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Message: 3900 - Contents - Hide Contents Date: Mon, 18 Feb 2002 04:47:22 Subject: Re: twintone, paultone From: clumma>> >:7:8 -> 7:8:10 >>"Lumma" maps this to [8,-4]:[1,2]:[0,3] -> [1,2]:[0,3]:[8,-3] >>> Ignoring consistency and using the lowest-rms approximations, that >> will be: >>You're not allowed to do this, since you are using the regular >temperament Lumma and must use what it gives you.Okay, so what you're saying is, you prefer to measure the error of mappings from just intonation, rather than of ets -- it is a mistake to say, 'such-and-such equal temperament has such-and-such an error in the such-limit'. Fine by me, but since mappings are consistent by nature, your use of them to support your statment that you "can't hear consistency" is fallacious -- in order to speak of the accuracy of ets, the notion of 'switching mappings' must be substituted for the notion of inconsistency. -Carl

Message: 3901 - Contents - Hide Contents Date: Mon, 18 Feb 2002 05:01:13 Subject: Re: twintone, paultone From: clumma> Okay, so what you're saying is, you prefer to measure the error > of mappings from just intonation, rather than of ets -- it is a > mistake to say, 'such-and-such equal temperament has such-and-such > an error in the such-limit'. Fine by me, but since mappings are > consistent by nature, your use of them to support your statment > that you "can't hear consistency" is fallacious -- in order to > speak of the accuracy of ets, the notion of 'switching mappings' > must be substituted for the notion of inconsistency.In fairness to you, Gene, Graham did seem to be artificially kicking out the 34-et tuning of the two 7-limit mappings it contains, for the simple reason there are two of them. That would be wrong. I did it too:>...even tempering some commas out while inventing news ones can >probably be interesting. But for me, as a composer, this is just >too confusing. Thus, I restrict myself to consistent ets.The anomalous comma problem wouldn't be grounds for this. But the fact that consistency serves as a badness measure might be. -Carl

Message: 3902 - Contents - Hide Contents Date: Mon, 18 Feb 2002 06:08:45 Subject: Re: Maximal consistent sets of odds for ETs From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> A similar message was posted to the tuning list: > > I think we are astoundingly ignorant of the rational identities > available in ETs where 3's or 5's are not included. > > Who can generate a list for all the ETs up to 2000, giving for each ET > the maximal sets of mutually-consistently-approximated odd numbers up > to 35?didn't carl write a program to do this or something very much like it?

Message: 3903 - Contents - Hide Contents Date: Mon, 18 Feb 2002 06:24:24 Subject: Re: twintone, paultone From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> The anomalous comma problem wouldn't be grounds for this. But > the fact that consistency serves as a badness measure might be.I still don't know what the problem is. If you go to a high enough n, any n-et will have more than one good mapping. Is that still a problem?

Message: 3904 - Contents - Hide Contents Date: Mon, 18 Feb 2002 20:59:38 Subject: ratio presentation From: jpehrson2 Well, here's a simple one that I need some help with. I don't understand the process of finding "common denominators" for some of the examples on the Monz I-IV-V7-I webpage. For example, we have here: "The I and IV triads are tuned to the 5-limit JI 4:5:6 proportions, and the V7 to 36:45:54:64 = 4:5:6|27:32." Now, *finally* I understand that Monz was simply trying to reduce the triad and show the size of the 7th with the use of the "slash," not trying to do any fancy kind of division... :) But, how do we get the 36:45:54:64?? The first three numbers 4:5:6 are multiplied by *9* and the 7th, 27:32 has each number multiplied by *2*. Whyzzat?? There must be some kind of "common denominator" or some such in 36:45:54:64 that I'm not seeing. Why is it presented this way (again?) JP

