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Message: 5200 Date: Thu, 12 Sep 2002 14:15:50 Subject: Re: A common notation for JI and ETs From: David C Keenan Hi Gene, Good to hear from you in this thread. I'm glad you're checking to make sure we don't betray your original concept of notating both rational tunings and ETs using accidentals representing one comma per prime. This is certainly still possible with the sagittal notation as it currently stands, and I intend it to always be posssible (for primes up to 31). To do this one takes certain prime-comma interpretations of certain symbols, and treats these symbols as atomic, taking no notice of the fact that they are composed of various "flags" or half-arrow-heads, which don't quite add up if considered as individual commas. In any case, the extent of their not-adding-up is less than 0.5 cents. In many cases, this way of using the notation will require multiple prime-comma accidentals against a single note, often pointing in opposite directions (in addition to any sharps or flats, the 3-comma symbols). Here they are: /| 5-comma 80;81 |) 7-comma 63;64 /|\ 11-diesis 32;33 /|) 13-diesis 1024;1053 ~| 17-comma 2176;2187 )| 19-comma 512;513 |~ 23-comma 729;736 (| 29-comma 256;261 )|\ 31-comma 243;248 There are also some symbols for alternative commas for some primes, e.g. ~|( for the 17'-comma 4096;4131. Of course the real symbols look much nicer than these ASCII representations of them, and can be found in several .bmp files in George's or my folder, in the files section of this yahoo group. However George and I have been concentrating on standardising the sagittal notation for 15-limit JI and all ETs up to about 76 (and many others up to about 300), in such a way that only _one_ sagittal accidental is ever required. This involves redefining certain symbols, including some high-prime comma symbols, as representing commas involving two (and occasionally three) primes greater than 3. These redefinitions never involve a change in value of more than 1 cent and are mostly less than 0.5 cents. For example, the 29-comma symbol (| is redefined as the 7:11-comma 45056;45927, only 0.34 cents smaller. As well as redefining a few high-prime symbols, several new symbols are introduced (but with no new flags). For example, |( is the 5:7-comma 5103;5120. (|( is the 5:11-comma 44;45 and sometimes the 7:13-comma 1664;1701 (which only differ by 0.84 cents). And we have defined double-shaft symbols as the apotome-complements of these symbols. This one-accidental-per-note notation is the most difficult to decide upon. It is then easy to decompose this into either dual-symbol, where at most one saggital is used in conjunction with a sharp or flat, or multi-symbol using one symbol per prime. Now to your question: Gene Ward Smith wrote: >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: > > Maybe we should round them all to the nearest 5 cents. > >Doesn't that invalidate the whole idea? I suspect you haven't been following recent discussions closely, and I don't blame you. This suggestion was in the context of a small digression from the main effort (one that seemed wise to persue now, since we couldn't reach agreement on 48-ET or 96-ET). This digression involved designing what we call the 12-R notation, a kind of bastard child of the proper sagittal notation. 12-R notation is only an approximate notation with a resolution of 5 cents (max error of +-2.5 cents). But as such, it does allow one to notate any tuning _relative_to_12_equal_ in a manner that agrees as much as possible with the proper sagittal notation for most n*12-ETs. We only want to do this because we figure people will try to use the sagittal symbols in this way anyhow, and we wanted to standardise it. Anyone who wants better than 5 cent resolution in a 12-relative notation, should write the cents near the noteheads. Anyone who wants precise notation of rational or ET tunings, should use the true sagittal notation (in one of the three mutually compatible forms described above). Hey George, Can you put together in one message, in numerical order notations for (1) all the ETs we agree on, and (2) your proposals for all the ETs we have yet to agree on, and I will respond? I'm going away for 4 months in 1.5 weeks time. Could you please list _all_ ETs in order and just write "as subset of <whatever>" against those that are not notated with their native fifth, and "prefer subset of <whatever>" when they have an optional native-fifth-based notation. