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Message: 6275 - Contents - Hide Contents Date: Sun, 02 Feb 2003 08:52:33 Subject: Re: A common notation for JI and ETs From: David C Keenan Another correction. I wrote: "the consequences of mistaking a wavy flag for a straight one are not very serious musically (about 15 cents), while mistaking a triple shaft for a double is very serious (about 100 cents)." That should have been: ... while mistaking a triple shaft for a double is serious (about 50 cents). -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 6276 - Contents - Hide Contents Date: Sun, 02 Feb 2003 14:22:14 Subject: Re: A common notation for JI and ETs From: David C Keenan Some suggestions for a term that means a diesis larger than a half-apotome. biesis (contraction of "big diesis") diesoma (ending somewhat like "comma", also -oma = growth, unfortunately an unnatural one) oediesis (swollen diesis) ediasis (same as oediasis but with modern spelling) Apparently the "di" in "diesis" doesn't mean two, but is "dia" meaning "through" or "across". And "esis" means something like "into". This is gleaned from the Shorter Oxford. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 6277 - Contents - Hide Contents Date: Mon, 03 Feb 2003 19:56:48 Subject: Re: A common notation for JI and ETs From: David C Keenan Now that we have all those single ASCII characters representing the most common single-shaft sagittals, it suggests we might use that for the keyboard mapping of the font. Shift could supply the apotome complement of each symbol and Ctrl could add an apotome to those. The less common symbols would be mapped in some other way and be available when Alt was pressed. What do you think? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 6278 - Contents - Hide Contents Date: Mon, 03 Feb 2003 20:11:14 Subject: Re: A common notation for JI and ETs From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> way and we would still have the advantage, when talking about the > development of sagittal, that only schisminas vanish.hmm . . . i think we've been over this before, but for any schismina, no matter how tiny, there'll be some excellent temperament where it doesn't vanish. does this matter?

