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Message: 400 - Contents - Hide Contents Date: Tue, 26 Jun 2001 04:07:27 Subject: Re: 41 "miracle" and 43 tone scales From: jpehrson@r... --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Yahoo groups: /tuning-math/message/366 * [with cont.]> > So Graham is right that, except for this inconspicuous little > tabulation, Partch does not mention 31-, 41- or 72-EDO. > Good detective work, Graham!!! >So the thought is that, possibly, something was "bothering" him about these temperaments... (??) _________ _________ ________ Joseph Pehrson
Message: 401 - Contents - Hide Contents Date: Tue, 26 Jun 2001 04:13:29 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > > But I find it *more* than very interesting that Partch > knew about 31-EDO's good approximations to a significant > percentage of his scale, and chose to say *nothing* about it!That's understandable, since 31-tET conflates pairs of ratios in his diamond, such as 9:8 and 10:9, and gives them both an error of 11 cents! Since these were primary consonances in Partch's system, and 11 cent errors were almost unthinkably large to Partch, the dismissal is not surprising. Plus, you might say, he was utterly predisposed to dismissing any ET on the principle of the thing.
Message: 402 - Contents - Hide Contents Date: Tue, 26 Jun 2001 04:17:19 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- In tuning-math@y..., jpehrson@r... wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > Yahoo groups: /tuning-math/message/366 * [with cont.] > >>>> So Graham is right that, except for this inconspicuous little >> tabulation, Partch does not mention 31-, 41- or 72-EDO. >> Good detective work, Graham!!! >> >> So the thought is that, possibly, something was "bothering" him about > these temperaments... (??) >That's kind of silly. He did include 31 in his table, and was unfamiliar with 41 and 72, both absent from the literature with which he was familiar. But yes, he was predisposed toward dismissing any ET, and probably wasn't in a hurry to go about finding a "good" one.
Message: 404 - Contents - Hide Contents Date: Mon, 25 Jun 2001 22:21:07 Subject: Re: 41 "miracle" and 43 tone scales From: monz> ----- Original Message ----- > From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, June 25, 2001 9:13 PM > Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >> >>>> But I find it *more* than very interesting that Partch >> knew about 31-EDO's good approximations to a significant >> percentage of his scale, and chose to say *nothing* about it! >> That's understandable, since 31-tET conflates pairs of ratios > in his diamond, such as 9:8 and 10:9, and gives them both an > error of 11 cents! Since these were primary consonances in > Partch's system, and 11 cent errors were almost unthinkably > large to Partch, the dismissal is not surprising.Paul, thanks so much for your insight into this. More below.> Plus, you might say, he was utterly predisposed to dismissing any > ET on the principle of the thing.Yes, I would have said something like this myself. I think "diamondic" is indeed the paradigm which best characterizes Partch's feelings about his scale. This whole thread about a possible MIRACLE intuition guiding Partch has made it abundantly clear to me that the literal structures embedded in the Tonality Diamond were of paramount importance to him. 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. If this is the case, then I find that to be a very valuable insight into Partch's _modus operandi_. It's also fascinating that Partch was more interested in expanding his harmonic resources along Pythagorean lines (pun intended) rather than the higher-prime relationships approximated by MIRACLE. 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. -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: 405 - Contents - Hide Contents Date: Tue, 26 Jun 2001 06:30:50 Subject: Re: ET's and unison vectors From: Dave Keenan --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:> You've probably seen or made charts of equal tunings sorted by the size of > their fifths and major thirds. > > http://www.io.com/~hmiller/png/et-scales.png - Type Ok * [with cont.] (Wayb.)Hey that's neat.
