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Message: 5200 - Contents - Hide Contents 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 * [with cont.] (Wayb.)

Message: 5201 - Contents - Hide Contents Date: Thu, 12 Sep 2002 08:01:29 Subject: Re: commas from wedgies (was: Proposal: a high-order septimal schisma) From: Gene Ward Smith --- In tuning-math@y..., "monz" <monz@a...> wrote:> at last, i finally understand how you're calculating wedgies!For the 7-limit, at least. :)> OK, so the syntonic comma (81/80) is there ... > but what happened to 126/125 and 225/224? why are > they not in this list, and why are the other ones there?Those four commas are special, in that they all leave out a prime; you should have written them as 4-vectors, not 3-vectors: [-4,4,-1,0] -- the 5-limit comma [-13,10,0,-1] -- the {2,3,7} comma [12,0,-10,4] -- the {2,5,7} comma [0,12,-13,4] -- the odd comma The "kernel" is generated by these commas but also by 81/80 and 126/125, etc; so they are all in the kernel, but 126/125 and 225/224 have all four 7-limit primes in their factorization.

Message: 5202 - Contents - Hide Contents Date: Thu, 12 Sep 2002 08:07:00 Subject: Re: A common notation for JI and ETs From: Gene Ward Smith --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> 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.As you surmised, I've lost track of what the two of you are up to. I keep expecting a report.> This one-accidental-per-note notation is the most difficult to decide upon.I can believe it.> Anyone who wants better than 5 cent resolution in a 12-relative notation, > should write the cents near the noteheads.Of course, there are other possibilities, such as my proposal to base things on a 9-note system and ennealimmal, and Graham's ideas. My ennealimmal plan can use ordinary musical notation software, and does far better than 5 cent resolution. Of course, getting people used to it is another matter!

Message: 5203 - Contents - Hide Contents Date: Fri, 13 Sep 2002 17:16:15 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4641]:> 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?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.>>> ...>>>>> 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 for4 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 alsoproposed 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 whynot 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.>>>>>> 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 (||).||) 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?>> ... (End of reply to your message #4532.) >> Hoorah! Well done!At least not burnt to a crisp!> 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.That's a relief!> 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.At least medium rare. I haven't found that there's very much above 217 that can be done reasonably well, anyway. For example, I would have expected 224 to be fairly easy, but it isn't. The best I could do is: 224: )| ~)| ~|( /| |) |\ ~|\ //| /|) /|\ (|) (|\ ~)|| ~||( /|| ||) ||\ ~||\ //|| /||) /||\ (MS) So I'm leaving those off for the time being. The list you requested will follow shortly, as soon as I update it with these latest changes. --George

Message: 5204 - Contents - Hide Contents Date: Fri, 13 Sep 2002 17:20:38 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4633]:> ... 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.And here it is: 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.) 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: |( ~| ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ ~|| ~|| ( /|| ||) ||\ (||( //|| /||) /||\ 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?) 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. 5: using 5 out of 7 naturals (or as subset of 50) 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) 49: |\ /| /|\ (|) ||\ /|| /||\ (or as subset of 147)?? 55a: ~|\ /|\ ~)|| /||\ (RC) 55b: /|( /|\ ~||~ /||\ (RC) 55c: /|( /|\ /||( /||\ (AC & MS) 55d: ~|\ /|\ ~||\ /||\ (MS) 56, 63: |) /| /|\ (|) ||\ ||) /||\ 67: /|( /|) (|\ /||) /||\ 68: |\ /| /|\ /|) (|) ||\ /|| /||\ if we permit /|\ < /|) 70: /| |\ /|\ (|) /|| ||\ /||\ 74: )|) /|) (|\ /||) /||\ 77: /| |) /|\ (|) ||) ||\ /||\ 80: )| /| (|~ /|\ (|) )|| ||\ (||~ /||\ [13'-(11-5)+23 = 11-19 diesis] 81, 88: )|) /|) (|\ /||) /||\ 87a: |~ /| ~|) /|\ (|) ||~ ||\ ~||) /||\ (RC) 94a: ~|( /| (|( /|\ (|) ~||( ||\ (||( /||\ (RC) 87b, 94b: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ (MM) 87c, 94c: |~ /| /|~ /|\ (|) ||~ ||\ /||~ /||\ (MM) 87d, 94d: |~ /| /|~ /|\ (|) ~|| ||\ ~||\ /||\ (MM) 99a: |~ /| ~|) /|) (|~ (|\ ||~ ||\ ~||) /||\ (RC) 99b: ~| /| ~|\ /|) (|~ (|\ ~|| ||\ ~||\ /||\ (MM) 99c: |~ /| /|~ /|) (|~ (|\ ||~ ||\ /||~ /||\ (MM) 104a: )| |) /| (| /|\ (|) )||~ ||\ ||) (||~ /||\ [|~ as 23 comma] (RC) 104b: )| |) /| (|~ /|\ (|) )|| ||\ ||) (||~ /||\ [|~ as 23 comma] (RC) 108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ (RC) 108b: /| |( |) /|) (|\ ||) )||) ||\ /||\ (MM) 111: ~| /| |\ ~|\ /|\ (|) ~|| /|| ||\ ~||\ /||\ 118: ~| /| |\ //| /|\ (|) ~|| /|| ||\ //|| /||\ 120: /| (| |) /|) /|\ (|\ ||) )||~ ||\ /||\ 125: ~|( /| |\ (|( /|\ (|) ~||( /|| ||\ (||( /||\ 128a: )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (||( (||~ /||\ (RC) 128b: )| ~|( /| (|( ~|\ /|\ (|) )|| ~||( ||\ (||( ~||\ /||\ (MM) 130: |( /| |) |\ /|) /|\ (|\ /|| ||) ||\ /||) /||\ 132a: ~|( /| |) |\ (|~ ~||( /|| ||) ||\ (||~ /||\ (MS) 132b: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||~ /||\ (MS) 135a: ~| |~ /| (| /|~ /|\ (|) ~|| ||~ ||\ (|| /||~ /||\ (MM) 135b: ~| ~|( /| (| /|~ /|\ (|) ~|| ~||( ||\ (|| /||~ /||\ (MM) 140 (70 ss.): )| ~|( /| )|\ ~|\ /|) (|~ (|\ )|| ~||( ||\ ) ||\ ~||\ /||\ (MM) 142: )| /| |) |\ /|) /|\ (|) (|\ /|| ||) ||\ /||) /||\ 144: ~|( /| )|) |\ /|) /|\ (|\ /|| )||) ||\ /||) /||\ 147: ~| ~|( /| |\ ~|\ /|) /|\ (|\ ~||( /|| ||\ ~||\ /||) /||\ 149: ~|( /| /|( |\ /|) /|\ (|) (|\ /|| /||( ||\ /||) /||\ 152a: )| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ (|| ( /||) /||\ (MS; 14deg AC) 152b: )| |~ /| |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||\ ~||) /||) /||\ (MS; 14deg AC) 152c: )| ~| /| |\ ~|) /|) /|\ (|) (|\ ~|| /|| ||\ ~||) /||) /||\ (MS; 10,13,14deg AC) 159: |( ~|( /| |\ ~|\ /|) /|\ (|) (|\ ~||( /|| ||\ ~||\ /||) /||\ 171: |( ~|( /| |) |\ ~|\ /|) /|\ (|\ ~||( /|| ||) ||\ ~||\ /||) /||\ 176a: |( |~ /| |) |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||) ||\ ~||) /||) /||\ (RC & MS) 176b: |( ~| /| |) |\ ~|) /|) /|\ (|) (|\ ~|| /|| ||) ||\ ~||) /||) /||\ (MS & MM) 181a: |( ~| |~ /| /|( ~|) /|~ /|) (|~ (|\ ||( ~|| ||~ ||\ /||( ~||) /||~ /||\ (MM) 181b: |( ~| ~|( /| /|( (| /|~ /|) (|~ (|\ ||( ~|| ~||( ||\ /||( (|| /||~ /||\ (MM) 183: |( ~|( /| |) |\ (|( /|) /|\ (|) (|\ ~||( /|| ||) ||\ /||~ /||) /||\ 186: can't be done, so 62 must be done with native fifth 193: )| ~| ~|( /| |\ ~|) ~|\ /|) /|\ (|) (|\ ~|| ~|| ( /|| ||\ ~||) ~||\ /||) /||\ 207: ~| ~|( /| /|( (| |\ ~|\ /|) /|\ (|) (|\ ~|| ( /|| /||( (|| ||\ ~||\ /||) /||\ So there it is. Do the best you can with it. --George

Message: 5205 - Contents - Hide Contents 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 * [with cont.] (Wayb.)

