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Message: 3825 - Contents - Hide Contents Date: Tue, 5 Feb 2002 14:07:49 Subject: Re: the missing consistencies From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, February 05, 2002 1:53 PM > Subject: [tuning-math] Re: the missing consistencies > >>>> They're not hiding, you mistook the .html extensions >>> for .txt. >> >>>> thanks, Manuel, it's fixed now. >> Definitions of tuning terms: consistent, (c) 1... * [with cont.] (Wayb.) >> >> >> paul, you had asked me to change something about my graphics >> here and in the "unique" definition ... what was it? >> well, before i remind you of that, which data exactly are you > plotting here? i would have expected only odd number limits, since > you're restricing yourself to equal temperaments with an integer > number of notes per octave . . .hmmm ... i simply copied the data from Manuel's webpages and pasted it into a spreadsheet, from which i made the graphs. so i guess those integer-EDOs are merely signposts along the x-axis. the plots correspond exactly with the data in the spreadsheet, so the EDOs plotted are not integer divisions. i admit that these graphs could be improved a lot -- this was just a very quick-and-dirty deal. i've never really understood these consistency tables well until now. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3826 - Contents - Hide Contents Date: Tue, 05 Feb 2002 22:17:26 Subject: Re: the missing consistencies From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> hmmm ... i simply copied the data from Manuel's webpages > and pasted it into a spreadsheet, from which i made the graphs. > > so i guess those integer-EDOs are merely signposts along > the x-axis. the plots correspond exactly with the data in > the spreadsheet, so the EDOs plotted are not integer divisions. > > i admit that these graphs could be improved a lot -- this > was just a very quick-and-dirty deal. i've never really > understood these consistency tables well until now.i don't think you're doing a good job of presenting this data visually, and it couldn't be much improved in this format. i think it would be far better to simply plot the odd-limit data for the integer et case, if you wanted one graph to illustrate 'consistency'. if you don't know how to read this information from manuel's table, you can find it on paul hahn's website, for example 1 1200.0| 498.0 * [with cont.] (Wayb.), which gives the maximum errors within each odd limit (the columns are 3, 5, 7, 9, 11, etc.) and ceases once inconsistency is reached.
Message: 3827 - Contents - Hide Contents Date: Tue, 5 Feb 2002 18:52:26 Subject: Re: the missing consistencies From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, February 05, 2002 2:17 PM > Subject: [tuning-math] Re: the missing consistencies > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> hmmm ... i simply copied the data from Manuel's webpages >> and pasted it into a spreadsheet, from which i made the graphs. >> >> so i guess those integer-EDOs are merely signposts along >> the x-axis. the plots correspond exactly with the data in >> the spreadsheet, so the EDOs plotted are not integer divisions. >> >> i admit that these graphs could be improved a lot -- this >> was just a very quick-and-dirty deal. i've never really >> understood these consistency tables well until now. >> i don't think you're doing a good job of presenting this data > visually, and it couldn't be much improved in this format. i think it > would be far better to simply plot the odd-limit data for the integer > et case, if you wanted one graph to illustrate 'consistency'. if you > don't know how to read this information from manuel's table, you can > find it on paul hahn's website, for example > 1 1200.0| 498.0 * [with cont.] (Wayb.), which gives the > maximum errors within each odd limit (the columns are 3, 5, 7, 9, 11, > etc.) and ceases once inconsistency is reached.yes, paul, i think you're probably right that that approach would work (and look) better. i'll check it out ... thanks. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3828 - Contents - Hide Contents Date: 5 Feb 2002 20:46:27 -0800 Subject: smallest interval between ratios in n-limit From: paul@xxxxxxxxxxxxx.xxx 3-limit: 4/3 : 3/2 5-limit: 6/5 : 5/4 7-limit: 7/5 : 10/7 ******* 9-limit: 10/9 : 9/8 11-limit: 12/11 : 11/10 13-limit: 14/13 : 13/12 15-limit: 16/15 : 15/14 17-limit: 18/17 : 17/16 19-limit: 20/19 : 19/18 21-limit: 26/21 : 21/17 ******* 23-limit: 24/23 : 23/22 25-limit: 26/25 : 25/24 27-limit 28/27 : 27/26 29-limit 29/21 : 40/29 ******* note how the 'pattern' is 'broken' at 7, 21, 29 . . . anyone got any explanations/comments?
