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Message: 2750 - Contents - Hide Contents Date: Wed, 26 Dec 2001 23:25:28 Subject: Re: Gene's notation & Schoenberg lattices From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Gene (or anyone who knows), can you please explain this in > excruciating detail, illuminating every step and revealing > what all those cryptic letters represent? What's the business > with the "round" function?"Round" means round off to the nearest integer; sometimes, as with 1/2 or -1/2, there isn't a nearest integer. Then we can define it as round down, round up, or round towards, or round away--that is, round 1/2 down to 0 and -1/2 down to -1; or round 1/2 up to 1 and -1/2 up to 0; or round -1/2 to 0 and also 1/2 to 0, or round 1/2 to 1 and -1/2 to -1. The last two systems are not good ones on mathematical groups, but get used anyway because they are easy to do and programmers are not always very sensative to good mathematical technique. How do you "choose" your denominator?> > If you can explain this in terms I understand, namely, the > matrix which lists the unison-vectors, then at least I can > begin to comprehend.If I start from <56/55,33/32,64/63,81/80,45/44> I can form the corresponding matrix: [ 3 0 -1 1 -1] [-5 1 0 1 1] [ 6 -2 0 -1 0] [-4 4 -1 0 0] [-4 4 -1 0 0] [-2 2 1 0 -1] 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] The columns of this are essentially a set of ets; they are maps from intervals to the integers, something I call a "val". The third column differs from the first by having a 19 instead of a 20 for the value that 7 gets mapped to. If I call the second column map "h12", I find that h12(56/55)=h12(64/63)=h12(81/80)=h12(45/44), but h12(33/32)=1; all of the columns are dual in this way to the rows of the previous matrix, by the definition of matrix inverse. For any non-zero I can define a scale by calculating for 0<=n<d step[n] = (56/55)^round(7n/d) (33/32)^round(12n/d) (64/63)^round(7n/d) (81/80)^round(-2n/d) (45/44)^round(5n/d) Choosing a value of d is what I meant by "chooing a denominator". If the value of d is not one of the values on the top row of the inverted matrix, which shows where 2 maps to, we may or may not get slightly goofy results such as repeated steps, but generally it seems to work reasonably well.
Message: 2751 - Contents - Hide Contents Date: Wed, 26 Dec 2001 23:26:36 Subject: Re: Paul's lattice math and my diagrams From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, December 26, 2001 2:58 PM >> Subject: [tuning-math] Re: Paul's lattice math and my diagrams >> >> >> But alas, you are not getting the infinite strips I refer to above. > >> And exactly how am I supposed to portray "infinite strips" on a > computer screen, other than leaving the "infinite" part to the > reader's imagination?!Well, at least have two or three repetition of the pitches along that direction, to *suggest* the infinitude . . . currently, you have only one instance of most of the pitches, and two for a few others.>>> The 5-limit periodicity blocks are bounded by 2 unison-vectors, >>> one of which is tempered out (the 81:80 syntonic comma) and >>> one of which isn't -- and that one is the one which appears >>> at the end of each meantone chain. >>>> Right -- but since the 81:80 _is_ tempered out, your lattices should >> be proceeding infinitely in the direction of the 81:80. > >> Well... this is the part of your post that I was least sure about. > My lattices obviously proceed infinitely in the direction of > the interval that's not tempered out.If you're latticing a periodicity block, they shouldn't -- you should hit a "wolf" at some point in that direction.> The syntonic comma is > the interval that sets the boundaries on the *other* two sides. > But *isn't* that how meantones work?No -- there are no boundaries in the direction of 81:80. These *other* two sides that you refer to -- they meet one another when you roll the lattice into a cylinder.>>>> Now let's go back to "any vector in the lattice". This vector, >>>> added to itself over and over, will land one back at a pitch >>>> in the same equivalence class as the pitch one started with, >>>> after N iterations (and more often if the vector represents >>>> a generic interval whose cardinality is not relatively prime >>>> with N). In general, the vector will have a length that is >>>> some fraction M/N of the width of one strip/layer/hyperlayer, >>>> measured in the direction of this vector (NOT in the direction >>>> of the chromatic unison vector). M must be an integer, since >>>> after N iterations, you're guaranteed to be in a point in the >>>> same equivalence class as where you started, hence you must be >>>> an exact integer M strips/layers/hyperlayers away. As a >>>> special example, the generator has length 1/N of the width >>>> of one strip/layer/hyperlayer, measured in the direction of >>>> the generator. >>> >>>>>> This is precisely what was in my mind when I came up with >>> these meantone lattices. >>>> Really? I don't see the strips, and I don't see how the generator >> could be said to have any property resembling this in your lattices. > >> I only show part of one periodicity-block. To do it properly, > I should have a nice big grid representing the infinite lattice, > then simply draw the unison-vectors as boundaries to the various > tiled periodicity-blocks. Then you'd see the strips, each one > at the same angle as the meantone chain itself, and each one as > wide as a syntonic comma.That's not correct. The strips I'm referring to would be as wide as the other comma -- the one that's not tempered out.