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Message: 1001 - Contents - Hide Contents Date: Mon, 15 Jan 2001 09:38:34 Subject: Re: [tuning] Re: badly tuned remote overtones From: monz I don't recall ever getting a response to this. Still interested ...> From: monz <joemonz@xxxxx.xxx> > To: <tuning@xxxxxxxxxxx.xxx>; <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, January 11, 2002 2:04 PM > Subject: [tuning-math] Re: [tuning] Re: badly tuned remote overtones > > > > First, I'd like to start this post off with a link to my > "rough draft" of a lattice of the periodicity-block Gene > calculated for Schoenberg's theory: > > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) > > This shows the 12-tone periodicity-block (primarily 3- and 5-limit, > with one 11-limit pitch), and its equivalent p-block cousins at > +/- each of the four unison-vectors. > > > Now to respond to Paul... > >>> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> >> To: <tuning@xxxxxxxxxxx.xxx> >> Sent: Friday, January 11, 2002 12:47 PM >> Subject: [tuning] Re: badly tuned remote overtones >> >> >> You seem to be brushing some of the unison vectors you had >> previously reported, and from which Gene derived 7-, 5-, and 2-tone >> periodicity blocks, under the rug. > >> Ah ... so then this, from Gene: ... >>> From: genewardsmith <genewardsmith@xxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Wednesday, December 26, 2001 3:25 PM >> Subject: [tuning-math] Re: Gene's notation & Schoenberg lattices >> >> ... This matrix is unimodular, meaning it has determinant +-1. >> If I invert it, I get >> >> [ 7 12 7 -2 5] >> [11 19 11 -3 8] >> [16 28 16 -5 12] >> [20 34 19 -6 14] >> [24 42 24 -7 17] >> >> ... actually *does* specify "7-, 5-, and 2-tone periodicity blocks". > Yes? > >>> Face it, Monz -- without some careful "fudging", Schoenberg's >> derviation of 12-tET as a scale for 13-limit harmony is not >> the rigorous, unimpeachable bastion of good reasoning that >> you'd like to present it as. > >> Your point is taken, but please try to understand my objectives > more clearly. I agree with you that "Schoenberg's derviation ... > is not the rigorous, unimpeachable bastion of good reasoning" etc. > I'm simply trying to get a foothold on what was in his mind when > he came up with his radical new ideas for using 12-tET to represent > higher-limit chord identities. > > I've seen it written (can't remember where right now) that without > the close personal attachment to Schoenberg that his students had, > it's nearly impossible to understand all the subtleties of his > teaching. I'm just trying to dig into that scenario a bit, and > in a sense to "get closer" to Schoenberg and his mind. > >>> The contradictions in Schoenberg's arguments were known at least >> as early as Partch's Genesis, and he isn't going to weasel his way >> out of them now :) If 12-tET can do what you and Schoenberg are >> trying to say it can, it can do _anything_, and there would be >> no reason ever to adopt any other tuning system. > >> Ahh ... well, I think you've put on finger on the crux of the matter. > > Schoenberg consciously rejected microtonality and also made a > conscious decision to use the 12-tET tuning as tho it *could* do > "_anything_". > > As I've documented again and again, he *did* have a favorable attitude > towards adopting other tuning systems, but was of the opinion that > only in the future would the time be right for that. With us now > living *in* that future, it seems to me that perhaps he was right > after all. Perhaps it's even possible that Schoenberg's actions > in adopting the "new version" of 12-tET ("atonality") helped to > precipitate the current trend towards microtonality and alternative > tunings. ...? > > > Always curious about these things, > > -monz _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 1002 - Contents - Hide Contents Date: Mon, 15 Jan 2001 09:59:52 Subject: Re: [tuning] Re: badly tuned remote overtones From: monz I also never got replies on my questions here, and am still waiting. I'm particularly curious about how 56/55 was added as a unison-vector. Thanks.> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, January 11, 2002 1:13 AM > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones > > > Hi Paul and Gene, > > >>> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> >> To: <tuning@xxxxxxxxxxx.xxx> >> Sent: Thursday, January 10, 2002 10:10 AM >> Subject: [tuning] Re: badly tuned remote overtones >> >> >> --- In tuning@y..., "monz" <joemonz@y...> wrote: >>>>> The periodicity-blocks that Gene made from my numerical analysis >>> of Schoenberg's 1911 and 1927 theories are a good start. >>>> Well, given that most of the periodicity blocks imply not 12-tone, >> but rather 7-, 5-, and 2-tone scales, it strikes me that Schoenberg's >> attempted justification for 12-tET, at least as intepreted by you, >> generally fails. No? > > >> I originally said: > >>> From: monz <joemonz@xxxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Tuesday, December 25, 2001 3:44 PM >> Subject: [tuning-math] lattices of Schoenberg's rational implications >> >> >> Unison-vector matrix: >> >> 1911 _Harmonielehre_ 11-limit system >> >> ( 1 0 0 1 ) = 33:32 >> (-2 0 -1 0 ) = 64:63 >> ( 4 -1 0 0 ) = 81:80 >> ( 2 1 0 -1 ) = 45:44 >> >> Determinant = 7 >> >> ... <snip> ... >> >> But why do I get a determinant of 7 for the 11-limit system? >> Schoenberg includes Bb and Eb as 7th harmonics in his description, >> which gives a set of 9 distinct pitches. But even when >> I include the 15:14 unison-vector, I still get a determinant >> of -7. And if I use 16:15 instead, then the determinant >> is only 5. > > >> But Paul, you yourself said: > >>> From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Thursday, July 19, 2001 12:43 PM >> Subject: [tuning-math] Re: lattices of Schoenberg's rational implications >> >> >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>> Could anyone out there do some periodicity-block >>> calculations on this theory and say something about that? >>>> It's pretty clear that Schoenberg's theory implies a 12-tone >> periodicity block. > >> That was quite a while ago ... have you changed your position > on that? I thought that Gene showed clearly that a 12-tone > periodicity-block could be constructed out of Schoenberg's > unison-vectors. > > >>> From: genewardsmith <genewardsmith@xxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Wednesday, December 26, 2001 12:27 AM >> Subject: [tuning-math] Re: lattices of Schoenberg's rational implications >> >> >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> Can someone explain what's going on here, and what candidates >>> may be found for unison-vectors by extending the 11-limit system, >>> in order to define a 12-tone periodicity-block? Thanks. >>>> See if this helps; >> >> We can extend the set {33/32,64/63,81/80,45/44} to an >> 11-limit notation in various ways, for instance >> >> <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5] >> >> where g7 differs from h7 by g7(7)=19. > >> Gene, how did you come up with 56/55 as a unison-vector? > Why did I get 5 and 7 as matrix determinants for the > scale described by Schoenberg, but you were able to > come up with 12? > >>> Using this, we find the corresponding block is >> >> (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12) >> (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9; the >> Pythagorean scale. We don't need anything new to find a >> 12-note scale; we get >> >> 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--19/9--15/8 >> >> or variants, the variants coming from the fact that 12 >> is even, by using 12 rather than 7 in the denominator. > >> Can you explain this business about variants in a little > more detail? I understand the general concept, having seen > it in periodicity-blocks I've constructed on my spreadsheet, > but I'd like your take on the particulars for this case. > > > > -monz _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 1003 - Contents - Hide Contents Date: Tue, 6 Feb 2001 04:26:36 Subject: exactly what is a xenharmonic bridge? From: monz please pardon the length of this ... i feel that this is very important to my own work, and i'm confused ... an Instant Message discussion i had with Paul ...> joemonz: i want to discuss this with you via IM: > the objections you were posting the other day about > my apparent reinterpretation of "xenharmonic bridges". > > paulerlich: oh, the LucyTuning-vs.-3/10-comma meantone? > that clearly deprives them of all value, at least any > value that could be relevant for finity. in order for > finity to begin to occur, two ways of constructing an > interval from the _same_ set of basis intervals (usually > primes for you, but can be anything) have to lead to the > same, or aurally indistinguishable results. but this > doesn't happen in your latest case. > > joemonz: i'm a little confused about this. i think that > my conception of "xenharmonic bridge" is probably a > more general umbrella type of thing, which includes > "unison-vector" under it as a more specific aspect. > my original idea about "xenharmonic bridges" was that, > for example, we'd hear something in some ET (which > approximates a certain type of JI) and interpret it > as being *in* that JI. > > paulerlich: ok . . . that's closer to what harmonic > entropy is good for. but neither definition would > seem to justify calling your LucyTuning-vs.