Message: 3829 - Contents - Hide Contents Date: Wed, 6 Feb 2002 11:03 +00 Subject: Re: question: partch scale as 41-tone periodicity block From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a3pkqi+cmvj@xxxxxxx.xxx> paulerlich wrote:> 1. is there a better choice, in terms of capturing more of Partch's > choices correctly?One way to find out would be to list the step vectors (I did this once, but can't find them) and then list all the intervals between pairs of step vectors. Take the simplest, or something like that.> 2. > > (a) if the answer to (1) is no, what is the linear temperament > defined by the unison vectors 896/891, 441/440 and 245/243? > (b) if the answer to (1) is yes, then . . .I have a CGI for getting temperaments from unison vectors as well. In this case, using your 100:99 as the chromatic unison vector, I get 8/41, 234.3 cent generator basis: (1.0, 0.19528912541079024) mapping by period and generator: [(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13)] mapping by steps: [(36, 5), (57, 8), (83, 12), (101, 14), (125, 17)] highest interval width: 30 complexity measure: 30 (31 for smallest MOS) highest error: 0.004454 (5.345 cents) unique which is interesting in that it's a 41 note scale with a 31 note MOS so it looks suspiciously like Miracle. In fact it isn't because it isn't consistent with h31. Meaning the mapping of 11-limit harmony to the nearest intervals of 31-equal doesn't fit the mapping you get from this temperament, which is [31, 49, 71, 87, 108] instead of [31, 49, 72, 87, 107] and that gives a g72 of [72, 114, 166, 202, 250] instead of [72, 114, 167, 202, 249] In fact, it's this part of the scale tree .5 31 . 36 . 41 67 . 46 77 103 98 so g31&h41 has contorsion, and g72 doesn't figure. Ah, so that is h41&h46 (taking the nearest-prime mapping of 46-equal). Graham
Message: 3830 - Contents - Hide Contents Date: Wed, 6 Feb 2002 04:33:51 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@s...> > 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: 3831 - Contents - Hide Contents Date: Wed, 06 Feb 2002 21:21:41 Subject: Re: question: partch scale as 41-tone periodicity block From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <a3pkqi+cmvj@e...> > paulerlich wrote: >>> 1. is there a better choice, in terms of capturing more of Partch's >> choices correctly? >> One way to find out would be to list the step vectors (I did this once, > but can't find them) and then list all the intervals between pairs of > step vectors. Take the simplest, or something like that.Interesting you should say this, because I just did this and was about to post about it. Here are the step vectors and their differences: [121/120, 100/99, 99/98, 81/80, 64/63, 56/55, 55/54, 50/49, 49/48, 45/44] [9801/9800, 3025/3024, 2401/2400, 4000/3993, 540/539, 441/440, 5120/5103] Here are the linear temperaments these give which appeared in the previous post: Hemiennealimmal wedgie [36, 54, 36, 18, 2, -44, -96, -68, -145, -74] Octoid wedgie [24, 32, 40, 24, -5, -4, -45, 3, -55, -71] Miracle wedgie [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] Unidec wedgie [12, 22, -4, -6, 7, -40, -51, -71, -90, -3] wedgie [12, 34, 20, 30, 26, -2, 6, -49, -48, 15] (Consistent with both 58 and 72--probably deserves a name.) wedgie [2, -4, -16, -24, -11, -31, -45, -26, -42, -12] Here are ones that did not appear--including Paul's parallel 29-et system, I note: wedgie [1, 33, 27, -18, 50, 40, -32, -30, -156, -144] map [[0, -1, -33, -27, 18], [1, 2, 16, 14, -4]] bad 285.4371097 rms 1.115729551 g 27.84651812 generators [.4144789553, 1] 497.3747464 1200. ets 41 70 82 111 152 193 234 345 wedgie [42, 47, 34, 33, -23, -64, -93, -53, -86, -25] map [[0, 42, 47, 34, 33], [1, -13, -14, -9, -8]] bad 249.5656038 rms .5795196051 g 38.06010286 generators [.3472619033, 1] 416.7142840 1200. ets 72 95 144 167 239 311 383 455 478 550 622 694 861 933 wedgie [6, 75, 39, 73, 105, 45, 