Message: 3905 - Contents - Hide Contents Date: Mon, 18 Feb 2002 23:41:27 Subject: Re: ratio presentation From: paulerlich --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:> Well, here's a simple one that I need some help with. > > I don't understand the process of finding "common denominators" for > some of the examples on the Monz I-IV-V7-I webpage. > > For example, we have here: > > "The I and IV triads are tuned to the 5-limit JI 4:5:6 proportions, > and the V7 to 36:45:54:64 = 4:5:6|27:32." > > Now, *finally* I understand that Monz was simply trying to reduce the > triad and show the size of the 7th with the use of the "slash," not > trying to do any fancy kind of division... :) > > But, how do we get the 36:45:54:64?? > > The first three numbers 4:5:6 are multiplied by *9* and the 7th, > 27:32 has each number multiplied by *2*. Whyzzat?? > > There must be some kind of "common denominator" or some such in > 36:45:54:64 that I'm not seeing. > > Why is it presented this way (again?)joseph, this presentation shows the intervals between *adjacent* notes. those intervals are 4:5, 5:6, and 27:32. just major third, just minor third, pythagorean minor third. now to get the entire chord in lowest terms . . . note that 4:5 and 5:6 can already be combined into 4:5:6. but 6 is not 27. the lowest common multiple of 6 and 27 is 54: 6*9 = 54 27*2 = 54 so we multiply the terms in 4:5:6 by 9, and multiply the terms in 27:32 by 2. we get 36:45:54, and 54:64. now we can stick them "back together", and get 36:45:54:64 that's all there is to it . . .

Message: 3906 - Contents - Hide Contents Date: Mon, 18 Feb 2002 21:25:24 Subject: Re: Maximal consistent sets of odds for ETs From: Carl Lumma>282-tET would do me fine and so would 19-limit, but I figure there are >others who would want more.Let them eat cake! :) I'll go to 25 for now, to get some ASSes. Once I get my ASS-complete chord progie going, I'll drop it to 19.>> I don't think any of this gets you just what you want, Dave... >> I suppose I could add code that would create as the input list >> all 17-tone subsets of the 35-limit otonality (an 18-ad), and >> if none of them pass, all 16-note subsets, and so on. In many >> cases there may be multiple "maximal" sets (sets of the same >> card that pass). How would you like me to proceed? >>I see my meaning of "maximal" is still unclear. Cardinality isn't >relevant. There's probably a better term. Gene? It's kinda-like >saturated. What I mean by a maximal-mutually-consistent-set, is one to >which no further odds (within the 35-limit) can be added without >introducing inconsistency. An ET may have several of these which >overlap.Not unclear; just wanted to be sure you knew about that last bit. Cardinality is relevant -- you want the largest sets that are still consistent.>Please proceed however you can. Thanks. And it would be good if more >than one person did it, so results could be checked. It's beyond my >spreadsheeting I'm afraid.I'm glad you got back to me before the work week started. I'll see if I can fire this off tonight. No promises. It should be easy to check the results by hand, though I'd love a 2nd opinion... actually, I fully expect to be the 2nd opinion, since others here can probably beat me to this. -Carl

Message: 3907 - Contents - Hide Contents Date: Tue, 19 Feb 2002 13:33:57 Subject: magma From: Carl Lumma may be of interest to some here: Magma Computational Algebra System Home Page * [with cont.] (Wayb.) forget if I ever gave this link: Linear Algebra Toolkit * [with cont.] (Wayb.) -Carl

Message: 3908 - Contents - Hide Contents Date: Tue, 19 Feb 2002 16:34:05 Subject: Re: magma From: Carl Lumma>> >ay be of interest to some here: >> Magma Computational Algebra System Home Page * [with cont.] (Wayb.) >

Message: 3909 - Contents - Hide Contents Date: Tue, 19 Feb 2002 00:36:53 Subject: Re: ratio presentation From: jpehrson2 --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/3343 * [with cont.]> > joseph, this presentation shows the intervals between > *adjacent* notes. > > those intervals are 4:5, 5:6, and 27:32. > > just major third, just minor third, pythagorean minor third. > > now to get the entire chord in lowest terms . . . > > note that 4:5 and 5:6 can already be combined into 4:5:6. > > but 6 is not 27. > > the lowest common multiple of 6 and 27 is 54: > > 6*9 = 54 > 27*2 = 54 > > so we multiply the terms in 4:5:6 by 9, and multiply the terms in > 27:32 by 2. > > we get 36:45:54, and 54:64. > > now we can stick them "back together", and get 36:45:54:64 > > that's all there is to it . . .***Well, this makes a lot of sense, and I'm glad that it didn't say in the introductory page of this forum that this had to be "Advanced" Tuning Math... :) However,> the lowest common multiple of 6 and 27 is 54: > > 6*9 = 54 > 27*2 = 54Is there a "method" for doing this? Surely one doesn't just go "multiplying around" until one arrives at the same number?? Or is that how to do it?? ?? Thanks!!!! JP

Message: 3910 - Contents - Hide Contents Date: Tue, 19 Feb 2002 01:15:44 Subject: Re: ratio presentation From: genewardsmith --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:> Is there a "method" for doing this? Surely one doesn't just > go "multiplying around" until one arrives at the same number??You can find the greatest common divisor of two numbers by Euclid's algorithm, which is related to continued fractions. Calling that gcd(a,b), we have lcm(a,b) = a*b/gcd(a,b).