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5205 Date: Fri, 13 Sep 2002 10:24:52 Subject: Re: A common notation for JI and ETs From: David C Keenan At 10:30 AM 12/09/2002 -0700, George Secor wrote: >From: George Secor, 9/12/2002 (#4638) >Subject: A common notation for JI and ETs > >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4633]: >You can see that all of the time we have spent discussing how many >schismas can vanish on the point of a flag has not gone to waste. Tee hee. > > This suggestion was in the context of a small digression from the >main > > effort (one that seemed wise to persue now, since we couldn't reach > > agreement on 48-ET or 96-ET). This digression involved designing what >we > > call the 12-R notation, a kind of bastard child of the proper >sagittal > > notation. 12-R notation is only an approximate notation with a >resolution > > of 5 cents (max error of +-2.5 cents). But as such, it does allow one >to > > notate any tuning _relative_to_12_equal_ in a manner that agrees as >much as > > possible with the proper sagittal notation for most n*12-ETs. > >I hadn't replied to this yet because I didn't want to make a hasty >response without thinking through the ramifications of this proposal. > >I don't like having obscure symbols such as )|) and (|~ in this scheme, >because 1) they don't represent any low-number ratios or even any >simple primes, for that matter; 2) neophytes who take the time to >memorize these might then become frustrated once they learn that these >symbols aren't even important, but were just put there to fill in some >gaps. Good point. Lose 'em. >Which brings me to the question, what is the purpose of having a 12-R >notation with 5-cent resolution, anyway? Certainly we don't think that >it would be very important to notate 240-ET (or any particular multiple >of 12 over 100, for that matter). If certain rational intervals are stacked then they will eventually fall in the gaps. But you're right, the inconsistencies of 240-ET would tie them in knots anyway. >What we are left with, then, is multiples of 12 through 96. For these, >the only symbols you need are: > >12: /||\ > 100 >24: /|\ /||\ > 50 100 >36: |) ||) /||\ > 33 67 100 >48: |) /|\ ||) /||\ > 25 50 75 100 >60: /| /|) (|\ ||\ /||\ > 20 40 60 80 100 >72: /| |) /|\ ||) ||\ /||\ > 17 33 50 67 83 100 >84: /| |) /|) (|\ ||) ||\ /||\ > 14 29 43 57 71 86 100 >96: /| |) /|) /|\ (|\ ||) ||\ /||\ > 13 25 38 50 63 75 88 100 >12-R: /| |) /|) /|\ (|\ ||) ||\ /||\ > 15 33 39 50 61 67 85 100 > >which requires nothing more than: > >Sym Approximate offset and Comma interpretation >------------------------------------------------ >/| 15 cents as 5:9, 3:5, 1:5, 1:15 commas > |) 33 cents as 7:9, 3:7, 1:7 commas >/|) 39 cents as 9:13, 3:13, 1:13 dieses >/|\ 50 cents as 9:11, 3:11, 1:11 dieses >(|\ 61 cents as large 9:13, 3:13, 1:13 dieses > >It would probably be desirable to include three more symbols that would >complete the 11 limit notation: > > |( 18 cents as 5:7 and 7:15 commas >(| 18 cents as 7:11 comma >(|( 36 cents as 5:11, 11:15 comma > >You will observe that, except for the two 13 diesis, all of these come >very close to 72-ET. > >And you might also want to include these, since they are simple enough >to comprehend: > >//| 27 cents as 5+5 comma > |\ 35 cents as 11-5 comma > >Anyway, I thought that //| would be a better option than: > >~|) 26 cents as 17 comma + 7 comma Yes. >I would say stop there and don't worry about whatever gaps remain. As >long as they can