>>>> Anyhow, each occurence of the vector will cross either >>>> floor(M/N) or ceiling(M/N) boundaries between >>>> strips/layers/hyperlayers. Now, each time one crosses >>>> one of these boundaries in a given direction, one shifts >>>> by a chromatic unison vector. Hence each specific occurence >>>> of the generic interval in question will be shifted by >>>> either floor(M/N) or ceiling(M/N) chromatic unison vectors. >>>> Thus there will be only two specific sizes of the interval >>>> in question, and their difference will be exactly 1 of the >>>> chromatic unison vector. And since the vectors in the chain >>>> are equally spaced and the boundaries are equally spaced, >>>> the pattern of these two sizes will be an MOS pattern. >>>>>> Isn't this exactly how my pseudo-code works? (posted here: >>> <Yahoo groups: /tuning-math/messages/2069? * [with cont.] expand=1>). >>>> Monz, I don't see anything in your pseudocode that would give you any >> of this -- have you actually managed to produce MOSs with it? > >> Um... well... I never actually checked that anything I my code > produced was MOS or had any other scalar property. I simply > compared my periodicity-block coordinates (and those of all > their contents) with the ones you, Gene, and Fokker produced > using the same unison-vectors and kept working on the code > until it produced the same results.Right, but you quoted a paragraph of mine and then asked "Isn't this exactly how my pseudo-code works?" -- so what was it in that paragraph that you think your pseudocode does? That paragraph was meant to demonstrate the truth of my hypothesis about the relationship between PBs and MOS, which appears to be original to me and not something Fokker ever mentioned.
Message: 2752 - Contents - Hide Contents Date: Wed, 26 Dec 2001 23:31:53 Subject: Re: Gene's notation & Schoenberg lattices From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> > If I start from <56/55,33/32,64/63,81/80,45/44> I can form the corresponding matrix: > > [ 3 0 -1 1 -1] > [-5 1 0 1 1] > [ 6 -2 0 -1 0] > [-4 4 -1 0 0] > [-4 4 -1 0 0] > [-2 2 1 0 -1]You put [-4 4 -1 0 0] in there twice.> > This matrix is unimodular, meaning it has determinant +-1.Funny, if I take out [-4 4 -1 0 0], I get a determinant of 35!
Message: 2753 - Contents - Hide Contents Date: Wed, 26 Dec 2001 23:38:36 Subject: Re: Gene's notation & Schoenberg lattices From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Funny, if I take out [-4 4 -1 0 0], I get a determinant of 35! Hmmm...try[ 3 0 -1 1 -1] [-5 1 0 0 1] [ 6 -2 0 -1 0] [-4 4 -1 0 0] [-2 2 1 0 -1]
Message: 2754 - Contents - Hide Contents Date: Wed, 26 Dec 2001 20:42:47 Subject: Re: Paul's lattice math and my diagrams From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 26, 2001 3:26 PM > Subject: [tuning-math] Re: Paul's lattice math and my diagrams > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> From: paulerlich <paul@s...> >>> To: <tuning-math@y...> >>> Sent: Wednesday, December 26, 2001 2:58 PM >>> Subject: [tuning-math] Re: Paul's lattice math and my diagrams >>> >>> >>> But alas, you are not getting the infinite strips I refer to > above. >> >>>> And exactly how am I supposed to portray "infinite strips" on a >> computer screen, other than leaving the "infinite" part to the >> reader's imagination?! >> Well, at least have two or three repetition of the pitches along that > direction, to *suggest* the infinitude . . . currently, you have only > one instance of most of the pitches, and two for a few others. >>>>> The 5-limit periodicity blocks are bounded by 2 unison-vectors, >>>> one of which is tempered out (the 81:80 syntonic comma) and >>>> one of which isn't -- and that one is the one which appears >>>> at the end of each meantone chain. >>>>>> Right -- but since the 81:80 _is_ tempered out, your lattices > should>>> be proceeding infinitely in the direction of the 81:80. >> >>>> Well... this is the part of your post that I was least sure about. >> My lattices obviously proceed infinitely in the direction of >> the interval that's not tempered out. >> If you're latticing a periodicity block, they shouldn't -- you should > hit a "wolf" at some point in that direction.Right, of course... they continue infinitely in the direction of the meantone chain if you don't close the chain somewhere. I *am* interested in closing it so that I get a periodicity-block. I'm almost finished with a lattice for the UV: (4 -1),(19 9) periodicity-block which contains a 55-tone 1/6-comma chain. I'll post it as soon as it's done.