Message: 3911 - Contents - Hide Contents Date: Tue, 19 Feb 2002 04:27:55 Subject: Re: Maximal consistent sets of odds for ETs From: dkeenanuqnetau --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>>> I think we are astoundingly ignorant of the rational identities >>> available in ETs where 3's or 5's are not included. >> There's some truth to this. >>>> Who can generate a list for all the ETs up to 2000, giving for >>> each ET the maximal sets of mutually-consistently-approximated >>> odd numbers up to 35? ...> Given a list of chords and an et (I could step through all ets > up to 2000, but I think 282 would be plenty enough), I can return:282-tET would do me fine and so would 19-limit, but I figure there are others who would want more. ...> I don't think any of this gets you just what you want, Dave... > I suppose I could add code that would create as the input list > all 17-tone subsets of the 35-limit otonality (an 18-ad), and > if none of them pass, all 16-note subsets, and so on. In many > cases there may be multiple "maximal" sets (sets of the same > card that pass). How would you like me to proceed?I see my meaning of "maximal" is still unclear. Cardinality isn't relevant. There's probably a better term. Gene? It's kinda-like saturated. What I mean by a maximal-mutually-consistent-set, is one to which no further odds (within the 35-limit) can be added without introducing inconsistency. An ET may have several of these which overlap. Please proceed however you can. Thanks. And it would be good if more than one person did it, so results could be checked. It's beyond my spreadsheeting I'm afraid.

Message: 3912 - Contents - Hide Contents Date: Tue, 19 Feb 2002 05:13:10 Subject: lowest common multiple From: jpehrson2 --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: Yahoo groups: /tuning-math/message/3345 * [with cont.]> --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote: >>> Is there a "method" for doing this? Surely one doesn't just >> go "multiplying around" until one arrives at the same number?? >> You can find the greatest common divisor of two numbers by Euclid'salgorithm, which is related to continued fractions. Calling that> gcd(a,b), we have lcm(a,b) = a*b/gcd(a,b). ***Thanks, Gene!See, on the other list, I *said* I was supposed to be studying Euclid!... :) the lowest common multiple of 6 and 27 is 54: 6*9 = 54 27*2 = 54 So, let's see: lcm(6,27) = 162/gcd(6,27) lcm = 162/3 lcm = 54 It rather looks like it works! :) JP

