> the right and left edges connect. Thus the centered >> meantone may imply either of any pair of pitches which >> would be separated by a comma on the flat lattice, and >> each of those pairs of points on the flat lattice map to >> the same point on a cylinder. -monz _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3619 - Contents - Hide Contents Date: Thu, 31 Jan 2002 02:02:38 Subject: 152-EDO as adaptive-JI (was: IM conversation with Monz) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 31, 2002 1:33 AM > Subject: Re: [tuning-math] IM conversation with Monz > > > That's cool! So in other words, a 38-tone subset of 152-EDO would > give you a nice adaptive-JI based on 1/3-comma meantone?Oops ... I should have been more specific. I meant a 38-tone subset of 152-EDO which is two 19-EDOs, ~1/3-comma apart. And 2^(1/152) is ~1/3-comma. So it's two 19-EDOs which are one 152-EDO degree apart. Got it. AWESOME! Now *THIS* looks like a tuning I'd like to explore more!! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3620 - Contents - Hide Contents Date: Thu, 31 Jan 2002 02:29:59 Subject: Re: 152-EDO as adaptive-JI (was: IM conversation with Monz) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 31, 2002 2:02 AM > Subject: [tuning-math] 152-EDO as adaptive-JI (was: IM conversation with Monz) > > >> [me, monz]>> That's cool! So in other words, a 38-tone subset of 152-EDO would >> give you a nice adaptive-JI based on 1/3-comma meantone? > >> Oops ... I should have been more specific. I meant a 38-tone > subset of 152-EDO which is two 19-EDOs, ~1/3-comma apart. > > And 2^(1/152) is ~1/3-comma. > > So it's two 19-EDOs which are one 152-EDO degree apart. > > Got it. AWESOME! Now *THIS* looks like a tuning I'd like > to explore more!!Ah ... 152-EDO gets more and more interesting the more I look at it! So, since 19-EDO is a meantone, the lattice wraps into a cylinder. And since 1 step of 152-EDO is ~1/3-comma, you only need *3* chains of 19-EDO each separated by 1 step of 152-EDO, in order to represent the entire infinite JI lattice! And you don't get comma problems! *WAY* cool !!!!!! So the 57-tone subset of 152-EDO which is three 19-EDOs starting on the 1st, 2nd, and 3rd 152-EDO degrees is really some sort of magical tuning for us adaptive-JI fans! It looks like this might be the tuning I've been looking for for my Mahler retunings. Thanks BIGTIME, Paul! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3621 - Contents - Hide Contents Date: Thu, 31 Jan 2002 02:54:10 Subject: 217-EDO as adaptive-JI (was: 152-EDO as adaptive-JI) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 31, 2002 2:29 AM > Subject: Re: [tuning-math] 152-EDO as adaptive-JI > > > So, since 19-EDO is a meantone, the lattice wraps into a cylinder. > > And since 1 step of 152-EDO is ~1/3-comma, you only need *3* > chains of 19-EDO each separated by 1 step of 152-EDO, in order > to represent the entire infinite JI lattice! And you don't > get comma problems! *WAY* cool !!!!!! > > So the 57-tone subset of 152-EDO which is three 19-EDOs starting > on the 1st, 2nd, and 3rd 152-EDO degrees is really some sort of > magical tuning for us adaptive-JI fans!And so, likewise, 217-EDO can perform this function for 31-EDO. 217 = 31 * 7. So 217-EDO is like 7 bicycle chains of 31-EDO. 1 step of 217-EDO is ~1/4-comma. 1/4-comma meantone, represented by 31-EDO, bends the JI lattice into a cylinder in which 4 chains of the meantone can cover the whole lattice. So the 124-tone subset of 217-EDO which is four 31-EDOs starting on the first four 217-EDO degrees also covers the whole meantone lattice. And of course, since 152- and 217- are both EDOs, they close the meantone system and allow even further punning/bridging! The 57-out-of-152 is much simpler, but my guess is that the 124-out-of-217 would be even more appropriate for the retuning of "common-practice" repertoire I'm interested in. This is so cool, I need a coat. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3622 - Contents - Hide Contents Date: Thu, 31 Jan 2002 04:08:38 Subject: Re: ET that does adaptive-JI? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> In a discussion from August, in which Bob Wendell was > looking for a nice consistent ET to supplement the use > of cents and the consensus was the 111-tET fit his > criteria, Paul wrote: > >>> tuning-math message 837 >> From: "Paul Erlich" <paul@s...> >> Date: Thu Aug 23, 2001 5:18 pm >> Subject: Re: EDO consistency and accuracy tables (was: A little > research...) >> >>>> Right . . . but I think the whole idea of 72 or 111 or 121 as a least- >> common-denominator way of describing ideal musical practice kind of >> falters when _adaptive JI_ comes into the picture . . . doesn't it? > >> I see that there are follow-ups which I haven't yet read, > but in case they don't quite address this question, I'm > very curious -- what kind of criteria would have to be set > to find an ET that *does* work for adaptive-JI? > > > > -monzThanks for asking this now, Monz. I have offered 152-tET as a Universal Tuning -- one reason for this is that it supports the wonderful adaptive JI system of two (or three, or more, if necessary) 1/3-comma meantone chains, tuned 1/3- comma apart. This gives you 5-limit adaptive JI with no drift problems, and the pitch shifts reduced to normally imperceptible levels. 1/152 oct. ~= 1/150 oct. = 8 cents. In addition, it acts as a strict 11-limit JI system, with the maximum error in the consonant intervals and in the Tonality Diamond pitches is 2.23 cents -- 0.68 cents through the 5-limit. Finally, and most importantly, it contains 76-tET, which contains all the linear temperaments that interest me most, as they support omnitetrachordal scales (and are 7-limit): 1. Meantone within 19-tET subsets 2. Pajara 3. Double-Diatonic within 38-tET subsets as well as the non-tetrachordal 4. Kleismic within 19-tET subsets (see 11 note chain-of-minor-thirds scale * [with cont.] (Wayb.)). The full 152-tET could probably support a large number of other interesting linear temperaments. Anyway, this is all just an example of how ETs can be used consistently by exploiting their inconsistency ;) though 152-tET _is_ consistent through the 11-limit (see above).