notate the multiples of 12 through 96 and an 11-limit >tonality diamond, I think a lot of people will be satisfied with this >as a start. Yes. Stop there. >In order to complete the 13 limit, you need no new symbols, only >additional uses for existing symbols that in 12-R are considerably >different in size: > > |( 11 cents as 11:13 comma (2nd usage) >(|( 28 cents as 7:13 comma (2nd usage) >//| 47 cents as 5:13 and 13:15 commas (2nd usage) Too confusing. Leave 'em out. >For these there is no question that you would need to write the number >of cents near the notehead for performers using 12-ET instruments. > > > We only want to do this because we figure people will try to use the > > sagittal symbols in this way anyhow, and we wanted to standardise it. > > > Anyone who wants better than 5 cent resolution in a 12-relative >notation, > > should write the cents near the noteheads. Anyone who wants precise > > notation of rational or ET tunings, should use the true sagittal >notation > > (in one of the three mutually compatible forms described above). > >My recommendation is to have the cents written above the notes in any >and every part for an instrument of flexible pitch. Those who don't >need them can ignore them, and those who do will be able to memorize >them as they become familiar with the notation. Fair enough. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5207 Date: Fri, 13 Sep 2002 11:48:13 Subject: Re: A common notation for JI and ETs From: David C Keenan At 10:30 AM 12/09/2002 -0700, George Secor wrote: >From: George Secor, 9/12/2002 (#4639) >Subject: A common notation for JI and ETs > >(This is a continuation of my message #4604, which is in reply to Dave >Keenan's message #4532.) > > >--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: > > >Now 55 is a real problem, because nothing is really very good for > > >1deg. The only single flags that will work are |( (17'-17) or (| >(as > > >the 29 comma), and the only primes that are 1,3,5,n-consistent are > > >17, 23, and 29. > > > > > >If I wanted to minimize the number of flags, I could do it by > > >introducing only one new flag: > > > > > >55: ~|\ /|\ ~|| /||\ > > > > > >so that 1deg55 is represented by the larger version of the 23' comma > > > >symbol. Or doing it another way would introduce only two new flags: > > > > > >55: ~|~ /|\ ~||~ /||\ > > > > > >The latter has for 1deg the 17+23 symbol, and its actual size (~25.3 > > > >cents) is fairly close to 1deg55 (~21.8 cents). Besides, the >symbols > > >are very easy to remember. So this would be my choice. > > > > I would not use a 23 comma to notate this when it can be done in >17-limit. Luckily ~|\ works for 1 step as the 17+(11-5) comma (which >also agrees with 2 steps of 110-ET). So I go for your first (min flags) >suggestion: > > > > 55: ~|\ /|\ > >I didn't do the complement properly for that one (what I gave was left >over from when we were doing inverse complements); ~|\ didn't even have >a rational complement defined. With the proposals that I made in the >previous message, ~|\ not only has a rational complement ~)||, but ~|\ >also is *both* the 17+(11-5) comma and the 23' comma. That would make >the symbol sequence: > >55a: ~|\ /|\ ~)|| /||\ (RC) > >The flags in ~)|| don't really add up to the proper amount, but we >aren't using )| in any other symbol, so there is no inconsistency in >symbol arithmetic created by "forcing" the complement. > >There is a possibility in which the symbols for both 1deg and 3deg are >rational complements consistent in 55: > >55b: /|( /|\ ~||~ /||\ (RC) > >but this uses more flags. Instead we could just use an alternate >complement to achieve matching symbols: > >55c: /|( /|\ /||( /||\ (AC & MS) > >But if we forget about rational or alternate complements, we can have >matching sequences, consistent symbol arithmetic, and a meaningful >symbol (23' comma) in the first apotome with a minimum of new flags: > >55d: ~|\ /|\ ~||\ /||\ (MS) > >Take your pick, but I would go with version d; I think it's