-3/10-comma > meantone thing a xenharmonic bridge. > > joemonz: well, that's what you've been maintaining all > along, but i'm having a hard time understanding why. > the xenharmonic bridge idea is that finity is an > essential part of music, because no single person > can comprehend all possible tonal relationships -- > different tunings quantize the pitch-continuum in > different ways, but *those differences are often not > audible*, and that's because of the bridging that's > going on all the time. perhaps there are exceptions > -- for example, La Monte Young-style strict JI or > barbershop, where essentially the listender *does* > hear the actual tuning. does that help you see where > i'm coming from? > > paulerlich: again, finity won't happen until two pitches > constructed from _the same set of basis intervals_ > are considered the same. all your examples of xenharmonic > bridging up to this point have been fine illustrations > of this -- you're about to send that all down the tubes > with this error in reasoning. > > joemonz: hmm ... but what i just wrote to you was the > *original* conception i had of xenharmonic bridging, > from 1998. i'm confused now. > > paulerlich: do you want it to aid toward finity, that is, > to reduce the number of dimensions of infinite extent > in the lattice? > > joemonz: well, yes, by definition xenharmonic bridges > limit infinity in at least two ways (i'll use a JI example): > 1) by reducing the total number of dimensions, > 2) by creating a periodicity *within* the remaining dimensions. > > paulerlich: ok . . . now how can either of those things > happen if the two intervals being 'bridged' between > have no basis intervals in common? > > joemonz: hmm ... i have the mathematical definitions of > "basis" (which i can't even read), and i'm pretty sure > that i have an intuitive grasp of the concept, but how's > about you try explaining it to me in plain english a little? > > paulerlich: i just mean the smallest units in the lattice > > joemonz: i suppose we'd better create an example ... > i'm always better with concrete examples and diagrams, > and not so good with abstract stuff. so create a > hypothetical situation -- let's say someone who's > never consciously been exposed to microtonal music > before gets a chance to hear a 13-limit Ben Johnston > quartet. my bet is that that person will probably > hear it mostly in terms of 12-edo, and *maybe* will > pick up some 5- and possibly 7-limit harmonic > structures, but will probably fail to comprehend > the 11- and 13-limit ones, because of lack of exposure > to them. feel free to argue about or change any of this > ... the whole purpose is to get me to understand a little > better where you're coming from. > > paulerlich: i don't see this as relevant at all so please > hold on to my previous explanation. anyway, they will hear > it in terms of 12-equal because of *categorical perception*, > which is a powerful phenomenon in all of psychology in its > own right. this has nothing to do, in my opinion, with > unison vectors or the like. and, if they simply 'fail to > comprehend' 11-limit and 13-limit harmonic structures, > rather than hearing them as something else, then there's > no problem to discuss, because they are not invoking any > bridges at all. > > joemonz: i don't see it that way. i'd say that there's a > whole slew of bridges in effect: 13==5, 11==5, 13==7, > 11==7, as well as the ones that bridge to 12-edo that i > don't have names for. they'd probably just hear the 11- > and 13-limit intervals as "out of tune", but they'd still > be comprehending them as 5/7-limit or 12-edo, which is > n o t what the music actually is. > > joemonz: please realize that i'm not approaching this as > a debate with you ... i believe that you probably have > a clearer grasp of this than i do, and i'm just trying > to understand your criticisms. > > paulerlich: 11=5? > > joemonz: an 11-limit interval that is close to, and is > being taken for, a 5-limit one. > > paulerlich: such as? > > paulerlich: and wouldn't that be hearing them as something > else, rather than failing to comprehend them? > > joemonz: exactly yes ... "hearing them as something else" > is exactly what i had in mind originally with "xenharmonic > bridges". ok ... here's an example of 11==5: suppose > there's a prominent 11:8 in a Johnston chord. especially > if there are a lot of 5-limit harmonies going on, i think > there's a very good chance that a lot of listeners will > perceive that 11:8 as a very out-of-tune 45:32. i'm even > willing to stick my neck out and claim that they might > perceive it as 5625:4096 = [2,3,5] [-12,2,4] = ~549.1648572 > cents -- i already know that you're going to argue that > that's highly unlikely, so we can stick with the 45:32 case. > > paulerlich: well, we're talking about three very > different things: > > paulerlich: (1) categorical perception > > paulerlich: (2) the hearing of a harmonic interval > as some other harmonic interval, isolated from context, > where harmonic entropy applies > > paulerlich: (3) the hearing of a tuning-system pitch as > some other tuning-system pitch, where all the pitches > are labeled with ratios in order to make their derivation > from consonant/basis intervals clear. this is probably > what you are talking about in the current example, and > is where finity applies. > > (paulerlich: it's kinda funny . . . in your book you argue > that the sharp 11 chord implies the 11th harmonic . . . > but here you are arguing the exact opposite!) > > joemonz: but can you see xenharmonic bridge as a very > general term that could encompass all three? in my mind, > they're all related to creating finity. > > paulerlich: these three are all related to creating finity, > yes, but they must be clearly distinguished from one > another, and mixed with much more care. > > (joemonz: re: 11 ... yeah, well, you know, 11 is the > oddball case, because it's nearly exactly between the > 12-edo F and F#. besides, i've grown a lot since i wrote > my book ... thanks largely to you! remember back when i > didn't say anything about meantone? ) > > joemonz: ok, well that's good for me to know. but i'm > still not clear on your answer to my question: is it > possible that "xenharmonic bridge" can serve as a > general term that covers all 3 aspects? if that's not > a good idea, then please, tell me why, and more importantly, > tell me exactly *what* "xenharmonic bridge" d o e s cover > ... would that simply be the case where it's like a > unison-vector but works *between* prime-factors rather > than *within* them (as a UV does)? > > paulerlich: can we use different symbol for ratios that > are built up from simpler basis intervals -- how about > a semicolon in the case of intervals and a backslash in > the case of pitches . . . margo suggested something > similar involving altering the order of the higher number > and the lower number, but i think this would be more clear. > > joemonz: oh yes, i've always been in full agreement with > adopting the convention of colon-for-interval and > slash-for-pitch, so this idea works for me. lay it on me. > > paulerlich: so we'd say 11:8 as opposed to 45;32 and > 5625;4096 in your example, or 11/8 as opposed to 45\32 > and 5625\4096 > > paulerlich: so as to your question . . . > > paulerlich: sense (3) is the sense that means unison vector. > if you like 'xenharmonic bridge' being a _subcategory_ > of unison vector, then it's awfully strange to apply it to > sense (2) or sense (1), let alone the lucy-tuning vs. > 3/10-comma meantone comparison, which is none of the above!and from an old post ...> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 03, 2002 9:04 PM > Subject: Re: the unison-vector<-->determinant relationship > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>>> There are other fraction-of-a-comma meantones >>>>>> which come closer to the center, and it seems >>>>>> to me that the one which *does* run exactly down >>>>>> the middle is 8/49-comma. >>>>>> >>>>>> Is this derivable from the [19 9],[4 -1] matrix? >>>>>>>>>> You should find that the interval corresponding >>>>> to (19 9), AS IT APPEARS in 8/49-comma meantone, >>>>> is a very tiny interval. >>>>>>>> Ah... so then 8/49-comma meantone does *not* run >>>> *exactly* down the middle. How could one calculate >>>> the meantone which *does* run exactly down the middle? >>> >>> It's 55-tET. >> >>>> Not if the periodicity-block is a parallelogram. >> 10/57-comma meantone is much closer to 55-EDO than >> 1/6-comma meantone, yet it is further away from the >> center of this periodicity-block. >> Hmm . . . the line you want is the vector (19 9). So any > generator 3^a/b * 5^c/d that is a solution to the equation > > a/b * 19 + c/d * 9 = 0 > > would work. This gives > > a/b*19 = -c/d*9 > > Does this help?it seems to me that what i'm getting at is that i think kernels should be definable with matrices of fractions as well as integers. i've had a hard time all along understanding why my plots of fraction-of-a-comma meantones