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Message: 6150 - Contents - Hide Contents Date: Tue, 21 Jan 2003 23:51:35 Subject: Re: Superparticular temperaments From: Carl Lumma>> > thought something had cast doubt on the poptimal stuff. >> And didn't Paul say the entire p range tends to minimax? >> didn't i say what?That as p goes to infinity you get minimax. -Carl
Message: 6151 - Contents - Hide Contents Date: Tue, 21 Jan 2003 23:54:52 Subject: Re: Superparticular temperaments From: Carl Lumma>> >ook ye above. >> yes, it's the exact opposite of what you thought -- if there's > only one comma, you're not missing or ignoring anything, because > straightness doesn't even exist (let alone come into play). OK.>>>> Adaptive JI. It should go without saying with me. I >>>> didn't even know about strict JI until joining this list, >>>> and it took 2 years of confusion before we figured it out!! >>>>> you're kidding me, right? >> >> No, why would you say that? >>you didn't refer to pitches as ratios?Nope. Well, I had seed scales in ratios, but I never even thought of the idea of a global pitch set.>i'm also not interested in the more accurate temperaments. >they're mathematical curiosities. what interests me are >temperaments where a "small-numbered" MOS already contains >a good deal of the desired harmonies, and you can develop >a grammar around the harmonic meaning of scalar alterations, >etc. Exactly! -Carl
Message: 6152 - Contents - Hide Contents Date: Wed, 22 Jan 2003 00:01:47 Subject: Re: heuristic and straightness From: Carl Lumma>> >o is this right: >> >> Straightness...LengthUVs...Length+/-UV...Badness >> Down...........Up..........Down..........Same >> Up.............Down........Up............Same >> >> ? >> if you replace "badness" with "error", it's right.I should have noted that this was for a given temperament, not for all temperaments, though I take it you took it that way. So what should the badness column be? -Carl
Message: 6154 - Contents - Hide Contents Date: Wed, 22 Jan 2003 02:33:07 Subject: Re: A 13-limit comma list From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> i'm mystified as to why comma size would either be a convenient, or a > useful, way to set a bound on the search . . .A comma always introduces its equivalences and sets a limit to accuracy. I don't see much merit in commas which are larger than the smallest consonance of the system, at any rate--do you? One of my "top" systems has a TM basis of [4/3, 7/5, 8/5], which I think is a little over the top for top. The cutoff I suggested would at least exclude commas larger than (n+1)/n for odd limit n--151 cents in the 11-limit. Do we need to go that high?
Message: 6155 - Contents - Hide Contents Date: Wed, 22 Jan 2003 22:31:32 Subject: Re: Superparticular temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> I thought something had cast doubt on the poptimal stuff. >>> And didn't Paul say the entire p range tends to minimax? >>>> didn't i say what? >> That as p goes to infinity you get minimax.well, yes, that's pretty well-known, so i don't think i even bothered saying it explicitly. but "as p goes to infinity you get minimax" sure sounds very different to me than "the entire p range tends to minimax" !
Message: 6156 - Contents - Hide Contents Date: Wed, 22 Jan 2003 03:04:19 Subject: The Dirty Dozen From: Gene Ward Smith Here are the 11-limit commas smaller than 12/11, larger than 50 cents, and with epimericity less than 0.35: [27/25, 15/14, 16/15, 35/33, 81/77, 21/20, 22/21, 25/24, 126/121, 80/77,28/27, 125/121, 33/32]
Message: 6157 - Contents - Hide Contents Date: Wed, 22 Jan 2003 22:32:55 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> So is this right: >>> >>> Straightness...LengthUVs...Length+/-UV...Badness >>> Down...........Up..........Down..........Same >>> Up.............Down........Up............Same >>> >>> ? >>>> if you replace "badness" with "error", it's right. >> I should have noted that this was for a given temperament, > not for all temperaments, though I take it you took it > that way.yes, because of the last column being all "same".> So what should the badness column be?well, i guess that's all "same" too!
