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Message: 525 - Contents - Hide Contents Date: Wed, 18 Jul 2001 23:21:26 Subject: Re: Naming intervals using Miracle From: Dave Keenan --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:> I don't know about calling 10:11 a neutral second, I'd have to take a > good listen. In the Scala list I called it a 4/5-tone. It's not that > far from the 10/9 whole tone.It's closer to 11:12. Ok. How about this argument. We agree that 11:12 is a (unmodified) neutral second and that 8:9 is a (unmodified) major second. Although you avoid the issue for major seconds by calling 9:10 a minor whole tone and 8:9 a major whole tone (which is fine), your ninths make it clear how they should be named as seconds. You have: 4:9 "major ninth" 9:20 "small ninth" (= narrow major ninth) So doesn't it have to be: 11:12 neutral second 10:11 wide neutral second 9:10 narrow major second 8:9 major second? If 10:11 is to be any kind of major second, then it would have to be a "very narrow major second" or some such. You call it Ptolemy's second (which is fine). Can you point me to any scale it is used in, where one can say that it definitely functions as a major second and not a neutral second. Regards, -- Dave Keenan
Message: 526 - Contents - Hide Contents Date: Thu, 19 Jul 2001 00:54:38 Subject: Re: Temperament catalog From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9j36rh+ipou@e...> > Dave Keenan wrote:>> You haven't told us what sort of interval the generator is, for some >> of them ... >> The mapping's enough to define the temperament.I totally agree, but that doesn't mean you shouldn't be kind to those without math degrees and give at least the interval-class (say from the Fokker/Miracle naming scheme) of the generator. It's not simple to extract the generator unless there's a 1 or -1 in the prime mapping, and many people wouldn't even know to look for that.>> You could add to the Miracle entry, the fact that Secor proposed it as >> a way of making sense of Partch's 43 note gamut (i.e. not out of a >> musical vacuum.) >> Too much information for a brief list.If you've got room to mention Arabic and Indian etc under Schismic, Diaschismic and Neutral Thirds, you've got room to mention Partch under Miracle. All I'm asking for is a phrase inserted, such as "Discovered by George Secor 1975 as an efficient mapping of Partch's 43 note gamut, rediscovered by ...">> Wilson's 13-limit mapping is: >> (-1, 8, 14, -23, -20), perfect fourth generator (about 498 c), octave >> period, 41 and 53-EDO. >> I can't see this one. Does his layout contradict his numbers?Maybe so, or I may be wrongly attributing this temperament to Wilson, but see page 7 of 404 Not Found * [with cont.] Search for http://www.anaphoria.com/tres.pdf in Wayback Machine Oops, that's labelled "Casandra". I'll have to have to study this layout again when I have time. Either way, (-1, 8, 14, -23, -20) does exist. But it may or may not be interesting. However it was wrong of me (and Wilson?) to say it was covered by 41 and 53-EDO. 53-EDO is not 13-limit consistent. It should be 41 and 94-EDO. -- Dave Keenan
Message: 527 - Contents - Hide Contents Date: Thu, 19 Jul 2001 20:39:35 Subject: Re: Temperament catalog From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:>> As for omissions -- How about BP, for one? >> Either a JI scale or an ET, but not a linear temperament, unless you know > better.Perhaps no one has viewed it as such yet, but what if you take the generator of the 9-tone BP diatonic scale, and call that the generator of the LT?
Message: 528 - Contents - Hide Contents Date: Thu, 19 Jul 2001 20:43:27 Subject: Re: lattices of Schoenberg's rational implications From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > Could anyone out there do some periodicity-block > calculations on this theory and say something about that?It's pretty clear that Schoenberg's theory implies a 12-tone periodicity block.