95, -120, -90, 70] map [[0, 6, 75, 39, 73], [1, -1, -30, -14, -28]] bad 664.6829813 rms 1.006667259 g 49.18550890 generators [.4309653303, 1] 517.1583964 1200. ets 58 181 239 420 478 601 wedgie [3, 17, -1, -13, 20, -10, -31, -50, -89, -33] map [[0, 3, 17, -1, -13], [1, 1, -1, 3, 6]] bad 253.7980928 rms 3.005389215 g 14.32031524 generators [.1953763085, 1] 234.4515702 1200. ets 5 41 46 82 87 92 128 133 169 174 wedgie [4, 50, 26, -31, 70, 30, -63, -80, -245, -177] map [[0, 4, 50, 26, -31], [1, 1, -5, -1, 8]] bad 441.6305312 rms .9569121300 g 39.67456904 generators [.1464510525, 1] 175.7412630 1200. ets 41 82 157 198 239 280 437 478 wedgie [18, 15, -6, 9, -18, -60, -48, -56, -31, 46] map [[0, 6, 5, -2, 3], [3, 0, 3, 10, 8]] bad 219.5497298 rms 1.357051968 g 21.15250745 generators [.2640465184, 1/3] 316.8558221 400. ets 15 30 57 72 87 102 144 159 174 231 wedgie [6, 46, 10, 44, 59, -1, 49, -106, -57, 89] map [[0, 3, 23, 5, 22], [2, 1, -12, 2, -9]] bad 314.3980722 rms 1.127989485 g 29.31601221 generators [.3618387335, 1/2] 434.2064802 600. ets 58 94 152 246 340 wedgie [2, 25, 13, 5, 35, 15, 1, -40, -75, -31] map [[0, 2, 25, 13, 5], [1, 1, -5, -1, 2]] bad 252.7955445 rms 3.221444274 g 13.70349278 generators [.2929392038, 1] 351.5270446 1200. ets 41 58 82 wedgie [10, 26, -34, -28, 18, -82, -79, -152, -155, 39] map [[0, 5, 13, -17, -14], [2, 1, -1, 13, 13]] bad 352.7792183 rms .8819853677 g 36.41035959 generators [.2171375651, 1/2] 260.5650781 600. ets 46 92 106 152 198 244 350 wedgie [0, 29, 29, 29, 46, 46, 46, -14, -33, -19] map [[0, 0, 1, 1, 1], [29, 46, 58, 72, 91]] bad 433.0072655 rms 2.285966134 g 23.25172805 generators [.3239193886, 1/29] 388.7032663 41.37931034 ets 29 58 87 145 174 232 wedgie [9, -7, -61, -10, -32, -122, -47, -122, 1, 183] map [[0, -9, 7, 61, 10], [1, 2, 2, 0, 3]] bad 401.1533812 rms .8397457939 g 40.50396806 generators [.4603708883e-1, 1] 55.24450660 1200. ets 87 152 174 239 326 391 413 478 565 wedgie [4, 50, 26, 68, 70, 30, 94, -80, -15, 101] map [[0, 2, 25, 13, 34], [2, 2, -10, -2, -13]] bad 544.1352214 rms 1.140709497 g 40.46868277 generators [.2929368122, 1/2] 351.5241746 600. ets 58 140 198 338 536
Message: 3832 - Contents - Hide Contents Date: Wed, 06 Feb 2002 21:25:14 Subject: Re: question: partch scale as 41-tone periodicity block From: paulerlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <a3pkqi+cmvj@e...> > paulerlich wrote: >>> 1. is there a better choice, in terms of capturing more of Partch's >> choices correctly? >> One way to find out would be to list the step vectors (I did this once, > but can't find them) and then list all the intervals between pairs of > step vectors. Take the simplest, or something like that.that doesn't do the trick. my original choice came from this approach and i had to keep using multiples or quotients of the intervals between the steps in order to capture more and more of partch's choices.> >> 2. >>>> (a) if the answer to (1) is no, what is the linear temperament >> defined by the unison vectors 896/891, 441/440 and 245/243? >> (b) if the answer to (1) is yes, then . . . >> I have a CGI for getting temperaments from unison vectors as well. > > In this case, using your 100:99 as the chromatic unison vector, I get > > 8/41, 234.3 cent generator > > basis: > (1.0, 0.19528912541079024) > > mapping by period and generator: > [(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13)] > > mapping by steps: > [(36, 5), (57, 8), (83, 12), (101, 14), (125, 17)] > > highest interval width: 30 > complexity measure: 30 (31 for smallest MOS) > highest error: 0.004454 (5.345 cents) > uniquewas this in gene's list of 35?> > which is interesting in that it's a 41 note scale with a 31 noteMOS so it> looks suspiciously like Miracle. In fact it isn't because it isn't > consistent with h31. Meaning the mapping of 11-limit harmony to the > nearest intervals of 31-equal doesn't fit the mapping you get from this > temperament, which is > > [31, 49, 71, 87, 108] > > instead of > > [31, 49, 72, 87, 107] > > and that gives a g72 of > > [72, 114, 166, 202, 250] > > instead of > > [72, 114, 167, 202, 249] > > In fact, it's this part of the scale tree > > .5 31 > . 36 > . 41 67 > . 46 77 103 98 > > so g31&h41 has contorsion, and g72 doesn't figure. > > Ah, so that is h41&h46 (taking the nearest-prime mapping of 46- equal).one day, all of this is going to have to be written up with precise definitions and the like. at least i hope so. otherwise, what a waste of intellect.