are not unique for each meantone, because i can see that if the exponents of the prime-factors are rational, the whole matrix can be multiplied by the gcd and the matrix will once again be composed of all integers. can we please start with an example which compares 19-edo to 1/3-comma meantone? here are both generators and their difference: ~cents [2 3 5] [ 1/3 -1/3 1/3] = 694.7862377 = 1/3cmt "5th" - [2 3 5] [ 11/19 0 0 ] = 694.7368421 = 19edo "5th" ---------------------------- [2 3 5] [-14/57 -1/3 1/3] = 0.0493956 = 1/3cmt==19edo bridge now if my goal is to plot b o t h of these temperaments along with the JI pitches all on the same lattice, with the same 0,0 origin = 1/1 for all three tunings, then why is this 1/3cmt==19edo bridge (that is, "1/3-comma meantone is equivalent to 19-EDO") not valid as a basis? n o t necessarily as a l a t t i c e m e t r i c , but as a mathematical basis for comparing the set of tunings nonetheless. as i said, the whole matrix could be multiplied by the gcd: [2 3 5] ( [ 19 -19 19] * 1/57) = 1/3cmt "5th" - [2 3 5] ( [ 33 0 0] * 1/57) = 19edo "5th" ---------------------------- [2 3 5] ( [-14 -19 19] * 1/57) = 1/3cmt==19edo bridge is that right? -- assuming that it is ... now to get the solution i'm seeking, don't we have to keep the same exponent of 2 for both the 19edo and the 1/3cmt? so, for this case, what's the solution for the a,b,c,d of paul's post? that would be: the rational exponent pair for prime-factors 3 and 5 which most closely approximates each generator step of 19-edo, while the exponent of 2 is kept the same for both that tuning and the equivalent 1/3cmt step. so what i want to do now is change the exponent numerator of 2 in the last matrix above to 19 for the 19edo "5th", and adjust the values of 3 and 5 accordingly. this would therefore plot that tiny "difference" between 19edo and 1/3cmt, and the vector between those two points would be a xenharmonic bridge. or ... if i'm restricting the meaning of xenharmonic bridge as Paul suggests i should, then it's something else that deserves a unique name in my theory, because it's expressing an important component of my theory. -monz _________________________________________________________ Do You Yahoo!? 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Message: 1004 - Contents - Hide Contents Date: Wed, 27 Jun 2001 11:03:40 Subject: [tuning] questions about Graham's matrices (was: 13-limit mappings) From: monz Thanks for the explanation, Graham. A lot of it is now much clearer. I started this post as a series of questions about what I still didn't understand, but by working thru it I've gotten most of it. I decided to post my working-out here in the hope that it will help others to understand. A few questions remain below...> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Wednesday, June 27, 2001 4:35 AM > Subject: [tuning] Re: 13-limit mappings > >>>> mapping by period and generator: >>> ([1, 0], ([0, 2, -1], [5, 1, 12])) >> The first two-element list shows the mapping of the octave. The second > element is always zero for both my scripts, as the period is always a > fraction of an octave.So in other words you always use the nearest integer here? I'm still confused about that "0".> So the first number tells you how many equal > parts the octave is being divided into. Here it's 1 which is the > simplest case.Confused about this too... I thought this example divided the octave into 41 parts? Again, is this pair of numbers expressing the nearest integer fraction of an octave? Also, I think it's confusing the way you give the "octave correction" first and the "number of generators" second in this line, but it's reversed in all the following lines, generator first and octave second.> > In more familiar terms, the generator is a 5:4 major third. 5 major > thirds are a 3:1 perfect twelfth.(2^(380.391/1200))^5 does indeed equal exactly 3. Following you so far...> An octave less two major thirds is a 9:7 supermajor third.OK... using the regular ratios (which I understand are only approximated by your generator)... In regular math: 2 / ((5/4)^2) = 2 * (16/25) = 32/25 In vector addition: 2/1 = [ 1 0 0] (5/4)^2 = [-2 0 1] * 2 = [-4 0 2] and [ 1 0 0] -[-4 0 2] ---------- [ 5 0 -2] = 32/25 32/25 is ~7.7 cents narrower than 9:7, but both of these are approximated well by your actual result of 2 / ((2^(380.391/1200))^2). So I follow this too. Now comes the tricky part...> (2*(5,0) - (12,-1) = (-2, 1))OK, so as I said above, ((2^(380.391/1200))^5) * (2^0) = 3 . The "2*" means that we square that, and so the first group stands for 3^2 = 9 . And ((2^(380.391/1200))^12) * (2^-1) = ~6.983305074 , which agrees with your definition above as ~7. The minus sign means we divide the terms, and... Voilą! ... ~9/7 . And checking the answer: ((2^(380.391/1200))^-2) * (2^1) does indeed equal the ~9/7. So you're putting an equivalence relationship in here. That was confusing... I had a hard time understanding how ~9/7 = "An octave less two major thirds". Now it's clear.>>> mapping by steps: >>> [(22, 19), (35, 30), (51, 44), (62, 53)] >> Each pair shows the size of a prime interval in terms of scale steps. > Call the steps x and y. An octave is 22x+19y. For the case where x=y, > you have 41-equal. Where x=0, you have 19-equal. Where y=0, you have > 22-equal. So 19, 22 and 41-equal are all members of this temperament > family. > > 3:1 is 35x+30y, 5:1 is 51x+44y and 7:1 is 62x+53y. You can get any > 7-prime limit interval in terms of x and y by combining these.OK, I understand all the math here, but I'm not quite following the logic which deterimines that they are "all members of this temperament family". How does your program find the 19, 22 and 41 in this example?> > For visualising the scale, it can be simpler to reduce each prime > interval to be within the octave. >>>> [(22, 19), (35, 30), (51, 44), (62, 53)] >> 3:2 is 13x + 11y Because[ 35 30] -[ 22 19] ---------- [ 13 11]> 5:4 is 7x + 6y[ 51 44] -[ 22 19] * 2 = [ 51 44] -[ 44 38] ---------- [ 7 6]> 8:7 is 4x + 4y[ 62 53] -[ 22 19] * 3 = [ 62 53] -[ 66 57] ----------- [- 4 - 4]> > and use simpler coordinates. Here, q=x+y and p=x > > 3:2 is 11q + 2p > 5:4 is 6q + p > 8:7 is 4qOops... now you lost me.> So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal > and p is 1 step in 41-or 22-equal, and no steps in 19-equal. Getting foggier... > > >>> unison vectors:>>> [[-10, -1, 5, 0], [5, -12, 0, 5]]So these are the ratios 3125/3072 and 537824/531441 ? -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 1005 - Contents - Hide Contents Date: Wed, 27 Jun 2001 19:53:23 Subject: Re: 41 "miracle" and 43 tone scales From: jpehrson@xxx.xxx --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Yahoo groups: /tuning-math/message/404 * [with cont.]> > Since arguably the thing the Diamond shows best is the > at-least-dual nature of each ratio, which is a property > Partch emphasized repeatedly was inherent in ratios (quite > obvious to my mind, since they're a relationship described > by two numbers, duh!), then it seems to me to follow that > this dual property was perhaps the primary conceptual focus > of his tuning system. > A question:In arithmetic and mathematics is the *numerator* of a fraction ever considered "more important" than the *denominator?* Or is that a silly question...? It seems to me in simple arithmetic, the numerator seems more "impressive..." maybe because the numbers are larger?? Just as in "otonal??" Hasn't the "otonal" series, on the overall, been considered *significantly* more important than the *utonal* over the years?? Or am I just "out to lunch..." _________ _______ ______ Joseph Pehrson
Message: 1006 - Contents - Hide Contents Date: Wed, 27 Jun 2001 19:59:20 Subject: Re: 41 "miracle" and 43 tone scales From: jpehrson@xxx.xxx --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: Yahoo groups: /tuning-math/message/407 * [with cont.]>> I'm interested now more than ever in knowing some of Daniel >> Wolf's knowledge and opinions on this subject. A full-scale >> analysis of the *non*-JI harmonies in Partch's compositions >> would reveal a ton of information. >> Yes indeed. We might be able to better answer the "schismic vs. > miracle" question based on that. > > -- Dave KeenanDoesn't this imply that, somehow, Partch was using the "non-JI" harmonies in a different way than his "JI" harmonies?? Personally, I would doubt that. Once he had his scale, he probably just used it "as is" regardless of the derivation of the notes.. ?? __________ ________ ________ Joseph Pehrson
Message: 1008 - Contents - Hide Contents Date: Wed, 27 Jun 2001 20:51:54 Subject: Re: ET's, unison vectors (and other equivalences) From: jpehrson@xxx.xxx --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Yahoo groups: /tuning-math/message/417 * [with cont.]> > ----- Original Message ----- > From: D.Stearns <STEARNS@C...> > To: <tuning-math@y...> > Sent: Tuesday, June 26, 2001 5:27 PM > Subject: Re: [tuning-math] Re: ET's, unison vectors (and other equivalences) > >>> Hi Paul and everyone, >> >> You can also use the 2d lattice as a basic model for plotting >> coordinates other that 3 and 5. >> >> Earlier I gave the [4,3] 7-tone, neutral third scale as an example >> with the unison vectors 52/49 >> = 2^2 * 7^-2 * 13^1 = [ 2 0 0 -2 0 1] > > >> and 28672/28561. >> = 2^12 * 7^1 * 13^-4 = [ 12 0 0 1 0 -4] > >>> This would be an example of plugging 13 and 7 into >> a 2D lattice space while retaining the diatonic matrix. >> As cab be seen at a glance in either of the prime-factor notations. >I can see a cab at a glance, but I don't get this vector notation... Could you please gently run it down again, or would that be for the *arithmetic* list?? Thanks! _________ ______ _____ Joseph Pehrson