Message: 6279 - Contents - Hide Contents Date: Mon, 03 Feb 2003 23:03:02 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote: Dave, I won't be able to reply to everything now, but I will address the comma-classification terminology that you proposed, and something about the shorthand ascii notation. Re: the 5' comma, 32768:32805:> ... I think there is an advantage in calling it the > 5-schisma. We could invent another term for the sub-cent ones like > "schismina" (pron. skizmeena) literally "a small schisma".As I indicated in another message, this sounds like a good name, although I thought that we should be more specific about a couple of things. After a more careful reading of your message, I see that you've done that by proposing a boundary, which sounds okay (unless some rational interval ~1.0 cent could be specified). The only question that remains is whether others would be willing to exclude the term "schisma" as applying to intervals of less than ~1 cent, and if not, then we would have to make "schismina" a subclass of "schisma".> The advantage relates to a wider consideration - how the symbols should be > pronounced when reading them out loud. You wisely replaced all my "sharp" > and "flat" with "up" and "down" and it is natural to want the whole name to > be as short as possible to say. This leads me to prefer "55-comma" to "11-5 > comma". And those "prime"s (as in "five prime comma") sound silly and don't > really tell you anything about the size.Yes, the word "prime" leaves something to be desired, since it is also liable to be confused with a "primary" (as opposed to secondary) comma role for a symbol.> It would be good if, when there are two notational commas available for a > given prime number N (or combination of primes), the smaller is called the > N-comma and the larger the N-diesis. This already occurs in many cases, so > one can drop the "prime" for them. But in the case of N = 5 we can't do > that so it would be good to call them 5-comma and 5-schisma.Yes, there's no question that this would easily be accepted by others.> This will also work for N = 17 and 19 although in the 17 case it would be > better to call the small one the 17-kleisma. Incidentally the traditional > kleisma is the 5^6-kleisma and the "septimal kleisma" is the 7:25- kleisma. > Both of these are notated '|( so commas notated as |( (5:7) should probably > be called kleismas too.That sounds like it'll work.> If we set the cutoff between a kleisma and comma at exactly half of a > Pythagorean comma or 11.73 cents, this will work in the maximum number of > cases.That boundary comes almost exactly between two interpretations of ~) |, -- the 17:19 comma (~11.352c) and the 17+19 comma (~12.108c) or a possible alternate 5:17 comma (135:136, ~12.777c, although I see that .~|( would be much better for this last one), so we might want to adjust it somewhat (see below). But the idea of a kleisma-comma boundary is good. Recall that I had something to say not too long ago (msg. #5202, 16 Dec 2002) about boundaries. I separated the eight flags into two groups, between which your proposed boundary falls: small flags: '| )| |( ~| are the schismas and kleismas, and large flags: |~ /| |) |\ (| are the commas. A diesis would be the sum of two large flags, i.e., two commas, but a kleisma plus a comma would still be a comma. (The exception would be /|~, ~38.051c, but we aren't using it in the notation.) So the largest two-flag comma (i.e., comma + kleisma) would be ~|\, ~40.496c, and the smallest remaining two-flag diesis would be //|, ~43.013c. This is consistent with your proposed upper boundary for a comma, < 125:128, ~41.058c, so I can agree to that. This isn't actually the boundary that I suggested in that message, which was anything larger than the 5:11 comma (~38.906c), which would make ~|\ a diesis, even though it is the sum of a kleisma and a comma (which I didn't notice). But this would give us the convenience of distinguishing between a 23 comma (729:736, ~16.544c) and a 23 diesis (16384:16767, ~40.004c), even if we shifted the upper boundary for a comma to anything infinitesimally smaller than 16384:16767, i.e., for all practical purposes 40 cents. Can we justify anything this small as a diesis? Well, yes:1deg31 (~38.710c) has been called a diesis. In fact it's below all of the comma-diesis boundaries that we've proposed, but it's a tempered interval, so I don't think that we should let that bother us, since the just intervals (or dieses) that it approximates are above the boundary. So the 40-cent boundary gets my vote. Now for the kleisma-comma boundary. Let me quote from that earlier message that I mentioned above (in which I refer to a kleisma as a small comma): << Another basis for establishing a boundary between large and small commas (which agrees with this) goes back to the original definition of comma: the difference in size between the two largest steps in a diatonic tetrachord. About the smallest that these steps can get is in Ptolemy's diatonic hemiolon, where they are 9:10 and 10:11, with a comma of 99:100 (~17.399 cents). The next smallest superparticular pair are 11:12 and 10:11, making a lesser comma of 120:121 (~14.367 cents, which is not only significantly smaller than 1deg72 (~16.667 cents), but also closer in size to 1deg94 (~12.766 cents), in which system both the 5 and 7 commas are 2deg (and 120:121 is only slightly more than one-half the size of a 7 comma.) So I think this is getting a bit small to be considered a comma in the original sense. What we really need is a separate name for commas smaller than ~1deg72, and I don't think "kleisma" fills the bill. >> I made this last remark about the term "kleisma", because I had the impression that the upper limit for a kleisma should probably be smaller, but perhaps I was mistaken. (And by the way, I thought that 99:100 might be a good interval to name "Ptolemy's comma", since Pythagoras, Didymus, and now Archytas each have one. 9:10 and 10:11 is also the largest pair of superparticular ratios that are the same number of degrees in 41-ET -- hence Ptolemy's comma vanishes in 41 just as Didymus' comma does in 19, 31, and meantone. But I digress.) The point here is that I thought that the comma (120:121, ~14.367c) between the next smaller pair of superparticular ratios (10:11 and 11:12) should be smaller than the lower size limit for a comma. If they were used as the two ("whole") tones in a tetrachord, their sum would be 5:6, which would leave 9:10 as the remaining interval (or "semitone") of the tetrachord. But to have a "semitone" in a tetrachord that is larger than either of the "whole" tones is absurd, hence a practical basis for a boundary. You want the boundary to be somewhere between what we have been calling the 17 comma (~8.7c) and 17' comma (~14.730c). To accommodate both of these requirements, we could put the lower boundary for a comma at infinitesimally above 120:121, >14.37c. Would this be too large an upper limit for a "kleisma?" If so, why? If not, then this would set both the upper and lower boundaries for the term "comma" based on both historical considerations and prime- number-comma size groupings. It would also be good to have input from others regarding what the upper size limit for a kleisma should be. Merely to state that the term has not previously been applied to anything as large as 14 cents would probably not be enough to disqualify its use -- I believe that it would be necessary to demonstrate some specific reason to insist on that, just as I have given a reason for the lower limit for a more specific usage of the term "comma" such as we require. (The word could still be used in a broader sense to cover all of these categories, as has been done in the past.)> The cutoff between schisma and kleisma doesn't matter too much for this > purpose since no combination of primes ever has two useful commas in this > range. So I go to Manuel's collected interval names in the file intnam.par > that comes with Scala. Having hauled one into a spreadsheet some time ago > and sorted it by size, I find that a 3.80 cent interval is the largest > referred to as a schisma (33554432:33480783, septimal) and the smallest > called a kleisma is 4.50 cents (384:385). Halfway between the 19- schisma )| > and the 5:7-kleisma |( would be 4.57 cents, so I propose that the cutoff > should be infinitesimally below 384:385 or at 4.50 cents.That looks good as far as I'm concerned. Again, it would be nice to have input from others about this.> The best cutoff between comma and diesis for this purpose would be exactly > half a pythagorean limma or 45.11 cents. However this would omit the > 25-diesis and THE diesis (125:128) so I propose placing the cutoff > infinitesimally below 125:128 or at 41.05 cents.I already addressed this (above).> We then need an easy-to-say way to distinguish large dieses from small for > things like the 11 and 13 dieses. This includes the 35, 5:49, 625 and 13:19 > dieses. The cutoff between these is obviously at exactly a half apotome or > 56.84 cents. Yes. > Any suggestions? Is there a common suffix meaning "big" that > we can tack on to "diesis"? I suppose we could just use "diesis" and "big > diesis".This terms "great" and "small" diesis have already been used in more than one way, so we may be adding to the confusion if we use those adjectives. On the other hand, there isn't an officially accepted usage for them, so there's nothing to stop us from defining them to suit our purposes. But I wonder whether we should put the upper limit on a diesis at half an apotome (~56.843c) and use another term for anything larger. My reason for this is that by the time you reach ~63 cents (27:28, 1deg19), the interval has a melodic effect much more like a small semitone (and a very effective one at that) than a quartertone (or diesis). By contrast, the single degree of 22-ET (~54.545c) can function either as a very small diatonic semitone *or* as a quartertone (i.e., 11 diesis), so I would consider an interval of this size to be at or near the borderline. If we want a rational interval, then the upper limit for a diesis could be the 5:49 diesis (392:405, ~56.482c) that I proposed to notate the hemififth family of temperaments. (We would still need to settle whether '(/| would be its rational symbol, as well as (/| for the 49 diesis. I am inclined to go with it, if only because of its accuracy.)> The upper limit for a big diesis would be 70.17 cents for our purposes.I wouldn't put any sort of boundary there for whatever we might call this interval class, and we really don't need one there, since there will be no larger class of single-shaft symbols from which to distinguish this one. I consider the ideal melodic and most harmonically dissonant "semitone" somewhere in the range of 63-78 cents. This is actually what would more accurately be called a third- tone (1 degree of 17, 18, or 19), the sort of interval that's melodically very effective in the enharmonic genus, which is what a label for this interval range might suggest. ("Limma" won't do; that's for the chromatic genus). Or perhaps a prefix or suffix to modify the word diesis, as was done to get schismina. Any ideas?> ...>> Why don't we do the smallest commas this way: >> >> '| ' 5'-comma sharp 32768:32805 >> .! . 5'-comma flat >> )| " 19-comma sharp 512:513 >> )! ; 19-comma flat >> |( ( 5:7-comma sharp 5103:5120 >> !( c 5:7-comma flat >> >> Since the 19-comma is about twice as large as the 5' comma, it would >> be appropriate if it were to use characters indicating an approximate >> double. (If you think that the colon would be better than the >> semicolon for 19-comma-down, that would be okay with me.) And the >> 5:7 comma gets a better deal in the process. This would also allow >> us to notate three more ETs with single characters: 80, 104, and 152 >> (which I'm sure Paul would appreciate). >> I'd like to have the 19-schisma in the single-ASCII, and if so ; and " > would be the obvious choice (semicolon, not colon for reasons I gave > earlier). Okay, agreed. > But I really don't like using ( for 5:7-kleisma up. > 1. It will get missed in text (i.e. parsed as an opening parenthesis). > 2. Folks are already used to thinking of ([<{ as meaning down and )] >] as > up. Scala uses ( for diesis down. > > I thought we already agreed not to use () purely for reason 1.I didn't agree not to use ( on account of reason 1, but only because ) was not a suitable opposite. If ( were used, it would always be as the rightmost character of a symbol, in which position it would never be an opening parenthesis, whereas an opening parenthesis would always be leftmost (since it is always preceded by a space). This is similar to why a period used as the 5' comma ascii symbol would never be confused with a period ending a sentence. As for reason 2, I don't think that using ( in someone else's ascii symbol system is a good enough reason to discard something that would work so well in ours. (More about this below.)> As you say, we're scraping the bottom of the barrel. It can't be an > uppercase character. In approximate keyboard order: It can't be > `,~!|@#%^&()+-{}{}\/'.";?<>.Hey, watch your language! ;-)> It can't be qwtyuosdfhjxcvbnm. Already used or > rejected for any use. > > That only leaves $*_=:eripagklz. > > A lowercase character shouldn't be used unless it has a descender, or > no-ascender and is open at the bottom. That eliminates eaklz leaving ripg. > p and g are too big to represent something that small. Cant use $ because > it is wavy not concave. I want to reserve colon for placing between notes > to form chords. _ is obviously down, not up. = suggests nodirection and is> utterly unlike an arrow. > > That leaves *ri. > > I note that k isn't a bad looking down symbol and might be paired with p > for some use, for lack of anything else to pair it with and because p's > obvious partner b is already taken. I also note that e and a or g and a > might make a pair, and possibly $ and z. But none of these suit a small > right concave flag. > > r is more like |) or )|). i looks like an inverted ! which should at least > make it an up symbol, but I'm inclined to go with * because of its > smallness and upwardness and because it seems better to use special > characters rather than letters when possible. > > Do we want to consider something other than c for its partner? k bears a > vague resemblance to *, but it seems a bit too big. What do you think?I still think that ( and c are best. It wasn't my intention to have a notation that should indefinitely *coexist* with other notations -- I wanted a notation that would be the best one possible -- one that would, in effect, be good enough to *replace* other notations that also use 7 nominals, so there would be no need for competing systems. I see no particular reason why ( and ) should have been chosen to represent a diesis in Scala, but we have a very good reason to use ( and c for the 5:7 comma rather than something else from the bottom of the barrel that everybody will have a much harder time remembering. Supposing that we're successful in getting a lot of others to adopt our notation, we'd later regret not making the best choice from the start (and having to justify a change -- over complaints and objections, such as, why didn't we do it right the first time?). And supposing that hardly anybody uses our notation, then what does it matter what we chose? --George