Message: 5206 - Contents - Hide Contents Date: Fri, 13 Sep 2002 21:25:32 Subject: Re: A common notation for JI and ETs From: gdsecor I just happened to notice that there is an error for 15deg183: --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> 183: |( ~|( /| |) |\ (|( /|) /|\ (|) (|\ ~||( /|| ||) > ||\ /||~ /||) /||\This should be: 183: |( ~|( /| |) |\ (|( /|) /|\ (|) (|\ ~||( /|| ||) ||\ (||( /||) /||\ I hope I didn't make any more mistakes. --George

Message: 5207 - Contents - Hide Contents 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 * [with cont.] (Wayb.)

Message: 5209 - Contents - Hide Contents Date: Fri, 13 Sep 2002 04:00:25 Subject: Re: A common notation for JI and ETs From: wallyesterpaulrus good work, gentlemen! for as yet unrelated reasons, i wanted to pose the following research question to the group: for each ET in each limit, was is the most "efficient" generator for geometrically aligning one with the consonant intervals in the (hyper- )triangular lattice?

Message: 5210 - Contents - Hide Contents Date: Fri, 13 Sep 2002 23:07:26 Subject: Re: New file uploaded to tuning-math From: Gene Ward Smith --- In tuning-math@y..., tuning-math@y... wrote: This is an old paper (my rejection letter is dated 1984, which means I completed it in 1983) which David Feldman scanned for me.

Message: 5211 - Contents - Hide Contents Date: Fri, 13 Sep 2002 04:21:11 Subject: Re: A common notation for JI and ETs From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <perlich@a...> wrote:> for each ET in each limit, was is the most "efficient" generator for > geometrically aligning one with the consonant intervals in the (hyper- > )triangular lattice?Sounds like a great question, but what does it mean? Why not give an example.

Message: 5212 - Contents - Hide Contents Date: Fri, 13 Sep 2002 05:22:40 Subject: Re: A common notation for JI and ETs From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <perlich@a...> wrote: >>> for each ET in each limit, what is the most "efficient" generator for >> geometrically aligning one with the consonant intervals in the (hyper- >> )triangular lattice? >> Sounds like a great question, but what does it mean? Why not give an example.basically, i mean the same thing as your geometric complexity measure for linear temperaments (except i think here the fact that you're allowed to "sneak around the other side" of the circle of generators might make a difference). didn't you do something like this for 72 in the 11-limit or something a while back?

Message: 5215 - Contents - Hide Contents 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 * [with cont.] (Wayb.)

Message: 5217 - Contents - Hide Contents 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 Le... * [with cont.] (Wayb.) Especially the two 10MB mpeg files are worth viewing. Manuel

Message: 5218 - Contents - Hide Contents 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 prefer111b: ~| /| |\ //| /|\ (|) ~|| /|| ||\ //|| /||\ which is the same as 118-ET below.>118: ~| /| |\ //| /|\ (|) ~|| /|| ||\ //|| /||\ Agreed. >120: /| (| |) /|) /|\ (|\ ||) )||~ ||\ /||\ Agreed. >125: ~|( /| |\ (|( /|\ (|) ~||( /|| ||\ (||( /||\ I prefer125b: |( /| |) //| /|\ (|) ~|| ||) ||\ /||) /||\>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 prefer140 (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 prefer159: ~| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ (||( /||) /||\ (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 prefer183b: |( ~|( /| |) |\ //| /|) /|\ (|) (|\ ~||( /|| ||) ||\ //|| /||) /||\>186: can't be done, so 62 must be done with native fifthAgreed. There's no symbol comma that is 2 steps.>193: )| ~| ~|( /| |\ ~|) ~|\ /|) /|\ (|) (|\ ~|| ~||( /|| > ||\ ~||) ~||\ /||) /||\ I prefer193b: )| ~| ~)| /| |\ (| ~|\ /|) /|\ (|) (|\ ~|| ~)|| /|| ||\ (|| ~||\ /||) /||\ 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 * [with cont.] (Wayb.)