the >simplest. Hmm. I think I like (c) the best, because /|( is close to its rational/Pythagorean value as the 5+(17'-17) comma, and because it's both AC & MS. But I could probably accept any of these. Which one most easily falls out of your spreadsheet, based on rules designed for other ETs? >69, 76: /|) )|\ (|\ /||\ (RC) OK. > > ... > > >> 67,74: ~|) /|) (|\ ~||( /||\ > > > > > >I'm certainly in agreement with the 2deg and 3deg symbols, and if >you > > >must do both ET's alike, then what you have for 1deg would be the > > >only choice (apart from (| as the 29 comma). We both previously > > >chose )|) for 1deg74 (see message #4412), presumably because it's >the > > >smallest symbol that will work, and I chose |( for 1deg67 (in >#4346), > > >which would give this: > > > > > >67: |( /|) (|\ /||) /||\ > > >74: )|) /|) (|\ (||( /||\ > > > > > >So what do you prefer? > > > > I prefer yours, but I'm uncertain about the complement used for 4 >steps of 74. > >Add to this your latest observation about 67: > ><< We agreed on |( for 1deg67 which is wrong (or at least not >1,3,5,7-consistently right) if |( is the 7-5 comma. I also proposed it >for >93-ET (3*31) but we didn't agree on a notation for that. >> > >I now propose these as most memorable (fewest flags): > >67: /|( /|) (|\ /||) /||\ (MM) >74: )|) /|) (|\ /||) /||\ (MM) I certainly agree with the first three symbols for each, but why not AC or RC? > > >> 81,88: )|) /|) (|\ (||( /||\ [13-commas] > > > > > >This is exactly what I have for 74, above. Should we do 67 as I did > > > >it above and do 74, 81, and 88 alike? > > > > Yes. > > > > >On the other hand, why wouldn't 88 be done as a subset of 176? > > > > I have a reason to do both 81 and 88 as subsets, apart from the fact >that they are 1,3,9-inconsistent. When using their native fifths they >need a single shaft symbol for 4 steps and none is available. > > > > >It is with some surprise that I find that |( is 1deg in both 67 and > > >81, so 81 could also be done the same way as I have for 67, above. > > > > Better to do it the same as 74 and 88 (or as a subset). > >Your observation that |( for 1deg is wrong if |( is the 7-5 comma also >holds here. I think that the simplest notation (fewest flags) for both >81 and 88 is: > >81, 88: )|) /|) (|\ )||\ /||\ [13 commas] (MM) Single-shafters agreed. Don't understand )||\ as complement of )|), except that it's consistent with the following flag values. )| -1 /| 0 |) 2 (| 0 |\ 3 2nd | 2 But so are ||) /||) and (||). > > >> 6 steps per apotome > > >> 37,44,51: )| /| /|) ||\ (||\ /||\ [13-commas] > > >> or > > >> 37,44,51: |) )|) /|) (||( ||) /||\ [13-commas] > > > > > So are you agreeing to one of these for 37 and 44? Presumably not the >second one because of |) not being the 7-comma. > >Yes I prefer the first one, but not with (||\ for 5deg (how did you get >that?). Beats me. Too long ago. Too many changes since. > With no new flags it could be: > >37a, 44a: )| /| /|) ||\ )||\ /||\ [13-commas] (MM) > > > And with rational complements? > >With rational complements we would have this: > >37b, 44b: )| /| /|) ||\ (||~ /||\ [13-commas] (RC) > >But I think I prefer version a -- fewer flags and easier to remember, >whereas the rational complementation of version b doesn't really >accomplish anything. I agree. Version a it is. > > ... > > >> 86,93,100: )|) |) )|\ (|\ (||( /||\ [13-commas] > > >> or > > >> 86,100: )|( |) )|\ (|\ (||) /||\ [13-commas] > > >> 93: |( |) )|\ (|\ /||) /||\ [13-commas] > > > > > >I would do 93-ET and 100-ET as subsets of 186-ET and 200-ET, > > >respectively. > > > > I can agree to that for 100-ET since there is no single-shaft symbol >for 5 steps, but it is of course 2*50, and 93 is 3*31, so the fifth >sizes are quite acceptable. > > > > >For 86, I wouldn't use |) by itself as anything other than the 7 > > >comma, as explained above, > > > > I totally agree we should avoid this in all cases. > > > > > but would use convex flags for symbols > > >that are actual ratios of 13. So this is how I