Message: 3623 - Contents - Hide Contents Date: Thu, 31 Jan 2002 04:17:58 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Not really. > >> Huh? Take a look at the "ratios" I post below. > 2^(2/10) * 3^(-2/10) * 5^(3/10) is *mathematically* > totally different from 2^(2pi+1)/4pi, even tho they're > acoustically identical.Again, Monz, you're looking too closely at the way the math looks, and not thinking about what it means. I could write pi in fifteen different ways that look totally mathematically different to you, but they're all be the same.>>> Again, I refer you to my (very vague but seemingly always >>> getting clearer) ideas on finity. Xenharmonic Bridges in >>> effect here. >>>> Can you elaborate, please? > > > Sure. >> (Well, I'll be cornswoggled -- that doesn't mean anything ... > I just realized I had explained this on an updated definition > I made for "LucyTuning", but I never uploaded it until now!) > > > It's all in the Dictionary now at > Definitions of tuning terms: LucyTuning, (c) 2... * [with cont.] (Wayb.) > > But I'll post it anyway: >>>> This [LucyTuning] generator or "5th" is composed of three >>> Large (3L) plus one small note (s), i.e. (3L+s) >>> = (~190.986*3) + (~122.535) = ~695.493 cents or ratio of >>> >>> 2^(3/2pi) * ( 2/(2^(5/2pi)) )^(1/2) >>> >>> = 2^( (2pi + 1) / 4pi ) >>> >>> = 2^( 1/2 + 1/4pi ) >>> >>> = ~1.494412. >>> >>> >>> This generator is audibly indistinguishable from that >>> of 3/10-comma quasi-meantone: >>> >>> 2^(2/10) * 3^(-2/10) * 5^(3/10) 3/10-comma quasi- meantone > "5th">>> - 2^(2pi+1)/4pi Lucytuning "5th" >>> ------------------------------------------ >>> 2^(-12pi-10)/40pi * 3^(-2/10) * 5^(3/10) = ~0.010148131cent = ~1/99> cent > > > That last number is the "ratio" of the tiny xenharmonic bridge > I'm talking about.That's no kind of unison vector, Monz. All that is is a tiny difference between two functionally identical (as meantones in 5- limit) tuning systems. A real unison vector expresses a relationship within _one_ tuning system that allows it to take a pitch in more than one sense (and thus lead you on the road to finity). This interval you're talking about does not lead you in that direction at all (as you claimed above, in case you're wondering what I'm rambling on about).> If I could find some way to represent LucyTuning on the flat > lattice (which means finding some way to represent pi in a > universe where everything is factored by 3 and 5), then bend > the lattice into a meantone cylinder, then warp the cylinder > so that the LucyTuning "5th" occupies the same point as the > 3/10-comma meantone "5th", I'd have it. > > The fact that pi is transcendental, irrational, whatever, > makes it hard for me to figure out how to do this.If you want people to help you figure this out, you'll have to determine what the above construction is all about. What does it mean? Why do we want to see it on our perfectly good lattices-wrapped- onto-cylinders? P.S. There are so many wonderful varieties of cylindrical tunings your JustMusic software could be helping people visualize -- they'll all look different -- why not let the meantones look pretty much the same -- compositionally, they pretty much are, wouldn't you say?> >>> Somewhere a long time ago, perhaps in the Mills times, I posted a >> proposed formula for these lengths. But I'm not too picky about it. >> Why not show mistuning as a tiny "break" in the consonant connections? > >> Hmm ... sounds interesting. You mean like on the recent Blackjack > lattice you posted? Please elaborate. > >>>>>> And I *still* don't understand how a note that I factor as, >>>>> for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the >>>>> 1/6-comma meantone "whole tone", can be represented as >>>>> anything else. There is no other combination of exponents >>>>> for 3 and 5 which will plot that point in exactly that spot. >>>>>>>> Sure there are -- they'd just be irrational exponents. But when >>>> some meantones, like LucyTuning, will require irrational exponents >>>> anyway in order to get a "spiral" happening for them, that >>>> implies to me that irrational exponents are just as meaningful >>>> as rational ones. . . . > >> I'm still not getting this. It seems to me that you're thinking > in terms of something other than the 3x5 plane within which I'm working. > > But I can certainly buy what you're saying about irrational exponents. > As I said, if I could figure out *how* to lattice them, I would. > (See the bit above about the xenharmonic bridge.) > > >>>> Hmmm ... yes, I'm thinking that the meantone-spiral