Message: 6280 - Contents - Hide Contents Date: Tue, 04 Feb 2003 01:36:43 Subject: Re: heuristic and straightness From: wallyesterpaulrus carl, here's an old message where i explained the error heuristic: Yahoo groups: /tuning-math/message/1437 * [with cont.] and you can see that gene, in his reply, was the one who actually suggested the word "heuristic" in connection with this . . . --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> the heuristics are only formulated for the one-unison-vector >>>> case (e.g., 5-limit linear temperaments), and no one has >>>> bothered to figure out the metric that makes it work exactly >>>> (though it seems like a tractable math problem). but they do >>>> seem to work within a factor of two for the current "step" >>>> and "cent" functions. "step" is approximately proportional to >>>> log(d), and "cent" is approximately proportional to >>>> (n-d)/(d*log(d)). >>>>>> Why are they called "step" and "cent"? How were they derrived? >>>> that's what gene used to call them. "step" is simply complexity, >> and "cent" is simply rms error. >> Now, look here. Maybe this was obvious to everyone but me, but > a single paragraph on the derrivation each of these would have > saved us much heartache... > > "There are complexity and error heuristics. They approximate > many different complexity and error functions (resp.) of > temperaments in which one comma is tempered out, through simple > math on the ratio, n/d (in lowest terms, n > d) representing > the comma that is tempered out. > > "The complexity heuristic is log(d). It works because ratios > sharing denominator d are confined to a certain radius on the > harmonic lattice. blah blah blah > > "The error heuristic is |n-d|/d*log(d). It works because it > reflects the size of the comma, per the number of consonant > intervals over which it must vanish. That is, ratios whose > n is far bigger than d are larger, and you'll recognize the > complexity heuristic underneath. blah blah blah > > "To apply the heuristics to temperaments where more than one > comma vanishes, we might consider each of them in turn, but > we must be careful to include the difference/sum vector, > because it too must vanish. A concept called "straightness" > measures the angle between the commas on the harmonic lattice, > and therefore the relative length of the difference/sum vector. > blah blah blah" > > > Just an example, and probably contains errors. I must fly > to lunch! > > -Carl

Message: 6281 - Contents - Hide Contents Date: Tue, 4 Feb 2003 16:46:31 Subject: Re: A common notation for JI and ETs From: manuel.op.de.coul@xxxxxxxxxxx.xxx George Secor wrote:>I wanted a notation that would be the best one possible -- one that >would, in effect, be good enough to *replace* other notations that >also use 7 nominals, so there would be no need for competing systems.I see your and Dave's notation as a complementary approach to the Scala notation. You are using lots of symbols and achieve a very accurate representation. Scala on the other hand is very economical with symbols, less accurate, but much more quickly to learn in my opinion. There's value in both, no problem with a bit of competition. Manuel

Message: 6282 - Contents - Hide Contents Date: Tue, 04 Feb 2003 04:46:03 Subject: Re: heuristic and straightness From: Carl Lumma> carl, here's an old message where i explained the error > heuristic: > > Yahoo groups: /tuning-math/message/1437 * [with cont.]Great, thanks! I hadn't seen this, as "heuristic" doesn't appear in it.> and you can see that gene, in his reply, was the one who > actually suggested the word "heuristic" in connection > with this . . .I do see that... you were already using the term for the complexity heuristic at that time, right? I understand everything but a few details...>log(n) + log(d) (hence approx. proportional to log(d)Is it any more proportional to log(d) than log(n) in this case? Since n~=d?>w=log(n/d) Got that. >w~=n/d-1How do you get this from that?

Message: 6283 - Contents - Hide Contents Date: Tue, 4 Feb 2003 18:13:37 Subject: Re: A common notation for JI and ETs From: manuel.op.de.coul@xxxxxxxxxxx.xxx Gene wrote:>Any name for 100/99 should be part of a pair with 99/98. >This is a problem with "small unidecimal comma" for 99/98; >if 99/98 is "small", what is 100/99--smaller?Then we have a pair now, Ptolemy's comma and small undecimal comma. 100/99 could also be called "2nd small undecimal comma" but George's idea is better. Manuel

Message: 6284 - Contents - Hide Contents Date: Tue, 04 Feb 2003 09:33:03 Subject: Re: heuristic and straightness From: Graham Breed Carl Lumma wrote:>> log(n) + log(d) (hence approx. proportional to log(d >> Is it any more proportional to log(d) than log(n) in this > case? Since n~=d?No, and the spreadsheet sorted by d is also sorted by n. And in that case it would have been easier to go straight to log(n*d).>> w=log(n/d) > > > Got that. > > >> w~=n/d-1 > >> How do you get this from that?It's the first order approximaton where n/d ~= 1. See (8) in Natural Logarithm -- from MathWorld * [with cont.] or check on your calculator.>> w~=(n-d)/d > > Ditto.That's subtracting fractions. Did you do fractions at school? 4/3 - 1 = 4/3 - 3/3 = (4-3)/3 = 1/3 5/4 - 1 = 5/4 - 4/4 = (5-4)/4 = 1/4 n/d - 1 = n/d - d/d = (n-d)/d Graham