Message: 3913 - Contents - Hide Contents Date: Tue, 19 Feb 2002 08:22:42 Subject: (unknown) From: genewardsmith I think a scale of nine tones to the octave is just about perfect, so I might want to consider what 9-polyhexes do for me. There are 6572 9-polyhexes (see Polyhex -- from MathWorld * [with cont.] ) so this is a big project. However, I'm most interested in polyhexes with a lot of internal nodes, corresponding to triads, so I start by looking at the paralleogram 9-polyhex. Placing my polyhex on the 5-limit lattice of classes, I get the scale Scale 0: 1--16/15--6/5--5/4--4/3--3/2--8/5--5/3--15/8 This is epimorphic from h9, and is a Fokker block from [128/125, 135/128]. I may rotate it by the transformation 2->2, 3->10/3, 5->16/3, giving me Scale 1: 1--10/9--6/5--5/4--4/3--3/2--8/5--5/3--9/5 This is *not* epimorphic, and hence not a Fokker block or (by my definition) any other kind of block, despite its block-like appearance. However, though its scale steps are a bit more irregular, it seems a perfectly fine scale, and can in fact also be gotten from the first scale by means of the major<->minor transformation. We therefore see that being a block is *not* invariant under major<->minor. Rotating again, we get Scale 2: 1--25/24--6/5--5/4--4/3--3/2--8/5--5/3--28/25 This now is epimorphic from h9, and a Fokker block from [128/125, 27/25]. We find no more scales by inversion or major<->minor transformation; this is the complete list of scales deriving from the paralleogram 9-polyhex. We might wonder, however, about the Fokker block [135/128, 27/25] which would seem to go with the other two. This is Scale 4: 1--9/8--6/5--5/4--4/3--3/2--8/5--5/3--16/9 This gives us a different 9-polyhex, with a central line five hexes long surrounded by two two-hex wings. I could go on and look at the other scales belonging to this polyhex, but I wasn't sure the process terminates any time soon. The first three scales have 16 5-limit intervals, and "Scale 4" 14; all of course are connected. This means that any mapping of these scales by a temperament will give us a 5-connected scale belonging to that temperament with at least 16 5-limit intervals. Below I show the number of 5,7,9 and 11-limit intervals for each of these scales for every "standard" equal temperament from 21 to 100 (starting at 21 insures that we have nine tones to the octave.) We can see from it evidence of interesting scales belonging to 81/80, 225/224, and so forth, which should be worth exploring. [1, 16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8] [1, 10/9, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 9/5] [1, 25/24, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 48/25] [1, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9] 21 [21, 27, 30, 36] [17, 24, 27, 31] [21, 27, 30, 36] [15, 22, 25, 31] 22 [16, 24, 29, 30] [17, 20, 27, 32] [16, 19, 26, 29] [14, 21, 27, 30] 23 [19, 24, 31, 31] [16, 23, 28, 31] [16, 21, 24, 27] [19, 24, 31, 31] 24 [21, 21, 27, 27] [20, 20, 27, 27] [21, 21, 27, 27] [20, 20, 27, 27] 25 [16, 21, 23, 30] [16, 21, 25, 32] [16, 17, 19, 29] [15, 24, 28, 30] 26 [16, 19, 24, 31] [19, 19, 26, 33] [16, 18, 23, 28] [19, 19, 26, 33] 27 [21, 27, 30, 30] [17, 23, 27, 31] [21, 27, 30, 30] [15, 23, 28, 30] 28 [16, 19, 22, 31] [16, 23, 26, 30] [16, 23, 25, 30] [14, 19, 22, 29] 29 [16, 21, 29, 30] [17, 17, 27, 32] [16, 16, 22, 27] [14, 16, 24, 28] 30 [21, 27, 30, 30] [18, 26, 32, 32] [21, 27, 30, 30] [16, 26, 30, 30] 31 [16, 22, 30, 30] [19, 22, 30, 30] [16, 21, 27, 28] [19, 22, 30, 30] 32 [16, 24, 29, 29] [16, 19, 24, 28] [16, 17, 22, 24] [14, 21, 27, 29] 33 [21, 27, 30, 30] [17, 24, 27, 28] [21, 27, 30, 30] [15, 22, 25, 26] 34 [16, 17, 23, 23] [16, 19, 26, 29] [16, 21, 26, 27] [14, 15, 22, 22] 35 [16, 20, 25, 31] [16, 19, 27, 29] [16, 21, 24, 27] [14, 15, 22, 28] 36 [21, 21, 27, 27] [20, 20, 27, 27] [21, 21, 27, 27] [20, 20, 27, 27] 37 [16, 21, 23, 27] [17, 20, 26, 32] [16, 19, 23, 29] [14, 21, 26, 30] 38 [16, 16, 21, 24] [19, 19, 26, 27] [16, 16, 19, 24] [19, 19, 26, 27] 39 [21, 21, 27, 27] [17, 17, 24, 24] [21, 21, 27, 27] [15, 15, 22, 22] 40 [16, 21, 29, 29] [16, 16, 25, 25] [16, 16, 25, 25] [14, 16, 24, 25] 41 [16, 21, 29, 30] [16, 16, 24, 30] [16, 18, 