Message: 5219 - Contents - Hide Contents Date: Wed, 18 Sep 2002 01:16:48 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> At 10:24 AM 13/09/2002 -0700, George Secor wrote:>> ET Notation Agreed Upon >> ----------------------- >> ...> Thanks for collecting those. I haven't checked them. > > I thought that in cases where we propose both a native fifthnotation 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?At this point, no. But I'm open to the possibility that my opinion may change once we get feedback from someone who actually tries to use the notation for one of these ETs and comes to a different conclusion.> 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.Well, D is the center of symmetry for the 7 naturals. But pitch standards are usually set for A or C, so how did you intend to handle that if the ET doesn't have either A or C in the notation? Do you have a particular pitch standard for D in mind that the rest of the world might be willing to accept? (Come to think about it, D isn't a bad choice for a pitch standard once you consider the 5 notes corresponding to the open strings of the violin family.)>>> ET Notation Proposals >> ---------------------Before I comment on any particular division, I want to discuss some of the principles which I used to select some of the symbols. I first assigned the 5 comma, 11 diesis, 7 comma, and 13 diesis, where possible, along with their respective rational complements (including the 11' and 13' dieses). For the larger divisions (for which we would want matching symbols in the half-apotomes) I also assigned the 11-5 comma, where possible. For the remaining degrees I evaluated other symbols in the following order for their suitability (along with their rational complements). Except for the first one, I have put these in pairs, which facilitates the process of achieving matching symbols in the half- apotomes. |( <--> /||) 5:7 comma and 11:13 comma (also 17'-17) (|( <--> ~||( 5:11 and 7:13 comma (also 11:17) ~|( <--> (||( 17' comma //| <--> ~|| 5+5, 25, and 5:13 comma ~| <--> //|| 17 comma (| <--> )||~ 7:11 comma (also 13:17) )|~ <--> (|| 19' comma )| <--> (||~ 19' comma (|~ <--> )|| various complex dieses |~ <--> ~||) 23 comma ~|) <--> ||~ 7+17 comma ~|\ <--> ~)|| 23' comma ~)| <--> ~||\ 17+19 comma /|( <--> ~||~ 5+(17'-17) comma ~|~ <--> /||( 17+23 comma If all of the degrees aren't assigned by this point, then desperation begins setting in, and I start looking for just about anything else that will work. My first choice for assignment is |(, for two reasons: 1) it has the simplest comma ratio of any of these (5:7), and 2) its rational complement has the same flags as the 13 diesis (which takes advantage of an opportunity to match flags in the half-apotomes). If |) is valid as both the 5:7 and 11:13 commas, then I will almost certainly assign it for the notation (and definitely if the 17'-17 comma is also the same number of degrees. Otherwise I will defer assignment of this symbol until I have evaluated the other alternatives. When I assign a comma, I will also assign its rational complement, in this case /||), if it is valid for the division. My next choice is (|(, which is the next simplest comma (5:11), which I will also check to see if it is valid in its other major role as the 7:13 comma. I will also check to see if its unidecimal-diesis complement ~|( is valid as /|\ minus (|(. If all of these are valid, I will assign both of these symbols. Otherwise the decision is deferred. Next will be //|, which I will consider similarly. As we discussed, the assignment of this symbol depends upon its being valid as the 5+5 comma. I will defer assignment if it is not also valid as both the 25 comma (i.e., 1,5,25 consistency) and the 5:13 comma. I will also check to see if ~| is valid as /|\ minus //|. If all of these are okay, then I will assign both //| and ~|, as well as their rational complements. So if there is a choice between (|( and //|, for example, it will come down to how many of their assigned roles they are able to play. The above order causes the 17 comma ~| to be considered after the 17' comma ~|(. Even though the 17-comma symbol is simpler in appearance, I consider the two to be approximately equal in priority, differing only in whether a note such as 17/16 is going to be notated as an altered sharp or flat. As I go down the list I find that the less desirable symbols have not only fewer but also less important roles to play, which makes their validation both easier and less critical. It is important to observe whether alternate interpretations of certain flags such as |) or )| result in different numbers of degrees and to make the symbol assignment arithmetically consistent. If satisfactory rational and unidecimal-diesis complements are not valid for a division, then I look for alternate complements that minimize the number of flags. In general, I would seek to retain symbols that are valid in all (or at least the most important) of their comma-roles and to replace the ones that don't fulfill those roles with alternate complement symbols. In setting up a spreadsheet to make these evaluations, I have not attempted to evaluate divisions differing by 7 simultaneously, so the process does not attempt to assign these divisions the same set of symbols. One thing that *does* result from this is that a division is not forced to accept a less desirable set of symbols that would be shared with a second division if a better set is possible for the first one. (This principle comes into effect in evaluating 87 vs. 94, discussed below.) Keeping these things in mind, I will now consider the following.>> 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.I really wasn't very happy with any of the choices for 3deg80. I agree that (|( is definitely the most familiar symbol, but I place a higher value on ratios of 13 than you do, and I wanted to use it only if it is valid as both a 5:11 and 7:13 comma. But the alternatives aren't really any better, so I guess I can go along with this. The choice that I made had something to do with what I have to say next. The problem I had with |) for 1deg80 was only indirectly related to its size relative to /|: this unusual placement results in using ||) for 8deg as a rational complement -- a two-degree discrepancy in symbol arithmetic (whereas I wanted to allow no more than one degree off, as we allowed for 72). I justified using the 19 comma because it's better represented in this division. That caused me to use (||~ as its rational complement for 8deg, and I used the (|~ for 3deg because it matched. With your proposal I don't know what to do for apotome complements. This isn't a very good division, and I personally don't care very much what we use for it. With so many problems involving the more familiar symbols, my solution was to use less familiar ones. I guess you could say that I thought that the division and the symbols deserved each other! So unless you have any more ideas, a decision on this one would best be deferred.> ...>> 87a: |~ /| ~|) /|\ (|) ||~ ||\ ~||) /||\ (RC) >> 94a: ~|( /| (|( /|\ (|) ~||( ||\ (||( /||\ (RC) >> 87b, 94b: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ (MM) >> 87c, 94c: |~ /| /|~ /|\ (|) ||~ ||\ /||~ /||\ (MM) >> 87d, 94d: |~ /| /|~ /|\ (|) ~|| ||\ ~||\ /||\ (MM) >> I'd prefer the single-shaft symbols to be > 87e, 94e: ~| /| (| /|\ (|) ? ||\ ? /||\Since |\ is not used, there is no opportunity to have matching symbols in the half-apotomes, so I assumed that rational complementation should be the organizing principle, if possible. For what you have, the following rational complements would be indicated: 87e, 94e: ~| /| (| /|\ (|) )|~ ||\ //|| /||\ Neither )|~ nor //|| is the correct number of degrees for the flags, whereas my 87a and 94a choices were determined on the basis of which pairs of symbols would work best as rational complements. However, I can appreciate your desire that the single-shaft symbol choices not be compromised by the need to get rational complements, so I will plead my case on that basis. For 1deg87 I now see that |~ is a rather poor choice; the only advantage it had was that it had a valid rational complement. But I won't pursue that any further. For 1deg94 ~|( is not as simple a symbol as ~|, but the two different 17 commas are equally useful. I chose ~|( because it has a good rational complement in 94, which, however, is of no use for 87. I would have to agree on ~| for 87, but I think ~|( is better for 94. However, I will keep ~| for 94 for now so I can continue to discuss the two divisions together. For 3deg I think that (|( has a distinct advantage over (| because it will be a more frequently used symbol (e.g., as one of those in the 217 standard set), especially since it is valid as *both* the 5:11 and 7:13 commas in *both* 87 and 94), whereas (| represents only the 7:11 comma. Besides this, its rational complement ~||( avoids the |~ flag in the notation, introduces no other additional flags, and is the correct number of degrees in both 87 and 94. So this would give us: 87f, 94f: ~| /| (|( /|\ (|) ~||( ||\ //|| /||\ (RC) The only problem I have with this is whether we can get away with forcing //|| as 8deg. If not, then I would use ~||\ as an alternate complement (valid in both 87 and 94): 87g, 94g: ~| /| (|( /|\ (|) ~||( ||\ ~||\ /||\ (8degAC) But if I consider 94 apart from 87, I would prefer my first version, because all of the flag usages, comma roles, and rational complements are free of any problems: 94a: ~|( /| (|( /|\ (|) ~||( ||\ (||( /||\ (RC) Should we let the lesser division drag the better one down?>>> 99a: |~ /| ~|) /|) (|~ (|\ ||~ ||\ ~||) /||\ (RC) >> 99b: ~| /| ~|\ /|) (|~ (|\ ~|| ||\ ~||\ /||\ (MM) >> 99c: |~ /| /|~ /|) (|~ (|\ ||~ ||\ /||~ /||\ (MM) >> I prefer 99a. Okay!