Message: 6285 - Contents - Hide Contents Date: Tue, 04 Feb 2003 19:10:35 Subject: Re: heuristic and straightness From: Carl Lumma>>> >nd in that case it would have been easier to go straight >>> to log(n*d). >>>> Straight to where (do you see log(n*d))? >>log(n*d) = log(n) + log(d)Of course... so we're coming from log(n*d), not going to it.>> The Mercator series?? And all the stuff on this page applies >> only to ln, not log in general (which is what I assume Paul >> meant), right? >>log2(x) = log(x)/log(2). In this case we're only estimating >the complexity, so the common factor if 1/log(2) isn't important.Ok, but I still don't get how the "Mercator series" shown in (8) dictates the rules for this approximation. -Carl

Message: 6286 - Contents - Hide Contents Date: Tue, 04 Feb 2003 10:08:37 Subject: Re: heuristic and straightness From: Carl Lumma>>> >og(n) + log(d) (hence approx. proportional to log(d >>>> Is it any more proportional to log(d) than log(n) in this >> case? Since n~=d? >>No, and the spreadsheet sorted by d is also sorted by n.So it could just as well be (n-d)/(d*log(n))?>And in that case it would have been easier to go straight >to log(n*d).Straight to where (do you see log(n*d))?>>> w=log(n/d) // >>> w~=n/d-1 >> >>>> How do you get this from that?Oh, (n/d)-1, not n/(d-1).>It's the first order approximaton where n/d ~= 1. See (8) in >Natural Logarithm -- from MathWorld * [with cont.]The Mercator series?? And all the stuff on this page applies only to ln, not log in general (which is what I assume Paul meant), right?>>> w~=(n-d)/d >> >> Ditto. >>That's subtracting fractions. Did you do fractions at school?Yes; I was still seeing n/(d-1). -Carl

Message: 6287 - Contents - Hide Contents Date: Tue, 04 Feb 2003 21:04:13 Subject: Re: heuristic and straightness From: Graham Breed Carl Lumma wrote:> Ok, but I still don't get how the "Mercator series" shown in (8) > dictates the rules for this approximation.Oh, I thought you had followed that. It's usually called the Taylor series. I don't know what Mercator's got to do with it. But anyway it's ln(1+x) = x - x**2/2 + x**3/3 + ... where x is small. Because x is small, x**2 must be even smaller, so you can use the first order approximation ln(1+x) =~ x In this case, 1+x is n/d, so x = n/d - 1 ln(n/d) =~ n/d - 1 If we really wanted the logarithm to base 2, that'd be log2(n/d) = ln(n/d)/ln(2) =~ (n/d - 1)/ln(2) or log2(n/d) ~ n/d - 1 where ~ is "roughly proportional to" which is all we need to know. And the same's true whatever base logarithm you use. Graham

Message: 6288 - Contents - Hide Contents Date: Tue, 04 Feb 2003 10:26:03 Subject: Re: heuristic and straightness From: Graham Breed Carl Lumma wrote:> So it could just as well be (n-d)/(d*log(n))?This is approximately log(n/d)/log(n) The order will be reversed. If you can calculate it, see if it holds the ordering. With numerators it's easy as they're already in the database.>> And in that case it would have been easier to go straight >> to log(n*d). >> Straight to where (do you see log(n*d))?log(n*d) = log(n) + log(d) The intervals start out in prime factor notation. So log(n*d) can be calculated as log(2)*abs(x_1) + log(3)*abs(x_2) + log(5)*abs(x_3) + log(7)*abs(x_4) where x_i is the ith prime component of n/d.> The Mercator series?? And all the stuff on this page applies > only to ln, not log in general (which is what I assume Paul > meant), right?log2(x) = log(x)/log(2). In this case we're only estimating the complexity, so the common factor if 1/log(2) isn't important. For estimating comma sizes using mental arithmetic, remembering that 1200/log(2) is approximately 1730 comes in handy. 1730/80 = 21.625, so a syntonic commma's around 22 cents. Graham

Message: 6289 - Contents - Hide Contents Date: Tue, 04 Feb 2003 21:18:35 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> carl, here's an old message where i explained the error >> heuristic: >> >> Yahoo groups: /tuning-math/message/1437 * [with cont.] >> Great, thanks! I hadn't seen this, as "heuristic" doesn't > appear in it. >>> and you can see that gene, in his reply, was the one who >> actually suggested the word "heuristic" in connection >> with this . . . >> I do see that... you were already using the term for the > complexity heuristic at that time, right?no, gene introduced the word "heuristic".> I understand everything but a few details... >>> log(n) + log(d) (hence approx. proportional to log(d) >> Is it any more proportional to log(d) than log(n) in this > case? no. > Since n~=d? yes.i prefer log(odd limit) over either log(n) or log(d).>> w=log(n/d) > > Got that. > >> w~=n/d-1 >> How do you get this from that?standard taylor series approximation for log . . . if x is close to 1, then log(x) is close to x-1 (since the derivative of log(x) near x=1 is 1/1 = 1).>> w~=(n-d)/d > > Ditto.arithmetic. n/d - 1 = n/d - d/d = (n-d)/d.

Message: 6290 - Contents - Hide Contents Date: Tue, 04 Feb 2003 10:31:51 Subject: Re: A common notation for JI and ETs From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote: (And by the way, I thought that> 99:100 might be a good interval to name "Ptolemy's comma", since > Pythagoras, Didymus, and now Archytas each have one.Any name for 100/99 should be part of a pair with 99/98. This is a problem with "small unidecimal comma" for 99/98; if 99/98 is "small", what is 100/99--smaller?

Message: 6291 - Contents - Hide Contents Date: Tue, 04 Feb 2003 21:23:11 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> log(n) + log(d) (hence approx. proportional to log(d >>>>>> Is it any more proportional to log(d) than log(n) in this >>> case? Since n~=d? >>>> No, and the spreadsheet sorted by d is also sorted by n. >> So it could just as well be (n-d)/(d*log(n))?a very different sorting. that would be heuristic error, not heuristic complexity. of course it's a very different sorting, since knowing log(n) or log(d) tells you nothing about (n-d).>> And in that case it would have been easier to go straight >> to log(n*d). >> Straight to where (do you see log(n*d))?meaning the sorting by log(n*d) would be virtually identical to the sorting by log(n) or by log(d).>>> It's the first order approximaton where n/d ~= 1. See (8) in >> Natural Logarithm -- from MathWorld * [with cont.] >> The Mercator series?? And all the stuff on this page applies > only to ln, not log in general (which is what I assume Paul > meant), right?i meant ln. i always use matlab, in which "log" means ln.