Message: 530 - Contents - Hide Contents Date: Fri, 20 Jul 2001 06:15:25 Subject: Re: Temperament catalog From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., graham@m... wrote: >>>> As for omissions -- How about BP, for one? >>>> Either a JI scale or an ET, but not a linear temperament, unless > you know >> better. >> Perhaps no one has viewed it as such yet, but what if you take the > generator of the 9-tone BP diatonic scale, and call that the > generator of the LT?So what are the generator, period, mapping from primes to generators and periods, example ED3s. i.e. Please write Grahams catalog entry for it.
Message: 531 - Contents - Hide Contents Date: Fri, 20 Jul 2001 07:26:17 Subject: Re: Temperament catalog From: Dave Keenan Graham, I've gone over 404 Not Found * [with cont.] Search for http://www.anaphoria.com/tres.pdf in Wayback Machine pages 6 & 7 again, and can confirm that you've got the name "Cassandra" on the wrong one in your Catalog, and that Cassandra (p7) (-1, 8, 14, -23, -20) (497.9c) covers 41 and 94-EDO, but not 53. The other unnamed one (p6) (-1, 8, 14, 18) (497.4c) is only given as an 11-limit mapping on that page, although Wilson may have used the full (-1, 8, 14, 18, 21) (497.2 c) elsewhere. -- Dave Keenan
Message: 533 - Contents - Hide Contents Date: Fri, 20 Jul 2001 09:57 +0 Subject: Re: Temperament catalog From: graham@m... In-Reply-To: <9j8mep+v7v2@e...> Dave Keenan wrote:> I've gone over 404 Not Found * [with cont.] Search for http://www.anaphoria.com/tres.pdf in Wayback Machine > pages 6 & 7 again, and can confirm that you've got the name > "Cassandra" on the wrong one in your Catalog, and that Cassandra (p7) > (-1, 8, 14, -23, -20) (497.9c) covers 41 and 94-EDO, but not 53.I said at the top I'm not bothered about consistency. This scale comes out if you plug 41 and 53 into my temperament algorithm.> The other unnamed one (p6) (-1, 8, 14, 18) (497.4c) is only given as > an 11-limit mapping on that page, although Wilson may have used the > full (-1, 8, 14, 18, 21) (497.2 c) elsewhere.Ah! Now I see my problem! Page 8 gives alternative mappings for 11 and 13, but I was only looking at the bottom and ignored the dotted lines. Graham
Message: 534 - Contents - Hide Contents Date: Fri, 20 Jul 2001 09:57 +0 Subject: Re: Temperament catalog From: graham@m... In-Reply-To: <006a01c110df$bb6cb6a0$f45ed63f@s...> Dan Stearns wrote:> I've posted a bunch on this.Dan, I know you've got a lot of similar ideas to me, but you don't present them in a digestible way! If you send a short writeup, I'll add it to the catalog. I also know that you and Mats have both used a lot of MOS scales. I'm going to have to leave them out unless they have some interesting just approximations. Otherwise I'll end up with the whole of the scale tree in there! Graham
Message: 536 - Contents - Hide Contents Date: Fri, 20 Jul 2001 07:58:19 Subject: Re: [tuning] Re: fractional exponents of prime-factors From: monz Hi Jay, You posted this question to the Tuning List, where I sent an abbreviated reply. But I feel it's more appropriate to send the full reply here (tuning-math, with a copy sent to justmusic).> From: J Gill <JGill99@i...> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Friday, July 20, 2001 5:06 AM > Subject: [tuning] Re: fractional exponents of prime-factors > > >>> Manuel wrote:>>> The consequence of this is that the same tone can show up in >>> multiple places in the lattice. That's the reason that the >>> III-V lattice of 12-tET gets warped into a torus shape. >> What is the meaning of such a periodicity (wrapping of the "III-V > lattice" (of 12-tones scales) lattice into a "torus" shape? > > Is this the same thing as a tone appearing in "multiple places" in a > lattice?Yes. Manuel is referring to the fact that in 1/4-comma