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Message: 2725 - Contents - Hide Contents Date: Wed, 26 Dec 2001 11:43:43 Subject: Re: a different example From: monz Hi Gene and Paul, I'm finally getting around to replying to the "different example" periodicity-blocks we created a couple of days ago. Gene, you must have been sleepy, because there are three errors in your post; I'll correct them in the quote.> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 3:31 PM > Subject: [tuning-math] a different example (was: coordinates from unison-vectors) > > > ... So anyway, I put in the [(3,5) unison-vector] matrix: > > ( 6 -14) > (-4 1) > > and could see that the resulting periodicity-block had a strong > correlation (in the sense of my meantone-JI implied lattices) > with the neighborhood of 2/7- to 3/11-comma meantone. > > The determinant is 50, so this agrees with my observation. > From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 8:04 PM > Subject: [tuning-math] Re: a different example > > > ... First I put the 2 back into the above commas, and get > > q1 = 2^23 3^6 5^-15 and q2 = 80/81.Er... there's a typo in that last exponent: it should be q1 = 2^23 3^6 5^-14.> ... I have one et, h50, such that h50(q1) = h50(q2) = 0. > Now I search for something where h(q1)=0 and h(q2)=1, > obtaining h19, h69, -h31 and -h81. The simplest of these > is h19, and I choose it. > > Next, I look for something such that h(q1)=1 and h(q2)=0, > and I get h16, -h34, -h84; I choose h16. > > Now I form a 3x3 matrix from these, and invert it: > > [ 50 19 16] > [ 79 30 25]^(-1) = > [116 44 37] > > [-10 -1 5] > [ 23 6 -14] > [ 4 -4 1] > > The rows of the inverted matrix correspond to the commas > q0 = 2^-10 3^-1 5 = 3125/3072 (small diesis),Typo: that should be q0 = 2^-10 3^-1 5^5> q1 = 2^23 3^6 5^14, and q2=80/80.Typo: according to the signs on the integers in the bottom row of the last matrix, q2 = 80/81. If that's not right, then it should be q2 = [-4 4 -1] = 81/80.> > Now I calculate the scale; the nth step is > > scale[n] = q0^n * q1^round(19n/50) * q2^round(16n/50), > > where "round" rounds to the nearest integer. It doesn't matter > which paticular hn I selected when I do this, or where I start > and end; though my definition of "nearest integer" does matter. > > I got in this way: > > 1, 3125/3072, 128/125, 25/24, 82944/78125, 16/15, 625/576, > 3456/3125, 10/9, 15625/13824, 144/125, 125/108, 18432/15625, > 6/5, 625/512, 768/625, 5/4, 15625/12288, 32/25, 125/96, > 20736/15625, 4/3, 3125/2304, 864/625, 25/18, 110592/78125, > 36/25, 625/432, 4608/3125, 3/2, 15625/10368, 192/125, 25/16, > 24576/15625, 8/5, 625/384, 1024/625, 5/3, 15625/9216, 216/125, > 125/72, 27648/15625, 9/5, 3125/1728, 1152/625, 15/8, 78125/41472, > 48/25, 125/64, 6144/3125 > > How does this compare with your results? > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 8:33 PM > Subject: [tuning-math] Re: a different example (was: coordinates from unison-vectors) > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > > With 50 notes, some arbitrary decision has to be made -- no note can > be exactly in the center, since 50 is an even number. But you should > be getting the following block or its reflection through the origin: > > > p5's M3's > ---- ----- > > 3 -7 > 4 -7 > 1 -6 > 2 -6 > 3 -6 > 4 -6 > 1 -5 > 2 -5 > 3 -5 > 0 -4 > 1 -4 > 2 -4 > 3 -4 > 0 -3 > 1 -3 > 2 -3 > 3 -3 > 0 -2 > 1 -2 > 2 -2 > -1 -1 > 0 -1 > 1 -1 > 2 -1 > -1 0 > 0 0 > 1 0 > -2 1 > -1 1 > 0 1 > 1 1 > -2 2 > -1 2 > 0 2 > -3 3 > -2 3 > -1 3 > 0 3 > -3 4 > -2 4 > -1 4 > 0 4 > -3 5 > -2 5 > -1 5 > -4 6 > -3 6 > -2 6 > -1 6 > -4 7Sorry about the long quote, but I wanted both of these sets of data to be together here, because after going thru all the trouble of prime-factoring Gene's set, I see that it's identical to Paul's. Now, how does this compare with my results? Well... the shape and size of the periodicity-block is exactly the same; the only difference is that my block is not quasi-centered on 1/1, as Gene/Paul's is and as I wanted mine to be. I'd appreciate it if you guys could fix my pseudo-code so that I can use my own spreadsheet method and still get the correct results. Again, my spreadsheet is at Yahoo groups: /tuning-math/files/monz/matrix m... * [with cont.] Thanks. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2726 - Contents - Hide Contents Date: Wed, 26 Dec 2001 01:37:58 Subject: Re: My top 5--for Paul From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> Yes. It is a fine example of the musical irrelevance of a flat > badness>> measure. I think musicians would rate it somewhere between 5 and >> infinity times as bad as the other four you listed. 50 notes for > one>> triad? The problem, as usual is that an error of 0.5 c is >> imperceptible and so an error of 0.0002 c is no better, and does > not>> compensate for a huge number of generators. Sorry if I'm sounding > like>> a stuck record. >> Let's not make decisions for musicians. Many theorists have delved > into systems such as 118, 171, and 612. We would be doing no harm to > have something to say about this range, even if we don't personally > feel that it would be musically useful.But Paul! You _are_ making decisions for musicians! You can't help but do so. Unless you plan to publish an infinite list of temperaments, the fact that you rate cases like this highly means that you will include fewer cases having more moderate numbers of gens per consonance. Shouldn't the question be rather whether you are making a _good_ decision for musicians? There's nothing terribly personal about the fact that an error of 0.5 c is imperceptible by humans. Theorists have delved into systems such as 118, 171, 612-tET, but has anything musical ever come of it? And if it has or does, surely we would be looking at subsets, not the entire 118 notes per octave etc. i.e. we'd be looking at temperaments within these ETs where consonant intervals are produced by considerably fewer than 50 notes in a chain (or chains) of generators.
Message: 2727 - Contents - Hide Contents Date: Wed, 26 Dec 2001 20:21:49 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> That seems to say there *is* no accepted definition. It seems to me >we want to consider at least two distinct properties--epimorphic and >convex. Yup. >Connectedness is another one, which would need to be defined, since >more than one choice might be made there.Yup -- 7-limit connected and 9-limit connected are two possibilities in the 3D case. Some of Fokker's blocks were disconnected despite being convex.
Message: 2728 - Contents - Hide Contents Date: Wed, 26 Dec 2001 02:49:57 Subject: Re: My top 5--for Paul From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> There's nothing terribly personal about the fact that an error of 0.5 > c is imperceptible by humans.It sure is if you're playing with a loud or distorted sound system -- my favorite!> Theorists have delved into systems such as 118, 171, 612-tET, but has > anything musical ever come of it? And if it has or does, surely we > would be looking at subsets, not the entire 118 notes per octave etc. > i.e. we'd be looking at temperaments within these ETs where consonant > intervals are produced by considerably fewer than 50 notes in a chain > (or chains) of generators.Perhaps not yet . . . but what harm comes from _informing_ musicians of these systems? I'd love it if a genius musician did make use of not considerably fewer than 50 notes per octave -- oh, wait a minute, my lips are a little partched today . . . and when I make lattices for these systems, you can be sure I'm going to start with the simplest and work my way up until the impenetrable thickets of notes make me decide a single line of data from Gene would be more appropriate. Hey Dave, why not look at Gene's list of 5-limit temperaments and see if he's missed anything?
Message: 2729 - Contents - Hide Contents Date: Wed, 26 Dec 2001 20:24:34 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> I'll probably add this to the definition too, unless you guide >> me into what should and shouldn't be included in existing Dictionary >> entries pertaining to the subject, or into what new definitions >> need to be added. >> What I would suggest is that rather than adding something >immediately, we agree on some precise definitions. To start with, >does Paul see convexity or connectedness as important, as I do?I see them as important if the block is left in JI. But again, even some of Fokker's blocks were disconnected (though still convex). There's nothing wrong with saying "convex periodicity block" or "9- limit-connected periodicity block" or whatever when that's the concept you want.