Message: 6158 - Contents - Hide Contents Date: Wed, 22 Jan 2003 03:19:30 Subject: Re: Calculating geometric complexity and badness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> probably gene's typos . . .Here are my Maple routines: qw11 := proc(q) # 11-limit weighted quadratic form local a,b,c,d; a := ordp(q,3); b := ordp(q,5); c := ordp(q,7); d := ordp(q,11); ln(3)^2*a^2+ln(5)^2*b^2+ln(7)^2*c^2+ln(11)^2*d^2+ ln(3)^2*a*b+ln(3)^2*a*c+ln(3)^2*a*d+ln(5)^2*b*c+ln(5)^2*b*d+ln(7)^2*c*d end: gc11 := proc(l) # complexity of 11-limit linear wedgie sqrt(37.4669556799257*l[1]^2+19.8578149236787*l[2]^2+13.8576504872655*l[3]^2+ 8.43459044261178*l[4]^2-9.24811506924099*l[2]*l[3]- 12.9637247061659*l[1]*l[2]-6.02856647730047*l[1]*l[3]- 6.41005295068753*l[3]*l[4]-2.95953128437572*l[1]*l[4]- 4.54006536578504*l[2]*l[4]) end: gpc11 := proc(l) # complexity of 11-limit planar wedgie sqrt(18.1878630999135*l[1]^2 +13.2165141603747*l[2]^2 +8.13090525300357*l[3]^2 +6.57565648053141*l[4]^2 +4.20601082222387*l[5]^2 +2.76216685071556*l[6]^2 +9.98976207451545*l[1]*l[2] -4.90415316714511*l[1]*l[3] +4.65474179366984*l[1]*l[4] -2.28509613536279*l[1]*l[5] -.721921985754273*l[1]*l[6] +6.45350417171744*l[2]*l[3] +5.37666377942441*l[2]*l[4] +1.44384397150855*l[2]*l[5] -1.56317414960855*l[2]*l[6] +3.00701812111700*l[3]*l[5] -1.56317414960855*l[3]*l[6] +3.84182937372243*l[4]*l[5] +2.39798540221390*l[4]*l[6] +2.39798540221390*l[5]*l[6]) end:
Message: 6159 - Contents - Hide Contents Date: Wed, 22 Jan 2003 22:39:56 Subject: Re: A 13-limit comma list From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>>> i'm mystified as to why comma size would either be a convenient, or a >> useful, way to set a bound on the search . . . >> A comma always introduces its equivalences ? > and sets a limit to >accuracy.lots of things set a limit to accuracy.> I don't see much merit in commas which are larger than the >smallest consonance of the system, at any rate--do you?i could rephrase this in terms of the complexity and/or error (possibly by way of the heuristic), and it would make for a more convincing "merit" criterion. i'm just wondering why, when doing the search, you'd bother calculating the JI size of the commas at all. btw, did you see my request to revisit the "ultimate 5-limit" list using the heuristic? perhaps you could guide me as to a good method to program the search, if you don't want to do it yourself. i think i'd like the "funky" ones all in there, so i can show them on the "q" equal temperament graph, the one that includes two instances of 2- equal.
Message: 6160 - Contents - Hide Contents Date: Wed, 22 Jan 2003 18:48:15 Subject: Re: A common notation for JI and ETs From: David C Keenan I suggest it's time to settle the issue of what single ASCII characters we recommend to stand for the more common single-shaft saggital symbols. I describe certain sagittal symbols below by giving the multi-ASCII-character approximation we've been using. But if you want to get involved in this, you should check out the real symbols in Yahoo groups: /tuning-math/files/Dave/SymbolsB... * [with cont.] or Yahoo groups: /tuning-math/files/secor/notatio... * [with cont.] or Yahoo groups: /tuning-math/files/secor/notatio... * [with cont.] some of which are a little out of date in regard to what symbols are in and what are out, but are fine for seeing the symbols mentioned in this message. I suggest we start with the ASCII sets used by Manuel in Scala, as far as they go, except that I think we should use < and > as an alternative to L and 7 for the 7-comma, to avoid confusion with 7 as an octave number or 7th chord, and to agree with the ASCII-Sims notation. And we should not use L or 7 for anything so at least we don't conflict with Scala in this regard. Note that Scala uses < and > as single step symbols for some ETs in Rapoport notations, where they are considered to represent a 1/n fraction of the 5-comma 80:81 for some small n. We have no need to symbolise such things. Some time ago I suggested that Manuel use -/ and \- for these in the same way he has used -) and (- for diesis fractions, but he understandably doesn't like to change established usages in Scala and likes < and > for this purpose because of their resemblance to the filled triangles that Rapoport proposed (See Xenharmonikon 16, 1995). However I'm prepared to ride roughshod over Rapoport's symbols since they appear to have been chosen almost at random, as opposed to the systematic symbol structure of sagittal. Unfortunately Scala has multiple uses for v and ^. When notating ETs in Rapoport's notation Scala uses them for the diaschisma 2025:2048. When notating JI it uses them for the 11-diesis 32:33 and when notating 72-ET and 144-ET in Richter Herf notation it uses them for the quartertone. These latter two usages are compatible with each other but not with the former. We must choose the JI/Richter-Herf usage due to the strong graphical resemblance to the relevant sagittal symbol /|\, sans shaft. Manuel uses v and ^ for the diaschisma due to their resemblance to those proposed by Rapoport. I