26, 29] [14, 16, 24, 26] 42 [21, 27, 30, 30] [17, 23, 27, 31] [21, 27, 30, 30] [15, 23, 28, 30] 43 [16, 22, 30, 30] [19, 22, 30, 30] [16, 21, 27, 28] [19, 22, 30, 30] 44 [16, 24, 29, 30] [17, 20, 27, 32] [16, 19, 26, 29] [14, 21, 27, 30] 45 [16, 19, 24, 31] [19, 19, 26, 33] [16, 18, 23, 28] [19, 19, 26, 33] 46 [16, 17, 23, 26] [16, 19, 26, 27] [16, 21, 24, 28] [14, 15, 22, 24] 47 [16, 17, 22, 28] [16, 19, 27, 29] [16, 21, 24, 26] [14, 15, 22, 28] 48 [16, 16, 21, 22] [16, 16, 23, 29] [16, 18, 23, 26] [14, 14, 21, 22] 49 [16, 21, 23, 27] [16, 19, 23, 30] [16, 17, 19, 26] [14, 21, 26, 28] 50 [16, 22, 30, 30] [19, 22, 30, 30] [16, 21, 27, 28] [19, 22, 30, 30] 51 [16, 21, 29, 30] [17, 17, 27, 32] [16, 16, 22, 27] [14, 16, 24, 28] 52 [16, 21, 29, 29] [16, 16, 25, 25] [16, 16, 25, 25] [14, 16, 24, 25] 53 [16, 21, 29, 29] [16, 16, 24, 24] [16, 16, 22, 22] [14, 16, 24, 24] 54 [16, 24, 29, 29] [16, 19, 24, 28] [16, 17, 22, 24] [14, 21, 27, 29] 55 [16, 16, 21, 24] [19, 19, 26, 27] [16, 16, 19, 22] [19, 19, 26, 27] 56 [16, 16, 22, 23] [16, 16, 23, 29] [16, 18, 23, 26] [14, 14, 21, 22] 57 [16, 16, 21, 21] [19, 19, 26, 26] [16, 16, 19, 19] [19, 19, 26, 26] 58 [16, 17, 23, 26] [16, 19, 26, 27] [16, 21, 24, 28] [14, 15, 22, 24] 59 [16, 21, 23, 27] [17, 20, 26, 32] [16, 19, 23, 29] [14, 21, 26, 30] 60 [16, 21, 29, 30] [16, 16, 24, 30] [16, 18, 26, 29] [14, 16, 24, 26] 61 [16, 17, 22, 25] [16, 19, 26, 27] [16, 21, 24, 28] [14, 15, 22, 23] 62 [16, 22, 30, 30] [19, 22, 30, 30] [16, 21, 27, 28] [19, 22, 30, 30] 63 [16, 21, 29, 30] [16, 16, 24, 30] [16, 18, 26, 29] [14, 16, 24, 26] 64 [16, 21, 29, 29] [16, 16, 25, 25] [16, 16, 25, 25] [14, 16, 24, 24] 65 [16, 17, 22, 22] [16, 19, 26, 26] [16, 21, 24, 25] [14, 15, 22, 23] 66 [16, 16, 22, 23] [17, 17, 26, 31] [16, 16, 19, 22] [14, 14, 21, 24] 67 [16, 16, 21, 24] [19, 19, 26, 27] [16, 16, 19, 22] [19, 19, 26, 27] 68 [16, 16, 22, 25] [16, 16, 23, 24] [16, 16, 19, 22] [14, 14, 21, 23] 69 [16, 16, 21, 21] [19, 19, 26, 26] [16, 16, 19, 20] [19, 19, 26, 26] 70 [16, 17, 23, 23] [16, 19, 26, 26] [16, 21, 24, 24] [14, 15, 22, 22] 71 [16, 16, 21, 22] [16, 16, 23, 29] [16, 18, 23, 26] [14, 14, 21, 22] 72 [16, 21, 29, 29] [16, 16, 24, 24] [16, 16, 22, 22] [14, 16, 24, 24] 73 [16, 17, 22, 25] [16, 19, 26, 27] [16, 21, 24, 28] [14, 15, 22, 23] 74 [16, 22, 30, 30] [19, 22, 30, 30] [16, 21, 27, 28] [19, 22, 30, 30] 75 [16, 21, 29, 29] [16, 16, 24, 24] [16, 16, 22, 22] [14, 16, 24, 24] 76 [16, 16, 21, 21] [19, 19, 26, 26] [16, 16, 19, 19] [19, 19, 26, 26] 77 [16, 17, 22, 25] [16, 19, 26, 27] [16, 21, 24, 28] [14, 15, 22, 23] 78 [16, 16, 22, 23] [16, 16, 23, 29] [16, 18, 23, 26] [14, 14, 21, 22] 79 [16, 16, 21, 21] [16, 16, 23, 23] [16, 16, 19, 19] [14, 14, 21, 21] 80 [16, 16, 22, 25] [16, 16, 23, 24] [16, 16, 19, 23] [14, 14, 21, 23] 81 [16, 22, 30, 30] [19, 22, 30, 30] [16, 21, 27, 28] [19, 22, 30, 30] 82 [16, 21, 29, 30] [16, 16, 24, 30] [16, 18, 26, 29] [14, 16, 24, 26] 83 [16, 16, 21, 24] [16, 16, 23, 24] [16, 16, 19, 22] [14, 14, 21, 22] 84 [16, 21, 29, 29] [16, 16, 24, 24] [16, 16, 22, 23] [14, 16, 24, 24] 85 [16, 21, 29, 30] [16, 16, 24, 30] [16, 18, 26, 29] [14, 16, 24, 26] 86 [16, 16, 21, 21] [19, 19, 26, 26] [16, 16, 19, 20] [19, 19, 26, 26] 87 [16, 16, 21, 21] [16, 16, 23, 23] [16, 16, 19, 19] [14, 14, 21, 22] 88 [16, 16, 21, 21] [19, 19, 26, 26] [16, 16, 19, 19] [19, 19, 26, 26] 89 [16, 17, 22, 25] [16, 19, 26, 27] [16, 21, 24, 28] [14, 15, 22, 23] 90 [16, 16, 22, 25] [16, 16, 23, 24] [16, 16, 19, 22] [14, 14, 21, 23] 91 [16, 21, 29, 29] [16, 16, 24, 24] [16, 16, 22, 22] [14, 16, 24, 24] 92 [16, 17, 23, 26] [16, 19, 26, 27] [16, 21, 24, 28] [14, 15, 22, 24] 93 [16, 22, 30, 30] [19, 22, 30, 30] [16, 21, 27, 27] [19, 22, 30, 30] 94 [16, 21, 29, 29] [16, 16, 24, 24] [16, 16, 22, 22] [14, 16, 24, 24] 95 [16, 16, 21, 24] [16, 16, 23, 24] [16, 16, 19, 23] [14, 14, 21, 22] 96 [16, 21, 29, 29] [16, 16, 24, 24] [16, 16, 22, 23] [14, 16, 24, 24] 97 [16, 16, 21, 22] [16, 16, 23, 29] [16, 18, 23, 26] [14, 14, 21, 22] 98 [16, 16, 21, 24] [19, 19, 26, 27] [16, 16, 19, 22] [19, 19, 26, 27] 99 [16, 16, 21, 24] [16, 16, 23, 24] [16, 16, 19, 22] [14, 14, 21, 22] 100 [16, 16, 21, 21] [19, 19, 26, 26] [16, 16, 19, 20] [19, 19, 26, 26]