Message: 6292 - Contents - Hide Contents Date: Tue, 04 Feb 2003 21:28:26 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> And in that case it would have been easier to go straight >>>> to log(n*d). >>>>>> Straight to where (do you see log(n*d))? >>>> log(n*d) = log(n) + log(d) >> Of course... so we're coming from log(n*d), not going to it.what do you mean, we're coming from log(n*d)??

Message: 6293 - Contents - Hide Contents Date: Tue, 04 Feb 2003 22:37:59 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" >>> --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> >> wrote: > Since the term schismina is not required in the _use_ of sagittal, but only > in describing the theory behind it, I don't think it matters much whether > others use the term at all, or whether they accept a boundary at > sqrt(32805/32768), or whether they consider schismina a subclass of > schisma, or indeed whether the _only_ thing they consider as a schisma is > 32768:32805 itself, which is, I think, where things stood before we > started. The term may never be used anywhere outside of the XH paper and > yet I think we should leave the possibility open that it may be used > elsewhere and therefore not define it purely as something that vanishes in > sagittal.Okay, I just wanted you to clarify this so that we would both be using the term the same way. The 1/2-of-32768:32805 upper limit for a schismina should be okay (unless we find something else).> ... > There are two slightly more popular pairs that would benefit from a higher > kleisma-comma boundary. > > N kleisma comma > 245 14.19 37.65 > 7:13 14.61 38.07 > > I note that neither of us is willing to bring the comma diesis boundary > down below 38.07 or 37.65 cents.But I'm willing to consider it if there's a good reason for it.> ... Otherwise:>> ... the 40-cent boundary gets my vote. >> Mine too. It does cause 11:19 to have two dieses 40.33 49.89 (andit has an> ediesis 63.79) but 11:19 is much further down the popularity list and there > are lots of other ratios that have two dieses and an ediasis or two. > > I'll use the term ediasis (pron. ed-I-as-is, not ee-DI-as-is) for a diasis > larger than a half apotome, until someone tells me they like something else > better.Okay. (Somebody, *please* suggest something better; ediasis sounds too much like a disease.)>> Now for the kleisma-comma boundary. ... >> Some things in the kleisma size range have been called semicommas.I just think that "kleisma" sounds better.>> ... >> The point here is that I thought that the comma (120:121, ~14.367c) >> between the next smaller pair of superparticular ratios (10:11 and >> 11:12) should be smaller than the lower size limit for a comma. If >> they were used as the two ("whole") tones in a tetrachord, their sum >> would be 5:6, which would leave 9:10 as the remaining interval >> (or "semitone") of the tetrachord. But to have a "semitone" in a >> tetrachord that is larger than either of the "whole" tones is absurd, >> hence a practical basis for a boundary. >> I find this argument interesting but not convincing. Why must the whole > tones be superparticular?They don't have to, but since there are actual examples of ancient Greek tetrachords with diatonic steps of 7:8 with 8:9 (Archytas), of 8:9 with 9:10 (Didymus), and of 9:10 with 10:11 (Ptolemy's hemiolon), these being all of the possible cases, I found that it was possible to draw a conclusion from them. Interestingly, the next larger pair, 6:7 with 7:8 -- difference of 48:49, ~35.697c -- adds up to an exact 3:4, leaving a semitone that vanishes. Their difference is still a bit smaller than we were considering for the comma-diesis boundary, which makes me wonder if 40 cents is still too large.> Why must they even be simple ratios?Because a comma, by definition, is the difference between two rational intervals similar in size. But let's get back to the kleisma-comma boundary discussion.> Anyway, some very small intervals have been called commas for a long time. > e.g. We have Mercator's comma at about 3.6 cents and Wuerschmidt's comma at > about 11.4 cents. These are from Scala's intnam.par.These are just examples that the term can have a broad or generic usage in addition to the more specific definition that we're seeking.>> You want the boundary to be somewhere between what we have been >> calling the 17 comma (~8.7c) and 17' comma (~14.730c). To >> accommodate both of these requirements, we could put the lower >> boundary for a comma at infinitesimally above 120:121, >14.37c. >> Would this be too large an upper limit for a "kleisma?" If so, why? >> No I can't really argue that, although it is getting close to double the > size of _the_ kleisma. I now want to put the boundary even a little higher > than you suggest, at just above 28431:28672 (or 14.614 c) so wehave a 7:13> kleisma and a 7:13 comma (38.07 c) as mentioned above. > > This does mean we have the 17-comma and the 7:13-kleisma being notated with > the same symbol ~|(But it doesn't make much sense, though.> but I can probably live with that. Or would you rather > have two 7:13 commas?This might be a good reason to make the comma-diesis boundary somewhere around 37 to 38 cents. This would then put 1deg31 in the diesis range (at the lower end) -- 1deg31 is also functions as a 7- comma, but I think its dual use demonstrates that it's appropriate to have the boundary somewhere around this size. If 1664:1701 (~38.073c) is the 7:13 diesis, then 1377:1408 (~38.543c) and 44:45 (~38.906c) would become the 11:17 and 5:11 dieses. Do you know of any potential problems with these designations?>> If not, then this would set both the upper and lower boundaries for >> the term "comma" based on both historical considerations and prime- >> number-comma size groupings. >> I think I missed something there. What upper limit for a comma do you get > from historical considerations?I didn't get it directly -- it was inferred as being coincident with the lower limit for a diesis.>> It would also be good to have input from others regarding what the >> upper size limit for a kleisma should be. > > Sure.IF ANYONE OUT THERE HAS ANY INPUT ABOUT THIS (upper limit of ~14.5 cents for a kleisma), PLEASE SAY SOMETHING SOON!>> Merely to state that the >> term has not previously been applied to anything as large as 14 cents >> would probably not be enough to disqualify its use -- >> No. I wouldn't try to argue that. >>> I believe that >> it would be necessary to demonstrate some specific reason to insist >> on that, just as I have given a reason for the lower limit for a more >> specific usage of the term "comma" such as we require. >> I think it's fine for us to just define it "for our purposes" and let > others worry about whether they want to also adopt it for their own purposes.If nobody else says anything, then that's what's going to happen. I can't see making another boundary between semicomma and kleisma without a good reason.>>> The best cutoff between comma and diesis for this purpose would be >> exactly>>> half a pythagorean limma or 45.11 cents. However this would omit >> the>>> 25-diesis and THE diesis (125:128) so I propose placing the cutoff >>> infinitesimally below 125:128 or at 41.05 cents. >>>> I already addressed this (above). >> There are many ratios with pairs of commas that add up to a limma. In most > cases these are not very close to the half limma so the 40 cent boundary > between comma and diesis serves to separate them. Here are the most popular > ones that don't get separated: > > diesis 1 diesis 2 > N cents cents > ------------------------------ > 5:13 43.83 46.39 > 37 42.79 47.43 > 11:19 40.33 49.89 > 25:77 44.66 45.56 > > It doesn't seem like a good idea to have a category that only covers the > range 40.00 to 45.11 cents, but that's what we need for the above.I'll have to look these over to see if there's really any musical need some of these. For example, the 46.39-cent 5:13 diesis would notate 10:13 as an interval of a third, but I can't imagine that anyone would want a third this large in a diatonic or heptatonic scale very often.> ...>> But I wonder whether we should put the upper limit on a diesis at >> half an apotome (~56.843c) and use another term for anything larger. >> Sure. e.g. diesis below, ediasis above. Again I think we can afford to be > vague about whether edieses are really a subset of dieses, but "for our > purposes" we should consider them disjoint. Okay.>> ... (We would still need to settle whether '(/| would be >> its rational symbol, as well as (/| for the 49 diesis. I am inclined >> to go with it, if only because of its accuracy.) >> OK. Lets drop |)) completely, in favour of (/|. Fine! > But I see no need for a rational boundary. sqrt(2187/2048) seems ideal.Since an apotome is so important in the scheme of things, I agree.>>> The upper limit for a big diesis would be 70.17 cents for our >> purposes. >>>> I wouldn't put any sort of boundary there for whatever we might call >> this interval class, and we really don't need one there, since there >> will be no larger class of single-shaft symbols from which to >> distinguish this one. >> Well no, it's just the boundary of what we are willing to notate with two > flags and a single shaft. As I mentioned before, 70.17 cents is the most > that '((| could possibly represent, but I'm happy to stay below an apotome > minus a half limma, 68.57 cents so we never have more than one ediasis for > a ratio. Okay. > ...>>> But I really don't like using ( for 5:7-kleisma up. >>> ... Folks are already used to thinking of ([<{ as meaning down and )] >>> ] as >>> up. Scala uses ( for diesis down. >>>>> ... I don't think that using ( in someone else's ascii >> symbol system is a good enough reason to discard something that would >> work so well in ours. ... >> Weeell, it's not just that it's used in someone elses system. It's that > everyone, whenever they want to use one of the brackets (){}[]<> as an > accidental just naturally takes ({[< as down symbols and )}]> as up > symbols. It's similar to the reason why / is up and \ is down. Because we > read from left to right, the left parentheses are taken as arrows pointing > to the left which is conventionally the negative direction. This would be > the case even if no one had actually used them before.Okay, I get your point. This would also allow user-defined symbols using any of the laterally mirrored pairs in conjunction with our sagittal shorthand ones, should that be desired.> If we can't agree on this, I'd prefer to roll back to where weused " and ;> for the 5:7-kleisma and had no symbols for the 19-schisma.No, I think we should include the 19 comma.> But here's another attempt at the 5:7-kleisma without using " or ; or *. > How about k for down and p for up?Maybe. Let me think about it. --George