pointed out that they better resemble Rapoport's Pythagorean comma symbols (which are however not used for notating ETs) and that u and n look more like Rapoport's diaschisma symbols. But I now want to use u and n for something else since the diaschisma symbol is rarely (if ever) used in sagittal notation. We should avoid conflict with the ASCII Sims notation too, except in the one area where it is unavoidable. The characters v and ^ must stand for the 11-diesis 32:33 in sagittal, as in Scala and in Monz's notation, not the 5-comma 80:81 or 1/12 tone as in the ASCII Sims. However, we might use [ and ] for the 11'-diesis 704:729 without too much of a problem since this also corresponds to 3deg72, although it is not used in the _standard_ sagittal notation for 72-ET. We should also avoid conflict with Monz's and Johnston's notations by not using + or - for anything. So here's what this gives us so far. /||\ # apotome sharp 2048:2187 \!!/ b apotome flat /| / 5-comma sharp 80:81 \! \ 5-comma flat |) > 7-comma sharp 63:64 !) < 7-comma flat /|\ ^ 11-diesis sharp 32:33 \!/ v 11-diesis flat (|) ] 11'-diesis sharp 704:729 (!) [ 11'-diesis flat Here's what Scala has for some higher prime commas. Note that each symbol can represent two different commas depending whether one is using the system JI1 or JI2. Note that half of these are two-characters and the other half do not really look like up and down pairs. I'm inclined to suggest some different symbol pairs, and simply to avoid clashing with these by not using any of those symbols for anything. | 27/26 1053/1024 ; 26/27 1024/1053 #' 17/16 2187/2176 b' 16/17 2176/2187 % 19/18 513/512 d 18/19 512/513 @ 24/23 736/729 * 23/24 729/736 #! 29/27 261/256 b! 27/29 256/261 |' 32/31 248/243 ;' 31/32 243/248 I suggest the following for our 13-diesis symbols /|) n 13-diesis sharp 1024:1053 \!) u 13-diesis flat (|\ } 13'-diesis sharp 26:27 (!/ { 13'-diesis flat Notice the correspondence with the 11' dieses. In both cases the smaller diesis is given by an up-down pair of (mostly) lowercase letters and the larger by a left-right pair of brackets. In the 11 case the up down pair v^ have two straight sides like the sagittal symbol (but no shaft) and in the 13 case the up down pair have a straight side and a curved side un (but not on the correct sides in the case of the u, and no shaft). With the brackets [] and curly braces {}, the resemblance is more to the number 11 (underlined and turned sideways) and the 3 of 13, than it is to the actual sagittal symbols. So we have a bit of a system here with the 5, 11 and 13 commas resembling their sagittal symbols with the omission of the shaft. We cant do that for the 7-comma using ( and ) because once the shaft is gone these could equally well be the 7:11, 19 or 5:7 comma symbols (see below) in various states of up or down. Best not to use ( or ) at all in the single-ASCII version of sagittal, unless they're for the same use that Manuel makes of them. ) 125-diesis sharp 125:128 ( 125-diesis flat The 7-comma symbols < and > are like the smaller 11 and 13 symbols in being a kind of bracket that resembles the appropriate prime number. Any comments/objections to the above? There are no more bracket pairs available. The only ASCII character pairs I can suggest for use as further common sagittals are the following down-up pairs, in my order of preference: jf yh wm dq o* &% J? Can anyone think of any others that don't clash with existing uses? I suggest we not have a single character for the 25'-diesis 6400:6561 //| but simply use \\ and //. Likewise << and >> for the |)) 49'-diesis 3969:4096. The next most common sagittals, and therefore those most deserving of a single character, are, in order of popularity: |( 5:7 comma 5103:5120 ~5.758c 11:13 comma 351:352 ~4.925c 7:25 comma (|( 5:11 comma 44:45 ~38.906c 7:13 comma 1664:1701 ~38.073c 11:17 comma 1377:1408 ~38.543c (| 7:11 comma 45056:45927 ~33.148c 13:17 comma 51:52 ~33.617c 29 comma 256:261 ~33.487c ~| 17 comma 2176:2187, ~8.730c ~|( 17' comma 4096:4131 ~14.730c Can anyone suggest a convincing way to map single ASCII character pairs to these? I'd be happy to get this far, since it would see 217-ET notated along with the 15-limit diamond and the first 17 odd harmonics. But if you were keen, you might push on to single ascii characters for: )| 19 comma 512:513 ~3.378c )|~ 19' comma 19456:19683 ~20.082c 19+23 comma* 432:437 ~19.922c |~ 23 comma 729:736 ~16.544c
Message: 6161 - Contents - Hide Contents Date: Wed, 22 Jan 2003 23:50:53 Subject: Re: Superparticular temperaments From: Carl Lumma> "as p goes to infinity you get minimax" > > sure sounds very different to me than > > "the entire p range tends to minimax" > > !Yeah, sorry 'bout that. Just in a hurry, or half asleep. -Carl