Message: 3914 - Contents - Hide Contents Date: Wed, 20 Feb 2002 00:24:00 Subject: Re: magma From: paulerlich --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> may be of interest to some here: > Magma Computational Algebra System Home Page * [with cont.] (Wayb.)are you sure that's the correct link?

Message: 3915 - Contents - Hide Contents Date: Wed, 20 Feb 2002 07:02:16 Subject: 4296 From: genewardsmith Joe has a table for 4296 on one of his new web pages, and takes note of the fact that it is absurdly well in tune in the 5-limit. In fact, if you go out to 200000 it turns out to have the best log-flat badness score: 1 .7369655945 2 .7439736471 3 .4245472985 4 .6797000080 5 .8728704449 7 .6706891205 12 .5418300617 15 .8735997285 19 .5083949041 31 .8578063580 34 .6488389972 53 .4527427539 65 .7839445968 118 .4134357960 171 .6499654469 289 .9207676000 441 .6622791000 559 .9976240155 612 .5675982650 730 .6113208564 1171 .7597497149 1783 .5008376597 2513 .5396476355 4296 .2748910261 6809 .7979361504 8592 .7775092347 16572 .9822272987 20868 .6543929561 25164 .5742465687 52841 .8091483956 73709 .8117671040 78005 .3351525026 151714 .7011086648 156010 .9479544293 Joe credits it to Mark Jones, but it would be interesting to know if it is much older than that--it could have been discovered by hand in the same way 612 and 730 were. I found it about 25 years back cranking a poor, abused TI58 programmable calculator for days at a time doing searchs, where I got up as far as 20868 in the 5-limit, but this brute force method is not really needed in the case of the 5-limit, where various algorithms would turn it up. The MT reduced basis for 4296 is <2^-90 3^-15 5^49, 2^71 3^-99 5^37>. The first comma is the smallest one on my list of best 5-limit temperaments, and gvies us the map [[0, 49, 15], [1,-6,0]]. This divides the 5 into 15 parts, and if we tempered 71 or 84 notes by it, we would get a lot of essentially just ratios. If Mark has no objection, perhaps the "Jones" would be a good name for this temperament; the Jones generator being 665/4296, slightly short of satanic. If we take the Jones comma and wedge it with 2^161 3^(-84) 5^(-12) we get h4296 in the 5-limit; this comma I also took note of and it has been mentioned on the main tuning list--by whom initially, I don't know. We can regard 4296 as the intersection of these two absurdly accurate temperaments on the charts of Paul and Joe--if 71 appears on the chart, then the line from 71 to 84, passing through the origin, would be the Jones line, by the way. So far as performance goes, I wonder if it even makes sense in physical terms to claim to have done this. 4296 gives a fifth which is .0003064 cents sharp, and a third which is .0008647 cents flat. Obviously we can't hear the difference between this and just intonation, but can we even accomplish it? If the answer is yes, we can always go on and ask the same question of 78005, whose MT reduced basis, in case anyone cares, is <2^-573 3^237 5^85, 2^140 3^-374 5^195>