>> 104a: )| |) /| (| /|\ (|) )||~ ||\ ||) (||~ /||\ [|~as 23 comma] (RC)>> 104b: )| |) /| (|~ /|\ (|) )|| ||\ ||) (||~ /||\ [|~as 23 comma] (RC)> > I prefer 104a. Okay!>> 108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ (RC) >> 108b: /| |( |) /|) (|\ ||) )||) ||\ /||\ (MM) >> I prefer 108a. Okay!>> 111: ~| /| |\ ~|\ /|\ (|) ~|| /|| ||\ ~||\ /||\ > > I prefer> 111b: ~| /| |\ //| /|\ (|) ~|| /|| ||\ //|| /||\ > which is the same as 118-ET below.Here //| is valid as the 5+5 and 7:13 commas, but 111 is not 1,5,25 consistent. My choice of ~|\ was based on using no new flags in addition to the fact that it is valid as the 23' comma. However, I am willing to go with 2 valid out of 3 comma roles for //|, plus the fact that it is also valid as the rational complement of ~||, and therefore //|| as RC of ~|. So 111b it is, with rational complementation and matching symbols!>> 118: ~| /| |\ //| /|\ (|) ~|| /|| ||\ //|| /||\ > > Agreed. >>> 120: /| (| |) /|) /|\ (|\ ||) )||~ ||\ /||\ > > Agreed. >>> 125: ~|( /| |\ (|( /|\ (|) ~||( /|| ||\ (||( /||\ > > I prefer> 125b: |( /| |) //| /|\ (|) ~|| ||) ||\ /||) /||\I notice that I passed over |(, which isn't valid in the secondary role as the 11:13 comma, yet I used (|(, which isn't valid in the secondary role of 7:13 comma, so I see that wasn't the reason for my choice. I now see that my objective was to have both matching symbols and rational complementation. In both of our versions rational complementation is maintained, but you forsook matching symbols by using a 7-comma symbol. I made it a principle that, if there were over 10 symbols to the apotome, that matching symbols should be used wherever possible. So now what is your preference?>>> 128a: )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (||( (||~ >> /||\ (RC) >> 128b: )| ~|( /| (|( ~|\ /|\ (|) )|| ~||( ||\ (||( ~||\ >> /||\ (MM) >> I prefer 128a. Okay.>> 130: |( /| |) |\ /|) /|\ (|\ /|| ||) ||\ /||) /||\ > > Agreed. >>> 132a: ~|( /| |) |\ (|~ ~||( /|| ||) ||\ (||~ /||\ (MS) >> 132b: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||~ /||\ (MS) >> I prefer 132b, but why not |( as 5:7-comma for 1deg132?I try to choose symbols that are as valid in as many roles as possible. |( is valid only as the 5:7 comma and not as the 11:13 or 17'-17 commas (1 out of 3), whereas ~|( needs to be valid only as the 17' comma (1 out of 1). This is another one that I don't have strong feelings about, and in the course of working on the spreadsheet I might change my mind. Even if we don't get any final agreement at this point about some of these less common divisions, at least our discussion of these will provide some examples from which I can arrive at general principles for choosing symbols.>>> 135a: ~| |~ /| (| /|~ /|\ (|) ~|| ||~ ||\(|| /||~ /||\ (MM)>> 135b: ~| ~|( /| (| /|~ /|\ (|) ~|| ~||( ||\(|| /||~ /||\ (MM)> > I prefer 135a. >>> 140 (70 ss.): )| ~|( /| )|\ ~|\ /|) (|~ (|\ )|| ~||(||\ )||\ ~||\ /||\ (MM)> > I prefer > 140 (70 ss.): )| ~| /| )|) ~|\ /|) (|~ (|\ )|| ~|| ||\ )||) ~||\ /||\ I guess that makes the 4deg and 5deg symbols easier to distinguish. Okay!>> 142: )| /| |) |\ /|) /|\ (|) (|\ /|| ||) ||\ /||) /||\ >> )| is wrong for 1deg142. How about > 142b: |( /| |) |\ /|) /|\ (|) (|\ /|| ||)||\ /||) /||\ (RC & MS) Yes, what you have is what I meant. I must have hit a wrong key by mistake.> ...>> 152a: )| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\(||( /||) /||\ (MS; 14deg AC)>> 152b: )| |~ /| |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||\~||) /||) /||\ (MS; 14deg AC)>> 152c: )| ~| /| |\ ~|) /|) /|\ (|) (|\ ~|| /|| ||\~||) /||) /||\ (MS; 10,13,14deg AC)> > I prefer 152b.I'm rather surprised by your choice -- one that uses both wavy flags and that prefers the 23 comma over either of the 17 commas. It looks very much like the set you chose in your message (#4272) of 15 May (which you quickly revised). So you need to explain this one to me. I discussed the above options in a previous message (#4596 of 28 Aug), which I will repeat here (with comma designations updated): << In version a, (|( as 6deg152 is valid as the 5:11 and 11:17 commas, but not the 7:13 comma. The replacements in version b result in higher primes and more flags; here ~|) is valid as both the 7+17 and 5+17 (or 5:17) commas. Version c uses the simplest matching symbols, and I am inclined to go with that. (I have reached the conclusion that if a set of symbols isn't close to flawless with rational complements, then we should just go for the most memorable set, with matching symbols in the half-apotomes where possible.) >> My thinking about this is still the same.>>> 159: |( ~|( /| |\ ~|\ /|) /|\ (|) (|\ ~||( /|| ||\~||\ /||) /||\> > I prefer > 159: ~| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ (||( /||) /||\ (RC & MS) The (| flag is not the same number of degrees in (|( and (|\, so (|( is not valid. I prefer |( because it is valid as the 5:7, 11:13, and 17'-17 commas, hence is more desirable for its lower-prime applications than a 17- comma symbol. In addition, it is consistent as the rational complement of /||). Neither of our options has rational complementation throughout.>> 171: |( ~|( /| |) |\ ~|\ /|) /|\ (|\ ~||( /|| ||)||\ ~||\ /||) /||\> > I think I prefer > 171b: |( ~|( /| |) |\ //| /|) /|\ (|\ ~||( /|| ||)||\ //|| /||) /||\ Okay!>> 176a: |( |~ /| |) |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||)||\ ~||) /||) /||\ (RC & MS)>> 176b: |( ~| /| |) |\ ~|) /|) /|\ (|) (|\ ~|| /|| ||)||\ ~||) /||) /||\ (MS & MM)> > Of those two, I prefer 176a, but I like these single-shafters better > 176c: |( |~ /| |) |\ //| /|) /|\ (|) (|\ > 176d: |( ~| /| |) |\ //| /|) /|\ (|) (|\This is another of the half-dozen larger divisions in which it is possible to have both matching symbols and complete rational complementation (version 176a), but it is at the price of using a couple of relatively unimportant symbols. Evidently you didn't care too much for them. Your versions differ only in using //| for 6deg. This time for //| it's only 1 out of 3: as the 5+5 comma, but not as the 25 or 5:13 commas. For the more nondescript symbol ~|) it's 1 out of 2: as the 7+17 comma, but not as the 5:17 comma; but this is of little significance -- it's just a symbol to match ~||), the rational complement of |~. For (|(, a symbol that neither of us chose, it's 3 out of 3: as 5:11, 7:13, and 11:17 commas, but its unidecimal-diesis complement ~|( does not have the same number of degrees for the |( flag, so ~|( can't be used. With this many degrees in the apotome I thought it advisable to use matching symbols, so if I were to pick the best single-shaft symbols and duplicate the flags in the double-shaft symbols, I would have this: 176e: |( ~| /| |) |\ (|( /|) /|\ (|) (|\ ~|| /|| ||) ||\ (||( /||) /||\ (MS) On the other hand, using the same single-shaft symbols along with their rational complements would give this: 176f: |( ~| /| |) |\ (|( /|) /|\ (|) (|\ ~||( /|| ||) ||\ //|| /||) /||\ (RC) I'm beginning to wonder whether it would be more meaningful to have rational complements (instead of matching flags) for the double-shaft symbols whenever there is a good set of single-shaft symbols. (I'll have to try experimenting with the second half-apotome of some of these larger divisions to see how often that will work without the symbol arithmetic going to pieces.) Anyway, what do you think of the single-shaft symbols in those last two?>> 181a: |( ~| |~ /| /|( ~|) /|~ /|) (|~ (|\ ||( ~||||~ ||\ /||( ~||) /||~ /||\ (MM)>> 181b: |( ~| ~|( /| /|( (| /|~ /|) (|~ (|\ ||( ~|| ~||( ||\ /||( (|| /||~ /||\ (MM)> > These are both wrong if |( is the 5:7 comma, since the 5:7 comma vanishes > in this tuning.You're right; what was I thinking of, anyway?> I prefer > 181c: )| ~| |~ /| )|) (| /|~ /|) (|~ (|\ )|| ~|| ||~||\ )||) (|| /||~ /||\ (MM) It appears that you're just trying to minimize the number of flags. However, |~ is not the 23 comma here, but that's what the symbol is supposed to indicate. I would rather use something else for 3deg. The best choice appears to be ~|(, which adds the |( flag back into the notation. With that, there doesn't seem to be any point in replacing /|( with )|). So now I get this: 181d: )| ~| ~|( /| /|( (| /|~ /|) (|~ (|\ )|| ~|| ~||( ||\ /||( (|| /||~ /||\ (MM)>> 183: |( ~|( /| |) |\ (|( /|) /|\ (|) (|\ ~||( /||||) ||\ (||( /||) /||\> > I prefer > 183b: |( ~|( /| |) |\ //| /|) /|\ (|) (|\ ~||( /||||) ||\ //|| /||) /||\ For 6deg my decision is a matter of which symbol is valid in the greater number of roles. For //| it's 2 out of 3: as 5+5 and 25 commas, but not as 5:13. For (|( it's 3 out of 3: as 5:11, 7:13, and 11:17 commas. That, plus the fact that ~|( <--> (||( and (|( <--> ~|| ( are rational complements, makes this one of the few larger divisions that can have both matching symbols and complete rational complementation.> ...>> 193: )| ~| ~|( /| |\ ~|) ~|\ /|) /|\ (|) (|\ ~|| ~||( /|| ||\ ~||) ~||\ /||) /||\> > I prefer > 193b: )| ~| ~)| /| |\ (| ~|\ /|) /|\ (|) (|\ ~|| ~)|| /|| ||\ (|| ~||\ /||) /||\> 193c: )| ~| ~|( /| |\ (| ~|\ /|) /|\ (|) (|\ ~|| ~||( /|| ||\ (|| ~||\ /||) /||\ Yes, (| will work here. I prefer 193c.> ... > 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.Yes, I think that there are too many problems.> > 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: )| |( )|( ~|( /| ~|~ |) (| |\ //| ~|\ /|)(|~ /|\ (|)> ~| ~)| |~ )|) ~|) (|( |~)(|( is a better choice than //| for the comma roles it fulfills. (|~ and ~|~ look like they may be a little shaky in the flag arithmetic for |~. (A wavy flag becomes a shaky flag?)> > 318 is notatable if you accept (/| (the 31' comma) for 15 steps.Neither 306 nor 318 are 7-limit consistent, so I don't see much point in doing these, other than they may have presented an interesting challenge. Anyway, thanks for going over all of these. I'll try to answer more of your messages before time runs out. --George