Message: 6162 - Contents - Hide Contents Date: Wed, 22 Jan 2003 12:11:35 Subject: Re: A common notation for JI and ETs From: manuel.op.de.coul@xxxxxxxxxxx.xxx Dave gave an accurate summary except for this:>Unfortunately Scala has multiple uses for v and ^. When notating ETs in >Rapoport's notation Scala uses them for the diaschisma 2025:2048. When >notating JI it uses them for the 11-diesis 32:33 and when notating 72-ET >and 144-ET in Richter Herf notation it uses them for the quartertone. These >latter two usages are compatible with each other but not with the former.The symbols for 32:33 in JI were changed some time ago to [ and ], which are also used in ET notations. I kept v and ^ for Richter Herf since this notation system stands on its own and they resemble the up and down arrows more.>We must choose the JI/Richter-Herf usage due to the strong graphical >resemblance to the relevant sagittal symbol /|\, sans shaft.No reason to change this as far as I'm concerned but of course you can if you want to avoid confusion over [ and ]. The graphical translation in Scala looks about the same. Press Ctl+F7 to see the graphical notation, it's a recently added dialog window. Manuel
Message: 6163 - Contents - Hide Contents Date: Wed, 22 Jan 2003 23:55:41 Subject: Re: heuristic and straightness From: Carl Lumma>>>> >traightness...LengthUVs...Length+/-UV...Badness >>>> Down...........Up..........Down..........Same >>>> Up.............Down........Up............Same >>>> >>>> ? >>>>>> if you replace "badness" with "error", it's right. >>>> I should have noted that this was for a given temperament, >> not for all temperaments, though I take it you took it >> that way. >> yes, because of the last column being all "same". >>> So what should the badness column be? >> well, i guess that's all "same" too!So if error is the same and badness is the same, then complexity is the same (which I suppose makes since, if the volume of the block is not to change). So is it safe to conclude that straightness is important for heuristically searching temperaments, but not for choosing a commatic basis for a given temperament? -Carl
Message: 6164 - Contents - Hide Contents Date: Wed, 22 Jan 2003 10:00:07 Subject: Re: A common notation for JI and ETs From: David C Keenan Here's a spreadsheet that calculates all the reasonable notational commas for the ratios in the popularity list. Yahoo groups: /tuning-math/files/Dave/Notation... * [with cont.] -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 6165 - Contents - Hide Contents Date: Thu, 23 Jan 2003 20:14:37 Subject: algebra question From: wallyesterpaulrus an eminent music theorist wrote me in an e-mail: "Paul, this is really basic algebra, and this time I suggest that you consult a mathematician or a textbook. In any event, once you have created equivalence classes, you can only refer to a given class *as a whole*--not to its individual members." can gene confirm or deny?
Message: 6166 - Contents - Hide Contents Date: Thu, 23 Jan 2003 00:19:57 Subject: Re: A 13-limit comma list From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> i'm just wondering why, when doing the search, you'd bother > calculating the JI size of the commas at all.I don't. I use them to generate wedgies.> btw, did you see my request to revisit the "ultimate 5-limit" list > using the heuristic? perhaps you could guide me as to a good method > to program the search, if you don't want to do it yourself.After getting a comma list it is easy, so are you asking how to get an initial list?
Message: 6167 - Contents - Hide Contents Date: Thu, 23 Jan 2003 01:27:31 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>>> Straightness...LengthUVs...Length+/-UV...Badness >>>>> Down...........Up..........Down..........Same >>>>> Up.............Down........Up............Same >>>>> >>>>> ? >>>>>>>> if you replace "badness" with "error", it's right. >>>>>> I should have noted that this was for a given temperament, >>> not for all temperaments, though I take it you took it >>> that way. >>>> yes, because of the last column being all "same". >>>>> So what should the badness column be? >>>> well, i guess that's all "same" too! >> So if error is the same and badness is the same, then > complexity is the same (which I suppose makes since, > if the volume of the block is not to change). So is it > safe to conclude that straightness is important for > heuristically searching temperaments,i think so.> but not for > choosing a commatic basis for a given temperament?the commatic basis with the shortest uvs will generally be the straightest one.
Message: 6168 - Contents - Hide Contents Date: Thu, 23 Jan 2003 01:29:04 Subject: Re: A 13-limit comma list From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>>> i'm just wondering why, when doing the search, you'd bother >> calculating the JI size of the commas at all. >> I don't. I use them to generate wedgies.so why does the size of the comma come it at all?>> btw, did you see my request to revisit the "ultimate 5-limit" list >> using the heuristic? perhaps you could guide me as to a good method >> to program the search, if you don't want to do it yourself. >> After getting a comma list it is easy,it is? how would i be sure that there isn't some comma with lower heuristic badness than the one in your list with highest heuristic badness?> so are you asking how to get an initial list? that too.