Message: 1009 - Contents - Hide Contents Date: Wed, 27 Jun 2001 21:03:39 Subject: Re: 41 "miracle" and 43 tone scales From: jpehrson@xxx.xxx --- In tuning-math@y..., "M. Edward (Ed) Borasky" <znmeb@a...> wrote: Yahoo groups: /tuning-math/message/425 * [with cont.]> On Wed, 27 Jun 2001 jpehrson@r... wrote: > >> A question: >>>> In arithmetic and mathematics is the *numerator* of a fractionever considered "more important" than the *denominator?*> > Not that I know of -- see the definition of the rational numbers asequivalence classes of ordered pairs of integers. In an ordered pair, *somebody's* gotta be number one and somebody else's gotta be number two :-). Ya ain't got no ordered pair otherwise :-).>Got it! Thanks, Ed!>>> Just as in "otonal??" Hasn't the "otonal" series, on the overall, >> been considered *significantly* more important than the *utonal*over the years??> > Outside of Partch, yes -- Otonal/Major is *musically* moreimportant than Utonal/Minor *in common practice Western music*. One of the things Partch was trying to do, after having defined Otonal and Utonal to begin with, was to treat them equally in his music and right what he considered to be a wrong in this respect. I haven't heard enough of his music to know whether Otonal and Utonal are in fact equally respected in his works. Gee... this is an interesting question, but Jon Szanto isn't on this list... Maybe I'll post something to the "biggie...">>> Or am I just "out to lunch..." >> Are you buying? :-)Sure! But, unfortunately... you're in Oregon at the moment.... :) _________ _______ _____ Joseph Pehrson
Message: 1010 - Contents - Hide Contents Date: Wed, 27 Jun 2001 23:37:11 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan --- In tuning-math@y..., jpehrson@r... wrote:> In arithmetic and mathematics is the *numerator* of a fraction ever > considered "more important" than the *denominator?*No. I don't think so. It's all completely dual.> Or is that a silly question...?No. Its a good question.> It seems to me in simple arithmetic, > the numerator seems more "impressive..." maybe because the numbers > are larger??In ordinary (non-musical) usage the numerator is just as likely to be smaller than the denominator.> Just as in "otonal??" Hasn't the "otonal" series, on the overall, > been considered *significantly* more important than the *utonal* over > the years??Yes. But this doesn't make the numerator or denominator special. It makes _the_smallest_of_the_two_numbers_ special. Several frequencies having their fundamentals ocurring as if they are the harmonics of a lower virtual fundamental, gives more consonance than several frequencies that each have one harmonic corresponding to a higher "guide-tone". In the case of octave-equivalent pitches we have a convention to put them in a form that is between 1/1 and 2/1 so they have positive logarithms. But for non octave-equivalent pitches we can have 2/3 different from 3/2. For intervals, octave equivalence doesn't matter. 2:3 describes exactly the same interval as 3:2. I have argued before for a convention of putting the small number first, as we do for "extended ratios" such as 4:5:6. But when we want to take its logarithm (to convert to cents) we will still enter it as 3/2, i.e. big number as numerator, so that we are dealing with positive logarithms. But remember these are only conventions or conveniences. The musical specialness is in "big number versus little number", not "numerator versus denominator". Regards, -- Dave Keenan
Message: 1012 - Contents - Hide Contents Date: Thu, 28 Jun 2001 12:00 +0 Subject: Re: questions about Graham's matrices (was: 13-limit mappin From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <002c01c0ff33$9751a860$4448620c@xxx.xxx> I'm replying to this here, as we've had complaints about over-precise cents values. Perhaps Monz could post back to TBL (The Big List) when he understands it.>>>> mapping by period and generator: >>>> ([1, 0], ([0, 2, -1], [5, 1, 12])) >>>> The first two-element list shows the mapping of the octave. The >> second >> element is always zero for both my scripts, as the period is always a >> fraction of an octave. >> So in other words you always use the nearest integer here? > I'm still confused about that "0".It isn't the nearest integer to anything: these lists are the definition of the temperament. All the temperaments I'm currently considering use a period that's an equal division of the octave. So you never need the generator to get the octave, and that parameter's always zero.>> So the first number tells you how many equal >> parts the octave is being divided into. Here it's 1 which is the >> simplest case. >> Confused about this too... I thought this example divided the > octave into 41 parts? Again, is this pair of numbers expressing > the nearest integer fraction of an octave?The octave is divided into 41 *unequal* parts. There are no equal divisions of the octave. Compare with this 13-limit temperament I gave before: 9/52, 103.897 cent generator basis: (0.5, 0.086580634742799478) mapping by period and generator: ([2, 0], ([3, 5, 7, 9, 10], [1, -2, -8, -12, -15])) mapping by steps: [(58, 46), (92, 73), (135, 107), (163, 129), (201, 159), (215, 170)] unison vectors: [[1, 2, -3, 1, 0, 0], [-4, 0, 2, 1, -1, 0], [1, -3, 2, 1, 0, -1], [2, -1, 0, 1, -2, 1]] highest interval width: 17 complexity measure: 34 (46 for smallest MOS) highest error: 0.004911 (5.893 cents) unique and diaschismic temperament: 2/11, 105.214 cent generator basis: (0.5, 0.087678135277931377) mapping by period and generator: ([2, 0], ([3, 5], [1, -2])) mapping by steps: [(12, 10), (19, 16), (28, 23)] unison vectors: [[11, -4, -2]] highest interval width: 3 complexity measure: 6 (8 for smallest MOS) highest error: 0.002716 (3.259 cents) unique Both of them divide the octave in 2 equal parts all the time.> Also, I think it's confusing the way you give the "octave correction" > first and the "number of generators" second in this line, but > it's reversed in all the following lines, generator first and > octave second.Octave is always first in the printouts. I explained the generator mapping first because it makes more sense that way round. As another thread shows, you can ignore octaves completely and define the scale using the generator mapping and number of equal divisions of the octave. You don't know how many notes the MOS will contain, or whether it's unique assuming octave invariance, but you can still work out this information.>> In more familiar terms, the generator is a 5:4 major third. 5 major >> thirds are a 3:1 perfect twelfth. >> (2^(380.391/1200))^5 does indeed equal exactly 3. > > Following you so far... Oh good! > So I follow this too. Now comes the tricky part... >>> (2*(5,0) - (12,-1) = (-2, 1)) >> OK, so as I said above, ((2^(380.391/1200))^5) * (2^0) = 3 . > The "2*" means that we square that, and so the first group > stands for 3^2 = 9 .Oh, that is putting the generator first, isn't it? That's wrong, you're right to pull me up on it. So it should be (2*(0, 5) - (12, -1)) = (1, -2)) I think you're complicating it by bringing ratios and exact pitches back into it. A 9:1 is two 3:1 steps, hence 2*(0, 5). Any ratio can be prime factorized, and worked out in terms of octaves and generators using the conversion matrix.> And ((2^(380.391/1200))^12) * (2^-1) = ~6.983305074 , > which agrees with your definition above as ~7. > > The minus sign means we divide the terms, and... > Voilą! ... ~9/7 .7:1 is written as (2,-1). You read that straight off. So 9:7 or (0 2 0 -1)H is 2*(0,5)-(-1,12). You could write that (0 2 0 -1)( 1 0) ( 0 5) ( 2 1) (-1 12)> And checking the answer: > ((2^(380.391/1200))^-2) * (2^1) does indeed equal the ~9/7. > > So you're putting an equivalence relationship in here.That's a check, yes, but a simpler one would be to use the relationship at the top of the printout, that the generator is 13 steps from 41. 41-2*13=41-26=15. And 9:7 does approximate to 15 steps from 41.> That was confusing... I had a hard time understanding how > ~9/7 = "An octave less two major thirds". Now it's clear.It means you can construct an augmented triad with two 5-limit and one 9-identity thirds.