>> The syntonic comma is >> the interval that sets the boundaries on the *other* two sides. >> But *isn't* that how meantones work? >> No -- there are no boundaries in the direction of 81:80. These > *other* two sides that you refer to -- they meet one another when you > roll the lattice into a cylinder. >>>>>> Now let's go back to "any vector in the lattice". This vector, >>>>> added to itself over and over, will land one back at a pitch >>>>> in the same equivalence class as the pitch one started with, >>>>> after N iterations (and more often if the vector represents >>>>> a generic interval whose cardinality is not relatively prime >>>>> with N). In general, the vector will have a length that is >>>>> some fraction M/N of the width of one strip/layer/hyperlayer, >>>>> measured in the direction of this vector (NOT in the direction >>>>> of the chromatic unison vector). M must be an integer, since >>>>> after N iterations, you're guaranteed to be in a point in the >>>>> same equivalence class as where you started, hence you must be >>>>> an exact integer M strips/layers/hyperlayers away. As a >>>>> special example, the generator has length 1/N of the width >>>>> of one strip/layer/hyperlayer, measured in the direction of >>>>> the generator. >>>> >>>>>>>> This is precisely what was in my mind when I came up with >>>> these meantone lattices. >>>>>> Really? I don't see the strips, and I don't see how the generator >>> could be said to have any property resembling this in your >>> lattices. >> >>>> I only show part of one periodicity-block. To do it properly, >> I should have a nice big grid representing the infinite lattice, >> then simply draw the unison-vectors as boundaries to the various >> tiled periodicity-blocks. Then you'd see the strips, each one >> at the same angle as the meantone chain itself, and each one as >> wide as a syntonic comma. >> That's not correct. The strips I'm referring to would be as wide as > the other comma -- the one that's not tempered out.That's how I understood it when I first read your post. So you mean that on your ideal lattice you'd have long (or I probably should say wide) strips of cylinders, right?>>>>> Anyhow, each occurence of the vector will cross either >>>>> floor(M/N) or ceiling(M/N) boundaries between >>>>> strips/layers/hyperlayers. Now, each time one crosses >>>>> one of these boundaries in a given direction, one shifts >>>>> by a chromatic unison vector. Hence each specific occurence >>>>> of the generic interval in question will be shifted by >>>>> either floor(M/N) or ceiling(M/N) chromatic unison vectors. >>>>> Thus there will be only two specific sizes of the interval >>>>> in question, and their difference will be exactly 1 of the >>>>> chromatic unison vector. And since the vectors in the chain >>>>> are equally spaced and the boundaries are equally spaced, >>>>> the pattern of these two sizes will be an MOS pattern. >>>>>>>> Isn't this exactly how my pseudo-code works? posted here: Yahoo groups: /tuning-math/messages/2069?expand=1 * [with cont.] >>>>>> Monz, I don't see anything in your pseudocode that would give >>> you any of this -- have you actually managed to produce MOSs >>> with it? >> >>>> Um... well... I never actually checked that anything I my code >> produced was MOS or had any other scalar property. I simply >> compared my periodicity-block coordinates (and those of all >> their contents) with the ones you, Gene, and Fokker produced >> using the same unison-vectors and kept working on the code >> until it produced the same results. >> Right, but you quoted a paragraph of mine and then asked "Isn't this > exactly how my pseudo-code works?" -- so what was it in that > paragraph that you think your pseudocode does?My code transforms the prime-axes to a right-angled unit cube, transforms the primary lattice metrics along the 3 and 5 axes to the unit metrics along those new axes, then iterates thru the unit cube to fill it with coordinates x,y, always bouncing to the other side (i.e., modulo) when it goes beyond the floor or ceiling values (i.e., 1/2 > x,y > -1/2), then transforms back to the original lattice coordinates. This is exactly how I understood your paragraph. Please correct.> That paragraph was meant to demonstrate the truth of my > hypothesis about the relationship between PBs and MOS, > which appears to be original to me and not something > Fokker ever mentioned.OK. I really wasn't even concerning myself directly with MOS, just trying to figure out how to have Excel calculate not only the boundaries of the periodicity-block, but all of the coordinates within it as well. (I was quite impressed with myself for getting the job done on my own, even tho it's still quite crude.