Message: 6169 - Contents - Hide Contents Date: Fri, 24 Jan 2003 15:24:58 Subject: Re: A common notation for JI and ETs From: manuel.op.de.coul@xxxxxxxxxxx.xxx Dave wrote: Oh, this file is out of date. I don't update it with each release, the latest version only comes with the program. The webfile is only meant to give an impression what the program can do. But maybe it's time for an update now.>I'm hoping now that we've settled on the sagittal symbols that you will >remove JI3 and replace it with whatever subset of the sagittal notation we >manage to come up with single ASCII characters for.Yes that might be done, sagittal ASCII notation for JI is probably surveyable. For more I don't have much hope since the current Scala notations were already a huge amount of work.>How does Scala use [] 32:33 for ET notations?Like the others, it's based on the best approximations to 3 and 11. Do you have a recent Scala? Do View->Staff->Select... and you can browse the notations. E74 is the lowest one with [ and ]. Manuel
Message: 6170 - Contents - Hide Contents Date: Fri, 24 Jan 2003 09:20:42 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e... wrote: >Dave gave an accurate summary except for this: >>> Unfortunately Scala has multiple uses for v and ^. When notating ETs in >> Rapoport's notation Scala uses them for the diaschisma 2025:2048. When >> notating JI it uses them for the 11-diesis 32:33 and when notating 72-ET >> and 144-ET in Richter Herf notation it uses them for the quartertone. >> These latter two usages are compatible with each other but not with the >> former. >>The symbols for 32:33 in JI were changed some time ago to [ and ], Hi Manuel,I apologise for not looking at the latest help file. Scala help * [with cont.] (Wayb.) But when I do, I see that notations JI and JI2 are still the same, and show that v and ^ _are_ used for 32:33. But you have added: QUOTE The symbols which differ in JI3 are:> septimal comma sharp, 64/63< septimal comma flat, 63/64 ] 33/32 [ 32/33 } 1053/1024 { 1024/1053 f 2187/2176 j 2176/2187 h 513/512 y 512/513 UNQUOTE I believe these were only ever used by Gene and I in February 2002 in the "Notating ETs with one comma per prime" thread. At that time we were more concerned with the semantics than the symbols, but were assuming they would be an extension of the Sims symbols. Then George Secor came along and convinced us otherwise. No one actually asked for JI3. I merely suggested some changes to JI and JI2. However I must acknowledge that I did not object when you added JI3. I'm hoping now that we've settled on the sagittal symbols that you will remove JI3 and replace it with whatever subset of the sagittal notation we manage to come up with single ASCII characters for. But of course this is not only for JI, and has symbols for more than one comma per prime in some cases. The full set of sagittal symbols is too large for single ASCII characters (although most symbols are rarely used). But we do have a system for this in which up to 5 ASCII characters are used for a single sagittal, and of course Scala can do proper graphical characters now.> which are also used in ET notations.How does Scala use [] 32:33 for ET notations? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 6171 - Contents - Hide Contents Date: Fri, 24 Jan 2003 20:49:47 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote [#5565]:> George, > > To help with the development of the outline font, could you put up a bitmap > or gif with all the _up_ symbols we've found a need for so far, showing (in > the single-shaft case) the keyboard character that you want them mapped to. > This would be an update of your Symbols3.bmp which is the latest I can find > on the tuning-math files area. Don't include any combinations with 5' > accents, just one with the acute accent beside a plain shaft.Here's the new version of Symbols: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Symbols6.gif As you can see by the filename, I've done quite a bit of work with symbols since the last one was posted. I have 30 different single-shaft symbols there in the main portion of the figure -- no three-flaggers and no 5' comma alterations. (Some 5' comma symbols are at the very bottom.) As you noted, below, we don't need /|(, but I already have it in the graphic, and if I remove it from there now, then we'll probably change our minds for some reason and will want it back. :-) So I'm leaving it there for the time being, even though I don't think we'll ever use it. However, recall that /||( was the apotome complement of ~|~, and ~||~ the complement of /|(. So it looks as if eliminating /|( would also require eliminating ~|~. Are we using ~|~ for anything? It was formerly the 5:19 comma, but )/|, which is exact, has replaced it for that purpose. We also agreed to use ~|~ for 7deg342 and 8deg388 (which decision still stands), but we could also replace it with )/| for both of those. This would then eliminate both /|( and ~|~ from the notation. So that brings me down to 28 symbols. There's one that I tried changing from what you had: ~)| -- 5th from left at the very top. Near the