meantone, the lattice-point representing the "major 3rd" must fall at the 5:4 position, but must also represent 4 tempered 3:2s. Thus, the lattice has to be twisted into a torus so that each 3:2 goes 1/4 of the way towards 5^1 while 5:4 goes towards 3^4. Can anyone provide the math for this? (see my formula below)> > How could one (algebraically) solve for "where" in a (non-integer > power) lattice the tone would appear (since multiple combinations of > primes to various powers could result in the same interval as a > result), where no algebraic process exists to do so. Iteration? > > Maybe I am misinterpreting or missing something here?Unfortunately, Jay, I myself don't understand algebra well enough to answer this one. I'm even more confused about it than you are. The Visual Basic code for my lattice formula is here Yahoo groups: /justmusic/message/48 * [with cont.] Basically, this is it: '----------- BEGIN CODE ------------ 'The apostrophe ' indicates a comment. ' DECLARE VARIABLES '------------------ p 'prime-factor ar(p) 'angle (in radians) of p exp(p) 'exponent of p (+ or -) op(x) 'horizontal origin-point op(y) 'vertical origin-point rp(x) 'horizontal ratio plot rp(y) 'vertical ratio plot 'CALCULATE ANGLE OF PRIME '------------------------- for p=2, 3, 5, 7, 11, 13 '...etc., the prime series ar(p)=((3-(LOG(p)*12/LOG(2)))*PI)/6 next p 'CALCULATE RATIO-PLOT '--------------------- rp(x)=op(x) rp(y)=op(y) for p=2, 3, 5, 7, 11, 13 '...etc., the prime series start-vector(p)=rp(x),rp(y) rp(x)=rp(x)+((cos(ar(p))*rp(x))*(exp_num(p)/exp_denom(p)))*p rp(y)=rp(x)+((sin(ar(p))*rp(y))*(exp_num(p)/exp_denom(p)))*p end-vector(p)=rp(x),rp(y) next p '--------------- END CODE ---------------- -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 537 - Contents - Hide Contents Date: Fri, 20 Jul 2001 16:30:28 Subject: Re: Temperament catalog From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> Ah! Now I see my problem! Page 8 gives alternative mappings for 11 and > 13, but I was only looking at the bottom and ignored the dotted lines.Ah! Now that you mention it. I was wrong too. Pages 9, 10 and 11 make it very clear that the name "Cassandra" refers to the tuning, and not to either of the keyboard mappings or their corresponding linear temperaments. He simply calls one "41-like" and the other "41- and 53-like". So I guess you should just call them "29 and 41" and "41 and 53". Your entries say "p6" and "pp7-8" but they don't say of what. Regards, -- Dave Keenan
Message: 538 - Contents - Hide Contents Date: Fri, 20 Jul 2001 13:52:28 Subject: Re: Lamothe Web Translation Update From: Pierre Lamothe Hi Monz, I don't find the words to express my gratitude for your offer. I was entering since few days in a reflexive process about the opportunity to communicate via a website. An important motivation for that is to join the active searchers on related fields and it seems that so few could read in French. Besides, I have well easiness and great pleasure in research but difficulty and pain in communication. So I have to be sure that there will be sufficiently interested readers before to make the necessary efforts to prepare documents. I will communicate with you privately about that. Thank you so much, Pierre
Message: 539 - Contents - Hide Contents Date: Sat, 21 Jul 2001 03:42:31 Subject: [tuning] Re: fractional exponents of prime-factors From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > Yes. Manuel is referring to the fact that in 1/4-comma meantone, > the lattice-point representing the "major 3rd" must fall at the > 5:4 position, but must also represent 4 tempered 3:2s. Thus, > the lattice has to be twisted into a torusNot a torus, but a cylinder. A torus is created if you further temper out an additional unison vector, creating a closed temperament.