Message: 5220 - Contents - Hide Contents Date: Wed, 18 Sep 2002 18:41:44 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4599]:> At 12:47 PM 30/08/2002 -0700, George Secor wrote:>>> In that case, the largest number of steps to need a single-shaft symbol in>>> an ET is given by >>> =TRUNC(MAX(steps_in_tone, steps_in_diatonic semitone)/2) >>> in some cases the largest number of steps will be catered forby the # or b>>> itself. >>> -- Dave Keenan >>>> I don't understand this at all. For 43, steps_in_tone=7 and >> diatonic_semitone=4, for which your formula gives 3. Did you mean >> TRUNC(MAX(steps_in_tone/2, steps_in_diatonic_semitone)/2), for which >> your formula gives 2? (However, I found that doesn't work either, >> because it gives 1 for 27, 34, and 41-ET, but we want 2.) >> TRUNC(steps_in_apotome/2), which gives 1, is what I think it should be; >> we can still notate 43 with single-shaft symbols using only |): >> >> 0 1 2 3 4 5 6 7 >> >> C C|) C#!) C# C#|) Cx!) Cx >> Dbb Dbb|) Db!) Db D!) D >> >> This is how it would be with the 13-comma symbols: >> >> C C/|) C(|\ C# C#/|) C#(|\ Cx >> Dbb Db(!\ Db/!) Db(!\ D/!) D >> >> I don't recall that we previously objected to having a 7 comma alter in >> the opposite direction in combination with a sharp or flat. >> >> So I am at a loss as to what to do. > > Sorry George, >> I screwed up. You nearly got it. What I meant to say was > =TRUNC(MAX(steps_in_apotome, steps_in_Pythagorean_limma)/2) > > apotome = 2187:2048 > Pythagorean limma = 243:256 > (i.e. the Pythagorean versions of the chromatic and diatonic semitones) > > and sure, it doesn't matter if you put the divide-by-twos before the MAX. > And there's certainly no objection to having a 7 comma alter in > the opposite direction in combination with a sharp or flat. > > By the way, you left out the Db|) in your first example and the Db in your > second. > > The way of thinking that will favour using saggitals in combination with # > and b, is one that thinks of C# as a single symbol, and would rather not > have to accept Db as being a different pitch. In this person's mind there > are not 7 but 12 basic symbols which are to be modified by the saggitals. > For example, when the key is nominally C or Am then the 12 symbols are Eb > Bb F C G D A E B F# C# G# > > So it could be: > > 0 1 2 3 4 5 6 7 > > C C|) C#!) C# C#|) C#(|\ > D(!/ D!) D > > So you see it's the 4 step _limma_ (between C# and D) that causes the > problem here. Similarly: > > 0 1 2 3 4 > > B B|) B(|\ > C(!/ C!) C > > -- Dave Keenan > Brisbane, Australia--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4600]:> I wrote: > > "The way of thinking that will favour using saggitals incombination with #> and b, is one that thinks of C# as a single symbol, and would rather not > have to accept Db as being a different pitch. In this person's mind there > are not 7 but 12 basic symbols which are to be modified by the saggitals. > For example, when the key is nominally C or Am then the 12 symbols are Eb > Bb F C G D A E B F# C# G#" > > I should have said "_One_ way of thinking that will favour using sagittals > in combination with # and b ...", since some folks will prefer it even > though they don't prescribe to this way of thinking. However I think that > many trained musicians, who have never before had to deal with tunings > other than 12-ET, will think this way, in particular keyboard players and > players of other fixed pitch instruments where all 12 equally- spaced > pitches are almost equally playable. I became convinced of this through > discussions with Paul Erlich and Joseph Pehrson. > > It's clear that you and I have trouble seeing things from this perspective, > immersed as we have been, in tuning theory, for many years. > > I realised after sending the previous message that I have not followed it > consistently either. A person who does not want to see C# and Db as > different pitches (and therefore should use only one of them at a time to > avoid inconsistencies) will need a single shaft symbol for > TRUNC(steps_in_Pythagorean_limma/2) even if this is the same as > steps_in_apotome and could therefore be symbolised by # or b, e.g in 19-ET, > 26-ET, 38-ET and 45-ET. > > I certainly wouldn't expect you to _replace_ /||\ and \!!/ with single > shaft symbols in these (the extreme meantones), but I do feel that we must > provide single-shaft _alternatives_ for them, when used with a > chain-of-twelve-fifths basis (as opposed to a chain-of-seven- fifths). The > same goes for 2deg43, with an alternative to ||). > > (|\ is a sensible alternative for 1deg19 and 1deg26, but 2deg38 presents a > problem. I can find no consistent candidate below the 23 limit, but it > seems like we should use (|\ on the basis that 2deg38 is the same as 1deg19. > > |) is 2deg45 but it doesn't seem wise to use this symbol for something that > large and again I fall back on (|\. Neither 38 nor 45 are > 1,3,13-consistent, but a 2 step shift does at least give the best 3:13 in > both cases. > > A single shaft alternative for ||) as 2deg43 is no problem. It's fine to > use both |) as 1deg43 and (|\ as 2deg43, since the 13-schisma vanishes. > > 2deg50 is already the single-shaft (|\ as standard. > > (|\ also works for 3deg62, 3deg67, 3deg69, 3deg74, 4deg86, 4deg91. > > But I can't see any possibility of meaningful single-shaft alternatives for: > 3deg52, 3deg57, 3deg64, 4deg76, 4deg81, 4deg88, 4deg93 etc., so I'm > prepared to give up on them. These ETs are all 1,3,9-inconsistent and will > be better notated as subsets anyway. > > Here's a proposed rule: > if TRUNC(steps_in_Pythagorean_limma/2) > TRUNC(steps_in_apotome/2) then > the alternative single-shaft symbol for > degree[TRUNC(steps_in_apotome/2) + 1] is (|\. > > Here's a slightly more restrictive version of it. > > if TRUNC(steps_in_Pythagorean_limma/2) - TRUNC(steps_in_apotome/2)= 1 then> the alternative single-shaft symbol for > degree[TRUNC(steps_in_Pythagorean_limma/2)] is (|\. > > Let me know what anomalies these produce, if any. I think 93-ET (3*31) > might be a problem. > > -- Dave Keenan > Brisbane, Australia Dave,On first reading these two messages, I found it a bit difficult to follow your line of reasoning, and I put them aside because we had other things to deal with. After reading through them again a couple of times (to make sure I understand you correctly), I'm ready to throw up my hands. I never imagined that anyone would have a problem with the notation of 19-ET, but now you're saying that sometimes a sharp won't do for 1 degree, so we will need a sagittal symbol for this in addition (and also for some other divisions). Okay, I can go along with that, but then my question is, why does it have to be a single-shaft symbol, for which we may have to take extraordinary measures to justify? Why not use a double-shaft symbol instead? To us it may seem strange to blend the single and double- symbol versions of the notation, but if we are going to have to deal with the problem that some people find it difficult to accept the fact that sharps and flats can differ in pitch, then why do we have to bend over backwards catering to their difficulties by using single- shaft symbols in highly unorthodox ways when we already have another alternative available. Won't /||\ serve at least as well as (if not better than) (|\ for 1deg19? And why not use /||\ for 3deg57, or /||) for 4deg81, or even (|||( for 3deg52? To anyone new to this notation they're new symbols, just like the others. (Or are you so set against the single-symbol notation that you'll go to great lengths to avoid it, even if it makes a lot of sense to use it for this?) I rest my case. By the way, in looking at some of the divisions you mentioned I happened to notice 100-ET: 100-ET (apotome=6, limma=10) requires 5 symbols 100: )|) /|) )|\ (|\ )||\ /||\ We're also doing 100 as a subset of 200, but I didn't give a notation for 200, so here it is: 200: |( ~| |~ /| |\ ~|) ~|\ /|) /|\ (|) (|\ ~|| ||~ /|| ||\ ~||) ~||\ /||) /||\ (MS) --George