bottom right I have an area labeled "experimental", where I have three versions of this symbol, and I chose the one in which I thought that the separate flags could most easily be identified (if you agree; however you propose to replace this, which I'll answer below). Here is what I now have for a keyboard layout (just tentative, easily subject to change, except for the top row). The most common symbols are at the top (standard 217 symbols in top row, plus 5' comma at far left), less common going downward; nothing is assigned to the bottom row, so there's plenty of room left, should we need it. Degrees of 217 and 494 are given to help in establishing a reasonable progression by size: ` 1 2 3 4 5 6 7 8 9 0 - = key '| |( ~| ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ symbol 1 1 2 3 4 5 6 7 8 9 10 11 12 deg217 1 2 4 6 9 11 13 16 18 20 22 25 27 deg494 Q W E R T Y U I O P [ ] \ )| )|( |~ )|~ )|) (| ~|) |~) (|~ (/| |\) 1 2 3 4 6 6 7 8 9 10 11 1 3 7 9 12 14 15 19 21 23 24 A S D F G H J K L ; ' ~)| )/| ~|~ )|\ ~|\ |)) 2 5 5 7 8 10 5 10 11 14 17 22 Z X C V B N M , . /> I'm thinking that we no longer need any 3-flag symbols (other than those > where the third "flag" is a 5' accent)The only two that I ever seriously considered are ~|() as the 7:17 comma (see discussion below) and )|)) as ~1/2-apotome for 15deg311 and 19deg400 (which would be its own complement), but the latter may be replaced with '|)), 392:405, which is also self-complementing.> and the only > 2-flags-on-the-same-side symbols that we need are //| and |)) and |\), > although I'm still wondering whether you think we'd be better off keeping > (/| instead of |)), since ||\) seems to be the only choice for its complement.I am very impressed by how well (/| and |\) work as complements, so I would hesitate to dump one of them. One problem is that (/| doesn't work in very many ETs as the 7^2 diesis (not valid in 270, 306, 311, 342, 364, or 494) where you might want to map this for JI; for all of these /|\ has the right number of degrees, but is almost 1.3 cents off. (/| works in 388, but 388 isn't 1,7,49-consistent. So (/| is fine for as the 49 diesis for JI, but it wouldn't be usable in most ETs. You would still need |)) in the font for '|)) for 19deg400, and if you have it there you might as well use it for 15deg311 and 29deg612 (see below). It also occurs in the hemififth notation I gave in msg. #5387.> We could replace > > ~)| 17:19 comma > and > |~) 13:19 comma > > which are way down the popularity list anyway (Nos 71 and 45, Ocurrences > 0.08% and 0.15%), > > with > > ')|( 17:19 commaI think not. The ~)| symbol is exact for the 17:19 comma and will therefore be valid in any ET that's 19-limit consistent. I think you need a really good reason to prefer something having a 5' comma over something without it that's exact. Besides, you would be removing a flag combination for apotome-complement pairs, ~)|| with ~|\ and ~)| with ~||\.> and > '//| 13:19 commaThis one you can make a better case for, because it works in most of the good divisions above 270 and the size is almost right on. It also solves the problem of bad symbol arithmetic using |~) for 19deg494. So I'll agree with '//| for the 13:19 diesis. But this doesn't necessarily eliminate the |~) symbol -- it presently has )||( as its apotome complement, and conversely )|( has ||~) as its apotome complement. It has also been proposed for use as 11deg306, 13deg342, and 13deg364 (although accented symbols could now be used for these). It would not be good to have a double-shaft symbol in the notation without a single-shaft version of it.> Whaddya think?I done thunk!> I understand we will never have a use for /|( since it is a synonym for |)I would tend to agree.> and we don't yet have a use for )|\ which is very close to ~|) and could be > replaced by '(|. '(| also has no known use so far, but we get it for free.I was using )|\ for 10deg364 (there was no other option) and also for the 31 comma, 243:243, but '(| will do very nicely for the 31 comma, while '|\ will take care of 10deg364. Since )|\ has no rational complement assigned, there is no problem with eliminating this symbol. So I agree.> So I count 26 single-shaft up symbols in all. > I'm thinking we may need to > revisit the apotome-complement issue again, with this symbol > reorganisation.Which I addressed as I went along. If we keep ~)| and also have |)) around for use as '|)), then I think that's 27 symbols, but you'd better check that.