Message: 3916 - Contents - Hide Contents Date: Wed, 20 Feb 2002 17:12 +0 Subject: Re: [tuning] Re: 2401:2400 (was:: Marc Jones EDOs) From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <013301c1ba18$428553e0$af48620c@xxx.xxx.xxx> monz wrote:> thanks, Graham. i asked quite a while ago if that notation > was correct or misleading etc., and no-one ever answered me.I thought I replied and said it wasn't. But that was in the context of matrices.> so how about if i write [2 3 5 7]**[-5 -1 -2 4] ? > > i prefer to keep the prime series in the notation if possible.There are programming languages that define the usual arithmetic operators to be element-wise. Such as Fortran90, Numeric Python and I think Matlab. So in such a context your notation would almost work. The only thing missing is the implied product to get you from [1/32, 1/3, 1/25, 2401] to 2401/2400. You could also write it explicitly as a matrix product .[log(2) log(3) log(5) log(7)][-5] . [-1] . [-2] . [ 4] or a wedge product val([log(2) log(3) log(5) log(7)]) ^ [-5 -1 -2 4] or even log(val([2 3 5 7])) ^ [-5 -1 -2 4] if you decide to define the logarithm of a val. Note that these versions all define the relative pitch rather than ratio of the interval. The nearest I can with my Python library as it stands is>>> (... Wedgable(map(temper.log2, (2, 3, 5, 7))).complement() ... ^ (-5, -1, -2, 4) ... ).invariant()[0] 0.00060099773453625716 I could add shortcuts for converting ratios and wedgies to pitches if you want them. Maybe even get scalar(val(pitches(2, 3, 5, 7)) ^ (-5, -1, -2, 4)) to work. You could always define a magic function Interval([2 3 5 7], [-5 -1 -2 4]) as part of your terminology. Graham

Message: 3917 - Contents - Hide Contents Date: Wed, 20 Feb 2002 13:36:47 Subject: Re: [tuning] Re: 2401:2400 From: monz hi Graham,> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, February 20, 2002 9:12 AM > Subject: [tuning-math] Re: [tuning] Re: 2401:2400 (was:: Marc Jones EDOs) > In-Reply-To: <013301c1ba18$428553e0$af48620c@xxx.xxx.xxx> > monz wrote: >>> so how about if i write [2 3 5 7]**[-5 -1 -2 4] ? >> >> i prefer to keep the prime series in the notation if possible. >> There are programming languages that define the usual arithmetic operators > to be element-wise. Such as Fortran90, Numeric Python and I think > Matlab. So in such a context your notation would almost work. The only > thing missing is the implied product to get you from [1/32, 1/3, 1/25, > 2401] to 2401/2400.thanks for explaining that ... but ... i don't see how the "implied product" is "missing". can you explain further?> You could also write it explicitly as a matrix product > > .[log(2) log(3) log(5) log(7)][-5] > . [-1] > . [-2] > . [ 4] > > or a wedge product > > val([log(2) log(3) log(5) log(7)]) ^ [-5 -1 -2 4] > > or even > > log(val([2 3 5 7])) ^ [-5 -1 -2 4] > > if you decide to define the logarithm of a val. Note that these versions > all define the relative pitch rather than ratio of the interval.exactly ... what i'm looking for is a way to simply specify the r a t i o using the matrix notation.> You could always define a magic function > > Interval([2 3 5 7], [-5 -1 -2 4]) > > as part of your terminology.hmmm ... that might work. i'd like to keep using something of the form [2 3 5 7]**[-5 -1 -2 4] if i can. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3918 - Contents - Hide Contents Date: Wed, 20 Feb 2002 00:43:39 Subject: Re: magma From: paulerlich --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>>> may be of interest to some here: >>> Magma Computational Algebra System Home Page * [with cont.] (Wayb.) >>>> are you sure that's the correct link? > > Yes. -C.well, i just get "The page cannot be displayed" carl, why won't you answer us on the tuning list? we're asking about the cd you made for me of various a capella groups. could you discuss them please on tuning?