Message: 406 - Contents - Hide Contents Date: Tue, 26 Jun 2001 06:38:24 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan --- In tuning-math@y..., jpehrson@r... wrote:> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:>> No. I didn't say that either. But he might not have noticed if > someone>> had substituted a scale which was MIRACLE-41 plus a couple of extra >> notes from MIRACLE-45.> I guess what I'm understanding is that some of the "fill in" notes > that Partch used to complete his 43-tone scale could be described by > the "miracle generator..." > > Am I on the right track??Yes. But I was referring to the early 43 toner in "Expositions on Monophony". It only applies to the later one in "Genesis" if Partch wouldn't have noticed you'd switched to his earlier scale. -- Dave Keenan
Message: 407 - Contents - Hide Contents Date: Tue, 26 Jun 2001 07:08:46 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Paul Erlich wrote:>> That's understandable, since 31-tET conflates pairs of ratios >> in his diamond.This is useful terminology. "<temperament> conflates ratios in <JI structure>" means the same as "<JI structure> overloads <temperament>".> This whole thread about a possible MIRACLE intuition guiding > Partch has made it abundantly clear to me that the literal > structures embedded in the Tonality Diamond were of paramount > importance to him. > > 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. > > If this is the case, then I find that to be a very valuable > insight into Partch's _modus operandi_. > > > It's also fascinating that Partch was more interested in > expanding his harmonic resources along Pythagorean lines > (pun intended) rather than the higher-prime relationships > approximated by MIRACLE.What do you mean here by "higher-prime". I hope you only mean 5, 7 and 11. But it seems that he went Miracle at first and then later changed only four notes for Pythagorean. He changed only 49/48 to 81/80 and 27/20 to 15/11 (and their inversions). So I can postulate 3 forces in historical order: First Diamondic, then Miracle (which simply means that he wanted to fill in the diamond gaps while minimising the number of extra notes and maximising the number of 11-limit consonances, both strict and with small errors) and finally the old Pythagorean/Diatonic reasserted itself sightly.> 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
Message: 408 - Contents - Hide Contents Date: Tue, 26 Jun 2001 10:54 +0 Subject: Re: 41 "miracle" and 43 tone scales From: graham@m... In-Reply-To: <9h8hfa+ki4n@e...> Dave Keenan wrote:>> In that case, you'd expect the result to look > something like>> a 41-note MOS of a good 11-limit temperament. The scale he ends up > with does>> fit schismic better than Miracle. >> Please give details. How many holes in a chain that encompasses it. > How big are the errors? Are there any overloads? Maybe on the other > list.The 43 notes become a 41 note schismic MOS, with duplicates exactly where you expect them. Wilson showed this. You get the same 41 note MOS with either the Exposition of Monophony or Genesis 43 note scales. Graham
Message: 409 - Contents - Hide Contents Date: Tue, 26 Jun 2001 07:50:56 Subject: Re: 41 "miracle" and 43 tone scales From: monz> ----- Original Message ----- > From: Dave Keenan <D.KEENAN@U...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, June 26, 2001 12:08 AM > Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales > > > This is useful terminology. > "<temperament> conflates ratios in <JI structure>" > means the same as > "<JI structure> overloads <temperament>". I agree.Any of you want to write a couple of good definitions for me?> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> It's also fascinating that Partch was more interested in >> expanding his harmonic resources along Pythagorean lines >> (pun intended) rather than the higher-prime relationships >> approximated by MIRACLE. >> What do you mean here by "higher-prime". I hope you only > mean 5, 7 and 11.Good catch, Dave... I should have been more clear about that myself. Yes, that's exactly what I mean. I was differentiating between "traditional" Pythagorean root-movement and the possibilities offered collectively by 5, 7, and 11.> > But it seems that he went Miracle at first and then later changed only > four notes for Pythagorean. He changed only 49/48 to 81/80 and 27/20 > to 15/11 (and their inversions). > > So I can postulate 3 forces in historical order: First Diamondic, then > Miracle (which simply means that he wanted to fill in the diamond gaps > while minimising the number of extra notes and maximising the number > of 11-limit consonances, both strict and with small errors) and > finally the old Pythagorean/Diatonic reasserted itself sightly.Hmmm... at this point, I think I really should dig out my copy of Richard Kassel's dissertation "The Evolution of Partch's Monophony". It explains in detail all the early and intermediate stages in his theory, including tabulations of all his different scales. -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: 411 - Contents - Hide Contents Date: Tue, 26 Jun 2001 19:05:41 Subject: Re: ET's, unison vectors (and other equivalences) From: Paul Erlich --- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:> Instead of starting endless threads in my own terms on this list, I figured > I'd just wait until something familiar showed up. It took a month, but now > I can finally answer one of Paul's questions.OK, I'm looking forward to it (which one)?> > Taking it literally, it seemed to me these should be > thought of as different classes of meantone since there was some "tone" > split into a "mean".Huh? Let's take the scales were the diesis (128:125) vanishes. What "tone" is being split into a "mean" there?> Take the paradox of four fifths equalling a just major third. > You would see this as "a unison vector of 81:80". > > Now take four fifths equalling a just major sixth. > You would see this as "a unison vector of 81:80". > > Yes?I don't think so. Did you mean three fifths equallying a just major sixth? Then yes.> > > > "4P5 = M3" > The temperaments where: > two octaves below four of the closest note to a 3:2 > equals the closest note to a 5:4 > ...are: > > 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105 and 117. > > > > "3P5 = M6" > The temperaments where: > one octave below twoYou mean three? of the closest note to a 3:2> equals the closest note to a 5:3 > ...are: > > 12, 19, 26, 31, 33, 40, 43, 45, 50, 55, 64, 69, 74, 81 and 88.Aha -- for me these two series would come out identical, because I would exclude the 5-limit inconsistent ones from both -- such as 33. In general, though, a given unison vector or set of unison vectors vanishing implies a temperament, not an ET, unless the number of unison vectors vanishing is equal to the number of dimensions. For example, in Herman's chart, each point lies at the intersection of two lines (actually more, but two is enough for an unambiguous location), corresponding to the two unison vectors that vanish for the ET represented by that point. And you can calculate the ET from the absolute value of the determinant of the matrix of unison vectors. The determinant is defined as (a*d - b*c) for a matrix [a b] [c d] For example, 12-tET lies at the intersection of the line representing the syntonic comma (4 -1) and the schisma (8 1). So the matrix is [4 -1] [8 1] and the determinant is (4*1 - 8*(-1)) = 4+8 = 12. Voila! Try some others . . . As for the "paradox" or "subtle distinction" you bring up, I don't put too much weight on it, since inconsistent ETs can be mapped to the consonant intervals in more than one way, and it's really the _mapping_, not the ET itself, that determines which unison vectors are involved.