>>>> mapping by steps: >>>> [(22, 19), (35, 30), (51, 44), (62, 53)] >>>> Each pair shows the size of a prime interval in terms of scale steps. >> Call the steps x and y. An octave is 22x+19y. For the case where >> x=y, >> you have 41-equal. Where x=0, you have 19-equal. Where y=0, you have >> 22-equal. So 19, 22 and 41-equal are all members of this temperament >> family. >> >> 3:1 is 35x+30y, 5:1 is 51x+44y and 7:1 is 62x+53y. You can get any >> 7-prime limit interval in terms of x and y by combining these. >> OK, I understand all the math here, but I'm not quite following the > logic which deterimines that they are "all members of this temperament > family". How does your program find the 19, 22 and 41 in this example?My program *starts* with 19 and 22, along with the prime intervals and odd limit, and works everything out from them. It gets 41 by adding 19 and 22. My other program starts with the unison vectors, but that's more complex. A third program could go from unison vectors to a mapping in terms of generators within a period that's a fraction of an octave. In that case, it'd have to get the equal temperaments by optimizing for the best generator/period ratio, and walking the scale tree.>> and use simpler coordinates. Here, q=x+y and p=x >> >> 3:2 is 11q + 2p >> 5:4 is 6q + p >> 8:7 is 4q >> Oops... now you lost me.Substitute in for p and q: 11q+2p = 11(x+y) + 2x = 13x + 11y 6q+p = 6(x+y) + x = 7x + 6y 4q = 4(x+y) = 4x + 4y>> So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal >> and p is 1 step in 41-or 22-equal, and no steps in 19-equal. > > Getting foggier...| 19= | 22= | 41= --------------------------------- x | 0 | 1 | 1 y | 1 | 0 | 1 p | 0 | 1 | 1 q | 1 | 1 | 2>>>> unison vectors: >>>> [[-10, -1, 5, 0], [5, -12, 0, 5]] > >> So these are the ratios 3125/3072 and 537824/531441 ?The might well be, you're as capable as me of working them out. I missed off the bottom of the file before:>> highest interval width: 12This is the maximum number of generators needed for an interval within the consonance limit.>> complexity measure: 12 (13 for smallest MOS)The complexity measure is the previous number multiplied by the number of equal divisions of the octave. The number of otonal or utonal chords is the number of notes in the scale minus this measure. Roughly. In this case there's a 13 note MOS that can hold that complete otonality.>> highest error: 0.004936 (5.923 cents)This comes from the minimax optimisation. It's a bit over-precise so it can be checked against another program, and because it's possible for the program to throw out *very* accurate temperaments. The temperaments are sorted assuming the fewer notes needed and the smaller the error the better.>> uniqueThis is either there or not. I added it to be able to assess keyboard mappings as well as temperaments. I have lists of keyboard mappings, not uploaded yet, that rate a mapping twice as badly if it isn't unique, and ignore the accuracy. Graham
Message: 1013 - Contents - Hide Contents Date: Thu, 28 Jun 2001 12:49 +0 Subject: Re: Hypothesis revisited From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9he0uv+d2ll@xxxxxxx.xxx> Paul wrote:>> Yes, they both give Miracle41, but a different Miracle41 each time/ >> Can you explain what you mean > by "different"? They're both > centered around the 1/1, so it's > not the mode that's different . . .One is 10+41n, the other 31+41n. The mapping by period and generator is the same both times. So they're both aspects of the same temperament. It depends on whether you take this "set of MOS scales" result seriously. It doesn't come out of the octave invariant method discussed below.>> If you invert and normalize the octave-invariant matrix, the left >> hand column >> gives you the prime intervals in terms of generators. >> Well that sounds like it solves the > Hypothesis in a demonstrative > fashion, yes?If you can prove it will always work. I can't, but am pleased it does. You can certainly always define the scale in terms of some kind of octave-invariant interval, and call that the generator. Perhaps that's all it comes down to. But I've always said this was obvious from the matrix technique. But showing that the unison vectors lead to a linear temperament is different from showing they give a CS periodicity block, or whatever it is you asked. The octave-specific method doesn't always give a result. It fails with the unison vectors I'm using for the multiple-29 temperament. But you can always define a temperament in terms of a pair of intervals, even if they aren't the ones you want for the MOS. The octave-invariant result for multiple-29, BTW, is this mapping: [0, 707281, 707281, 707281, 707281] when I wanted [0, 29, 29, 29, 29] Incidentally, an alternative octave-specific case would be to define an extra chromatic unison vector instead of the octave. The the two left hand columns of the inverse will be the mapping by scale steps. Graham
Message: 1014 - Contents - Hide Contents Date: Thu, 28 Jun 2001 18:29:07 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9he0uv+d2ll@e...> > Paul wrote: >>>> Yes, they both give Miracle41, but a different Miracle41 each time/ >>>> Can you explain what you mean >> by "different"? They're both >> centered around the 1/1, so it's >> not the mode that's different . . . >> One is 10+41n, the other 31+41n.What do you mean by this notation?> The mapping by period and generator is > the same both times. So they're both aspects of the same temperament. > It depends on whether you take this "set of MOS scales" result seriously.I'm not following you.> It doesn't come out of the octave invariant method discussed below.What's "It" in this sentence?>>>> If you invert and normalize the octave-invariant matrix, the left >>> hand column >>> gives you the prime intervals in terms of generators. >>>> Well that sounds like it solves the >> Hypothesis in a demonstrative >> fashion, yes? >> If you can prove it will always work. I can't, but am pleased it does. > You can certainly always define the scale in terms of some kind of > octave-invariant interval, and call that the generator. Perhaps that's > all it comes down to.Yes, but this choice should be unique . . . there should only be one (octave-invariant) generator.> But I've always said this was obvious from the > matrix technique. But showing that the unison vectors lead to a linear > temperament is different from showing they give a CS periodicity block, > or whatever it is you asked.Well there may be some differences in our understanding of this, as the above (different miracle-41s) may be indicating. But I think we're on the right track . . . ?> > The octave-specific method doesn't always give a result.Uh-oh. So maybe I can convince you to switch over to octave-invariant?> It fails with > the unison vectors I'm using for the multiple-29 temperament. But you > can always define a temperament in terms of a pair of intervals, even if > they aren't the ones you want for the MOS.Don't they _have_ to be the generator and the interval of repetition?> > The octave-invariant result for multiple-29, BTW, is this mapping: > > [0, 707281, 707281, 707281, 707281] > > when I wanted > > [0, 29, 29, 29, 29]Can you explain how the number 707281 comes about?
Message: 1015 - Contents - Hide Contents Date: Thu, 28 Jun 2001 12:01:49 Subject: Re: questions about Graham's matrices (was: 13-limit mappin From: monz Hi Graham,> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, June 28, 2001 4:00 AM > Subject: [tuning-math] Re: questions about Graham's matrices (was: 13-limit mappin > > > I'm replying to this here, as we've had complaints about over-precise > cents values. Perhaps Monz could post back to TBL (The Big List) when he > understands it.Yup - I should have posted the other one here only, and not also on TBL. (Thanks for that convenient abbreviation).>>>>> [Graham] >>>>> mapping by period and generator: >>>>> ([1, 0], ([0, 2, -1], [5, 1, 12])) >>> >>> [Graham]>>> The first two-element list shows the mapping of the octave. >>> The second element is always zero for both my scripts, as >>> the period is always a fraction of an octave. >> >> [me, monz]>> So in other words you always use the nearest integer here? >> I'm still confused about that "0". > > [Graham]> It isn't the nearest integer to anything: these lists are the definition > of the temperament. > > All the temperaments I'm currently considering use a period that's an > equal division of the octave. So you never need the generator to get the > octave, and that parameter's always zero.OK, I think I understand that now. A counter-illustration using a period that's an unequal division, which would "need the generator to get the octave", would help.>>> So the first number tells you how many equal >>> parts the octave is being divided into. Here it's 1 which is the >>> simplest case. >>>> Confused about this too... I thought this example divided the >> octave into 41 parts? Again, is this pair of numbers expressing >> the nearest integer fraction of an octave? >> The octave is divided into 41 *unequal* parts. There are no equal > divisions of the octave.This was a bit confusing, because I got it tangled with what you said above about "a period that's an equal division of the octave". Now I see the difference. The scale under consideration is an temperament which is an *unequal* division of the octave (the period of equivalence), because it's a result of multiples of the generator and not of an equal division of anything. So all thru the rest of the explanation when you refer to "steps of 19=, 22=, 41="... they're all approximations to the generated scale. Right? I think what's been confusing me is that you refer to both ratios and EDOs as approximations of the scale resulting from your generator, and perhaps I've been taking them more literally than I should have been. I realize now that every interval is to be understood in terms of this temperament's approximations to the basic prime intervals 2, 3, 5, 7. So your matrices are presenting a set of transformations.