> I don't think that an un-accented symbol should ever have a > complement that is accented or vice versa. Is this possible?Yes, I think that will work. An easy-to-use rule for 5' symbol complements that would follow logically from this is: if a|b and c||d are are apotome complements, then 'a|b and .c||d should also be, and also .a|b and 'c||d. The principle that applies here is that flags in the second half-apotome (i.e., those used for double-shaft symbols) are arrived at according to the definition of apotome complements, such that: a|b equals /||\ minus c||d, where /||\ is the apotome, and it does not necessarily follow in the notation for any particular ET that a||b equals (|) plus a|b, only that this is highly desirable, so that symbol arithmetic in the second half-apotome is usually consistent (and never very obvious if it isn't). To ensure that this is generally true, we have defined apotome complements with very small offsets, i.e., a|b ~ /||\ minus c||d. I don't know whether the foregoing has been explicity stated in our previous discussions, but I thought that this would be a good time to do it. I don't think we need to be overly concerned about whether interlacing of complements is strictly maintained with the addition of the 5' rule, although it would be interesting to see just how many exceptions there are.> It might be a > good idea to try notating 612-ET and 624-ET before settling this.Okay. Here's a go at it: 612: '| )| |( '|( )|( ~)| .~|( ~|( '~|( ./| /| '/| .|) |) '|) |\ (| '(| .(|( (| ( .//| //| '//| ./|) /|) ./|\ /|\ (/| '|)) |\) (|) '(|) (|\ 624: '| )| |( '|( )|( ~)| .~|( ~|( '~|( ./| /| '/| .|) |) '|) .|\ |\ '|\ .(|( (| ( .//| //| '//| ./|) /|) '/|) ./|\ /|\ '/|\ .(|) (|) '(|) .(|\ (|\ And maybe we should do a few others besides. ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS, HERE'S A QUESTION: What other ETs above 494 besides 612 and 624 would you want to notate -- ones in which the 5' comma (a.k.a, historical 5- schisma, 32768:32805) is either a single degree of the ET or vanishes?> By the way, in your otherwise excellent quick reference, I must object to > the line > 7:17 diesis 448:459 ~41.995c (for 217 mapping) > > A correct symbol for this diesis would be either .~|\ or (not quite) '//|This one continues to be a problem (also, this was your turn to get the 5' symbols mixed up). '~|\ is just the right size, but it's not valid in most of the best 17-limit consistent ETs: 217, 311, 494; the only important one that handles it is 388. .//| is around a cent off, but it works in 311 and 494, but not in 217 or 388. I hesitate to use a symbol containing 5' that isn't even valid in half of the ETs into you might want to map JI. With ~|() we have no problem, because it's exact and therefore valid everywhere. And I think the symbol looks pretty good -- see the experimental section of my latest file: third column of symbols from the right. And if this is the ony non-5'-three-flagger that we allow, then nobody will be able confuse it with anything else. However, if we adopt ~|(), then there is the problem of an apotome complement for it, and I really can't see a good choice for that -- ) ||( is the only thing that's valid in 494, and it's not valid in 217, 270, 311, or 388 -- plus the fact that we would then need to find a complement for ~|\. So this just opens up a can of worms, in addition to the issue of a 3-flagger. Okay, I'm convinced that we should eliminate ~|() from consideration, so there will be no unaccented 3-flag symbols in the notation. For the 7:17 diesis symbol we must then make a choice between '~|\ and .//|. Points in favor of '~|\: 1) It's almost exactly the right size (as the 23' comma +5' comma) 2) It's less than 1/2 cent from the right size (as the sum of the 3 flags) 3) It contains a 17-comma flag (as a memory aid) Points against: 1) It's not valid in 217, 311, or 494 (but is in 388) Points in favor of .//|: 1) It's valid in both 311 and 494 2) It's less than a cent from the proper size 3) It uses a more familiar symbol (same as for the 125-diesis) Points against: 1) It's not valid 217 or 388 2) It's nearly a cent from the proper size Almost looks like a tossup, so I think we should look at this purely from a JI perspective and use the one that's closest in size -- therefore '~|\ gets my vote for the 7:17 diesis.> I don't think the fact that //| can represent this 7:17 diesis in 217-ET is > relevant here, where the universal comma roles of the symbols are being > defined.Yes. I'll delete it from the table when I add a listing of ET notations, and hopefully we'll have agreed on which of the above to use by then. --George
Message: 6172 - Contents - Hide Contents Date: Fri, 24 Jan 2003 03:08:38 Subject: Re: algebra question From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> an eminent music theorist wrote me in an e-mail: > > "Paul, this is really basic algebra, and this time I suggest that you > consult > a mathematician or a textbook. In any event, once you have created > equivalence classes, you can only refer to a given class *as a > whole*--not > to its individual members." > > can gene confirm or deny?I'm a little reluctent to enter a conversation without knowing what it's about, but I won't let that stop me. If S is an equivalence class, we refer to the class itself as S, and would refer to an element x being a member of the class S in one of two ways: (1) x \in S (where "\in" is TeX for the "element of" symbol) or (2) x S s, where s is known to be an element of S, or is used to represent S. I think it's neat a distingished theorist is writing to you; I was impressed when a distinguished composer emailed me yesterday, but a theorist is someone we could try to drag onto Yahoo and find out what the conversation is about.