Message: 5221 - Contents - Hide Contents Date: Wed, 18 Sep 2002 21:28:09 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> At 10:14 AM 27/08/2002 -0700, George Secor wrote:>> ... I thought that it was most productive to start with >> rational intervals, find the most useful schismas that can vanish, and >> then look for ETs that are consistent with those schismas. Working >> backwards by starting with a large-number ET and then finding the >> schismas that vanish in that ET is something that I don't have much >> experience with, and I have a feeling that we're not going to find >> anything better in 282 that will be useful in devising a notation that >> offers a better economy of symbols. >> I dusted off a spreadsheet I made way back near the start of this project. > It comes at it from the direction you suggested. I figure a schisma is > unlikely to be useful for notation if any prime has too high apower or if> it involves too many primes (with non-zero powers). So I first found all > the 31 limit schismas smaller than 1 cent that have no exponent with an > absolute value greater than 1 for the primes 7 thru 31, and none greater > than 2 for the prime 5. I then whittled that down to those wherethe sum of> the absolute exponents of the primes 5 to 31 is no greater than 4. I then > look at a selection of ETs to see in which of them each schisma vanishes. > Let me know if you want a copy of it.I'd like a copy of it. --George

Message: 5223 - Contents - Hide Contents Date: Thu, 19 Sep 2002 14:43:55 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> At 10:24 AM 13/09/2002 -0700, George Secor wrote: >>> ET Notation Proposals >> ---------------------I missed one of these:>>> 135a: ~| |~ /| (| /|~ /|\ (|) ~|| ||~ ||\(|| /||~ /||\ (MM)>> 135b: ~| ~|( /| (| /|~ /|\ (|) ~|| ~||( ||\(|| /||~ /||\ (MM)> > I prefer 135a.Now that I've had a chance to look this over again, I will agree with you for the single-shaft symbols in 135a, but I want to change the others to this: 135c: ~| |~ /| (| /|~ /|\ (|) )||( )||~ ||\ ~||) //|| /||\ (RC/AC) Since there is no opportunity to match the symbol sequence, we should be trying to maximize the rational complementation. I realized that the rational complement of (| is )||~, not ~||(, and that the RC of |~ can also be used: ~||). The symbol arithmetic is correct for both of these. For the rational complement of ~|, I am using //||; the symbol arithmetic is not correct for the two straight flags, but there is no /|| symbol here to conflict with it, so I think I might be able to justify forcing the symbol into use here, as I attempted with 87 and 94 a couple of messages ago: 87f, 94f: ~| /| (|( /|\ (|) ~||( ||\ //|| /||\ (RC) For 5deg135 there is no choice but to use /|~. This doesn't have a rational complement, and the two alternate complements nearest in size, ||~ and )||~ are the wrong number of degrees, and you will notice that )||~ is already being used. That leaves, in order of nearest size, )||(, ~||, and ||(. I hesitate to use ~|| because it gives the impression that it would represent /||\ minus //|, which is not valid as an apotome less either a 5+5 or 5:13 comma. I decided to use )||(, not only because it has the closest size, but also because its obscurity is comparable to that of /|~, i.e., both symbols are practically meaningless from a harmonic standpoint, so it is fitting that they complement one another. This sort of complementation is a bit different from what I have previously done. I am starting to concentrate more on the goal that the double-shaft symbols should function in more of these divisions as true rational complements, so that they can be remembered by their association with a harmonic function rather than their position in a symbol sequence. I am beginning to see that, as we did for ||) in 72- ET, an occasional single-degree discrepancy in symbol arithmetic can be tolerated, as long as it is not so noticeable as to be disruptive. Do you think I'm on the right track here? I just read your message #4662, so I want to quote and respond to the following portion before sending this: --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> At 06:19 PM 17/09/2002 -0700, George Secor wrote:>> ... So this would give us: >> >> 87f, 94f: ~| /| (|( /|\ (|) ~||( ||\ //|| /||\ (RC) >> >> The only problem I have with this is whether we can get away with >> forcing //|| as 8deg. If not, then I would use ~||\ as an alternate >> complement (valid in both 87 and 94): >> >> 87g, 94g: ~| /| (|( /|\ (|) ~||( ||\ ~||\ /||\ (8degAC) >> >> But if I consider 94 apart from 87, I would prefer my first version, >> because all of the flag usages, comma roles, and rational complements >> are free of any problems: >> I accept 87g.The above discussion of 135 assumed that the relaxing of symbol arithmetic to justify the use of //|| as a rational complement of ~| in 87 would be acceptable. I really would prefer 87f, because I think that //|| as /||\ minus ~| is a much more meaningful symbol than ~||\. So does your response mean that you didn't accept 87f? And after this further discussion is that still the case? (If this has any bearing on the matter, I read this:> So we see that the addition of that one new symbol |~) for the 13:19 comma > and the acceptance of a fuzzy right wavy flag, lets the maximum notatable > ET leap from 217 to 494, more than double!and am favorable to allowing fuzzy-right-wavy-flag logic if that will give us 494.) --George