>>> (2*(5,0) - (12,-1) = (-2, 1)) >>>> OK, so as I said above, ((2^(380.391/1200))^5) * (2^0) = 3 . >> The "2*" means that we square that, and so the first group >> stands for 3^2 = 9 . >> Oh, that is putting the generator first, isn't it? That's wrong, you're > right to pull me up on it. So it should be > > (2*(0, 5) - (12, -1)) = (1, -2))Oops!... your bad. You didn't reverse (12, -1) into (-1, 12) as you meant to do. I really think it's much more intuitive to have it the other way around (your mistake here shows the persistence of that way of thinking). Put the generator first and the octave second, consistently. I agree with you that the number of generators is the more important figure, and to me it makes sense to *see* that number first. (I think your unconscious switch in the original post shows that.) So reverse the "correction" you made here and put it back like it was, and reverse the *other* lines to agree with these. So your illustrated calcuation (call it "two fifths less a seventh") translates into approximate ratios as ~(3:2)^2 / ~7:4 = ~9:7 , and would look like: (2*(5, 0) - (12, -1)) = (-2, 1)) . And the "octave less two major thirds" translates into approximate ratios as ~2:1 / ~(5/4)^2 = ~32/25 . When I did the matrix calculation I got (0, 1) - 2*(1, 2) = (-2, -3) . Hmmm... the important number, the generator, works out to be the same -2, which is correct. But why is the period calculation not working out when the octave is included? Is is because there is no zero period? So your opening lines, with extra labels, would look like: basis (generator, period) as fraction of octave: (0.31699250014423125, 1.0) mapping by [generator], [period] (~2:1, (~3:2, ~5:4, ~7:4)) : ([0, 1], ([5, 1, 12], [0, 2, -1])) Actually, I think that second line would be better rearranged to agree with the octave notation: mapping by (generator, period) [~2:1, ~3:2, ~5:4, ~7:4]: [(0, 1), (5, 0), (1, 2), (12, -1)] To me, that's as plain as day. The octave can still be seen as set apart by virtue of being first/leftmost on the list.> > I think you're complicating it by bringing ratios and exact pitches back > into it.Agreed... but using the ratios allowed *me* to do the math in an Excel spreadsheet so that I could follow your reasoning. I went thru it step by step, looking at the cents values all along the way. If I understood better how to manipulate the matrices, I certainly would have done it that way too. I can see that it's *much* more elegant that way, even tho I've been having trouble understanding it. This is along the lines of what I was trying to get Paul to understand a couple of different times in the past. It's not necessary to always use prime-factors as the basis for lattice metrics... any numbers that give even, consistent divisions of the pitch-space *in SOME way* will do. The different ways of dividing (and multiplying) produce different kinds of lattices. (Of course Paul already understands all this, *and* the math to manipulate it, as do you. But I don't think he was following my reasoning when I was trying to make that point... probably because *I don't* understand the math! I'm not speaking the same language you guys are... altho I'm trying hard...)> A 9:1 is two 3:1 steps, hence 2*(0, 5).Yes, now that's *very* clear. I caught it, but certainly didn't explain it as elegantly as this.> Any ratio can be prime factorized, and worked out in terms of > octaves and generators using the conversion matrix.*That's* what's been giving me such trouble! Relating the calculations given in terms of this temperament to the approximate ratios really did confuse me, even tho in hindsight now I think it helped in the process of understanding.> >>> And ((2^(380.391/1200))^12) * (2^-1) = ~6.983305074 , >> which agrees with your definition above as ~7. >> >> The minus sign means we divide the terms, and... >> Voilą! ... ~9/7 . >> 7:1 is written as (2,-1).Oops! Your bad again... you meant (12, -1). ... well, actually you meant (-1, 12). (Boy, this interval sure keeps giving you the slip!)> You read that straight off. > So 9:7 or (0 2 0 -1)H is 2*(0,5)-(-1,12). You could write that > > (0 2 0 -1)( 1 0) > ( 0 5) > ( 2 1) > (-1 12) >Thanks, Graham. Seeing the matrix conversion broken down like this helps me a lot. So keeping to my (generator, period) reversal of your notation, (0 2 0 -1)H = 2*(5,0)-(12,-1) because ~3 = (5,0) and ~7 = (12,-1). Of course, in the case of an octave-invariant scale like this it's much simpler to just omit the period. So ignoring the first column in "H" because it's powers of 2, ratio prime vector 380.391-cent generators 2:1 = ( 1 0 0 0)H = ~ 0 3:2 = (-1 1 0 0)H = ~ 5 5:4 = (-2 0 1 0)H = ~ 1 7:4 = (-2 0 0 1)H = ~12 9:7 = ( 0 2 0 -1)H = ~(2*5)-12 = ~-2.>> And checking the answer: >> ((2^(380.391/1200))^-2) * (2^1) does indeed equal the ~9/7. >> >> So you're putting an equivalence relationship in here. >> That's a check, yes, but a simpler one would be to use the relationship > at the top of the printout, that the generator is 13 steps from 41. > 41-2*13=41-26=15. And 9:7 does approximate to 15 steps from 41.This helps a lot too. Thanks. So here you're back again to "an octave less two major thirds". ~2:1 / ~(5/4)^2 = ~32/25 or (0, 1) - 2*(1, 2) = (-2, -3) .>> That was confusing... I had a hard time understanding how >> ~9/7 = "An octave less two major thirds". Now it's clear. >> It means you can construct an augmented triad with two 5-limit and one > 9-identity thirds.Er... this is a little confusing, because a triad is constructed of only two intervals. You mean that if one measured all the intervals in an augmented triad *and its inversions*, the result would be two ~5:4s and one ~9:7.>>>>> mapping by steps: >>>>> [(22, 19), (35, 30), (51, 44), (62, 53)] >>>>>> Each pair shows the size of a prime interval in terms of scale steps. >>> Call the steps x and y. An octave is 22x+19y. For the case where >>> x=y, you have 41-equal. Where x=0, you have 19-equal. Where y=0, >>> you have 22-equal. So 19, 22 and 41-equal are all members of this >>> temperament family. >>> >>> 3:1 is 35x+30y, 5:1 is 51x+44y and 7:1 is 62x+53y. You can get any >>> 7-prime limit interval in terms of x and y by combining these. >>>> OK, I understand all the math here, but I'm not quite following the >> logic which deterimines that they are "all members of this temperament >> family". How does your program find the 19, 22 and 41 in this example? >> My program *starts* with 19 and 22, along with the prime intervals and > odd limit, and works everything out from them. It gets 41 by adding 19 > and 22.Ahhh!... that certainly explains *that*!> My other program starts with the unison vectors, but that's more complex. > A third program could go from unison vectors to a mapping in terms of > generators within a period that's a fraction of an octave. In that case, > it'd have to get the equal temperaments by optimizing for the best > generator/period ratio, and walking the scale tree.Hmmm.... that last algorithm sounds like a good one! Exactly the kind of thing I always wanted to include in my JustMusic software, applicable to rational systems as well as irrational.>>> and use simpler coordinates. Here, q=x+y and p=x >>> >>> 3:2 is 11q + 2p >>> 5:4 is 6q + p >>> 8:7 is 4q >>>> Oops... now you lost me. >> Substitute in for p and q: > > 11q+2p = 11(x+y) + 2x = 13x + 11y > 6q+p = 6(x+y) + x = 7x + 6y > 4q = 4(x+y) = 4x + 4yAh... if only I had been paying attention in algebra class... OK, now it's perfectly clear.>>> So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal >>> and p is 1 step in 41-or 22-equal, and no steps in 19-equal. >> >> Getting foggier... >> | 19= | 22= | 41= > --------------------------------- > x | 0 | 1 | 1 > y | 1 | 0 | 1 > p | 0 | 1 | 1 > q | 1 | 1 | 2 >Uh-oh... still not getting this part. Please elaborate. For some reason I'm not seeing the connection between p and q and the steps of the EDOs. Too many layers of abstraction for me to follow ...>>>>> unison vectors: >>>>> [[-10, -1, 5, 0], [5, -12, 0, 5]] >> >>>> So these are the ratios 3125/3072 and 537824/531441 ? >> The[y] might well be, you're as capable as me of working them out.OK... I was simply double-checking with you that the numbers stand for exponents of the prime-factors 2, 3, 5, and 7. So I guess they do. So your parenthetical lists are elegant, but IMO could use a little bit more of a legend explaining what those lists represent. Otherwise one has to learn the sequences beforehand and keep them in mind. I suggest adding a label giving the parameter list before each line. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 1016 - Contents - Hide Contents Date: Thu, 28 Jun 2001 19:31:55 Subject: Re: questions about Graham's matrices (was: 13-limit mappin From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > Er... this is a little confusing, because a triad is constructed of > only two intervals.Last time I checked, a triad had three intervals, a tetrad six, a pentad ten, and a hexad fifteen.