Message: 540 - Contents - Hide Contents Date: Fri, 20 Jul 2001 22:59:42 Subject: Re: [tuning] Re: fractional exponents of prime-factors From: monz> From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, July 20, 2001 8:42 PM > Subject: [tuning-math] [tuning] Re: fractional exponents of prime-factors > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> Yes. Manuel is referring to the fact that in 1/4-comma meantone, >> the lattice-point representing the "major 3rd" must fall at the >> 5:4 position, but must also represent 4 tempered 3:2s. Thus, >> the lattice has to be twisted into a torus >> Not a torus, but a cylinder. A torus is created if you further temper > out an additional unison vector, creating a closed temperament.Oops... my bad! Thanks for catching that, Paul. I knew that, but got my shapes mixed up with the wrong types of temperaments. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 541 - Contents - Hide Contents Date: Mon, 23 Jul 2001 09:16 +0 Subject: Re: Temperament catalog From: graham@m... In-Reply-To: <9j9mb4+j8lb@e...> Dave Keenan wrote:> Ah! Now that you mention it. I was wrong too. Pages 9, 10 and 11 make > it very clear that the name "Cassandra" refers to the tuning, and not > to either of the keyboard mappings or their corresponding linear > temperaments.I've called them *both* Cassandra now.> He simply calls one "41-like" and the other "41- and 53-like". So I > guess you should just call them "29 and 41" and "41 and 53". Your > entries say "p6" and "pp7-8" but they don't say of what.Good proof reading! I'd formatted the links e-mail style instead of HTML-style, and so they weren't showing through at all. Graham
Message: 542 - Contents - Hide Contents Date: Mon, 23 Jul 2001 20:27:04 Subject: Re: Temperament catalog From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:>> Perhaps no one has viewed it as such yet, but what if you take the >> generator of the 9-tone BP diatonic scale, and call that the >> generator of the LT? >> So what are the generator, 5 > period, 3 > example ED3s.13, 88, 271 . . .
Message: 543 - Contents - Hide Contents Date: Tue, 24 Jul 2001 02:16:10 Subject: BP linear temperament (was: Temperament catalog) From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >>>> Perhaps no one has viewed it as such yet, but what if you take > the>>> generator of the 9-tone BP diatonic scale, and call that the >>> generator of the LT? >>>> So what are the generator, > > 5That can't be right. The generator has to be a supermajor third (approx 7:9) MA optimum around 440 cents (1/3-BP-comma wide). BP-comma is 243:245 = 3^5 : 5^1 * 7^2. So the mapping is Prime No. Generators ----- -------------- 5 2 7 -1 and possibly 2 7 (the optimum moves to 442 cents if you include this)>> period, > > 3Yes. As in approx 1902 cents.>> example ED3s. >> 13, 88, 271 . . .So these are wrong too. Only 13-ED3 really makes any sense. Ignoring ratios of two, the next better one is 160-ED3. If we include ratios of 2 the next better one is 43-ED3. So any ED3 whose cardinality is of the form 13n+4 where n is between 4 and 12, may be of interest. MOS cardinalities for the BP linear-temperament go (not strictly proper in paren.) 4 (5) (9) 13 (17) (30) .... -- Dave Keenan
Message: 544 - Contents - Hide Contents Date: Tue, 24 Jul 2001 02:51:38 Subject: Re: BP linear temperament (was: Temperament catalog) From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >>>>>> Perhaps no one has viewed it as such yet, but what if you take >> the>>>> generator of the 9-tone BP diatonic scale, and call that the >>>> generator of the LT? >>>>>> So what are the generator, >> >> 5 >> That can't be right. The generator has to be a supermajor third > (approx 7:9) MA optimum around 440 cents (1/3-BP-comma wide). BP- comma > is 243:245 = 3^5 : 5^1 * 7^2. So the mapping is > > Prime No. Generators > ----- -------------- > 5 2 > 7 -1I'm glad I got you to find the right answer (my excuse for not thinking about this myself is that I have had strep bacteria for two weeks, but was only informed of this today, and just started penicillin).> and possibly > 2 7 (the optimum moves to 442 cents if you include this)No, you should absolutely not include this. There are no factors of 2 to be approximated by the BP scale as it is understood by any of its creators.> >>> example ED3s. >>>> 13, 88, 271 . . . >> So these are wrong too.With respect to the 9-tone diatonic BP scale, I see what you mean. These are simply ED3s that approximate 2-less JI well. Good work!