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2755 - Contents - Hide Contents Date: Wed, 26 Dec 2001 21:12:53 Subject: Re: Gene's notation & Schoenberg lattices From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 26, 2001 3:38 PM > Subject: [tuning-math] Re: Gene's notation & Schoenberg lattices > > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Funny, if I take out [-4 4 -1 0 0], I get a determinant of 35! > > Hmmm...try >> [ 3 0 -1 1 -1] > [-5 1 0 0 1] > [ 6 -2 0 -1 0] > [-4 4 -1 0 0] > [-2 2 1 0 -1]This gives me an adjoint of: [ 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] So that: h12(2) = 12 h12(3) = 19 h12(5) = 28 h12(7) = 34 h12(11) = 42 and h5(2) = 5 h5(3) = 8 h5(5) = 12 h5(7) = 14 h5(11) = 17 Looks correct to me, altho the approximations of 5-EDO to 5 and especially 11 are rather far off. But I also see that: h7(2) = 7 h7(3) = 11 h7(5) = 16 h7(7) = 19 and 20 h7(11) = 24 If I recall, there is some significance to the fact that h7(7) = both 19 and 20. This is telling us that 7-EDO gives excellent approximations to 2, 3, and 5, and a pretty good one to 11, but 7 lies approximately midway between the two closest approximations. Correct? (How am I doing on grokking your terminology, Gene?) What about the column beginning with -2? h-2(2)=-2, h-2(3)=-3, and h-2(11)=-7 all look OK, but -5 is closer to h-2(6) than to h-2(5), and -6 is exactly h-2(8). ...??? And then... what does any of this have to do with finding the 12-tone periodicity-block which Schoenberg had in mind? I don't get it. Also, I don't see any 7-limit ratios on the lattice... that's because of the 56:55 and 64:63, right? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2756 - Contents - Hide Contents Date: Thu, 27 Dec 2001 05:17:43 Subject: Re: My top 5--for Paul From: clumma> None whatever.That's been true since the 15th century.>I often don't use a scale. However, using a temperament differs >from not using one;You bet. Also, using a scale differs from not using one.>> That's the one! Namely the limitation is in human cognition. The >> composer can have computer assistance, but the listener can't. >>The listener doesn't need to sort out 612 notes, this is a red >herring.The listener will sort out notes one way or the other, and far fewer of them 171 or 612. Dave's point, I think, is that he or she wouldn't be able to tell the difference. -Carl
Message: 2757 - Contents - Hide Contents Date: Thu, 27 Dec 2001 21:38:24 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Did I miss any, Gene? >> Apparently not; however I notice that Orwell came close to not >making the cut. As probably the only person with experience using >it, I can tell you it's a lot more practical than that would suggest.In the 5-limit??
Message: 2758 - Contents - Hide Contents Date: Thu, 27 Dec 2001 05:29:37 Subject: Re: Paul's lattice math and my diagrams From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, December 26, 2001 3:26 PM >> Subject: [tuning-math] Re: Paul's lattice math and my diagrams >> >> >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>>> From: paulerlich <paul@s...> >>>> To: <tuning-math@y...> >>>> Sent: Wednesday, December 26, 2001 2:58 PM >>>> Subject: [tuning-math] Re: Paul's lattice math and my diagrams >>>> Right -- but since the 81:80 _is_ tempered out, your lattices >> should>>>> be proceeding infinitely in the direction of the 81:80. >>> >>>>>> Well... this is the part of your post that I was least sure about. >>> My lattices obviously proceed infinitely in the direction of >>> the interval that's not tempered out. >>>> If you're latticing a periodicity block, they shouldn't -- you should >> hit a "wolf" at some point in that direction. > >> Right, of course... they continue infinitely in the direction > of the meantone chain if you don't close the chain somewhere. > I *am* interested in closing it so that I get a periodicity-block.No, sir, I'm afraid you're completely misunderstanding me. If 81:80 is tempered out, then you can keep moving by as many 81:80s as you want in the lattice, and you're still within the strip! In terms of the cylinder, all you're doing is making a full circle around the cylinder in the same direction over and over again.> So you mean that on your ideal lattice you'd have long > (or I probably should say wide) strips of cylinders, right?Wide strips, _or_ a single cylinder.