Message: 6294 - Contents - Hide Contents Date: Tue, 04 Feb 2003 23:17:37 Subject: Re: A common notation for JI and ETs From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:> IF ANYONE OUT THERE HAS ANY INPUT ABOUT THIS (upper limit of ~14.5 > cents for a kleisma), PLEASE SAY SOMETHING SOON!these cutoffs are totally arbitrary and need not even be based strictly on JI cents considerations. but you guys may be interested in studying the following links (which, however, concern 5-limit only): Yahoo groups: /tuning/database? * [with cont.] method=reportRows&tbl=10&sortBy=5&sortDir=up (be sure through scroll through all the pages) Onelist Tuning Digest # 483 message 26, (c)200... * [with cont.] (Wayb.)

Message: 6295 - Contents - Hide Contents Date: Tue, 04 Feb 2003 23:30:11 Subject: Re: A common notation for JI and ETs From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> (be sure through scroll through all the pages)sorry -- that should be *click* through all the pages!

Message: 6296 - Contents - Hide Contents Date: Wed, 05 Feb 2003 01:06:01 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> >wrote: >As I indicated in another message, this sounds like a good name, >although I thought that we should be more specific about a couple of >things. After a more careful reading of your message, I see that >you've done that by proposing a boundary, which sounds okay (unless >some rational interval ~1.0 cent could be specified). The only >question that remains is whether others would be willing to exclude >the term "schisma" as applying to intervals of less than ~1 cent, and >if not, then we would have to make "schismina" a subclass >of "schisma".Since the term schismina is not required in the _use_ of sagittal, but only in describing the theory behind it, I don't think it matters much whether others use the term at all, or whether they accept a boundary at sqrt(32805/32768), or whether they consider schismina a subclass of schisma, or indeed whether the _only_ thing they consider as a schisma is 32768:32805 itself, which is, I think, where things stood before we started. The term may never be used anywhere outside of the XH paper and yet I think we should leave the possibility open that it may be used elsewhere and therefore not define it purely as something that vanishes in sagittal.>Yes, the word "prime" leaves something to be desired, since it is >also liable to be confused with a "primary" (as opposed to secondary) >comma role for a symbol. Indeed.>> This will also work for N = 17 and 19 although in the 17 case it >would be>> better to call the small one the 17-kleisma. Incidentally the >traditional>> kleisma is the 5^6-kleisma and the "septimal kleisma" is the 7:25- >kleisma.>> Both of these are notated '|( so commas notated as |( (5:7) should >probably>> be called kleismas too. >>That sounds like it'll work. >>> If we set the cutoff between a kleisma and comma at exactly half of >a>> Pythagorean comma or 11.73 cents, this will work in the maximum >number of >> cases. >>That boundary comes almost exactly between two interpretations of ~) >|, -- the 17:19 comma (~11.352c) and the 17+19 comma (~12.108c)Not almost exactly, but exactly. Good point. But only the smaller of these is of interest. The larger has too high a slope and too high a power of 3. There are many combinations of primes whose useful "commas" come in pairs whose absolute values sum to a Pythagorean comma. Here's an even worse example. An 11:35 above C can be described as either a Pythagorean G# + 11.82 cents or a Pythagorean Ab - 11.64 cents. These are so close it doesn't matter if you get the wrong one. So they could both be called the 11:35-kleismas.>or a >possible alternate 5:17 comma (135:136, ~12.777c, although I see >that .~|( would be much better for this last one), so we might want >to adjust it somewhat (see below).Yes .~|( would notate it exactly. And since the other one is 36.24 cents I agree it would be good to call them kleisma and comma. There are two slightly more popular pairs that would benefit from a higher kleisma-comma boundary. N kleisma comma 245 14.19 37.65 7:13 14.61 38.07 I note that neither of us is willing to bring the comma diesis boundary down below 38.07 or 37.65 cents.>But the idea of a kleisma-comma >boundary is good. Recall that I had something to say not too long >ago (msg. #5202, 16 Dec 2002) about boundaries. I separated the >eight flags into two groups, between which your proposed boundary >falls: > small flags: '| )| |( ~| are the schismas and kleismas, and > large flags: |~ /| |) |\ (| are the commas. Right! >A diesis would be the sum of two large flags, i.e., two commas, but a >kleisma plus a comma would still be a comma. (The exception would >be /|~, ~38.051c, but we aren't using it in the notation.) So the >largest two-flag comma (i.e., comma + kleisma) would be ~|\, >~40.496c, and the smallest remaining two-flag diesis would be //|, >~43.013c. This is consistent with your proposed upper boundary for a >comma, < 125:128, ~41.058c, so I can agree to that. This isn't >actually the boundary that I suggested in that message, which was >anything larger than the 5:11 comma (~38.906c), which would make ~|\ >a diesis, even though it is the sum of a kleisma and a comma (which I >didn't notice). But this would give us the convenience of >distinguishing between a 23 comma (729:736, ~16.544c) and a 23 diesis >(16384:16767, ~40.004c), even if we shifted the upper boundary for a >comma to anything infinitesimally smaller than 16384:16767, i.e., for >all practical purposes 40 cents.Yes. I'll go with a boundary at 16384:16767 - 1/oo so that so we have a 23-comma and a 23-diesis. It's a pity the thing about only two large commas making a diesis doesn't quite work, but I don't think that's important.>Can we justify anything this small as a diesis? Well, yes:1deg31 >(~38.710c) has been called a diesis. In fact it's below all of the >comma-diesis boundaries that we've proposed, but it's a tempered >interval, so I don't think that we should let that bother us, since >the just intervals (or dieses) that it approximates are above the >boundary.I agree this is a red herring.> So the 40-cent boundary gets my vote.Mine too. It does cause 11:19 to have two dieses 40.33 49.89 (and it has an ediesis 63.79) but 11:19 is much further down the popularity list and there are lots of other ratios that have two dieses and an ediasis or two. I'll use the term ediasis (pron. ed-I-as-is, not ee-DI-as-is) for a diasis larger than a half apotome, until someone tells me they like something else better.>Now for the kleisma-comma boundary. Let me quote from that earlier >message that I mentioned above (in which I refer to a kleisma as a >small comma): > ><< Another basis for establishing a boundary between large and small >commas (which agrees with this) goes back to the original definition >of comma: the difference in size between the two largest steps in a >diatonic tetrachord. About the smallest that these steps can get is >in Ptolemy's diatonic hemiolon, where they are 9:10 and 10:11, with a >comma of 99:100 (~17.399 cents). The next smallest superparticular >pair are 11:12 and 10:11, making a lesser comma of 120:121 (~14.367 >cents, which is not only significantly smaller than 1deg72 (~16.667 >cents), but also closer in size to 1deg94 (~12.766 cents), in which >system both the 5 and 7 commas are 2deg (and 120:121 is only slightly >more than one-half the size of a 7 comma.) So I think this is >getting a bit small to be considered a comma in the original sense. >What we really need is a separate name for commas smaller than >~1deg72, and I don't think "kleisma" fills the bill. >>Some things in the kleisma size range have been called semicommas.>I made this last remark about the term "kleisma", because I had the >impression that the upper limit for a kleisma should probably be >smaller, but perhaps I was mistaken. (And by the way, I thought that >99:100 might be a good interval to name "Ptolemy's comma", since >Pythagoras, Didymus, and now Archytas each have one. 9:10 and 10:11 >is also the largest pair of superparticular ratios that are the same >number of degrees in 41-ET -- hence Ptolemy's comma vanishes in 41 >just as Didymus' comma does in 19, 31, and meantone. But I digress.) > >The point here is that I thought that the comma (120:121, ~14.367c) >between the next smaller pair of superparticular ratios (10:11 and >11:12) should be smaller than the lower size limit for a comma. If >they were used as the two ("whole") tones in a tetrachord, their sum >would be 5:6, which would leave 9:10 as the remaining interval >(or "semitone") of the tetrachord. But to have a "semitone" in a >tetrachord that is larger than either of the "whole" tones is absurd, >hence a practical basis for a boundary.I find this argument interesting but not convincing. Why must the whole tones be superparticular? Why must they even be simple ratios? Anyway, some very small intervals have been called commas for a long time. e.g. We have Mercator's comma at about 3.6 cents and Wuerschmidt's comma at about 11.4 cents. These are from Scala's intnam.par.