would do it: > > > > > >86: ~|~ /|) (|~ (|\ ~||~ /||\ [13-commas and 23-comma] > > > > > >The two best primes are 13 and 23, so there is some basis for > > >defining |~ as the 23 flag. In any event, I believe that (|~ can be > > > >a strong candidate for half an apotome if neither /|\ nor /|) nor >(|\ > > >can be used. > > > > I have no argument about the even steps (they agree with 43 and >50-ET). But again I don't see the need to use a 23-comma. We have >already used )|\ for a half-apotome in the case of 69 and 76-ETs. It >works here too. 86-ET is 1,3,7,13,19-consistent. So why not: > > > 86,93,100: )|) /|) )|\ (|\ ?? /||\ [13-commas] > >I see that (|~ will work here for 86, but not 93 or 100. But I agree >that the )|\ symbol is better for minimizing the flags and especially >for keeping commonality over the three divisions when there is no >reason not to. For 5deg )||\ works for all three without adding any >new flags: > >86, 93, 100: )|) /|) )|\ (|\ )||\ /||\ (MM) Agreed. > > We can now consider the 31-ET family. > > > > 31: /|\ /||\ > > 62: /|) /|\ (|\ /||\ [13-commas] > > 93: )|) /|) )|\ (|\ ?? /||\ [13-commas] > >And we can fill in the blank with )||\ if you agree. Yes. > > and compare it to the 19-ET family > > > > 19: /||\ > > 38: /|\ /||\ > > 57: /|) (|\ /||\ [13-commas] > > 76: /|) )|\ (|\ /||\ [13-commas] > > > > Whew! > >And whew! to you, too. (End of reply to your message #4532.) Hoorah! Well done! And by the way, I agree with your latest pyramid for the 12-ET family. In particular, I now agree with your 48, 60 and 96 notations. I think that means we've agreed on all those with up to 6 steps per apotome (and some others). We only have to get up to 27 steps to the apotome (282-ET). Sigh. But I guess they get fairly rare by that time. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5215 Date: Sun, 15 Sep 2002 08:53:17 Subject: Re: A common notation for JI and ETs From: David C Keenan At 10:23 AM 13/09/2002 -0700, George Secor wrote: > > >55a: ~|\ /|\ ~)|| /||\ (RC) > > > > > >The flags in ~)|| don't really add up to the proper amount, but we > > >aren't using )| in any other symbol, so there is no inconsistency in > > >symbol arithmetic created by "forcing" the complement. > > > > > >There is a possibility in which the symbols for both 1deg and 3deg >are > > >rational complements consistent in 55: > > > > > >55b: /|( /|\ ~||~ /||\ (RC) > > > > > >but this uses more flags. Instead we could just use an alternate > > >complement to achieve matching symbols: > > > > > >55c: /|( /|\ /||( /||\ (AC & MS) > > > > > >But if we forget about rational or alternate complements, we can >have > > >matching sequences, consistent symbol arithmetic, and a meaningful > > >symbol (23' comma) in the first apotome with a minimum of new flags: > > > > > >55d: ~|\ /|\ ~||\ /||\ (MS) > > > > > >Take your pick, but I would go with version d; I think it's the > > >simplest. > > > > Hmm. I think I like (c) the best, because /|( is close to its > > rational/Pythagorean value as the 5+(17'-17) comma, and because it's >both > > AC & MS. But I could probably accept any of these. Which one most >easily > > falls out of your spreadsheet, based on rules designed for other ETs? > >I don't know yet, because I haven't gotten that far; these are less >common symbols, and I couldn't code any decisions in the spreadsheet >involving these until I had discussed them. I'm determining the rules >based on specific examples on which we have agreed, and these will be >subject to review whenever I find any ambiguities in our selections. >So we can leave a final determination for 55 until later, once we have >done the other ETs. OK. > > >67: /|( /|) (|\ /||) /||\ (MM) > > >74: )|) /|) (|\ /||) /||\ (MM) > > > > I certainly agree with the first three symbols for each, but why not >AC or RC? > >For 4deg /||( is not valid in either, and ~||~ is valid in 74 but not >in 67, although the symbol could be "forced" into use, since neither >wavy flag is used elsewhere. But that's just the point -- we would be >introducing two new flags -- better to use /||), which matches /|) and >is valid in both 67 and 74. OK. > > >Your observation that |( for 1deg is wrong if |( is the 7-5 comma >also > > >holds here. I think that the simplest notation (fewest flags) for >both > > >81 and 88 is: > > > > > >81, 88: )|) /|) (|\ )||\ /||\ [13 commas] (MM) > > > > Single-shafters agreed. Don't understand )||\ as complement of )|), >except > > that it's consistent with the following flag values. > > )| -1 > > /| 0 > > |) 2 > > (| 0 > > |\ 3 > > 2nd | 2 > > > > But so are ||) /||) and (||). > >||) is not the proper number of degrees for /||\ minus |), and (||) has >never been used, since it shouldn't be less than /||\. However, you >have a good point about /||) for 4deg, since it's valid in both 81 and >88, so then I think we should do it this way: > >81, 88: )|) /|) (|\ /||) /||\ [13 commas] (MS) > >Okay? OK. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5217 Date: Mon, 16 Sep 2002 17:59:17 Subject: Escher and mathematics From: manuel.op.de.coul@xxxxxxxxxxx.xxx Not about tuning, so I'll keep it short: Escher and the Droste effect - Universiteit Leiden * Especially the two 10MB mpeg files are worth viewing. Manuel
Message: 5218 Date: Mon, 16 Sep 2002 17:29:52 Subject: Re: A common notation for JI and ETs From: David C Keenan At 10:24 AM 13/09/2002 -0700, George Secor wrote: >ET Notation Agreed Upon >----------------------- > >Those divisions that are to be notated as subsets of a larger division >closely follow what you specified in message #4188. (One exception is >that I have proposed that 62-ET be notated only with its native fifth, >inasmuch as I have not been able to notate 186 adequately with the >present set of symbols.) Fine. I can't see any way to notate 186-ET either, and who cares. >2 (subset of 12) >3 (subset of 12) >4 (subset of 12) >6 (subset of 12) >8 (subset of 24) >9 (subset of 27) >11 (subset of 22) >12: /||\ >13 (subset of 26) >16 (subset of 48) >17: /|\ /||\ >18 (subset of 36) >19: /||\ >20 (subset of 60) >22: /| ||\ /||\ >23 (subset of 46) >24: /|\ /||\ >25 (subset of 50) >26: /||\ >27: /| /|) ||\ /||\ >28 (subset of 56) >29: /| ||\ /||\ >30 (subset of 60) >31: /|\ /||\ >32: (|( /|\ (|) ~||( /||\ > (or as subset of 96) >33 (subset of 99) >34: /| /|\ ||\ /||\ >35 (subset of 70) >36: |) ||) /||\ >37: )| /| /|) ||\ )||\ /||\ > (or as subset of 111) >38: /|\ /||\ >39: /| /|\ (|) ||\ /||\ >40 (subset of 80) >41: /| /|\ ||\ /||\ >42 (subset of 84) >43: /|) (|\ /||\ >44: )| /| /|) ||\ )||\ /||\ > (or as subset of 132) >45: /|) /||\ > (or as subset of 135) >46: /| /|\ (|) ||\ /||\ >47 (subset of 94) >48: |) /|\ ||) /||\ >50: /|) (|\ /||\ >51: |) /| /|) ||\ ||) /||\ >52: (|( /||\ > (or as subset of 104) >53: /| /|\ (|) ||\ /||\ >54 (subset of 108) >57: /|) (|\ /||\ > (or as subset of 171) >58: /| |\ /|\ /|| ||\ /||\ >59 (subset of 118) >60: /| /|) (|\ ||\ /||\ >61 (subset of 183) >62: /|) /|\ (|\ /||\ >64: /|) (|\ /||\ > (or as subset of 128) >65: /| |) /|\ ||) ||\ /||\ >66 (subset of 132) >69: /|) )|\ (|\ /||\ (using )|\ as half-apotome) > (or as subset of 207) >71 (subset of 142) >72: /| |) /|\ ||) ||\ /||\ >76: /|) )|\ (|\ /||\ (using )|\ as half-apotome) > (or as subset of 152) >79: /| |) /|\ ||) ||\ /||\ >84: /| |) /|) (|\ ||) ||\ /||\ >86: )|) /|) )|\ (|\ )||\ /||\ (using )|\ as half-apotome) >93: )|) /|) )|\ (|\ )||\ /||\ (using )|\ as half-apotome) >96: /| |) /|) /|\ (|\ ||) ||\ /||\ >100: )|) /|) )|\ (|\ )||\ /||\ (using )|\ as half-apotome) >217: |( ~| ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ ~|| ~||( >/|| ||) ||\ (||( //|| /||) /||\ Thanks for collecting those. I haven't checked them. I thought that in cases where we propose both a native fifth notation and a subset notation for the same ET, we agreed that we would indicate which was preferred. I also thought we agreed to always prefer the subset notation. Do you have a reason to change this? I also realise we need to say _which_ subset to use. I think we should always