>>>>>> Anyhow, each occurence of the vector will cross either >>>>>> floor(M/N) or ceiling(M/N) boundaries between >>>>>> strips/layers/hyperlayers. Now, each time one crosses >>>>>> one of these boundaries in a given direction, one shifts >>>>>> by a chromatic unison vector. Hence each specific occurence >>>>>> of the generic interval in question will be shifted by >>>>>> either floor(M/N) or ceiling(M/N) chromatic unison vectors. >>>>>> Thus there will be only two specific sizes of the interval >>>>>> in question, and their difference will be exactly 1 of the >>>>>> chromatic unison vector. And since the vectors in the chain >>>>>> are equally spaced and the boundaries are equally spaced, >>>>>> the pattern of these two sizes will be an MOS pattern. >>>>>>>>>> Isn't this exactly how my pseudo-code works? posted here: > Yahoo groups: /tuning-math/messages/2069?expand=1 * [with cont.] >>>>>>>> Monz, I don't see anything in your pseudocode that would give >>>> you any of this -- have you actually managed to produce MOSs >>>> with it? >>> >>>>>> Um... well... I never actually checked that anything I my code >>> produced was MOS or had any other scalar property. I simply >>> compared my periodicity-block coordinates (and those of all >>> their contents) with the ones you, Gene, and Fokker produced >>> using the same unison-vectors and kept working on the code >>> until it produced the same results. >>>> Right, but you quoted a paragraph of mine and then asked "Isn't this >> exactly how my pseudo-code works?" -- so what was it in that >> paragraph that you think your pseudocode does? > >> My code transforms the prime-axes to a right-angled unit cube, > transforms the primary lattice metrics along the 3 and 5 axes > to the unit metrics along those new axes, then iterates thru > the unit cube to fill it with coordinates x,y, always bouncing > to the other side (i.e., modulo) when it goes beyond the > floor or ceiling values (i.e., 1/2 > x,y > -1/2), then > transforms back to the original lattice coordinates. > > This is exactly how I understood your paragraph. Please correct.You appear to have the correct picture of how to create periodicity blocks. I was saying much more than that, but if that's all you were looking for, then you're fine.>> That paragraph was meant to demonstrate the truth of my >> hypothesis about the relationship between PBs and MOS, >> which appears to be original to me and not something >> Fokker ever mentioned. > >> OK. I really wasn't even concerning myself directly with MOS, > just trying to figure out how to have Excel calculate not only > the boundaries of the periodicity-block, but all of the > coordinates within it as well. > > (I was quite impressed with myself for getting the job done > on my own, even tho it's still quite crude.)If there's still any confusion, part 3 of the Gentle Introduction should clear it up.
Message: 2759 - Contents - Hide Contents Date: Thu, 27 Dec 2001 21:39:24 Subject: Re: Weighting From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> The idea of optimizing the intervals of a complete odd-limit chord >makes sense to me, and I can see some justification in counting >repeated intervals, but I don't see that, for instance, the major >thirds/minor sixths should weigh much more heavily than the minor >thirds/major sixths.They shouldn't!!! I agree with you 100%!!!!!!!!
Message: 2760 - Contents - Hide Contents Date: Thu, 27 Dec 2001 05:35:43 Subject: Re: Gene's notation & Schoenberg lattices From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> [ 3 0 -1 1 -1] >> [-5 1 0 0 1] >> [ 6 -2 0 -1 0] >> [-4 4 -1 0 0] >> [-2 2 1 0 -1] > >> This gives me an adjoint of: > > [ 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]In this case, that's also the inverse.> If I recall, there is some significance to the fact > that h7(7) = both 19 and 20.h7(7)=20 by my definition, which says to round to the nearest integer. I called the map g7 which maps to 19: g7(7)=19. This is telling us that> 7-EDO gives excellent approximations to 2, 3, and 5, > and a pretty good one to 11, but 7 lies approximately > midway between the two closest approximations. Correct?It's telling us that this particular set of commas does not work to give ussomething consistent so far as the mapping of 7 goes, since h7 and g7 are the same except for how they map 7. Schoenberg's choice is a peculiar one, but interesting because of it.> What about the column beginning with -2? h-2(2)=-2, > h-2(3)=-3, and h-2(11)=-7 all look OK, but -5 is closer > to h-2(6) than to h-2(5), and -6 is exactly h-2(8). ...???It's h-2(7)=-6; only the maps of primes are shown, since everything else can be determined from them by addition.> Also, I don't see any 7-limit ratios on the lattice... > that's because of the 56:55 and 64:63, right?Right; 56/55 64/63 = 512/495, which has no factor of 7. We can remove a row and column from the matrix of vals, and get [ 7 12 -2 5] [11 19 -3 8] [16 28 -5 12] [24 42 -7 17] The inverse of this is [ 9 -2 -2 -2] [-5 1 0 1] [-4 4 -1 0] [-2 2 1 -1] which corresponds to <512/495, 33/32, 81/80, 45/44>. Everything now is in a2^a 3^b 5^c 11^d system without 7s, and it works equivalently for finding blocks.