>You want the boundary to be somewhere between what we have been >calling the 17 comma (~8.7c) and 17' comma (~14.730c). To >accommodate both of these requirements, we could put the lower >boundary for a comma at infinitesimally above 120:121, >14.37c. >Would this be too large an upper limit for a "kleisma?" If so, why?No I can't really argue that, although it is getting close to double the size of _the_ kleisma. I now want to put the boundary even a little higher than you suggest, at just above 28431:28672 (or 14.614 c) so we have a 7:13 kleisma and a 7:13 comma (38.07 c) as mentioned above. This does mean we have the 17-comma and the 7:13-kleisma being notated with the same symbol ~|( but I can probably live with that. Or would you rather have two 7:13 commas?>If not, then this would set both the upper and lower boundaries for >the term "comma" based on both historical considerations and prime- >number-comma size groupings.I think I missed something there. What upper limit for a comma do you get from historical considerations?>It would also be good to have input from others regarding what the >upper size limit for a kleisma should be. Sure. >Merely to state that the >term has not previously been applied to anything as large as 14 cents >would probably not be enough to disqualify its use --No. I wouldn't try to argue that.> I believe that >it would be necessary to demonstrate some specific reason to insist >on that, just as I have given a reason for the lower limit for a more >specific usage of the term "comma" such as we require.I think it's fine for us to just define it "for our purposes" and let others worry about whether they want to also adopt it for their own purposes.>> The best cutoff between comma and diesis for this purpose would be >exactly>> half a pythagorean limma or 45.11 cents. However this would omit >the>> 25-diesis and THE diesis (125:128) so I propose placing the cutoff >> infinitesimally below 125:128 or at 41.05 cents. >>I already addressed this (above).There are many ratios with pairs of commas that add up to a limma. In most cases these are not very close to the half limma so the 40 cent boundary between comma and diesis serves to separate them. Here are the most popular ones that don't get separated: diesis 1 diesis 2 N cents cents ------------------------------ 5:13 43.83 46.39 37 42.79 47.43 11:19 40.33 49.89 25:77 44.66 45.56 It doesn't seem like a good idea to have a category that only covers the range 40.00 to 45.11 cents, but that's what we need for the above. There's a similar problem with the apotome complements of these with the new boundary required near 68.57 cents. So we would end up needing four categories of what used to be just dieses, except that we can keep it down to 3 by refusing to notate anything bigger than 68.57 cents (we've not wanted to so far.>But I wonder whether we should put the upper limit on a diesis at >half an apotome (~56.843c) and use another term for anything larger.Sure. e.g. diesis below, ediasis above. Again I think we can afford to be vague about whether edieses are really a subset of dieses, but "for our purposes" we should consider them disjoint.> My reason for this is that by the time you reach ~63 cents (27:28, >1deg19), the interval has a melodic effect much more like a small >semitone (and a very effective one at that) than a quartertone (or >diesis). By contrast, the single degree of 22-ET (~54.545c) can >function either as a very small diatonic semitone *or* as a >quartertone (i.e., 11 diesis), so I would consider an interval of >this size to be at or near the borderline. If we want a rational >interval, then the upper limit for a diesis could be the 5:49 diesis >(392:405, ~56.482c) that I proposed to notate the hemififth family of >temperaments. (We would still need to settle whether '(/| would be >its rational symbol, as well as (/| for the 49 diesis. I am inclined >to go with it, if only because of its accuracy.)OK. Lets drop |)) completely, in favour of (/|. But I see no need for a rational boundary. sqrt(2187/2048) seems ideal.>> The upper limit for a big diesis would be 70.17 cents for our >purposes. >>I wouldn't put any sort of boundary there for whatever we might call >this interval class, and we really don't need one there, since there >will be no larger class of single-shaft symbols from which to >distinguish this one.Well no, it's just the boundary of what we are willing to notate with two flags and a single shaft. As I mentioned before, 70.17 cents is the most that '((| could possibly represent, but I'm happy to stay below an apotome minus a half limma, 68.57 cents so we never have more than one ediasis for a ratio.> I consider the ideal melodic and most >harmonically dissonant "semitone" somewhere in the range of 63-78 >cents. This is actually what would more accurately be called a third- >tone (1 degree of 17, 18, or 19), the sort of interval that's >melodically very effective in the enharmonic genus, which is what a >label for this interval range might suggest. ("Limma" won't do; >that's for the chromatic genus). Or perhaps a prefix or suffix to >modify the word diesis, as was done to get schismina. Any ideas? Done.>> But I really don't like using ( for 5:7-kleisma up. >> 1. It will get missed in text (i.e. parsed as an opening >parenthesis).>> 2. Folks are already used to thinking of ([<{ as meaning down and )] >> ] as >> up. Scala uses ( for diesis down. >> >> I thought we already agreed not to use () purely for reason 1. >>I didn't agree not to use ( on account of reason 1, but only >because ) was not a suitable opposite. If ( were used, it would >always be as the rightmost character of a symbol, in which position >it would never be an opening parenthesis, whereas an opening >parenthesis would always be leftmost (since it is always preceded by >a space). This is similar to why a period used as the 5' comma ascii >symbol would never be confused with a period ending a sentence.Alright. I can agree with that argument.>As for reason 2, I don't think that using ( in someone else's ascii >symbol system is a good enough reason to discard something that would >work so well in ours. (More about this below.)Weeell, it's not just that it's used in someone elses system. It's that everyone, whenever they want to use one of the brackets (){}[]<> as an accidental just naturally takes ({[< as down symbols and )}]> as up symbols. It's similar to the reason why / is up and \ is down. Because we read from left to right, the left parentheses are taken as arrows pointing to the left which is conventionally the negative direction. This would be the case even if no one had actually used them before.>> As you say, we're scraping the bottom of the barrel. It can't be an >> uppercase character. In approximate keyboard order: It can't be >> `,~!|@#%^&()+-{}{}\/'.";?<>. >>Hey, watch your language! ;-) Sorry. :-)>> ... I'm inclined to go with * because of its >> smallness and upwardness and because it seems better to use special >> characters rather than letters when possible. >> ...>I still think that ( and c are best. It wasn't my intention to have >a notation that should indefinitely *coexist* with other notations -- >I wanted a notation that would be the best one possible -- one that >would, in effect, be good enough to *replace* other notations that >also use 7 nominals, so there would be no need for competing systems. Dream on. >I see no particular reason why ( and ) should have been chosen to >represent a diesis in Scala,It's because they look most like the symbols in Rapoport's paper, but as I say it's not that they are used in Scala that is important but just that they are so obviously paired in people's minds and it's so obvious what directions they should mean.> but we have a very good reason to use ( >and c for the 5:7 comma rather than something else from the bottom of >the barrel that everybody will have a much harder time remembering.I'm sorry. I just think that they will have a hard time remembering that "(" is up, not down, and that ")" is not its partner.>Supposing that we're successful in getting a lot of others to adopt >our notation, we'd later regret not making the best choice from the >start (and having to justify a change -- over complaints and >objections, such as, why didn't we do it right the first time?). And >supposing that hardly anybody uses our notation, then what does it >matter what we chose?If we can't agree on this, I'd prefer to roll back to where we used " and ; for the 5:7-kleisma and had no symbols for the 19-schisma. But here's another attempt at the 5:7-kleisma without using " or ; or *. How about k for down and p for up? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 6297 - Contents - Hide Contents Date: Wed, 05 Feb 2003 12:58:43 Subject: Schisminas and rational notation From: Gene Ward Smith If the four "schisminas" 2401/2400, 3025/3024, 4375/4374 and 9801/9800 vanish, we are in the Hemiennealimmal temperament or something compatible with it, such as the 612 et. I proposed some time back a notation for 11-limit JI based on nine nominals for which these are ignored, and Graham has a similar 10-nominal system based on 494. Is there some kind of convergence going on here?

