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Message: 6150 Date: Tue, 21 Jan 2003 23:51:35 Subject: Re: Superparticular temperaments From: Carl Lumma > > 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. -Carl
Message: 6151 Date: Tue, 21 Jan 2003 23:54:52 Subject: Re: Superparticular temperaments From: Carl Lumma > > Look 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 Date: Wed, 22 Jan 2003 00:01:47 Subject: Re: heuristic and straightness From: Carl Lumma > > 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. So what should the badness column be? -Carl
Message: 6160 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/SymbolsBySize3.bmp * or Yahoo groups: /tuning-math/files/secor/notation/AdaptJI.gif * or Yahoo groups: /tuning-math/files/secor/notation/Symbols3.bmp * 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 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 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 Date: Wed, 22 Jan 2003 23:55:41 Subject: Re: heuristic and straightness From: Carl Lumma > > > > 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, but not for choosing a commatic basis for a given temperament? -Carl
Message: 6164 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/NotationalCommas.xls.zip * -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 6169 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: >I apologise for not looking at the latest help file. >Scala help * 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 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 * 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 *
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