Message: 2730 - Contents - Hide Contents Date: Wed, 26 Dec 2001 07:35:11 Subject: Re: The epimorphic property From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> The only published articles on PBs are Fokker's. Inferring strict > definitions from these articles would suggest that a parallelepiped > (or N-dimensional equivalent) are the only accepted shape (thus I > call these _Fokker_ periodicity blocks, or FPBs), and that if there > is an even number of notes, one needs to produce two alternative > versions so that symmetry about 1/1 is maintainted.That seems to say there *is* no accepted definition. It seems to me we want to consider at least two distinct properties--epimorphic and convex. Connectedness is another one, which would need to be defined, since more than one choice might be made there.
Message: 2731 - Contents - Hide Contents Date: Wed, 26 Dec 2001 20:39:20 Subject: Re: Microtemperament and scale structure From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: [snipped old stuff from Gene]> Thanks for using easy examples to explain your methods here. > I'm finally beginning to understand what you write. > > This is an interesting result for 24-EDO, but doesn't really describe > accurately the case Paul used as his illustration. Why not? > It appears here: > <Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>, about > 2/3 of the way down the page. I've added graphs of the pitch-height > of the notes in the scale, and of the step-sizes between each degree. > > The periodicity-block I derived there is a JI system where the > unison-vectors which are not tempered out are not quarter-tones,Nor were they in the (snipped) example above.> but rather the syntonic comma 81:80 = (-4 4 -1) = ~21.5 cents, > and the diaschisma = (11 -4 -2) = ~19.55 cents.Monz, you're equivocating! On the page you reference, you write, "I had determined the finity of this pitch-set by the unison vectors which form the intervals of a skhisma [= 3^8 * 5^1 = ~2.0 cents] and a diesis [= 5^-3 = ~41.1 cents]." These are exactly the unison vectors of the (snipped) example above!> The numbers in the homomorphism still work to count the scale-steps > which represent the prime-factors,No they don't -- for example, look at all the 3:2s in the scale and count how many scale steps they subtend.> but the scale-steps are not > equal.Why should they be? They wouldn't be equal for any JI periodicity block. Arranged in descending pitch-height:> > ratio ~cents > > 256/135 1107.821284 > 15/8 1088.268715 > 9/5 1017.596288 > 16/9 996.0899983 > 27/16 905.8650026 > 5/3 884.358713 > 8/5 813.6862861 > 405/256 794.1337173 > 3/2 701.9550009 > 40/27 680.4487113 > 64/45 609.7762844 > 45/32 590.2237156 > 27/20 519.5512887 > 4/3 498.0449991 > 81/64 407.8200035 > 5/4 386.3137139 > 6/5 315.641287 > 32/27 294.1349974 > 9/8 203.9100017 > 10/9 182.4037121 > 16/15 111.7312853 > 135/128 92.17871646 > 81/40 21.5062896 > 1/1 0 > > > So the four step-sizes between degrees of the scale, > in ~cents, are: > > 90.225 _purana_ sruti > 70.672 _nyuna_ sruti > 21.506 } both of these are conflated > 19.553 } into the _pramana_ sruti > > and with an anomalous interval of ~92.179 cents between the > highest degree and the "8ve".I'd call that five step-sizes, then. Why is one "anomalous"? It also occurs between 81/64 and 4/3, for example.> If you didn't before, I think now you can see why Paul calls > this "not a well-behaved periodicity-block"... at any rate, > now *I* finally understand! You do? > The homomorphism column vector resulting from your calculations > is also off for this scale; here, it should be: > > [24] > [39] > [56] > > where the middle value is [39] instead of your [38]. > What does this mean? I can see that it's because in this > scale the 38th degree is much farther away from 3/2 than > is the 39th degree, whereas in 24-EDO it's the other > way around. Is there a way to adjust your formula so > that it works for this JI case?I think you're *not* understanding why this is "not a well-behaved periodicity-block". Sorry!