Message: 6298 - Contents - Hide Contents Date: Wed, 05 Feb 2003 01:37:46 Subject: Re: A common notation for JI and ETs From: David C Keenan At 03:37 AM 4/02/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:>--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus ><wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> >wrote: >>> way and we would still have the advantage, when talking about the >> development of sagittal, that only schisminas vanish. >>hmm . . . i think we've been over this before, but for any schismina, >no matter how tiny, there'll be some excellent temperament where it >doesn't vanish. does this matter?No. I believe it doesn't matter. Take 2400:2401 (the 5^2:7^4 schismina) which, when untempered is only 0.72 cents and so cannot itself be notated in sagittal. It may correspond to a step of some temperament which needs to be notated, however unless the step really is that small it is most likely that the step will also correspond to other commas which are larger when untempered and which therefore have saggital symbols. But we'd appreciate it if you'd find us some tunings that you think may cause difficulties for sagittal. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 6299 - Contents - Hide Contents Date: Wed, 05 Feb 2003 13:21:50 Subject: Mercator From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> On the mathworld page, it says "the Mercator series gives a > Taylor series for the natural logarithm", and in fact makes > it look like the Taylor series is the Mercator series.It is in this case--it's a special name for this series alone, sort of like "Gregory/Leibniz" for the arctangent (or the special value when x=1, namely pi/4.) This is "our" Mercator, of the Mercator comma, incidentally, and not the map Mercator.

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