Message: 6173 - Contents - Hide Contents Date: Fri, 24 Jan 2003 22:31:43 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote [#5599]:> I suggest it's time to settle the issue of what single ASCII characters we > recommend to stand for the more common single-shaft saggital symbols. > ... > So here's what this gives us so far. > > /||\ # apotome sharp 2048:2187 > \!!/ b apotome flat > /| / 5-comma sharp 80:81 > \! \ 5-comma flat > |) > 7-comma sharp 63:64 > !) < 7-comma flat > /|\ ^ 11-diesis sharp 32:33 > \!/ v 11-diesis flat > (|) ] 11'-diesis sharp 704:729 > (!) [ 11'-diesis flatI never liked the < and > symbols for the 7-comma, inasmuch as they're laterally confusable and point laterally rather than vertically. Besides, they don't look anything like the sagittal symbols that they're supposed to be representing. Instead I suggest the following: |) f 7-comma sharp 63:64 !) t 7-comma flat In addition to resembling vertically mirrored symbols and having convex curves in the right places, they don't conflict with any existing usage of these letters, as far as I can tell. For the 11' diesis (for the same reasons), why not use characters that are round on both sides (letter O's): (|) O 11'-diesis sharp 704:729 (!) o 11'-diesis flat And for the convex left flag: (| ? 7:11-comma sharp 45056:45927 (! j 7:11-comma flat The above is 11-limit. Now we move on to 13.> ... > I suggest the following for our 13-diesis symbols > > /|) n 13-diesis sharp 1024:1053 > \!) u 13-diesis flat > (|\ } 13'-diesis sharp 26:27 > (!/ { 13'-diesis flatI like n and u for the 13-diesis, but for 13' I would prefer something else. How about m and w, because they're something like n and u, only bigger, for these (the largest single-shaft symbols)? But since the tridecimal schisma 2048:2049 vanishes, you could just as well use these pairs: /|) /f 13-diesis sharp 1024:1053 \!) \t 13-diesis flat (|\ ?\ 13'-diesis sharp 26:27 (!/ j/ 13'-diesis flat Oops! Looks like I cheated here with the 13' diesis -- those straight flags are really supposed to be 11-5 commas, 54:55, but perhaps this would be okay, because sagittal notation never uses a 5 comma going up with a 7 comma down, or vice versa. 11-5 commas would still have to be covered. How about a couple of the unused pairs you gave that might do the trick: |\ & 11-5-comma sharp 54:55 !/ % 11-5-comma flat With just the above symbols (representing only straight and convex flags) you can notate all of the ETs in Table 3 of my paper, so a limited amount of shorthand ascii can cover a lot of ground. Of course, there would be no problem combining 5' commas with any of the above, if you wanted to extend the shorthand a little: '| ' 5'-comma sharp 32768:32805 .! . 5'-comma flat So the 125 diesis (125:128) up would be .// and down would be '\\> Can anyone suggest a convincing way to map single ASCII character pairs to > these? I'd be happy to get this far, since it would see 217-ET notated > along with the 15-limit diamond and the first 17 odd harmonics. ...For 217-ET that's 12 pairs of characters. I don't think I would want to see ) paired with (, for example. For the 5:7 comma why not ( for up and for down, and the 19 comma would be ) and }. Maybe a 17 comma up could be S and down s, or would that be better for the 23 comma? After that it gets more difficult. Do we really need shorthand ascii notation for anything more than straight and convex-flag symbols? (Or are you intending to combine those single-character ascii symbols in any way?) I think that it would get pretty complicated (hence difficult to remember), and the result would have very resemblance to what sagittal symbols look like. Wouldn't it be more productive just to use the sagittal ascii system that we're already using for these things? It would be easy to parse something with software to determine that it isn't shorthand, because it would contain at least one |, !, X, or x. --George
Message: 6174 - Contents - Hide Contents Date: Fri, 24 Jan 2003 03:15:27 Subject: Re: algebra question From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> an eminent music theorist wrote me in an e-mail:> If S is an equivalence class, we refer to the class itself as S, and would refer to an element x being a member of the class S in one of two ways: > > (1) x \in S (where "\in" is TeX for the "element of" symbol) or > > (2) x S s, where s is known to be an element of S, or is used to represent S.This should be (2) x R s, where R is an equivalence relation for which S is an equivalence class, and s represents S. That is S = {x | x R s}
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