Message: 2732 - Contents - Hide Contents Date: Wed, 26 Dec 2001 00:16:09 Subject: Re: The epimorphic property From: monz Hi Gene and Paul,> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, December 25, 2001 11:35 PM > Subject: [tuning-math] Re: The epimorphic property > > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> The only published articles on PBs are Fokker's. Inferring strict >> definitions from these articles would suggest that a parallelepiped >> (or N-dimensional equivalent) are the only accepted shape (thus I >> call these _Fokker_ periodicity blocks, or FPBs), and that if there >> is an even number of notes, one needs to produce two alternative >> versions so that symmetry about 1/1 is maintainted. >> That seems to say there *is* no accepted definition.If there isn't, then we're on our way to getting one. I've already added this paragraph of Paul's to my Tuning Dictionary entry for "periodicity-block" <Definitions of tuning terms: periodicity block... * [with cont.] (Wayb.)>, as well as a new entry for "Fokker Periodicity Block" <Definitions of tuning terms: FPB, (c) 2001 by ... * [with cont.] (Wayb.)>.> It seems to me we want to consider at least two distinct > properties--epimorphic and convex. Connectedness is another one, > which would need to be defined, since more than one choice might > be made there.I'll probably add this to the definition too, unless you guide me into what should and shouldn't be included in existing Dictionary entries pertaining to the subject, or into what new definitions need to be added. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2733 - Contents - Hide Contents Date: Wed, 26 Dec 2001 13:05:29 Subject: Re: Microtemperament and scale structure From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 26, 2001 12:39 PM > Subject: [tuning-math] Re: Microtemperament and scale structure > >>> The numbers in the homomorphism still work to count the scale-steps >> which represent the prime-factors, >> No they don't -- for example, look at all the 3:2s in the scale and > count how many scale steps they subtend. >>>> So the four step-sizes between degrees of the scale, >> in ~cents, are: >> >> 90.225 _purana_ sruti >> 70.672 _nyuna_ sruti >> 21.506 } both of these are conflated >> 19.553 } into the _pramana_ sruti >> >> and with an anomalous interval of ~92.179 cents between the >> highest degree and the "8ve". >> I'd call that five step-sizes, then. Why is one "anomalous"? It also > occurs between 81/64 and 4/3, for example.Oh... when I looked I saw it ocurring only in the one place I described.>> If you didn't before, I think now you can see why Paul calls >> this "not a well-behaved periodicity-block"... at any rate, >> now *I* finally understand! > > You do? >>> The homomorphism column vector resulting from your calculations >> is also off for this scale; here, it should be: >> >> [24] >> [39] >> [56] >> >> where the middle value is [39] instead of your [38]. >> What does this mean? I can see that it's because in this >> scale the 38th degree is much farther away from 3/2 than >> is the 39th degree, whereas in 24-EDO it's the other >> way around. Is there a way to adjust your formula so >> that it works for this JI case? >> I think you're *not* understanding why this is "not a well-behaved > periodicity-block". Sorry!OK, I thought I was beginning to get it. Can you please explain in a little more detail why what I wrote here is wrong? Gene gave an example which resulted in "quartertones", and this periodicity-block doesn't do that! Lost in the fog... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2734 - Contents - Hide Contents Date: Wed, 26 Dec 2001 08:27:08 Subject: Re: lattices of Schoenberg's rational implications From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Can someone explain what's going on here, and what candidates > may be found for unison-vectors by extending the 11-limit system, > in order to define a 12-tone periodicity-block? Thanks.See if this helps; We can extend the set {33/32,64/63,81/80,45/44} to an 11-limit notation in various ways, for instance <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5] where g7 differs from h7 by g7(7)=19. Using this, we find the corresponding block is (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12) (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9; the Pythagorean scale. We don't need anything new to find a 12-note scale; we get 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--19/9--15/8 or variants, the variants coming from the fact that 12 is even, by using 12 rather than 7 in the denominator.
Message: 2735 - Contents - Hide Contents Date: Wed, 26 Dec 2001 21:14:09 Subject: Re: Microtemperament and scale structure From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> OK, I thought I was beginning to get it. Can you please explain > in a little more detail why what I wrote here is wrong? Gene > gave an example which resulted in "quartertones", and this > periodicity-block doesn't do that!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. 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. 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.> > Lost in the fog... > > > > -monz > > > > > > _________________________________________________________ > Do You Yahoo!? > Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2736 - Contents - Hide Contents Date: Wed, 26 Dec 2001 08:29:15 Subject: Re: The epimorphic property From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> I'll probably add this to the definition too, unless you guide > me into what should and shouldn't be included in existing Dictionary > entries pertaining to the subject, or into what new definitions > need to be added.What I would suggest is that rather than adding something immediately, we agree on some precise definitions. To start with, does Paul see convexity or connectedness as important, as I do?