Message: 1017 - Contents - Hide Contents Date: Thu, 28 Jun 2001 20:00:54 Subject: Re: Hypothesis revisited From: Graham Breed>> >ne is 10+41n, the other 31+41n. >> What do you mean by this notation?Temperements including the ETs with 10+41n or 31+41n notes, where n is a non-negative integer.>> The mapping by period and generator is >> the same both times. So they're both aspects of the same > temperament.>> It depends on whether you take this "set of MOS scales" result > seriously. >> I'm not following you.I explained this before. When you generate the scales from a set of unison vectors, one of them chromatic, the natural result is something like 10+41n rather than a single MOS or the full range of temperaments defined by the commatic unison vectors.>> It doesn't come out of the octave invariant method discussed below. >> What's "It" in this sentence?The restricted set of temperaments. But in fact I was wrong there. In fact, the second column of the normalized octave-specific inverse is the same as the first column of the octave-invariant one, but with an extra zero. I didn't notice it was the generator mapping before, but managed to get the right results anyway :)>> If you can prove it will always work. I can't, but am pleased it > does.>> You can certainly always define the scale in terms of some kind of >> octave-invariant interval, and call that the generator. Perhaps > that's>> all it comes down to. >> Yes, but this choice should be unique . . . there should only be one > (octave-invariant) generator.This brings us back to """" The determinant is -41, and the inverse is [ 1 0 0 0 0 ] [ 65/41 6/41 -2/41 -1/41 -2/41] [ 95/41 -7/41 16/41 8/41 16/41] [ 115/41 -2/41 28/41 14/41 -13/41] [ 142/41 15/41 -5/41 -23/41 -5/41]> The left hand two columns should be > > [[ 41 0] > [ 65 -6] > [ 95 7] > [115 2] > [142 -15]]Up to a minus sign, yes.> > If they are, the two sets of unison vectors give exactly the same > results. They don't! """There are aways two generators that will work. The minus sign differentiates them.>> But I've always said this was obvious from the >> matrix technique. But showing that the unison vectors lead to a > linear>> temperament is different from showing they give a CS periodicity > block,>> or whatever it is you asked. >> Well there may be some differences in our understanding of this, as > the above (different miracle-41s) may be indicating. But I think > we're on the right track . . . ? Oh, unquestionably.>> The octave-specific method doesn't always give a result. >> Uh-oh. So maybe I can convince you to switch over to octave-invariant?I think it would be worth writing a script that only uses them. It would mean altering the code in temper.py to accept a mapping by generators, so it's abit of work.>> It fails with >> the unison vectors I'm using for the multiple-29 temperament. But > you>> can always define a temperament in terms of a pair of intervals, > even if>> they aren't the ones you want for the MOS. >> Don't they _have_ to be the generator and the interval of repetition?No. If you take this matrix at face value:> [[ 41 0] > [ 65 -6] > [ 95 7] > [115 2] > [142 -15]]/41it defines Miracle using one 41st part of an octave, and a 41st part of the usual generator. That works, but it isn't efficient.>> The octave-invariant result for multiple-29, BTW, is this mapping: >> >> [0, 707281, 707281, 707281, 707281] >> >> when I wanted >> >> [0, 29, 29, 29, 29] >> Can you explain how the number 707281 comes about?It's 29^4. I'm sure it means I chose the chromatic unison vector wrongly. The interesting thing is that the generator matrix is a multiple of what it should be. In fact, the whole matrix has a common factor, which may be the clue that something's wrong. Although dividing through by that common factor won't work. Also, this is a case where the inverse of the octave-specific matrix doesn't get the generator mapping right. If the method almost works with an arbitrary chroma, that means we're a step towards getting it to work with only commatic unison vectors, which should be possible. Graham "I toss therefore I am" -- Sartre
Message: 1018 - Contents - Hide Contents Date: Thu, 28 Jun 2001 20:21:01 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., Graham Breed <graham@m...> wrote:>>> One is 10+41n, the other 31+41n. >>>> What do you mean by this notation? >> Temperements including the ETs with 10+41n or 31+41n notes, where n is a > non-negative integer.I'm still confused about how there can be two different MIRACLE-41s. Are there two different Canastas too, or does the divergence only happen at 41?>>>> The mapping by period and generator is >>> the same both times. So they're both aspects of the same >> temperament.>>> It depends on whether you take this "set of MOS scales" result >> seriously. >>>> I'm not following you. >> I explained this before. When you generate the scales from a set of unison > vectors, one of them chromatic, the natural result is something like 10+41n > rather than a single MOS or the full range of temperaments defined by the > commatic unison vectors.A single MOS is what I expect. The number of notes in that MOS normally equals the determinant of the matrix of unison vectors, including the chromatic one. So where are we disagreeing?>> >> Yes, but this choice should be unique . . . there should only be one >> (octave-invariant) generator. >> This brings us back to > > """" > The determinant is -41, and the inverse is > [ 1 0 0 0 0 ] > [ 65/41 6/41 -2/41 -1/41 -2/41] > [ 95/41 -7/41 16/41 8/41 16/41] > [ 115/41 -2/41 28/41 14/41 -13/41] > [ 142/41 15/41 -5/41 -23/41 -5/41] >>> The left hand two columns should be >> >> [[ 41 0] >> [ 65 -6] >> [ 95 7] >> [115 2] >> [142 -15]] >> Up to a minus sign, yes. >>>> If they are, the two sets of unison vectors give exactly the same >> results. > > They don't! > """ >> There are aways two generators that will work. The minus sign differentiates > them.But if you center the resulting scale around 1/1, either the plus- sign or the minus-sign generator should give the same results. So that can't account for the difference we saw.>> >> Don't they _have_ to be the generator and the interval of repetition? >> No. If you take this matrix at face value: >>> [[ 41 0] >> [ 65 -6] >> [ 95 7] >> [115 2] >> [142 -15]]/41 >> it defines Miracle using one 41st part of an octave, and a 41stpart of the> usual generator. That works, but it isn't efficient.How does it work? Certainly the scale doesn't repeat itself every 41st of an octave.> Also, this [the 29th-of-an-octave thing] is a case where theinverse of the octave-specific matrix> doesn't get the generator mapping right. :( > > If the method almost works with an arbitrary chroma, that meanswe're a step> towards getting it to work with only commatic unison vectors, which should be > possible.Well you _should_ be able to find the generator without specifying the chroma, but you need the chroma to select a particular MOS.
Message: 1019 - Contents - Hide Contents Date: Thu, 28 Jun 2001 01:23:43 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., Graham Breed <graham@m...> wrote:> Yes, they both give Miracle41, but a different Miracle41 each time/Can you explain what you mean by "different"? They're both centered around the 1/1, so it's not the mode that's different . . .> > If you invert and normalize the octave-invariant matrix, the left hand column > gives you the prime intervals in terms of generators.Well that sounds like it solves the Hypothesis in a demonstrative fashion, yes?