Message: 3833 - Contents - Hide Contents Date: Wed, 06 Feb 2002 21:31:57 Subject: Re: exactly what is a xenharmonic bridge? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> 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.my opinion: this is sheer nonsense, the intellectual equivalent of driving off a cliff in terms of understanding temperaments. gene?> 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?i don't know what you mean by basis here. your basis is [2 3 5], as i see it.> 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.i hope gene will come in with his opinion. this could work out wonderfully, i don't know.
Message: 3834 - Contents - Hide Contents Date: Wed, 06 Feb 2002 21:37:03 Subject: Re: question: partch scale as 41-tone periodicity block From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., graham@m... wrote: >> In-Reply-To: <a3pkqi+cmvj@e...> >> paulerlich wrote: >>>>> 1. is there a better choice, in terms of capturing more of Partch's >>> choices correctly? >>>> One way to find out would be to list the step vectors (I did this once, >> but can't find them) and then list all the intervals between pairs of >> step vectors. Take the simplest, or something like that. >> Interesting you should say this, because I just did this and wasabout to post about it.> > Here are the step vectors and their differences: > > [121/120, 100/99, 99/98, 81/80, 64/63, 56/55, 55/54, 50/49, 49/48, 45/44] > > [9801/9800, 3025/3024, 2401/2400, 4000/3993, 540/539, 441/440, 5120/5103]note that this does not even produce the unison vectors i used to make the block! 245:243 and 896:891 are missing, for example. and how about my original question above? no one seems interested in my questions :(
Message: 3836 - Contents - Hide Contents Date: Wed, 6 Feb 2002 20:06:03 Subject: Re: exactly what is a xenharmonic bridge? From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, February 06, 2002 5:12 PM > Subject: [tuning-math] Re: exactly what is a xenharmonic bridge? > > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> 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. >>> my opinion: this is sheer nonsense, the intellectual equivalent of >> driving off a cliff in terms of understanding temperaments. gene? >> It isn't nonsense, but I don't see what value it has. One can > define kernels for mappings of finite-dimentional vector spaces > over the rational numbers Q. This produces non-finitely-generated > abelian group structures, whose musical meaning I don't see. > If Monz can explain why it makes sense, the math would not be > a problem.thanks, Gene. i'm trying hard to explain this to the rest of you, but Paul and i have been arguing about it for 3 years, so i won't be surprised if it takes a little longer for me to make my ideas clear. i'm sorry that i can't do this abstractly, because i don't have anywhere near enough understanding of the algebra. the only way i can even attempt to make my points is by using an example, and i'd like to stick with the one i already proposed, which is a comparison of 1/3-comma meantone and 19-edo. so ...