Message: 2737 - Contents - Hide Contents Date: Wed, 26 Dec 2001 21:29:31 Subject: Re: Microtemperament and scale structure From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> 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.And I was wrong--I turned the crank of an alorithm, and the 5-limit 24-et popped out, but it wasn't correct. You can't define it in terms of commas in the same way as the 12-et, which leads to the question (which we got into again lately) if we even want to bother with it. This example is where the whole business of "torsion" came from--if you try to define something in terms of commas, you don't get the 24-et, you get a goofball system with two 12-ets separated by an interval whose square is a unison--in other words, it isn't a comma but the square of it is.
Message: 2738 - Contents - Hide Contents Date: Wed, 26 Dec 2001 08:31:10 Subject: Re: lattices of Schoenberg's rational implications From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--19/9--15/8This should be 16/9 rather than 19/9, of course.
Message: 2739 - Contents - Hide Contents Date: Wed, 26 Dec 2001 13:41:25 Subject: Re: Microtemperament and scale structure From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > 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. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2740 - Contents - Hide Contents Date: Wed, 26 Dec 2001 00:36:10 Subject: Re: The epimorphic property From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 26, 2001 12:29 AM > Subject: [tuning-math] Re: The epimorphic property > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> I'll probably add this to the definition too, unless you guide >> me into what should and shouldn't be included in existing Dictionary >> entries pertaining to the subject, or into what new definitions >> need to be added. >> What I would suggest is that rather than adding something immediately, > we agree on some precise definitions. To start with, does Paul see > convexity or connectedness as important, as I do?Thanks, Gene. In addition, I'd like to start adding some definitions of Pierre Lamothe's terms, if he or any of you could help with those. -monz> > > > > To unsubscribe from this group, send an email to: > tuning-math-unsubscribe@xxxxxxxxxxx.xxx > > > > Your use of Yahoo! Groups is subject to Yahoo! Terms of Service * [with cont.] (Wayb.) > _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2741 - Contents - Hide Contents Date: Wed, 26 Dec 2001 21:43:36 Subject: Re: Microtemperament and scale structure From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> 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. >> And I was wrong--I turned the crank of an alorithm, and the 5-limit >24-et popped out, but it wasn't correct. You can't define it in >terms of commas in the same way as the 12-et, which leads to the >question (which we got into again lately) if we even want to bother >with it. This example is where the whole business of "torsion" came >fromWhat's the matter Gene, do you only read the first paragraph of my posts?
Message: 2742 - Contents - Hide Contents Date: Wed, 26 Dec 2001 08:40:58 Subject: Re: Keenan green Zometool struts From: dkeenanuqnetau --- In tuning-math@y..., paul@s... wrote: Zome folks haven't implemented your green strut idea yet.> > But I recently saw a kit called "Advanced Mathematics" which didcontain green struts.> > Did your ideas in fact help this product to be developed?Yes. I get an acknowledgment, among others, on the doco with the green line kit.> Should I buy the kit? It's between 100 and 200 US$.It might be better to get the creator kit with a different set of greens more later.
Message: 2743 - Contents - Hide Contents Date: Wed, 26 Dec 2001 21:45:19 Subject: Re: Microtemperament and scale structure From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> OK, I'm only up to late August, so there's obviously some more > important stuff coming up.Monz, it means a lot to me that you are taking the time and trouble to go through this stuff, and hopefully incorporate as much of it as possible into your webpages. It all feels a little more "meaningful", somehow . . .
Message: 2744 - Contents - Hide Contents Date: Wed, 26 Dec 2001 09:44:14 Subject: Re: My top 5--for Paul From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Theorists have delved into systems such as 118, 171, 612-tET, but has > anything musical ever come of it?The 118 et is a good one for schismic temperament, which we hear on the tuning list seems to have been hit on at one time, so I'd hardly dismiss it out of hand. Things such as ennealimmal haven't been tried, but until quite recently, neither had miracle been tried. And if it has or does, surely we> would be looking at subsets, not the entire 118 notes per octave etc. > i.e. we'd be looking at temperaments within these ETs where consonant > intervals are produced by considerably fewer than 50 notes in a chain > (or chains) of generators.This is the 21st century--there is no particular obstacle to using 612 notes, other than that is lot of notes to get around to. I just took a look at the 118 et, and find it has the ragisma (4375/4374) and the shisma as commas, and also what Manuel calls the "gamelan residue" of 1029/1024, and 3136/3125. Does anyone know where the name "gamelan resiude" comes from? In any case each of these separately, or more than one in combination, produce some interesting temperaments.