Message: 2761 - Contents - Hide Contents Date: Thu, 27 Dec 2001 21:44:44 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> Dave, I'll do the arithmetic for you and give you the RMS optima for >> the ones with RMS less than 20 cents and g<8. Is there anything >> _missing_ that is as good as one of these? >> >> Generator 522.86¢, Period 1 oct. >> Generator 505.87¢, Period 1/4 oct. >> Generator 163.00¢, Period 1 oct. >> Generator 491.20¢, Period 1/3 oct. >> Generator 379.97¢, Period 1 oct. >> Generator 503.83¢, Period 1 oct. >> Generator 494.55¢, Period 1/2 oct. >> Generator 442.98¢, Period 1 oct. >> Generator 387.82¢, Period 1 oct. >> Generator 271.59¢, Period 1 oct. >> Generator 317.08¢, Period 1 oct. >> Generator 498.27¢, Period 1 oct. >> Thanks Paul. First note that I am only looking at those with a period > of 1 octave at this stage.What do you mean? You did a whole spreadsheet with graphs for the 1/2- octave period case.> As a 5-limit approximation the 522.86c > generator is junk. It has an error of 25 c in the 2:3.On behalf of Herman Miller, Margo Schulter, Bill Sethares, and the entire island of Java, let me just say #(@*$& ?@#>$, and then let me just say, go play with this scale for a while. 'Junk' my &$$.> So there are > plenty of other temperaments as good as this.By _as good as_, I mean having an equal or lower RMS error ANS and equal or lower 'gens' measure.> If you omitted this and > 163.00 c I would agree with the list, and I would probably only find a > few more as good as 163.00 c. I'll let you know what these are when I > have more time.I'd appreciate it. You'd be making a positive contribution.
Message: 2762 - Contents - Hide Contents Date: Thu, 27 Dec 2001 06:49:32 Subject: Re: Microtemperament and scale structure From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, December 26, 2001 1:14 PM >> Subject: [tuning-math] Re: Microtemperament and scale structure >> >> >> First of all, Gene was talking about the case where the two unison >> vectors are _tempered out_. He initially thought that the result of >> doing that, for this periodicity block (the SAME ONE as your >> srutiblock), would be 24-tET. But you _aren't_ tempering out the >> unison vectors, so you wouldn't see quartertones anyway, even if Gene >> were right. > >> Well... I understood that. But I think the reason I got confused > is because... > >>> Secondly, you need to continue where you left off in the archives. I >> ended up convincing Gene that this does not in fact result in >> quartertones. He then realized that his error was because of >> something called "torsion". If you temper out the unison vectors, you >> actually get 12-tone equal temperament, not 24-tone equal temperament. > >> OK, I'm only up to late August, so there's obviously some more > important stuff coming up. > >>> Another reason that this is not a well-behaved periodicity block is >> that the 3:2, for example, is not subtended by the same number of PB >> steps everywhere it occurs -- even if you only look at 3:2s within >> the Indian Diatonic scale. Thus, I feel that your PB interpretation >> of the Indian sruti system is quite poor, because the sruti system >> prides itself on being able to represent all the 3:2s (at least the >> commonly used ones) by the same number of srutis. > >> Thanks, Paul, this helps a lot.Would you mind mentioning the torsion bit, and the point in the last paragraph above, in your srutiblock page? It would seem appropriate.
Message: 2763 - Contents - Hide Contents Date: Thu, 27 Dec 2001 21:51:26 Subject: Re: Paul's lattice math and my diagrams From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, December 26, 2001 9:29 PM >> Subject: [tuning-math] Re: Paul's lattice math and my diagrams >> > >>> [me, monz]>>> Right, of course... they continue infinitely in the direction >>> of the meantone chain if you don't close the chain somewhere. >>> I *am* interested in closing it so that I get a periodicity- block. >>>> No, sir, I'm afraid you're completely misunderstanding me. If 81:80 >> is tempered out, then you can keep moving by as many 81:80s as you >> want in the lattice, and you're still within the strip! In terms of >> the cylinder, all you're doing is making a full circle around the >> cylinder in the same direction over and over again. > >> Oh, OK Paul, I've got you now. My description really is based > on the planar representation,The wrong planar representation, in my opinion.> while you were talking about the > cylindrical representation._Or_ a planar representation, like the ones in _The Forms Of Tonality_.> Cool. But even tho it works, there still is something wrong > with the mathematics in my spreadsheet. I'd appreciate some > error correction.Since I think part 3 of the Gentle Introduction should answer the mathematics part of your question, I'm not currently inclined to decipher the meaning of the mathematical method you've come up with. I'd be happy to work with you on understanding and implementing the method in part 3 of the GI so that you may do what you're trying to do. P.S. How can you include W. A. Mozart under 55-EDO on your Equal Temperament definition page? I could understand if you wanted to put Mozart on a meantone page, but 55? Totally unjustified. Come on, let's not just make things up.