Message: 5224 - Contents - Hide Contents Date: Thu, 19 Sep 2002 21:12:19 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote (#4662):> At 06:19 PM 17/09/2002 -0700, George Secor wrote:>> From: George Secor (9/17/02, #4626) >> Subject: A common notation for JI and ETs >> >> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>>> At 10:24 AM 13/09/2002 -0700, George Secor wrote:>>>> ET Notation Agreed Upon >>>> ----------------------- >>> ...>>> I also realise we need to say _which_ subset to use. I think we should >>> always specify the subset that contains D natural, for reasonsI expect are>>> obvious to you. >>>> Well, D is the center of symmetry for the 7 naturals. But pitch >> standards are usually set for A or C, so how did you intend to handle >> that if the ET doesn't have either A or C in the notation? Do you have >> a particular pitch standard for D in mind that the rest of the world >> might be willing to accept? >> Yes, the D of 12-equal when its A is 440 Hz.An irrational-number frequency as a tuning standard? I thought that we could do better than that.>> (Come to think about it, D isn't a bad >> choice for a pitch standard once you consider the 5 notes corresponding >> to the open strings of the violin family.) >> Yes except cello is CGDA (not a problem). Guitars are GDAEB (not a problem).I was thinking of the violin family overall: CGDAE, for which D is in the middle.>>>>>>> ET Notation Proposals >>>> --------------------- >>>> Before I comment on any particular division, I want to discuss some of >> the principles which I used to select some of the symbols. >> >> I first assigned the 5 comma, 11 diesis, 7 comma, and 13 diesis, where >> possible, along with their respective rational complements (including >> the 11' and 13' dieses). For the larger divisions (for which we would >> want matching symbols in the half-apotomes) I also assigned the 11- 5 >> comma, where possible. >> >> For the remaining degrees I evaluated other symbols in the following >> order for their suitability (along with their rational complements). >> Except for the first one, I have put these in pairs, which facilitates >> the process of achieving matching symbols in the half-apotomes. >> >> |( <--> /||) 5:7 comma and 11:13 comma (also 17'-17) >> >> (|( <--> ~||( 5:11 and 7:13 comma (also 11:17) >> ~|( <--> (||( 17' comma >> >> //| <--> ~|| 5+5, 25, and 5:13 comma >> ~| <--> //|| 17 comma >> >> (| <--> )||~ 7:11 comma (also 13:17) >> )|~ <--> (|| 19' comma >> >> )| <--> (||~ 19' comma >> You mean 19 comma.Right. For something like this I frequently copy and paste a previous line, then edit it, which works fine as long as I edit everything that needs to be edited. This time I unfortunately find that I was unknowingly past my prime.>> (|~ <--> )|| various complex dieses >> The 11:19 comma, 171;176, is probably the simplest of them. But really this > symbol is just the half-apotome of last resort. I agree it's pointless to > give its comma value. We can always come up with one if challenged. >>> |~ <--> ~||) 23 comma > > Also 19'-19.I would prefer not to use this flag alone as the 19'-19 comma, because it's not going to find any practical use that way and will compromise the meaning of this symbol as the 23 comma. This is the same reason I don't want to see |) by itself as the 13-5 comma (if it's not also valid as the 7 comma) or |( as the 17'-17 comma (if it's not also valid as the 5:7 comma).>> ~|) <--> ||~ 7+17 commaI should start calling this the 5:17 comma.>> ~|\ <--> ~)|| 23' comma >> Also the 11':19' comma, 297;304.Although this symbol can be used to notate 19/11 (as A~|\ for C=1/1), it won't be used very often, since it treats 11:19 as a sixth (which corresponds to 16:19 as a raised second) rather than the more usual seventh, as Bb(!~, which corresponds to 16:19 as a lowered third. And at least half of the good higher-numbered ETs don't even allow the symbols to be used for this, including 217, 224, 311, and 494. So I'm not giving that role much importance.>> ~)| <--> ~||\ 17+19 comma >> Can also be described as 17:19 comma for what that's worth.Yes, although that way it is actually the 19'-17 comma (152:153, ~11.352 cents), which is not necessarily the same number of degrees as the symbols would indicate (e.g., different in 217 and 311, but same in 270 and 494).>> /|( <--> ~||~ 5+(17'-17) comma >> ~|~ <--> /||( 17+23 comma >> Primarily the 5:19 comma.For ~|~, that is. I'll have to get in the habit of using the names based on the symbol's practical harmonic function.>> If all of the degrees aren't assigned by this point, then desperation >> begins setting in, and I start looking for just about anything else >> that will work. >> >> My first choice for assignment is |(, for two reasons: 1) it has the >> simplest comma ratio of any of these (5:7), and 2) its rational >> complement has the same flags as the 13 diesis (which takes advantage >> of an opportunity to match flags in the half-apotomes). If |) is valid >> as both the 5:7 and 11:13 commas, then I will almost certainly assign >> it for the notation (and definitely if the 17'-17 comma is also the >> same number of degrees. Otherwise I will defer assignment of this >> symbol until I have evaluated the other alternatives. When I assign a >> comma, I will also assign its rational complement, in this case /||), >> if it is valid for the division. > > Sounds good. >>> My next choice is (|(, which is the next simplest comma (5:11), which I >> will also check to see if it is valid in its other major role as the >> 7:13 comma. I will also check to see if its unidecimal-diesis >> complement ~|( is valid as /|\ minus (|(. If all of these are valid, I >> will assign both of these symbols. Otherwise the decision is deferred. >> Sounds good, except ... >>> Next will be //|, which I will consider similarly. As we discussed, >> the assignment of this symbol depends upon its being valid as the 5+5 >> comma. I will defer assignment if it is not also valid as both the 25 >> comma (i.e., 1,5,25 consistency) and the 5:13 comma. >> I would still assign it if it is not the 5:13 comma. 5*13 is much greater > than 1*25. And stacked major thirds are common enough that people should > get the //| symbol for them if no less-complex comma symbol can been used. > I'd also assign this before (|( since 1*25 < 5*11. >>> I will also check >> to see if ~| is valid as /|\ minus //|. If all of these are okay, then >> I will assign both //| and ~|, as well as their rational complements. >> >> So if there is a choice between (|( and //|, for example, it will come >> down to how many of their assigned roles they are able to play. >> I disagree. I think that //| is so obvious a symbol for a double 5 comma, > and double 5 commas will be in far greater demand than any ratio of 11, > that I think it should have priority. I'm even prepared to use it when it > isn't the 25-comma, i.e. when the ET isn't 1,5,25 consistent.I can hardly even imagine doing microtonality without going to at least the 11 or 13 limit, because it's there that you get the unusual intervals that make it clearly evident that this isn't a 12-tone octave you're using. But I guess I'm of the school of thought that says I want to do something different, whereas I view the need for a double 5 comma to be more in line with the need to have 5-limit harmony in better intonation. But I'm not arguing about which interpretation of a symbol is more important, because I think they're both important, since one composer may favor one and another composer the other. And likewise I think that this holds for any symbol having multiple roles, which is why I would like to choose symbols that fulfill all or most of their roles over ones that don't, so that the recommended symbol sets are the ones that are most valid for use in a truly *general* sense. In choosing so-called "standard" symbols for ETs, I don't think that we should be sending the message that these are the ones to use, and no others. All of the symbols have meanings in any ET, and perhaps we should list recommended alternate symbols below the standard ones for optional use, where appropriate and/or helpful for indicating particular harmonic functions. This practice would in fact be very useful for notating compositions that could, under certain conditions, be "ported" from one tuning to another with minimal need to make adjustments to the symbols. In summary, what I am trying to arrive at with these recommended symbol sets are the "safest" choices for the composer who doesn't care to be bothered with mathematical ratios, but just wants a decent way to get the intervals in an ET down on paper.> ...>>>> 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. >>>> I really wasn't very happy with any of the choices for 3deg80. I agree >> that (|( is definitely the most familiar symbol, but I place a higher >> value on ratios of 13 than you do, and I wanted to use it only if it is >> valid as both a 5:11 and 7:13 comma. But the alternatives aren't >> really any better, so I guess I can go along with this. The choice >> that I made had something to do with what I have to say next. >> >> The problem I had with |) for 1deg80 was only indirectly related to its >> size relative to /|: this unusual placement results in using ||) for >> 8deg as a rational complement -- a two-degree discrepancy in symbol >> arithmetic (whereas I wanted to allow no more than one degree off, as >> we allowed for 72). >> I think that's generally a good rule, except I wouldn't let complements > dictate the single-shaft symbols, so I suppose I just wouldn't use ||) as > its complement. >>> I justified using the 19 comma because it's better >> represented in this division. That caused me to use (||~ as its >> rational complement for 8deg, and I used the (|~ for 3deg because it >> matched. >> >> With your proposal I don't know what to do for apotome complements. >> This isn't a very good division, and I personally don't care very much >> what we use for it. With so many problems involving the more familiar >> symbols, my solution was to use less familiar ones. I guess you could >> say that I thought that the division and the symbols deserved each >> other! >> >> So unless you have any more ideas, a decision on this one would best be >> deferred. >> 80-ET is of interest for being the smallest 19-limit-consistent division,A dubious honor, considering that three odd harmonics (7, 9, 15) deviate by over 40 percent of a degree.