Message: 545 - Contents - Hide Contents Date: Tue, 24 Jul 2001 03:18:26 Subject: Hey Carl From: Paul Erlich Hey Carl, just wanted to let you know that my "Hypothesis" is dedicated to you. You asked what a 2D and higher-dimensional generalization of an MOS might be. There was a lot of talk on the Tuning list about trivalent scales (scales where each generic interval has exactly three specific step sizes) for a while but those seem too rarefied to be the "right" answer. I believe we're on the right track with the Hypothesis. Namely: periodicity block with 1 unison vector not tempered out --> MOS periodicity block with 2 unison vectors not tempered out --> 2DGMOS periodicity block with 3 unison vectors not tempered out --> 3DGMOS etc. Examples in the second category would include the JI major scale, where neither of the 2 unison vectors are tempered out; and Dave Keenan's 31-tone 11-limit planar microtemperament, where 2 of the 4 unison vectors are tempered out.
Message: 548 - Contents - Hide Contents Date: Tue, 24 Jul 2001 11:16:03 Subject: Monzo trip to Europe From: monz Hello all, Sorry about all the cross-posting, but I want to reach as many tuning folk as possible. I will be in Europe from September 4 to 23, and would love to meet with as many of you as possible. I already have plans for some specific meetings, but would welcome the opportunity to see as many of you as I can fit into my itinerary. The centerpiece of the trip is the ISMA (International Symposium on Musical Acoustics) 2001 Conference in Perugia, Italy, for which I am an Invited Speaker. My lecture on "John Dowland's Lute: An Early Well-Temperament" will be presented in the Special Demo Session after the Workshop (Gestural Interfaces and Control of Expressiveness in Microtonality) on Thursday, September 13. Wim Hoogewerf will be joining me in the presentation, as he will perform some pieces by Dowland in Dowland's preferred tuning, on his Vogt guitar with moveable frets. My presentation will be an expansion of my Dowland webpage: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) The funds and time I have available for this trip are both extremely limited, and I have had to make my visit shorter and much less extensive than I had originally hoped. I will be flying into Paris on September 5 and spending the first few days there. The Conference will be the entire week of September 10-15, and I hope to be able to visit Rome and Florence either before or after, since they are both quite close to Perugia. Then after the Conference I will be visiting Amsterdam and Den Haag in the Netherlands, where Manuel Op de Coul, Robert Walker (and possibly also Graham Breed) and I will meet, primarily to focus on development of the JustMusic software. Lastly, I will spend a few more days in Paris, from where I leave on September 23. If anyone living in Europe can provide me with suggestions, advice, or assistance regarding inexpensive hotels or other places to stay, and regarding transportation, it would be most appreciated. I am particularly interested in being able to visit some other places outside the cities I have indicated, if it is possible to travel to them. Please respond privately. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 549 - Contents - Hide Contents Date: Tue, 24 Jul 2001 20:10:18 Subject: Re: BP linear temperament (was: Temperament catalog) From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Paul Erlich wrote, > > <<No, you should absolutely not include this. There are no factors of > 2 to be approximated by the BP scale as it is understood by any of its > creators.>> > > I really don't know why I bother, but whatever... as I mentioned > before I had a brief off-list correspondence with Heinz Bohlen that > would seem to contradict the severity of Paul's -- "absolutely not" -- > view, see: > > <Yahoo groups: /tuning/message/25735 * [with cont.] >Perhaps my view was too severe, but it definitely seems to contradict the _spirit_ of the near-just approximations to all simple ratios of odd numbers, since the approximations to simple ratios involving even numbers are so poor.> > And while I'm at it... here's another previous post mentioning the > linear temperament that 'no one's thought of BP as yet': > > <Yahoo groups: /tuning/message/20880 * [with cont.] >You thought of it first! Why didn't you say so?
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