Message: 412 - Contents - Hide Contents Date: Tue, 26 Jun 2001 19:39:18 Subject: Re: pairwise entropy minimizer From: carl@l...>> >ounds like you're reading me to say there was a problem >> with harmonic entropy. Maybe I should have said, that's >> a property of harmonic entropy which makes the optimization >> problem difficult. >> Again, can you name a dissonance function which would make the > optimization problem easier?No, and I have no desire to; I want harmonic entropy! Maybe I should have said, "that's a problem of dissonance that makes the optimization problem difficult." You've completely misunderstood me here Paul, and it's unfortunate. I am merely trying to summarize what _you_ once told _me_ for Ed, who was not a part of the original thread. Now you're playing the part I was in the original thread? Bizarre.>> (1) Are there results for scales with numbers of tones >> other than five? >> Yes . . . see the archives.Obviously, I've already been there. I was unable to find them.>> I think the fact that meantone pentatonics won is fairly >> interesting, but IIRC they didn't win by very much. >> >> In the long run, I'd be interested in finding scales where >> the total entropy is low >> Meaning total dyadic entropy, or entropy of larger chords? Dyadic.>> and the entropy of the modes are nearly the same. >> How could they not be the same?By modes here, I mean "the set of dyads measured from a given scale member". In the original thread, I argued that harmonic series segments would get high marks with total pairwise entropy (I can't remember if this was a guess, or if this scales really did show up). But one mode would be lower than the others, in most cases. -Carl
Message: 413 - Contents - Hide Contents Date: Tue, 26 Jun 2001 20:03:47 Subject: Re: pairwise entropy minimizer From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:>>> Sounds like you're reading me to say there was a problem >>> with harmonic entropy. Maybe I should have said, that's >>> a property of harmonic entropy which makes the optimization >>> problem difficult. >>>> Again, can you name a dissonance function which would make the >> optimization problem easier? >> No, and I have no desire to; I want harmonic entropy! Maybe > I should have said, "that's a problem of dissonance that makes > the optimization problem difficult." You've completely > misunderstood me here Paul, and it's unfortunate. I am merely > trying to summarize what _you_ once told _me_ for Ed, who was > not a part of the original thread. Now you're playing the part > I was in the original thread? Bizarre.Just wanted to make things clearer for _everyone_.>>>> (1) Are there results for scales with numbers of tones >>> other than five? >>>> Yes . . . see the archives. >> Obviously, I've already been there. I was unable to find > them.Sorry. You'll have to keep looking.>>> and the entropy of the modes are nearly the same. >>>> How could they not be the same? >> By modes here, I mean "the set of dyads measured from a given > scale member".I didn't do that -- I added all the dyads -- but Robert Valentine's approach is to do what you've described.