> however its 7s are relatively bad, so I could accept > 80c: )| /| (|( /|\ (|) )|| ||\ (||( /||\ (MM) > 80d: )| /| (|( /|\ (|) (||~ ||\ ~||( /||\ (RC)Okay, 80d will work, except that it should be: 80d: )| /| (|( /|\ (|) ~||( ||\ (||~ /||\ (RC)> ...>>>> 125: ~|( /| |\ (|( /|\ (|) ~||( /|| ||\ (||( /||\ >>> >>> I prefer>>> 125b: |( /| |) //| /|\ (|) ~|| ||) ||\ /||) /||\ >>>> I notice that I passed over |(, which isn't valid in the secondary role >> as the 11:13 comma, yet I used (|(, which isn't valid in the secondary >> role of 7:13 comma, so I see that wasn't the reason for my choice. I >> now see that my objective was to have both matching symbols and >> rational complementation. >> >> In both of our versions rational complementation is maintained, but you >> forsook matching symbols by using a 7-comma symbol. I made it a >> principle that, if there were over 10 symbols to the apotome, that >> matching symbols should be used wherever possible. >> >> So now what is your preference? >> 125-ET has very good 7s. Wouldn't it be a travesty not to provide the 7 > comma symbol, and instead to use a symbol that only means 11 comma minus 5 > comma? Likewise, if //| fulfills all its possible roles, how can we use (|( > in its place? Okay. > Even when the mutishaft version of the notation is used, surely the > single-shaft symbols will be used more often than any others and so should > not be made less useful or memorable thereby. Isn't this just an > application of your own principle that you mustn't make the simple things > more complex in the process of making the complex things simpler?I seem to remember saying something like that.> Do what you like with the complements but I still prefer the single shaft > symbols in 125b.Suppose I give you 2 out of 3 by choosing the single-shaft symbols that are usable for all of their possible roles, along with the 7- comma, and use all rational complements: 125c: ~|( /| |) //| /|\ (|) ~|| ||) ||\ (||( /||\ (RC)> ...>>>> 152a: )| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /||||\ (||( /||) /||\ (MS; 14deg AC)>>>> 152b: )| |~ /| |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||\~||) /||) /||\ (MS; 14deg AC)>>>> 152c: )| ~| /| |\ ~|) /|) /|\ (|) (|\ ~|| /|| ||\~||) /||) /||\ (MS; 10,13,14deg AC)>>> >>> I prefer 152b. >>>> I'm rather surprised by your choice -- one that uses both wavy flags >> and that prefers the 23 comma over either of the 17 commas. It looks >> very much like the set you chose in your message (#4272) of 15 May >> (which you quickly revised). So you need to explain this one to me. >> I think I screwed up. >>> I discussed the above options in a previous message (#4596 of 28 Aug), >> which I will repeat here (with comma designations updated): >> >> << In version a, (|( as 6deg152 is valid as the 5:11 and 11:17 commas, >> but not the 7:13 comma. The replacements in version b result in higher >> primes and more flags; here ~|) is valid as both the 7+17 and 5+17 (or >> 5:17) commas. Version c uses the simplest matching symbols, and I am >> inclined to go with that. >> Now I'm liking this one. > 152d: )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ /||||\ /||~ /||) /||\ (MS)> or maybe this > 152e: )| |~ /| |\ (|( /|) /|\ (|) (|\ ||~ /|| ||\ (||~( /||) /||\ (MS) It looks as if an extraneous character got in there. For matching symbols it would have to be: 152d: )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ /|| ||\ /||~ /||) /||\ (MS) 152e: )| |~ /| |\ (|( /|) /|\ (|) (|\ ||~ /|| ||\ (|| ( /||) /||\ (MS) I'm still trying to figure out why you're using a 23 comma symbol for 2deg when either 17 comma will work, and /|~ is an even more obscure choice than ~|). Maybe you should tell me why you're liking those now. (Could it have something to do with using only one kind of wavy flag?) If I were doing it to get the most useful single-shaft symbols for each degree that fulfilled all of their comma roles and used their rational complements, then it would be one of these: 152f: )| ~|( /| |\ ~|( /|) /|\ (|) (|\ ||~ /|| ||\ (||( (||~ /||\ (RC) 152g: )| ~| /| |\ ~|( /|) /|\ (|) (|\ ||~ /|| ||\ //|| (||~ /||\ (RC) But if I replace ~|( with (|( inasmuch as it is good for 2 out of 3 roles (in addition to having good symbol arithmetic for its rational complement), then I get these: 152h: )| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ (|| ( (||~ /||\ (RC) 152i: )| ~| /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ //|| (||~ /||\ (RC) In these last two, the single-shaft symbols differ from your version e only in the 2deg position. Version h almost has matching symbols, which would probably make it easier to remember than version i. If I modify version h to give matching symbols, I get: 152a: )| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ (|| ( /||) /||\ (MS; 14deg AC) which was the one I started with some 3 weeks ago. But please let me know why you prefer the 23 comma.>> (I have reached the conclusion that if a set >> of symbols isn't close to flawless with rational complements, then we >> should just go for the most memorable set, with matching symbols in the >> half-apotomes where possible.) >> >> I think I agree with that principle. But choose the best single- shafters > first and only let complements alter that choice if it does very little damage.But now that I'm finding more leeway with the rational complement symbol arithmetic, "most memorable" is starting to translate into "most harmonically meaningful.">>>> 159a: |( ~|( /| |\ ~|\ /|) /|\ (|) (|\ ~||( /||||\ ~||\ /||) /||\>>> >>> I prefer >>> 159b: ~| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /||||\ (||( /||) /||\ (RC & MS)>> >> The (| flag is not the same number of degrees in (|( and (|\, so (| ( is >> not valid. >> >> I prefer |( because it is valid as the 5:7, 11:13, and 17'-17 commas, >> hence is more desirable for its lower-prime applications than a >> 17-comma symbol. In addition, it is consistent as the rational >> complement of /||). Neither of our options has rational >> complementation throughout. >> OK. I'll go with yours. 159a. >>>>> 176a: |( |~ /| |) |\ ~|) /|) /|\ (|) (|\ ||~ /||||) ||\ ~||) /||) /||\ (RC & MS)>>>> 176b: |( ~| /| |) |\ ~|) /|) /|\ (|) (|\ ~|| /||||) ||\ ~||) /||) /||\ (MS & MM)>>> >>> Of those two, I prefer 176a, but I like these single-shafters better >>> 176c: |( |~ /| |) |\ //| /|) /|\ (|) (|\ >>> 176d: |( ~| /| |) |\ //| /|) /|\ (|) (|\ >>>> This is another of the half-dozen larger divisions in which it is >> possible to have both matching symbols and complete rational >> complementation (version 176a), but it is at the price of using a >> couple of relatively unimportant symbols. Evidently you didn't care >> too much for them. >> >> Your versions differ only in using //| for 6deg. This time for //| >> it's only 1 out of 3: as the 5+5 comma, but not as the 25 or 5:13 >> commas. For the more nondescript symbol ~|) it's 1 out of 2: as the >> 7+17 comma, but not as the 5:17 comma; but this is of little >> significance -- it's just a symbol to match ~||), the rational >> complement of |~. >> >> For (|(, a symbol that neither of us chose, it's 3 out of 3: as 5:11, >> 7:13, and 11:17 commas, but its unidecimal-diesis complement ~|( does >> not have the same number of degrees for the |( flag, so ~|( can't be >> used. With this many degrees in the apotome I thought it advisable to >> use matching symbols, so if I were to pick the best single-shaft >> symbols and duplicate the flags in the double-shaft symbols, I would >> have this: >> >> 176e: |( ~| /| |) |\ (|( /|) /|\ (|) (|\ ~|| /|| ||)||\ (||( /||) /||\ (MS)>> >> On the other hand, using the same single-shaft symbols along with their >> rational complements would give this: >> >> 176f: |( ~| /| |) |\ (|( /|) /|\ (|) (|\ ~||( /||||) ||\ //|| /||) /||\ (RC)>> >> I'm beginning to wonder whether it would be more meaningful to have >> rational complements (instead of matching flags) for the double- shaft >> symbols whenever there is a good set of single-shaft symbols. (I'll >> have to try experimenting with the second half-apotome of some of these >> larger divisions to see how often that will work without the symbol >> arithmetic going to pieces.) >> >> Anyway, what do you think of the single-shaft symbols in those last >> two? >> I like em.I thought so. I think I'll defer a decision on the double-shaft symbols for a little while, because I don't know which ones I prefer.> ... > The following is from a different message of yours but it seemed best to > address it here. >>> By the way, in looking at some of the divisions you mentioned I >> happened to notice 100-ET: >> >> 100-ET (apotome=6, limma=10) requires 5 symbols >> 100: )|) /|) )|\ (|\ )||\ /||\ > > Agreed. >>> We're also doing 100 as a subset of 200, but I didn't give a notation >> for 200, so here it is: >> >> 200: |( ~| |~ /| |\ ~|) ~|\ /|) /|\ (|) (|\ ~||||~ /|| ||\ ~||) ~||\ /||) /||\ (MS)> > I prefer these symbols for 5 and 6 degrees because they represent much > simpler commas. > 200b: |( ~| |~ /| |) (|( ~|\ /|) /|\ (|) (|\ ~||||~ /|| ||) (||( ~||\ /||) /||\ (MS) I was about to say that your choices for these are excellent, and then I noticed that |) is 5deg by itself, but was already used as 4deg in /|). Also, |( is 1deg by itself, but in (|( it would have to vanish, because (| is already 6deg, since (|) is 10deg. Anyway, nice try! I will have to leave the rest of these (the ones above 200) until I have more time to look at them. Things start getting very complicated with these, and I don't want to draw any hasty conclusions. --George

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