thing really >>> might be a useful new contribution to tuning theory. Thanks for >>> the acknowledgement! >>>> You got it, though I'm not banking on the "useful" part :) >> Particularly irksome is that you must choose a 1/1, such as C, which >> flies in the face of the true nature of meantone as a transposible >> system. > >> Ah! ... Paul, this is where you'd understand my ideas a little > better if you finally succumb to my always begging you to join > my justmusic group! <Yahoo groups: /justmusic * [with cont.] > > > > The idea is that the user could: > > 1) Choose meantone as the type of tuning desired: JustMusic > then draws the syntonic-comma based cylinder on the lattice. > > 2) Define which meantone by fraction-of-a-comma (or Lucy or Golden, > if I ever figure out how): JustMusic draws the spiral around > the cylinder. > > 3) Use the mouse to roll the cylinder around on the lattice > to get whichever key-center is desired, or simply input the > key and let JustMusic do the rolling. > > > That's just a brief outline. > > Sure do hope to see you over there! (and anyone else on this > list who is intrigued but hasn't signed up yet) > > > > -monz > > > > > > _________________________________________________________ > Do You Yahoo!? > Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3624 - Contents - Hide Contents Date: Thu, 31 Jan 2002 05:23:07 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 30, 2002 8:17 PM > Subject: [tuning-math] Re: new cylindrical meantone lattice > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > >>> Not really. >> >>>> Huh? Take a look at the "ratios" I post below. >> 2^(2/10) * 3^(-2/10) * 5^(3/10) is *mathematically* >> totally different from 2^(2pi+1)/4pi, even tho they're >> acoustically identical. >> Again, Monz, you're looking too closely at the way the math looks, > and not thinking about what it means. > > I could write pi in fifteen different ways that look totally > mathematically different to you, but they're all be the same. > > > ... >>>>> 3/10-comma quasi-meantone "5th" >>>> "-" Lucytuning "5th" >>>> --------------------------------- >>>> ~0.010148131 cent = ~1/99 cent >>>> >>>> = >>>> >>>> 2^(2/10) * 3^(-2/10) * 5^(3/10) >>>> - 2^(2pi+1)/4pi >>>> ------------------------------------------ >>>> 2^(-12pi-10)/40pi * 3^(-2/10) * 5^(3/10) >> >>>> That last number is the "ratio" of the tiny xenharmonic bridge >> I'm talking about. >> That's no kind of unison vector, Monz. All that is is a tiny > difference between two functionally identical (as meantones in 5- > limit) tuning systems. A real unison vector expresses a relationship > within _one_ tuning system that allows it to take a pitch in more > than one sense (and thus lead you on the road to finity). This > interval you're talking about does not lead you in that direction at > all (as you claimed above, in case you're wondering what I'm rambling > on about). >>> If I could find some way to represent LucyTuning on the flat >> lattice (which means finding some way to represent pi in a >> universe where everything is factored by 3 and 5), then bend >> the lattice into a meantone cylinder, then warp the cylinder >> so that the LucyTuning "5th" occupies the same point as the >> 3/10-comma meantone "5th", I'd have it. >> >> The fact that pi is transcendental, irrational, whatever, >> makes it hard for me to figure out how to do this. >> If you want people to help you figure this out, you'll have to > determine what the above construction is all about. What does it > mean? Why do we want to see it on our perfectly good lattices-wrapped- > onto-cylinders?Paul, *you* were the one who kept nagging me about "... but how are you going to lattice LucyTuning or Golden Meantone on this kind of lattice?". So if your point is that LucyTuning and 3/10-comma meantone are "functionally identical", then I can simply lattice LucyTuning *as* 3/10-comma meantone and call it a day. Now, about that bit where I wrote: "... finding some way to represent pi in a universe where everything is factored by 3 and 5", I had an idea for a simpler beginning approach. Let's start with meantone-like EDOs instead. As an example, we want to lattice 1/3-comma meantone alongside 19-EDO. Can you devise some formula that would find the fractional powers of 3 and 5 that would be needed to plot 19-EDO on a trajectory that would follow closely alongside the 1/3-comma trajectory? Now, I think *that* would be a meaningful lattice! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

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