Message: 414 - Contents - Hide Contents Date: Tue, 26 Jun 2001 21:58:44 Subject: Re: Hypothesis revisited From: Graham Breed Paul wrote:>> If they are, the two sets of unison vectors give exactly the same >> results. > > They don't! >>> I think they must be, because I remember checking the >> determinant before, and any chroma that gives a determinant of 41 > when>> placed with Miracle commas should give this result. >> Something must be wrong with one of your assumptions.Yes, they both give Miracle41, but a different Miracle41 each time/>> You most certainly do need octave-specific matrices. Otherwise, > that>> left-hand column won't be there. >> I see that as a good thing . . . don't you?No, it helps to define the temperament. If you invert and normalize the octave-invariant matrix, the left hand column gives you the prime intervals in terms of generators. If there's a common factor, divide through by it, and call it the octave division. The only problems are those anomalous cases where the determinant is a multiple of the temperament you want. So octave-specific are still winning.>> There may be an algorithm that works with octave >> invariant matrices, but it's easier to upgrade them to be >> octave-specific, and use a common or garden inverse. > > ?Okay, inverting the octave invariant matrices still tells you something. So how do we spot the anomalies? I'll update to <Unison vector to MOS script * [with cont.] (Wayb.)> if I remember. The unison vector finder is slightly improved in that it finds something for the multiple-29 scale now. Graham "I toss therefore I am" -- Sartre
Message: 416 - Contents - Hide Contents Date: Tue, 26 Jun 2001 23:11:29 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> This is useful terminology. >> "<temperament> conflates ratios in <JI structure>" >> means the same as >> "<JI structure> overloads <temperament>". > > I agree.> Any of you want to write a couple of good definitions for me?Two or more notes (ratios) of the JI structure become a single note of the temperament. For example 9/8 and 10/9 are replaced by a single "D" in meantone temperaments. So we say that meantone conflates 9/8 with 10/9 or that any JI structure containing _both_ 9/8 and 10/9 overloads meantone.>> But it seems that he went Miracle at first and then later changed only >> four notes for Pythagorean. He changed only 49/48 to 81/80 and 27/20 >> to 15/11 (and their inversions).Sorry. That should have been "changed ... 15/11 to 27/20". I typed that pair back to front.>> So I can postulate 3 forces in historical order: First Diamondic, then >> Miracle (which simply means that he wanted to fill in the diamond gaps >> while minimising the number of extra notes and maximising the number >> of 11-limit consonances, both strict and with small errors) and >> finally the old Pythagorean/Diatonic reasserted itself sightly. >> Hmmm... at this point, I think I really should dig out my copy of > Richard Kassel's dissertation "The Evolution of Partch's Monophony". > It explains in detail all the early and intermediate stages in his > theory, including tabulations of all his different scales. Sounds great.-- Dave Keenan
Message: 417 - Contents - Hide Contents Date: Tue, 26 Jun 2001 16:51:51 Subject: Re: ET's, unison vectors (and other equivalences) From: monz ----- Original Message ----- From: D.Stearns <STEARNS@C...> To: <tuning-math@xxxxxxxxxxx.xxx> 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. -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: 418 - Contents - Hide Contents Date: Wed, 27 Jun 2001 04:08:33 Subject: Re: pairwise entropy minimizer From: jpehrson@r... --- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote: Yahoo groups: /tuning-math/message/403 * [with cont.]> Or should that be "pleasant but weird"?? :-)Of course Ed, as I'm sure you're aware, these are ENTIRELY value judgements. When I hear the works of some so-called "contemporary" composers which use hackneyed "traditional" 12-tET progressions, the effect is anything but pleasant. What you are doing would probably strike me, therefore, as just "pleasant." :) ______ ________ _______ Joseph Pehrson
Message: 421 - Contents - Hide Contents Date: Tue, 26 Jun 2001 23:44:07 Subject: Re: ET's, unison vectors (and other equivalences) From: monz> ----- Original Message ----- > From: D.Stearns <STEARNS@C...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, June 27, 2001 12:28 AM > Subject: Re: [tuning-math] Re: ET's, unison vectors (and other equivalences) > > > Hi Joe, > > In this case I don't think "2^2 * 7^-2 * 13^1", or, "[ 2 0 0 -2 0 > 1]" is any easier to swallow than the plain ol' ratio of 52/49.Yup... you got me there. I just added the vector notation for that one to be consistent... the reason I really started translating your unison-vector ratios into prime-factor vector notation was because those other ratios had *huge* integers and I couldn't tell what was going on until I factored them.> > Also seeing as how this example is explicitly 2D, and just the > diatonic periodicity block in some colorful new party clothes, I think > the most direct shorthand would probably be, > > | -1 2 | > | 4 -1 |Right again, Dan. This notation really *is* simply a shorthand for what I wrote as: [ 0 0 2 0 -1] [ 0 0 -1 0 4] (Here, I've inverted the signs as you have, and ignore powers of 2 as I usually would but didn't before; I've also decided that I like the square brackets better than the bar because there's less chance of confusion with "1".) I'm curious tho... why did you put the 13-column first and the 7-column second in your matrix? -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: 422 - 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@m...> > 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: 423 - Contents - Hide Contents Date: Wed, 27 Jun 2001 19:53:23 Subject: Re: 41 "miracle" and 43 tone scales From: jpehrson@r... --- 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: 424 - Contents - Hide Contents Date: Wed, 27 Jun 2001 19:59:20 Subject: Re: 41 "miracle" and 43 tone scales From: jpehrson@r... --- 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
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