> 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 > > ... > > 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?so, i s it correct? i can't proceed until i know that what i've already done is working right. all i want to know here is if i'm using the correct notation. can the numbers in the first matrix be legitimately changed to the numbers in the second matrix, without altering their values? if this is incorrect, is there another way to express what i'm doing here? or is my whole procedure wrong?>> [me, monz] >> 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? > > [paul]> i don't know what you mean by basis here. your basis is > [2 3 5], as i see it.ok, i t h i n k that what i'm trying to do here is use a simple transformation to change my basis from [2 3 5] to [2/57 3/57 5/57] -- is that right? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3837 - Contents - Hide Contents Date: Thu, 7 Feb 2002 09:59 +00 Subject: Re: any ideas? From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <002601c1af92$f08c5820$a061d63f@stearns> D.Stearns wrote:> 1) Most rooted--as in the rotation most closely resembling a power of > 2 harmonic series. I guess this would only apply to RI scales. One way > to do this would be to simply take the LCM of the rotations' > denominators, and the one that best matches a power of 2 wins. In the > classic just and Pythagorean scales that would be the Lydian mode.In fact, I have little script that tries to follow Terhardt's formula for finding the root of a chord. That might do the trick. I'll dig it out sometime. Graham
Message: 3838 - Contents - Hide Contents Date: Thu, 7 Feb 2002 03:46:55 Subject: Re: exactly what is a xenharmonic bridge? From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, February 06, 2002 9:36 PM > Subject: [tuning-math] Re: exactly what is a xenharmonic bridge? > > > questions for monzo:this is unbelievable ... i've just spent hours writing a really detailed response, with lots of calculations of xenharmonic bridges illustrating one good example, and before i finished it my computer froze and now it's gone. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3839 - Contents - Hide Contents Date: Thu, 7 Feb 2002 03:50:18 Subject: Re: exactly what is a xenharmonic bridge? From: monz> this is unbelievable ... i've just spent hours writing a > really detailed response, with lots of calculations of > xenharmonic bridges illustrating one good example, > and before i finished it my computer froze and now > it's gone.before i do too much more, does anyone have any ideas on how i might retrieve that message? i was using Microsoft Outlook Express, and just as i had copied the message and pasted it into Notepad so that i could save it, the PC froze and i had to reboot. i don't see anything that would allow me to retrieve a previous item that i never saved, and i've used "Find" to search for all files saved over the last day, and it didn't turn up there either. help! :( -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3840 - Contents - Hide Contents Date: Thu, 7 Feb 2002 05:46:14 Subject: Re: exactly what is a xenharmonic bridge? From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, February 06, 2002 9:29 PM > Subject: [tuning-math] Re: exactly what is a xenharmonic bridge? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> ok, i t h i n k that what i'm trying to do here is use >> a simple transformation to change my basis from [2 3 5] to >> [2/57 3/57 5/57] -- is that right? >> . . . > > this was all based on the premise that simple ratios involving small > numbers of 2s, 3s, and 5s are 'understood' by the ear-brain system > for various reasons. > > and it has led to an ability to characterize and categorize tuning > systems by their 'equivalencies' -- in a way that has a direct > geometric interpretation in terms of that same, originally infinite, > lattice. > > now replace the [2 3 5] with [2/57 3/57 5/57]. what premise is that > based on?i'm not even sure if that [2 3 5] --> [2/57 3/57 5/57] transformation is correct ... in fact, i'm quite sure that it's not. Gene or Graham, can either of you help? i wrote:> ... a comparison of 1/3-comma meantone and 19-edo. so ... > > 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 > > ... > > 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?i'm attempting to change the basis, right? into what? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3841 - Contents - Hide Contents Date: Thu, 7 Feb 2002 06:00:08 Subject: xenharmonic bridges in the 12edo comma pump (was: exactly what...) From: monz ok, i've redone as much as i could of the post that disappeared. actually, some of it turned out better this time. enjoy!> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, February 06, 2002 9:36 PM > Subject: [tuning-math] Re: exactly what is a xenharmonic bridge? > > > questions for monzo: > > (note i'm using the new notation now) > > are 80;81 and 128;125 and 648;625 and 2048;2025 and 32805;32768 > xenharmonic bridges in 12-tET? > > is 80;81 a xenharmonic bridge in all meantones? > > just trying to figure out what you mean by xenharmonic bridge.it's hard for me to reason about this stuff abstractly, because i don't know enough about the algebra. so i'll have to use an example to illustrate. how about if i pick what's probably the most meaningful example? -- the "comma pump" progression in 12edo. comma pump I - vi - ii - V - I C - Am - Dm - G - C where "ratio" = 2^(x/12), the chords in this progression are all subsets of notes in the 12edo diatonic scale: x= B 11 A 9 G 7 F 5 E 4 D 2 C 0 C [0 4 7] I \ \ Am [9 0 4] vi ` - . Dm [2 5 9] ii ` - . G [7 11 2] V ` - . C [0 4 7] I i've used slashes and other marks to show the common-tone relationships between pairs of chords. the fact that every pair of chords here has at least one common-tone is what enables the circular progression in 12edo: I = C _______ .-' 0 `-. ,' 11 .. 1 `. / . . \ / 10 . . 2 \ ; . . : | 9 . . 3 | : . . ; \ 8 . . 4 / \ . ' / `. 7 5,' `-.___6___.-' vi = Am _______ .-' 0 `-. ,' 11 . . 1 `. / . . \ / 10 . . 2 \ ; . . : | 9 . . 3 | : ' . . ; \ 8 ' . 4 / \ / `. 7 5,' `-.___6___.-' ii = Dm _______ .-' 0 `-. ,' 11 1 `. / \ / 10 . 2 \ ; . ' ' : | 9 .' ' 3 | : . ' ; \ 8 . ' 4 / \ . ' / `. 7 ' 5,' `-.___6___.-' V = G _______ .-' 0 `-. ,' 11 . 1 `. / . ' . \ / 10 . ' 2 \ ; . ' : | 9 . ' 3 | : . ' ; \ 8 . ' 4 / \ . ' / `. 7 5,' `-.___6___.-' I = C _______ .-' 0 `-. ,' 11 .. 1 `. / . . \ / 10 . . 2 \ ; . . : | 9 . . 3 | : . . ; \ 8 . . 4 / \ . ' / `. 7 5,' `-.___6___.-' this comma pump progression would need two different D's to be tuned beat-free in JI: a 10/9 for the Dm chord (= 5/3 utonality) and a 9/8 for the G chord (= 3/2 otonality): 10:9----5:3-----5:4----15:8 D A E B \ / \ / \ / \ \ / \ / \ / \ 4:3-----1:1-----3:2-----9:8 F C G D so assuming that the musical context implies this JI structure, we note that since 12edo offers only one D, the syntonic comma 81:80 is being tempered out: 3==5 bridge [2 3 5] [-3 2 0] = 9:8 Pythagorean D - [2 3 5] [ 1 -2 1] = 10:9 JI D -------------------- [2 3 5] [-4 4 -1] = 81:80 syntonic comma but then we must also note that there are two xenharmonic bridges in effect for the relationship of the 12edo D to each of the JI D's : 12edo==Pythagorean bridge for 9/8 [2 3 5] [ -3 2 0] = 9/8 Pythagorean D - [2 3 5] [ 1/6 0 0] = 12edo D ----------------------- [2 3 5] [-19/6 2 0] = ~3.910001731 cents 12edo==JI bridge for 10/9 [2 3 5] [ 1/6 0 0] = 12edo D - [2 3 5] [ 1 -2 1] = 10/9 JI D ---------------------- [2 3 5] [-5/6 2 -1] = ~17.59628787 cents so our matrix for these three bridges is: [2 3 5] [-4 4 -1] = 81:80 syntonic comma [-19/6 2 0] = 12edo==9/8 bridge = ~3.910001731 cents [ -5/6 2 -1] = 12edo==10/9 bridge = ~17.59628787 cents and note that these three bridges are linearly dependent. now, most likely in "common-practice" repertoire a meantone harmonic paradigm is intended at least part of the time. the equivalent meantone to 12edo is 1/11-comma meantone, and the two tunings are very close indeed: 12edo==1/11cmt bridge [2 3 5] [ 1/6 0 0 ] = 12edo