Message: 2764 - Contents - Hide Contents Date: Thu, 27 Dec 2001 07:55:20 Subject: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich> This would be a lot easier for me if he would deign to give the > optimum generator (whether rms or max-absolute), in cents.Dave, I'll do the arithmetic for you and give you the RMS optima for the ones with RMS less than 20 cents and g<8. Is there anything _missing_ that is as good as one of these? Generator 522.86¢, Period 1 oct. Generator 505.87¢, Period 1/4 oct. Generator 163.00¢, Period 1 oct. Generator 491.20¢, Period 1/3 oct. Generator 379.97¢, Period 1 oct. Generator 503.83¢, Period 1 oct. Generator 494.55¢, Period 1/2 oct. Generator 442.98¢, Period 1 oct. Generator 387.82¢, Period 1 oct. Generator 271.59¢, Period 1 oct. Generator 317.08¢, Period 1 oct. Generator 498.27¢, Period 1 oct. Did I miss any, Gene? Thanks for checking, Dave.
Message: 2765 - Contents - Hide Contents Date: Thu, 27 Dec 2001 21:54:58 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> Apparently not; however I notice that Orwell came close to not >> making the cut. As probably the only person with experience using >> it, I can tell you it's a lot more practical than that would suggest. >> In the 5-limit??Weeel...I stayed mostly in the 7-limit, with excursions into the 11-limit. With a generator of 7/6, it's pretty hard to treat Orwell simply as a 5-limit system. However, the point I was making is that setting g<8 may be too low even if you have pretty strict ideas about what is practical.
Message: 2766 - Contents - Hide Contents Date: Thu, 27 Dec 2001 08:03:20 Subject: Re: Keenan green Zometool struts From: paulerlich "The only problem with the Advanced Math kit is that it doesn't have any short whole greens (G0). I recommend adding 48 of these (at US$9.60)." I can't figure out how to order 48 short whole greens from the website. Can you help?
Message: 2767 - Contents - Hide Contents Date: Thu, 27 Dec 2001 22:20:59 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Weeel...I stayed mostly in the 7-limit, with excursions into the > 11-limit. Well, then. >With a generator of 7/6, it's pretty hard to treat Orwell >simply as >a 5-limit system.Gee whiz, well you can't really think of it as 7/6 if you're talking about 5-limit systems, can you? It's a one-third-of-a-minor-sixth generator, really, in this context.>However, the point I was making is that >setting g<8 may be too low >even if you have pretty strict ideas >about what is practical.I just did that because Dave Keenan so far has tended to look at chains of up to 14 generators within a period of one octave, and chains of up to 8 generators within a period of 1/2 pctave. So, since I was asking him if anything was missing, it would have been useless to include the more complex systems, which I agree are valuable for certain musical styles, just as the simple systems with large errors are.
Message: 2768 - Contents - Hide Contents Date: Thu, 27 Dec 2001 08:14:02 Subject: Zometool yahoogroup From: paulerlich Yahoo groups: /ZomeWorld/ * [with cont.] Someone actually has an outstanding question -- maybe you can handle it, Dave?
Message: 2769 - Contents - Hide Contents Date: Thu, 27 Dec 2001 22:55:00 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> However, the point I was making is that >setting g<8 may be too low >> even if you have pretty strict ideas >about what is practical. >> I just did that because Dave Keenan so far has tended to look at > chains of up to 14 generators within a period of one octave, and > chains of up to 8 generators within a period of 1/2 pctave. So, since > I was asking him if anything was missing, it would have been useless > to include the more complex systems, which I agree are valuable for > certain musical styles, just as the simple systems with large errors > are.I'm now looking at up to 36 generators per prime, with an octave period.
Message: 2770 - Contents - Hide Contents Date: Thu, 27 Dec 2001 08:59:56 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Did I miss any, Gene?Apparently not; however I notice that Orwell came close to not making the cut. As probably the only person with experience using it, I can tell you it's a lot more practical than that would suggest.