specify the subset that contains D natural, for reasons I expect are obvious to you. >ET Notation Proposals >--------------------- > >RC = rational complementation >AC = alternate complementation >MS = matching symbol sequence >MM = most memorable sequence > >I've put in a few proposals for some very simple ETs (containing >circles of 5 and 7 fifths) to start the list (including three in which >the apotome vanishes) which are much simpler than doing them as subsets >of considerably higher numbers. (After all, shouldn't the very >simplest divisions be really simple?) I now agree we can provide the native fifth notation as an option in those cases where the apotome merely vanishes (rather than becoming negative), i.e. the n*5-ET and n*7-ET families. Whether a person prefers the subset notation or the native fifth notation will depend, I expect, on whether or not they have any interest in what just intervals are approximated. I still think we should say that we prefer the subset notation in these cases. >If you aren't able to go through all of these, can we at least get some >agreement on some of the best (and easiest) larger divisions, including >94, 99, 118, 130, 142, 152, 171, 176, and 183, and you can pick out >some others that you think deserve priority. Due to time constraints I've looked mainly at the single shaft symbols below, leaving the complements mostly up to you. >5: using 5 out of 7 naturals > (or as subset of 50) Agreed, but lets specify the 5 naturals as C G D A E in the interests of standardisation. Same for 10-Et and 15-ET. And, as an example of what I said about ten paragraphs back, the subset notation would be specifically A/|) C(!/ D E(|\ G\!) >7: using 7 naturals > (or as subset of 56) >10: /|\ /||\ using 5 out of 7 naturals > (or as subset of 50) >14: |) using 7 naturals > (or as subset of 56) >15: /| ||\ /||\ using 5 out of 7 naturals > (or as subset of 60) >21: |) |\ using 7 naturals > (or as subset of 63) Agreed. >49: |\ /| /|\ (|) ||\ /|| /||\ > (or as subset of 147)?? Agreed. We need to include "subset of 147" since we're invoking prime 11, and 49 is not 1,3,9-consistent, and it's pretty awful having |\ smaller than /|. >55a: ~|\ /|\ ~)|| /||\ (RC) >55b: /|( /|\ ~||~ /||\ (RC) >55c: /|( /|\ /||( /||\ (AC & MS) >55d: ~|\ /|\ ~||\ /||\ (MS) Yeah. Decide later. Currently my favourite is (c), yours is (d). >56, 63: |) /| /|\ (|) ||\ ||) /||\ Agreed. >67: /|( /|) (|\ /||) /||\ Agreed. >68: |\ /| /|\ /|) (|) ||\ /|| /||\ if we permit /|\ < /|) Agreed. This will work for 75-ET too. >70: /| |\ /|\ (|) /|| ||\ /||\ Agreed. >74: )|) /|) (|\ /||) /||\ Agreed. >77: /| |) /|\ (|) ||) ||\ /||\ Agreed. >80: )| /| (|~ /|\ (|) )|| ||\ (||~ /||\ [13'-(11-5)+23 = >11-19 diesis] I'd prefer the single-shaft symbols to be 80b: |) /| (|( /|\ (|) ? ||\ ? /||\ since it stays within the 11-limit. It isn't nice to have |) smaller than |\, but we've done it elsewhere. >81, 88: )|) /|) (|\ /||) /||\ Agreed >87a: |~ /| ~|) /|\ (|) ||~ ||\ ~||) /||\ (RC) >94a: ~|( /| (|( /|\ (|) ~||( ||\ (||( /||\ (RC) >87b, 94b: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ (MM) >87c, 94c: |~ /| /|~ /|\ (|) ||~ ||\ /||~ /||\ (MM) >87d, 94d: |~ /| /|~ /|\ (|) ~|| ||\ ~||\ /||\ (MM) I'd prefer the single-shaft symbols to be 87e, 94e: ~| /| (| /|\ (|) ? ||\ ? /||\ >99a: |~ /| ~|) /|) (|~ (|\ ||~ ||\ ~||) /||\ (RC) >99b: ~| /| ~|\ /|) (|~ (|\ ~|| ||\ ~||\ /||\ (MM) >99c: |~ /| /|~ /|) (|~ (|\ ||~ ||\ /||~ /||\ (MM) I prefer 99a. >104a: )| |) /| (| /|\ (|) )||~ ||\ ||) (||~ /||\ [|~ as >23 comma] (RC) >104b: )| |) /| (|~ /|\ (|) )|| ||\ ||) (||~ /||\ [|~ as >23 comma] (RC) I prefer 104a. >108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ (RC) >108b: /| |( |) /|) (|\ ||) )||) ||\ /||\ (MM) I prefer 108a. >111: ~| /| |\ ~|\ /|\ (|) ~|| /|| ||\ ~||\ /||\ I prefer 111b: ~| /| |\ //| /|\ (|) ~|| /|| ||\ //|| /||\ which is the same as 118-ET below. >118: ~| /| |\ //| /|\ (|) ~|| /|| ||\ //|| /||\ Agreed. >120: /| (| |) /|) /|\ (|\ ||) )||~ ||\ /||\ Agreed. >125: ~|( /| |\ (|( /|\ (|) ~||( /|| ||\ (||( /||\ I prefer 125b: |( /| |) //| /|\ (|) ~|| ||) ||\ /||) /||\ >128a: )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (||( (||~ >/||\ (RC) >128b: )| ~|( /| (|( ~|\ /|\ (|) )|| ~||( ||\ (||( ~||\ >/||\ (MM) I prefer 128a. >130: |( /| |) |\ /|) /|\ (|\ /|| ||) ||\ /||) /||\ Agreed. >132a: ~|( /| |) |\ (|~ ~||( /|| ||) ||\ (||~ /||\ (MS) >132b: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||~ /||\ (MS) I prefer 132b, but why not |( as 5:7-comma for 1deg132? >135a: ~| |~ /| (| /|~ /|\ (|) ~|| ||~ ||\ (|| /||~ /||\ >(MM) >135b: ~| ~|( /| (| /|~ /|\ (|) ~|| ~||( ||\ (|| /||~ /||\ > (MM) I prefer 135a. >140 (70 ss.): )| ~|( /| )|\ ~|\ /|) (|~ (|\ )|| ~||( ||\ >)||\ ~||\ /||\ (MM) I prefer 140 (70 ss.): )| ~| /| )|) ~|\ /|) (|~ (|\ )|| ~|| ||\ )||) ~||\ /||\ >142: )| /| |) |\ /|) /|\ (|) (|\ /|| ||) ||\ /||) /||\ )| is wrong for 1deg142. How about 142b: |( /| |) |\ /|) /|\ (|) (|\ /|| ||) ||\ /||) /||\ (RC & MS) >144: ~|( /| )|) |\ /|) /|\ (|\ /|| )||) ||\ /||) /||\ Agreed. >147: ~| ~|( /| |\ ~|\ /|) /|\ (|\ ~||( /|| ||\ ~||\ /||) >/||\ Agreed. >149: ~|( /| /|( |\ /|) /|\ (|) (|\ /|| /||( ||\ /||) /||\ Agreed. >152a: )| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ (||( >/||) /||\ (MS; 14deg AC) >152b: )| |~ /| |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||\ ~||) >/||) /||\ (MS; 14deg AC) >152c: )| ~| /| |\ ~|) /|) /|\ (|) (|\ ~|| /|| ||\ ~||) >/||) /||\ (MS; 10,13,14deg AC) I prefer 152b. >159: |( ~|( /| |\ ~|\ /|) /|\ (|) (|\ ~||( /|| ||\ ~||\ >/||) /||\ I prefer 159: ~| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ (||( /||) /||\ (RC & MS) >171: |( ~|( /| |) |\ ~|\ /|) /|\ (|\ ~||( /|| ||) ||\ >~||\ /||) /||\ I think I prefer 171b: |( ~|( /| |) |\ //| /|) /|\ (|\ ~||( /|| ||) ||\ //|| /||) /||\ >176a: |( |~ /| |) |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||) ||\ >~||) /||) /||\ (RC & MS) >176b: |( ~| /| |) |\ ~|) /|) /|\ (|) (|\ ~|| /|| ||) ||\ >~||) /||) /||\ (MS & MM) Of those two, I prefer 176a, but I like these single-shafters better 176c: |( |~ /| |) |\ //| /|) /|\ (|) (|\ 176d: |( ~| /| |) |\ //| /|) /|\ (|) (|\ >181a: |( ~| |~ /| /|( ~|) /|~ /|) (|~ (|\ ||( ~|| ||~ ||\ > /||( ~||) /||~ /||\ (MM) >181b: |( ~| ~|( /| /|( (| /|~ /|) (|~ (|\ ||( ~|| ~||( >||\ /||( (|| /||~ /||\ (MM) These are both wrong if |( is the 5:7 comma, since the 5:7 comma vanishes in this tuning. I prefer 181c: )| ~| |~ /| )|) (| /|~ /|) (|~ (|\ )|| ~|| ||~ ||\ )||) (|| /||~ /||\ (MM) >183: |( ~|( /| |) |\ (|( /|) /|\ (|) (|\ ~||( /|| ||) ||\ > (||( /||) /||\ I prefer 183b: |( ~|( /| |) |\ //| /|) /|\ (|) (|\ ~||( /|| ||) ||\ //|| /||) /||\ >186: can't be done, so 62 must be done with native fifth Agreed. There's no symbol comma that is 2 steps. >193: )| ~| ~|( /| |\ ~|) ~|\ /|) /|\ (|) (|\ ~|| ~||( /|| > ||\ ~||) ~||\ /||) /||\ I prefer 193b: )| ~| ~)| /| |\ (| ~|\ /|) /|\ (|) (|\ ~|| ~)|| /|| ||\ (|| ~||\ /||) /||\ 193c: )| ~| ~|( /| |\ (| ~|\ /|) /|\ (|) (|\ ~|| ~||( /|| ||\ (|| ~||\ /||) /||\ >207: ~| ~|( /| /|( (| |\ ~|\ /|) /|\ (|) (|\ ~||( /|| >/||( (|| ||\ ~||\ /||) /||\ Agreed. >So there it is. Do the best you can with it. Good work! Here are some others for your consideration: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 282: )| ~| ~)| |~ /| |) )|) (| (|( //| /|) (|~ /|\ (|) |( ~|( /|~ ~|\ |~) 11deg282 is the difficult one. /|) is only correct as the 5-comma + 7-comma, not the 13-comma, and |~) is a two-flags-on-the-same-side symbol I'm proposing to stand for the 13:19-comma (and possibly the 5:13-comma). But if you'd rather, I'll just accept that 282-ET and 294-ET are not notatable. However, 306-ET _is_ notatable without using any two-flags-on-the-same-side symbols. Alternatives for some degrees are given on the line below. 306: )| |( )|( ~|( /| ~|~ |) (| |\ //| ~|\ /|) (|~ /|\ (|) ~| ~)| |~ )|) ~|) (|( |~) 318 is notatable if you accept (/| (the 31' comma) for 15 steps. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
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