D - [2 3 5] [-25/11 14/11 2/11] = 1/11cmt D ------------------------------- [2 3 5] [161/66 -14/11 -2/11] = ~0.000232741 = ~1/4300 cent and here are the 1/11-comma meantone bridges to the two JI pitches: 1/11cmt==Pythagorean bridge for 9/8 [2 3 5] [ -3 2 0 ] = 9/8 Pythagorean D - [2 3 5] [-25/11 14/11 2/11] = 1/11cmt D ------------------------------- [2 3 5] [ -8/11 8/11 -2/11] = ~3.910234472 cents 1/11cmt==JI bridge for 10/9 [2 3 5] [-25/11 14/11 2/11] = 1/11cmt D - [2 3 5] [ 1 -2 1 ] = 10/9 JI D ------------------------------- [2 3 5] [-36/11 36/11 -9/11] = ~17.59605512 cents and again, note that these last two bridges and the syntonic comma are linearly dependent. so here's the entire list of bridges which i would say are in effect for the comma pump in 12edo: 2 3 5 ~cents [ -4 4 -1 ] = 81:80 syntonic comma = 21.5062896 [ -5/6 2 -1 ] = 12edo==10/9 bridge = 17.59628787 [-36/11 36/11 -9/11] = 1/11cmt==10/9 bridge = 17.59605512 [ -8/11 8/11 -2/11] = 1/11cmt==9/8 bridge = 3.910234472 [-19/6 2 0 ] = 12edo==9/8 bridge = 3.910001731 [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge = 0.000232741 now, i'm not claiming that any listener is consciously aware of all of these xenharmonic bridges at any given time. but any intelligent harmonic analysis of 12edo performance of "common-practice" repertoire (a good example is the thousands of MIDI files of this repertoire -- without any pitch-bend -- which are in existence), must take these bridges into account. so if anyone wants to say something to m e about music in 12edo which features the comma pump, it would be a good idea to mention something about this batch of intervals. so paul, should i be using the new notation (\ and ;) for the 81:80 here? i could find similar sets of dependent vectors using 128;125, 648;625, 2048;2025, and 32805;32768 instead of the syntonic comma. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3842 - Contents - Hide Contents Date: Thu, 07 Feb 2002 01:12:42 Subject: Re: exactly what is a xenharmonic bridge? From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> 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.> my opinion: this is sheer nonsense, the intellectual equivalent of > driving off a cliff in terms of understanding temperaments. gene?It isn't nonsense, but I don't see what value it has. One can define kernels for mappings of finite-dimentional vector spaces over the rational numbers Q. This produces non-finitely-generated abelian group structures, whose musical meaning I don't see. If Monz can explain why it makes sense, the math would not be a problem.
Message: 3843 - Contents - Hide Contents Date: Thu, 07 Feb 2002 01:15:36 Subject: Re: question: partch scale as 41-tone periodicity block From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> note that this does not even produce the unison vectors i used to > make the block! 245:243 and 896:891 are missing, for example. and how > about my original question above? no one seems interested in my > questions :(That's because it wasn't (and isn't) clear to me what the question is. It isn't finding temperaments of dimension 0 or 1, since we tried that; do you want to find a block containing Genesis?
Message: 3844 - Contents - Hide Contents Date: Thu, 07 Feb 2002 01:28:31 Subject: Re: question: partch scale as 41-tone periodicity block From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> [(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13)] >> >> mapping by steps: >> [(36, 5), (57, 8), (83, 12), (101, 14), (125, 17)] >> >> highest interval width: 30 >> complexity measure: 30 (31 for smallest MOS) >> highest error: 0.004454 (5.345 cents) >> unique >> was this in gene's list of 35?No, but it was on my second list of temperaments derived from Genesis itself. It's done quite well by 17/87 as a generator.>> Ah, so that is h41&h46 (taking the nearest-prime mapping of 46- > equal).There you go--41+46=87.