Message: 2771 - Contents - Hide Contents Date: Thu, 27 Dec 2001 23:36:55 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> Thanks Paul. First note that I am only looking at those with a > period>> of 1 octave at this stage. >> What do you mean? You did a whole spreadsheet with graphs for the 1/2- > octave period case.I just haven't got to that yet.>> As a 5-limit approximation the 522.86c >> generator is junk. It has an error of 25 c in the 2:3. >> On behalf of Herman Miller, Margo Schulter, Bill Sethares, and the > entire island of Java, let me just say #(@*$& ?@#>$, and then let me > just say, go play with this scale for a while. 'Junk' my &$$.Paul, I think you're severely distorting what I wrote. I didn't say pelog is junk. I said "as a 5-limit approximation ..." Is there really any evidence that pelog is a 5-limit temperament? Specifically that it exists (even partly) because it is a 7 note chain of generators such that a single generator approximates a 2:3, -3 generators approximates a 4:5 and 4 generators approximates a 5:6? e.g. Do they play a lot of the approximate 1:3:5 triads in this temperament? Or is it perhaps an approximate 7-tET, for mostly melodic reasons, with inharmonic timbres to make the "fifths" sound ok, and little or no importance placed on any approximate ratios of 5? I'm only guessing. I know very little about pelog. I'm just very wary of JI "explanations" for things like pelog and slendro.>> So there are >> plenty of other temperaments as good as this. >> By _as good as_, I mean having an equal or lower RMS error ANS and > equal or lower 'gens' measure.Why can't I use my own criteria for "as good as"?
Message: 2772 - Contents - Hide Contents Date: Thu, 27 Dec 2001 10:05:13 Subject: Weighting From: genewardsmith The idea of optimizing the intervals of a complete odd-limit chord makes sense to me, and I can see some justification in counting repeated intervals,but I don't see that, for instance, the major thirds/minor sixths should weigh much more heavily than the minor thirds/major sixths. Why so, and by how much?
Message: 2773 - Contents - Hide Contents Date: Thu, 27 Dec 2001 23:39:32 Subject: Ennealimmal & co From: genewardsmith 2^38 3^-2 5^-15 Parameantone Map: [ 0 1] [-15 4] [ 2 2] Generators: a = 71.00086/441 = 193.1996 cents (meantone); b = 1 badness: 138 rms: .0608 g: 13.14 errors: [.0508, .0855, .0347] 2^-16 3^35 5^-17 Minor Whole Tone (10/9)^17 ~ 6 Map: [ 0 1] [17 -1] [35 -3] Generators: a = 26.00142/171 = 182.4661 cents (~10/9); b = 1 badness: 386 rms: .02547 g: 24.75 errors: [-.03149, -.00059, .03089] 2 3^-27 5^18 Ennealimmal (27/25)^9 ~ 2 Map: [ 0 9] [-2 19] [-3 28] Generators: a = 160.9945/612 = 315.6755 cents (~6/5); b = 1/9 = 133.3333 cents badness: 188 rms .0256 g: 19.44 errors: [.0274, -.0068, -.0342] 2^91 3^-12 5^-31 Limmal Generators: a = 12.999127/118 = 132.1945 cents (~27/25); b = 1 badness: 463 rms: .0150 g: 31.38 errors: [.0152, .0204, .0052]
Message: 2774 - Contents - Hide Contents Date: Thu, 27 Dec 2001 11:13:36 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Dave, I'll do the arithmetic for you and give you the RMS optima for > the ones with RMS less than 20 cents and g<8. Is there anything > _missing_ that is as good as one of these? > > Generator 522.86¢, Period 1 oct. > Generator 505.87¢, Period 1/4 oct. > Generator 163.00¢, Period 1 oct. > Generator 491.20¢, Period 1/3 oct. > Generator 379.97¢, Period 1 oct. > Generator 503.83¢, Period 1 oct. > Generator 494.55¢, Period 1/2 oct. > Generator 442.98¢, Period 1 oct. > Generator 387.82¢, Period 1 oct. > Generator 271.59¢, Period 1 oct. > Generator 317.08¢, Period 1 oct. > Generator 498.27¢, Period 1 oct.Thanks Paul. First note that I am only looking at those with a period of 1 octave at this stage. As a 5-limit approximation the 522.86c generator is junk. It has an error of 25 c in the 2:3. So there are plenty of other temperaments as good as this. If you omitted this and 163.00 c I would agree with the list, and I would probably only find a few more as good as 163.00 c. I'll let you know what these are when I have more time.
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