Message: 3919 - Contents - Hide Contents Date: Thu, 21 Feb 2002 21:51:13 Subject: proxomitron From: Carl Lumma>It's not true of Proxomitron, at least, since that is a program, not >a service:Clever; the program itself is a proxy server running locally on your own machine. Other ad blocking programs I've seen use a remote proxy server. Ad blocking software still doesn't address many of my concerns. -Carl

Message: 3920 - Contents - Hide Contents Date: Thu, 21 Feb 2002 01:01:30 Subject: [tuning] Re: 2401:2400 From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> hmmm ... that might work. i'd like to keep using something > of the form [2 3 5 7]**[-5 -1 -2 4] if i can.Of course the most obvious way to do that is 2^-5 3^-1 5^-2 7^4, but that does not emphasize the vector aspect, and is a little clumbersome. We could steal the idea of bra and ket vectors from the physicists, and write |-5 -1 -2 4> (a ket vector) for the intervals, and <12 19 28 34| (a bra vector) for vals. Then the bra-ket inner product becomes <12 19 28 34|-5 -1 -2 4> = 12*-5 + 19*-1 + 28*-2 + 34*4 = h(12, 2401/2400) = 1.

Message: 3921 - Contents - Hide Contents Date: Thu, 21 Feb 2002 02:45:26 Subject: Jacks and jumping jacks to the 13 limit From: genewardsmith I obtained the following list of jacks--superparticular ratios which are ratios of two 13-limit consonances, but which are not themsevles consonances for the prime limit they define: 169/168, 144/143, 121/120, 100/99, 99/98, 91/90, 81/80, 78/77, 66/65, 65/64, 64/63, 56/55, 55/54, 50/49, 49/48, 45/44, 40/39, 36/35, 33/32, 28/27, 27/26, 26/25, 25/24, 22/21, 21/20, 16/15, 15/14, 10/9, 9/8 Superparticular ratios of jacks now give us the following jumping jacks: 9801/9800, 4225/4224, 4096/4095, 3025/3024, 2401/2400, 2080/2079, 729/728, 676/675, 625/624, 540/539, 441/440, 385/384, 352/351, 351/350, 225/224, 176/175, 81/80, 64/63, 28/27 We might note the following: (1) Some superparticular commas of importance such as 126/125 and 243/242 don't make the list, but see (3) below. (2) A number of interesting temperaments (eg, miracle, orwell, 72-et, 31-et etc.) have a basis consisting of jumping jacks--in other words, they can be obtained as wedge products of jumping jacks. (3) 123201/123200 = (351/350)/(352/351), which I dissed for not being a jumping jack, turns out to be a triple jack--the superparticular ratio of two jumping jacks which is not itself a jumping jack. Others of that ilk include 126/125, 4375/4374, and 243/242.

Message: 3923 - Contents - Hide Contents Date: Thu, 21 Feb 2002 08:24:41 Subject: Re: 4296 From: genewardsmith --- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:> The octave, 4296 is sort of burned into the brain.The same thing happened to me with 612; I thought about using 4296 among other possibilities as a system for representing intervals but it seemed like overkill, and too 5-limit.> (Also the fourth, 1783 would come up a lot, obviously if 2513 and 4296 are > accurate enough to show up in algorithms, then their difference would.)This sort of thing seems to happen a lot for some reason.> See, the whole idea of doing it in cents screwed me up. Because eventually > you have to work in a couple decimal points places and even then if you add > or subtract or multiply things too much you still might be a number off.Right; that's why I switched to 612. Now that I have to deal with other people I switched back to cents again, though.> I ran the Brun-o-matic for about three days years ago doing up to I think > 13th limit convergences...It's possible to miss things using such algorithms; brute force is safer, actually.> I'll say it again. NICE to see someone mention 20868!!!Hey, and what about 78005? :)

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