Message: 1020 - Contents - Hide Contents Date: Thu, 28 Jun 2001 22:18 +0 Subject: Re: questions about Graham's matrices From: graham@xxxxxxxxxx.xx.xx monz wrote:>> All the temperaments I'm currently considering use a period that's an >> equal division of the octave. So you never need the generator to get >> the >> octave, and that parameter's always zero. >> OK, I think I understand that now. A counter-illustration > using a period that's an unequal division, which would "need the > generator to get the octave", would help.As you never need it, it's difficult to find a counter-example... My schismic fourth keyboard mapping would be an example, where the period is a fourth. So the octave would be two periods plus a generator I think. But I've never worked that out with matrices.>> The octave is divided into 41 *unequal* parts. There are no equal >> divisions of the octave. >> This was a bit confusing, because I got it tangled with what you > said above about "a period that's an equal division of the octave". > Now I see the difference. > > The scale under consideration is an temperament which is an > *unequal* division of the octave (the period of equivalence), > because it's a result of multiples of the generator and not of > an equal division of anything.It's an unequal division of the period, which is an equal division of the interval of equivalence (in this case the trivial division of one) which for these example is always an octave.> So all thru the rest of the explanation when you refer to > "steps of 19=, 22=, 41="... they're all approximations to > the generated scale. Right?There isn't "a generated scale". The generator can be used for a whole family of scales. 19, 22 and 41-equal are merely special cases of the scales it can generate.> I think what's been confusing me is that you refer to both ratios > and EDOs as approximations of the scale resulting from your generator, > and perhaps I've been taking them more literally than I should > have been. I realize now that every interval is to be understood > in terms of this temperament's approximations to the basic prime > intervals 2, 3, 5, 7. So your matrices are presenting a set of > transformations.Yes, matrices are all about transformations. In this case between the harmonic and melodic ways of looking at things. It's a one-way transformation because you loose information in going from the just to tempered mapping.>> (2*(0, 5) - (12, -1)) = (1, -2)) > >> Oops!... your bad. You didn't reverse (12, -1) into (-1, 12) > as you meant to do. > > I really think it's much more intuitive to have it the other way around > (your mistake here shows the persistence of that way of thinking). > > Put the generator first and the octave second, consistently. > I agree with you that the number of generators is the more important > figure, and to me it makes sense to *see* that number first. > (I think your unconscious switch in the original post shows that.)No, because that would contradict the usual way of writing vectors from low to high primes.> And the "octave less two major thirds" translates into > approximate ratios as ~2:1 / ~(5/4)^2 = ~32/25 . > When I did the matrix calculation I got > > (0, 1) - 2*(1, 2) = (-2, -3) . > > Hmmm... the important number, the generator, works out to be > the same -2, which is correct. But why is the period calculation > not working out when the octave is included? Is is because > there is no zero period?You're using 5:1 instead of 5:4.>> I think you're complicating it by bringing ratios and exact pitches >> back >> into it. >> Agreed... but using the ratios allowed *me* to do the math in > an Excel spreadsheet so that I could follow your reasoning. > I went thru it step by step, looking at the cents values all > along the way. > > If I understood better how to manipulate the matrices, I certainly > would have done it that way too. I can see that it's *much* more > elegant that way, even tho I've been having trouble understanding it.If you've got Excel, you can do that! I explain it on my website. You use MINVERSE, MDETERM and MMULT, pressing CTRL-SHIFT-RETURN to enter the formulae.> This is along the lines of what I was trying to get Paul to > understand a couple of different times in the past. It's not > necessary to always use prime-factors as the basis for lattice > metrics... any numbers that give even, consistent divisions > of the pitch-space *in SOME way* will do. The different ways > of dividing (and multiplying) produce different kinds of lattices.Yes, I think this is what Pierre Lamothe was trying to get across before he left the list as well.> Of course, in the case of an octave-invariant scale > like this it's much simpler to just omit the period. So > ignoring the first column in "H" because it's powers of 2, > > ratio prime vector 380.391-cent generators > > 2:1 = ( 1 0 0 0)H = ~ 0 > 3:2 = (-1 1 0 0)H = ~ 5 > 5:4 = (-2 0 1 0)H = ~ 1 > 7:4 = (-2 0 0 1)H = ~12 > 9:7 = ( 0 2 0 -1)H = ~(2*5)-12 = ~-2.If you're thinking octave invariantly, you can simplify it further. ratio prime vector 380.391-cent generators 3:2 = (1 0 0)H = ~ 5 5:4 = (0 1 0)H = ~ 1 7:4 = (0 0 1)H = ~12 9:7 = (2 0 -1)H = ~(2*5)-12 = ~-2.>>> That was confusing... I had a hard time understanding how >>> ~9/7 = "An octave less two major thirds". Now it's clear. >>>> It means you can construct an augmented triad with two 5-limit and one >> 9-identity thirds. >> Er... this is a little confusing, because a triad is constructed of > only two intervals. > > You mean that if one measured all the intervals in an augmented triad > *and its inversions*, the result would be two ~5:4s and one ~9:7.Oh, however you count it, I was thinking of the octave as part of the chord.>> My other program starts with the unison vectors, but that's more >> complex. >> A third program could go from unison vectors to a mapping in terms of >> generators within a period that's a fraction of an octave. In that >> case, >> it'd have to get the equal temperaments by optimizing for the best >> generator/period ratio, and walking the scale tree. >> Hmmm.... that last algorithm sounds like a good one! Exactly the > kind of thing I always wanted to include in my JustMusic software, > applicable to rational systems as well as irrational.Try it. Type the octave-invariant unison vectors into the spreadsheet as an array, with the chromatic one at the top. Then select an equal sized square, type "=minverse(?:?)*mdeterm(?:?)" where ?:? is that original array, and the left hand column will be the mapping by generators.>>>> So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal >>>> and p is 1 step in 41-or 22-equal, and no steps in 19-equal. >>> >>> Getting foggier... >>>> | 19= | 22= | 41= >> --------------------------------- >> x | 0 | 1 | 1 >> y | 1 | 0 | 1 >> p | 0 | 1 | 1 >> q | 1 | 1 | 2 >> > >> Uh-oh... still not getting this part. Please elaborate. > For some reason I'm not seeing the connection between > p and q and the steps of the EDOs. > > Too many layers of abstraction for me to follow ...If you tune to a given ET, the table shows you how many steps will be in each interval.> So your parenthetical lists are elegant, but IMO could use > a little bit more of a legend explaining what those lists > represent. Otherwise one has to learn the sequences beforehand > and keep them in mind. I suggest adding a label giving the > parameter list before each line.The "parenthetical lists" are the stringifications of the objects the program uses. I could get it to write out an HTML file for each temperament, but for the moment it's cutting-edge data. Graham
Message: 1021 - Contents - Hide Contents Date: Thu, 28 Jun 2001 01:25:20 Subject: Re: ET's, unison vectors (and other equivalences) From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Hi Paul and everyone, > > You can also use the 2d lattice as a basic model for plotting > coordinates other that 3 and 5.We've done that before, for example Margo and my discussions (about 22-tET, among other things) in the (3,7) plane.
Message: 1022 - Contents - Hide Contents Date: Thu, 28 Jun 2001 14:24:15 Subject: Re: questions about Graham's matrices (was: 13-limit mappin From: monz> From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, June 28, 2001 12:31 PM > Subject: [tuning-math] Re: questions about Graham's matrices (was: 13-limit mappin > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> Er... this is a little confusing, because a triad is constructed of >> only two intervals. >> Last time I checked, a triad had three intervals, a tetrad six, a > pentad ten, and a hexad fifteen.Oops... my bad this time! I compounded my correction of Graham's not-entirely-correct statement by introducing an actual error. Yes, Paul, of course a triad *does* have three intervals. I was speaking specifically of the types of "3rds". Thanks. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 1023 - Contents - Hide Contents Date: Thu, 28 Jun 2001 01:34:29 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- In tuning-math@y..., jpehrson@r... wrote:> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: > > Yahoo groups: /tuning-math/message/407 * [with cont.] >>>> I'm interested now more than ever in knowing some of Daniel >>> Wolf's knowledge and opinions on this subject. A full-scale >>> analysis of the *non*-JI harmonies in Partch's compositions >>> would reveal a ton of information. >>>> Yes indeed. We might be able to better answer the "schismic vs. >> miracle" question based on that. >> >> -- Dave Keenan >> Doesn't this imply that, somehow, Partch was using the "non-JI" > harmonies in a different way than his "JI" harmonies??Well a question can't imply a fact. But if you mean, doesn't it _assume_ that, then no. In fact, the more Partch used them in the same way, the easier it will be to decide which unison vectors he may have accepted.> > Personally, I would doubt that. Once he had his scale, he probably > just used it "as is" regardless of the derivation of the notes.. > > ?? >Unfortunately, that may be a bit too much to hope for. Partch devised an involved compositional apparatus in _Genesis_ based on JI harmonies, and I would be shocked if this didn't still guide his later works somewhat.
Message: 1024 - Contents - Hide Contents Date: Thu, 28 Jun 2001 22:29 +0 Subject: Re: Hypothesis revisited From: graham@xxxxxxxxxx.xx.xx Paul Erlich wrote:> I'm still confused about how there can be two different MIRACLE-41s. > Are there two different Canastas too, or does the divergence only > happen at 41?There are two Canstas, 10+31n and 21+31n.> A single MOS is what I expect. The number of notes in that MOS > normally equals the determinant of the matrix of unison vectors, > including the chromatic one. So where are we disagreeing?It's not clear to me if the duality is real or not.>> There are aways two generators that will work. The minus sign > differentiates >> them. >> But if you center the resulting scale around 1/1, either the plus- > sign or the minus-sign generator should give the same results. So > that can't account for the difference we saw.Are the FPBs different in this sense? For the matrices, it's because the mapping to steps in the MOS is always the same.>> No. If you take this matrix at face value: >>>>> [[ 41 0] >>> [ 65 -6] >>> [ 95 7] >>> [115 2] >>> [142 -15]]/41 >>>> it defines Miracle using one 41st part of an octave, and a 41st> part of the>> usual generator. That works, but it isn't efficient. >> How does it work? Certainly the scale doesn't repeat itself every > 41st of an octave.Yes, it would do. If you try tuning a 12-note meantone in cents relative to 12-equal, you'll see the pattern.>> If the method almost works with an arbitrary chroma, that means> we're a step>> towards getting it to work with only commatic unison vectors, which > should be >> possible. >> Well you _should_ be able to find the generator without specifying > the chroma, but you need the chroma to select a particular MOS.Indeed so! But the octave invariant matrix doesn't give you that particular MOS. Although it gives you enough of a clue to work it out from the determinant, the main result is the mapping in terms of generators. Graham
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