Message: 2745 - Contents - Hide Contents Date: Wed, 26 Dec 2001 14:49:30 Subject: Paul's lattice math and my diagrams (was: Hey Carl [old]) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 26, 2001 1:45 PM > Subject: [tuning-math] Re: Microtemperament and scale structure > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> OK, I'm only up to late August, so there's obviously some more >> important stuff coming up. >> Monz, it means a lot to me that you are taking the time and trouble > to go through this stuff, and hopefully incorporate as much of it as > possible into your webpages. It all feels a little more "meaningful", > somehow . . .Thanks, Paul! Most people around here (any of the tuning lists, but especially this one and HE) hold you and your work in pretty high repute... so coming from you, this feels like some cachet being bestowed upon me. :) So now, if you don't mind backtracking a bit further, I've saved this one for last and think you'll get something out of it...> From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, July 28, 2001 10:40 AM > Subject: [tuning-math] Re: Hey Carl > > > Here's why the hypothesis should work. > > Take an n-dimensional lattice, and pick n independent > unison vectors. Use these to divide the lattice into > parallelograms or parallelepipeds or hyperparallelepipeds, > Fokker style. Each one contains an identical copy of a single > scale (the PB) with N notes. Any vector in the lattice now > corresponds to a single generic interval in this scale no > matter where the vector is placed (if the PB is CS, which > it normally should be). Now suppose all but one of the > unison vectors are tempered out. The "wolves" now divide > the lattice into parallel strips, or layers, or hyper-layers. > The "width" of each of these, along the direction of the > chromatic unison vector (the one that remains untempered), > is equal to the length of exactly one of this chromatic > unison vector.Paul, this is *exactly* what's going on in my "meantone acoustical rational implications lattices" here: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) 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. This is not really clear in the diagrams that exist on the webpage now, because they all have arbitrary limits of +/- 27 generators. I'm preparing some new diagrams which *do* show proper Fokker periodicity-blocks for these (and other) meantones.> > 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.> 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.] >).> QED -- right?Is that "Quod Erat Demonstrandum" or "Quite Easily Done"? (or "Quickly Ends Dandruff"? ;-) ) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2746 - Contents - Hide Contents Date: Wed, 26 Dec 2001 08:56:51 Subject: Re: Microtemperament and scale structure From: monz Hi Gene! I'm only just now getting around to studying what's been discussed on tuning-math from August and September, shortly after you came on board (and when I was planning a trip and then traveling in Europe). I've only recently gotten back into the groove here, so much of the last 5 months has slipped past me. I have some comments for you. Please, in reading any of my posts to this list, realize that I am one of the most math-challenged posters here. I somehow passed Algebra 1 in 9th grade without learning a thing, and spent most of my time in Geometry class in 10th grade sitting in the back row studying my score to Neilsen's 4th Symphony (the terrific "Inextinguishable"... my, how I'd like to do a microtonal computer realization of *that* some day!...). Then, after moving and getting the chair of the new school's math department for my Algebra 2 teacher, and being forced to pay attention and learn, and struggling heroically to earn a final "D", I chose no more math classes during the rest of my formal education. This is a lack which I sorely feel now, being so caught up in tuning math.> From: <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, August 18, 2001 2:18 PM > Subject: [tuning-math] Re: Microtemperament and scale structure > > > My way of saying things has the advantage of being standard > mathematical terminology, which allows one to bring relevant concepts > into play. I had noticed that unison vectors seemed to have something > to do with the kernel, but I couldn't tell if it meant generators of > the kernel or any element of the kernel, and I see by your comments > that no one has really decided! > > The kernel of some homomorphism h is everything sent to the identity-- > if this is the set of unison vectors then for instance 1 is *always* > a unison vector, since h(1) = 0. On the other hand, unison vectors > could be elements of a minimal set of generators for the kernel. In > this case 81/80, 128/125 and 2048/2025 would all belong in the same > kernel generated by any two of them. Depending on which set of > generators you picked, two of them would be unison vectors and the > other one would not be. 1 and 32805/32768 would also both be in the > kernel, but neither would be unison vectors. > > Probably the simplest solution at this point would be to drop the > terminology, but if you don't you need to decide what exactly it > means.This is an important suggestion for me, since I'm the guy who created and maintains the Tuning Dictionary. What have the rest of you heavyweights decided about this? I've been slugging around the regular periodicity-block and unison-vector terminology a lot here lately, and no-one has complained. Do we go with "kernel"?> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> We're well aware that any valid set comes from any other >> valid set simply by "adding" and "subtracting" the >> unison vectors from one another. But we've run >> into some pathological cases -- for example, the >> small diesis (128/125) and the schisma (32805/ >> 32768), while they can be derived from the same >> two unison vectors, define a periodicity block with >> 24, instead of 12, notes . . . and not a well-behaved >> periodicity block at that. Any insights? >> It's true that anything in the kernel is obtained by adding and > subtracting, since the kernel of an abelian group homomorphism is an > abelian group. It's not true that any linearly independent set of > kernel elements which span the corresponding vector space over the > rationals as a basis is also a minimal set of generators for the > kernel, and that is what you have discovered. > > Let's take 81/80 and 128/125 to start with. I may write these > additively, so that 81/80 = 2^-4 * 3^4 * 5^-1 is written [-4, 4, -1] > and 128/125 becomes [7, 0, -3]. We can put these together into a 2x3 > matrix, giving us > > [-4 4 -1] > [ 7 0 -3]I used to use [ ] around a matrix because I thought that was the correct notation. Then switched to | |, then Graham told me that that is wrong because it's for the determinant, and he uses ( ) so I switched to that. Now I see you using [ ], and my sense is that your methodology on this list is more highly respected than anyone else's, so I'm inclined to follow (and go back to what I felt was right in the first place!). What's the deal on this?> If we take the absolute value of the determinants of the minors of > this matrix, we recover the homomorphism: > > abs(det([[4, -1],[0,-3]]) = 12, abs(det([[-4, -1],[7,-3]]) = 19, and > abs(det([-4,4],[7,0]))= 28, recovering the homomorphism column vector > > [12] > [19] > [28] > > from the generators of the kernel. This computation shows these > two "unison vectors" do in fact generate the kernel. If we perform a > similar computation for 128/125 and 32805/32768 we first get the > matrix > > [7 0 -3] > [-15 8 1] > > The column vector we get from the absolute values of the determinants > of the minors of this is: > > [24] > [38] > [56] > > In other words, these two define the kernel of a homomorphism to the > 24 et division of the octave, which in the 5-limit is "pathological" > in the sense that you have two separate 12 divisions a quarter-tone > apart, and we cannot get from one to the other using relationships > taken from 5-limit harmony, as all of the numbers in the above > homomorphism are even.Thanks for using easy examples to explain your methods here. I'm finally beginning to understand what you write. This is an interesting result for 24-EDO, but doesn't really describe accurately the case Paul used as his illustration. It appears here: <Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>, about 2/3 of the way down the page. I've added graphs of the pitch-height of the notes in the scale, and of the step-sizes between each degree. The periodicity-block I derived there is a JI system where the unison-vectors which are not tempered out are not quarter-tones, but rather the syntonic comma 81:80 = (-4 4 -1) = ~21.5 cents, and the diaschisma = (11 -4 -2) = ~19.55 cents. The numbers in the homomorphism still work to count the scale-steps which represent the prime-factors, but the scale-steps are not equal. Arranged in descending pitch-height: ratio ~cents 256/135 1107.821284 15/8 1088.268715 9/5 1017.596288 16/9 996.0899983 27/16 905.8650026 5/3 884.358713 8/5 813.6862861 405/256 794.1337173 3/2 701.9550009 40/27 680.4487113 64/45 609.7762844 45/32 590.2237156 27/20 519.5512887 4/3 498.0449991 81/64 407.8200035 5/4 386.3137139 6/5 315.641287 32/27 294.1349974 9/8 203.9100017 10/9 182.4037121 16/15 111.7312853 135/128 92.17871646 81/40 21.5062896 1/1 0 So the four step-sizes between degrees of the scale, in ~cents, are: 90.225 _purana_ sruti 70.672 _nyuna_ sruti 21.506 } both of these are conflated 19.553 } into the _pramana_ sruti and with an anomalous interval of ~92.179 cents between the highest degree and the "8ve". If you didn't before, I think now you can see why Paul calls this "not a well-behaved periodicity-block"... at any rate, now *I* finally understand! The homomorphism column vector resulting from your calculations is also off for this scale; here, it should be: [24] [39] [56] where the middle value is [39] instead of your [38]. What does this mean? I can see that it's because in this scale the 38th degree is much farther away from 3/2 than is the 39th degree, whereas in 24-EDO it's the other way around. Is there a way to adjust your formula so that it works for this JI case? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2747 - Contents - Hide Contents Date: Wed, 26 Dec 2001 22:58:24 Subject: Re: Paul's lattice math and my diagrams (was: Hey Carl [old]) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Here's why the hypothesis should work. >> >> Take an n-dimensional lattice, and pick n independent >> unison vectors. Use these to divide the lattice into >> parallelograms or parallelepipeds or hyperparallelepipeds, >> Fokker style. Each one contains an identical copy of a single >> scale (the PB) with N notes. Any vector in the lattice now >> corresponds to a single generic interval in this scale no >> matter where the vector is placed (if the PB is CS, which >> it normally should be). Now suppose all but one of the >> unison vectors are tempered out. The "wolves" now divide >> the lattice into parallel strips, or layers, or hyper-layers. >> The "width" of each of these, along the direction of the >> chromatic unison vector (the one that remains untempered), >> is equal to the length of exactly one of this chromatic >> unison vector. > >> Paul, this is *exactly* what's going on in my "meantone > acoustical rational implications lattices" here: > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)But alas, you are not getting the infinite strips I refer to above.> 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.>> >> 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.>> 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?
Message: 2748 - Contents - Hide Contents Date: Wed, 26 Dec 2001 10:17:36 Subject: Gene's notation & Schoenberg lattices From: monz> From: <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, August 17, 2001 7:06 PM > Subject: [tuning-math] Re: Microtemperament and scale structure > > > ... For another easy example, the > rank 3 free group G can be sent to a rank 1 free group by a > homomorphism h12(2) = 12, h12(3) = 19, h12(5) = 28. > ... > ... In the case of h12, the kernel is > spanned by 81/80 (the diatonic comma) and 128/125 (the great diesis), > where we have h12(81/80) = h12(128/125) = 0. > ... > Consider the system h72(2) = 72, h72(3) = 114, h72(5) = 167, h72(7) = > 202, h72(11) = 249 ... h72(81/80) = 1 and h72(128/125) = 3... > h31(2) = 31, h31(3) = 49, h31(5) = 72, h31(7) = 87 and h31(11) = 107 > *does* have the property that h31(81/80) = 0; and while h31(128/125) = 1 > we still find h31 is much closer in structre to h12 than is h72.At last! I understand this!> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 26, 2001 12:27 AM > Subject: [tuning-math] Re: lattices of Schoenberg's rational implications > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> Can someone explain what's going on here, and what candidates >> may be found for unison-vectors by extending the 11-limit system, >> in order to define a 12-tone periodicity-block? Thanks. >> See if this helps; > > We can extend the set {33/32,64/63,81/80,45/44} to an 11-limit > notation in various ways, for instance > > <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5] > > where g7 differs from h7 by g7(7)=19. Using this, we find > the corresponding block is > > (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12) > (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9; > the Pythagorean scale. We don't need anything new to find a > 12-note scale;But alas! (pun intended) I have no idea what this means. 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? 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. Can you tell me how the pseudo-code which I posted here Yahoo groups: /tuning-math/messages/2069?expand=1 * [with cont.] can be expanded to calculate higher-than-2-D periodicity-blocks? (I know how to do the matrix stuff, but not the coordinates) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2749 - Contents - Hide Contents Date: Wed, 26 Dec 2001 15:15:05 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 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?!>> 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. The syntonic comma is the interval that sets the boundaries on the *other* two sides. But *isn't* that how meantones work? I seem to be missing something in this...>>> >>> 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. The actual lateral width of each strip would vary as the meantone chain's axis angle varies in relation to the fixed vector of the syntonic comma.>>> 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. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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