Message: 3845 - Contents - Hide Contents Date: Thu, 07 Feb 2002 01:45:40 Subject: Re: question: partch scale as 41-tone periodicity block From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> note that this does not even produce the unison vectors i used to >> make the block! 245:243 and 896:891 are missing, for example. and how >> about my original question above? no one seems interested in my >> questions :( >> That's because it wasn't (and isn't) clear to me what the question >>s. It isn't finding temperaments of dimension 0 or 1, since we tried >that;actually, i was very enthusiastic about those answers so far. thanks to both of you.>do you want to find a block containing Genesis?that was one of the questions, yes.
Message: 3846 - Contents - Hide Contents Date: Thu, 07 Feb 2002 01:47:18 Subject: Re: question: partch scale as 41-tone periodicity block From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> [(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13)] >>> >>> mapping by steps: >>> [(36, 5), (57, 8), (83, 12), (101, 14), (125, 17)] >>> >>> highest interval width: 30 >>> complexity measure: 30 (31 for smallest MOS) >>> highest error: 0.004454 (5.345 cents) >>> unique >>>> was this in gene's list of 35? >> No, but it was on my second list of temperaments derived from Genesis itself. It's done quite well by 17/87 as a generator. >>>> Ah, so that is h41&h46 (taking the nearest-prime mapping of 46- >> equal). >> There you go--41+46=87.well, i'm officially behind you guys in understanding. i really want to get the whole clifford algebra thing down . . . maybe i should shut up until i do.
Message: 3847 - Contents - Hide Contents Date: Thu, 07 Feb 2002 05:29:32 Subject: Re: exactly what is a xenharmonic bridge? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> ok, i t h i n k that what i'm trying to do here is use > a simple transformation to change my basis from [2 3 5] to > [2/57 3/57 5/57] -- is that right?ok try to listen to what i am saying now . . . wwwwwhat does that basis *mean* about what we *hear*? I mean, you and i agree that simple ratios involving small numbers of 2s, 3s, and 5s are 'understood' by the ear-brain system for various reasons. this in itself implies the infinite ji lattice. however, slowly add the consideration of 'xenharmonic bridging' in the full sense of unison vectors, and infinity begins to edge toward finity. either through slight tempering of the intervals in the basis, or through near-coincidences of multiples of 2, 3, and 5 (such as 80 to 81 and 32805 to 32768), we find equivalencies we can use, and the lattice begins to become finite. this was all based on the premise that simple ratios involving small numbers of 2s, 3s, and 5s are 'understood' by the ear-brain system for various reasons. and it has led to an ability to characterize and categorize tuning systems by their 'equivalencies' -- in a way that has a direct geometric interpretation in terms of that same, originally infinite, lattice. now replace the [2 3 5] with [2/57 3/57 5/57]. what premise is that based on?
Message: 3848 - Contents - Hide Contents Date: Thu, 07 Feb 2002 05:36:36 Subject: Re: exactly what is a xenharmonic bridge? From: paulerlich questions for monzo: (note i'm using the new notation now) are 80;81 and 128;125 and 648;625 and 2048;2025 and 32805;32768 xenharmonic bridges in 12-tET? is 80;81 a xenharmonic bridge in all meantones? just trying to figure out what you mean by xenharmonic bridge.
Message: 3849 - Contents - Hide Contents Date: Fri, 08 Feb 2002 06:14:03 Subject: The convex hull of the Genesis scale From: genewardsmith This turns out to be a complicated, literally multifacted structure in the four-dimensional space of octave equivalence classes. There are 22 vertices, 21 interior points, and 38 facets, or cells. The vertices (extreme points) are 81/80, 33/32, 21/20, 16/15, 11/10, 32/27, 11/9, 14/11, 9/7, 21/16, 7/5, 10/7, 32/21, 14/9, 11/7, 18/11, 27/16, 20/11, 15/8, 40/21, 64/33, 160/81 The interior points are 1, 12/11, 10/9, 9/8, 8/7, 7/6, 6/5, 5/4, 4/3, 27/20, 11/8, 16/11, 40/27, 3/2, 8/5, 5/3, 12/7, 7/4, 16/9, 9/5, 11/6 This is a little more complicated than I had hoped for, in that it makes analyzing Genesis from this point of view something of a chore.
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