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Message: 4700 - Contents - Hide Contents Date: Sat, 20 Apr 2002 04:46:53 Subject: huh From: Carl Lumma (kpsk '(3 2)) => 1 (kpsk '(5 3 2)) => 0 (kpsk '(7 5 3 2)) => 1 (kpsk '(11 7 5 3 2)) => 0 (kpsk '(13 11 7 5 3 2)) => 1 (kpsk '(17 13 11 7 5 3 2)) => 0 (kpsk '(19 17 13 11 7 5 3 2)) => 1 (kpsk '(23 19 17 13 11 7 5 3 2)) => 0 (kpsk '(29 23 19 17 13 11 7 5 3 2)) => 1
Message: 4702 - Contents - Hide Contents Date: Sat, 20 Apr 2002 18:24 +0 Subject: Re: More 12-tone JI scale comparisons From: graham@xxxxxxxxxx.xx.xx genewardsmith wrote:> I put together the ones Carl mentioned with the ones I cooked up, and > compared them using some of my measures and some from Scala. Playing > about with them, I got the impression that high lumma stability, > propriety, and CS (which seemed to go together) were good things for > scales to have, so that the ones with the most harmony did not > necessarily sound the best melodically. I'm still trying to figure out > all the arcane measures Carl and Graham are tossing at each other; > maybe they could explain using these scales as examples.The main ones you're missing are Rothenberg stability and efficiency. The former is the proportion of ambiguous intervals in the scale. Ambiguous intervals have the same size, but belong to different interval classes. The canonical example is the tritone in the 12-equal diatonic, which is both a fourth and a fifth. Rothenberg stability is undefined for improper scales, and unity for strictly proper scales. As all your examples fall into these categories (all JI scales do) they can't be used as examples for it. Unless you interpret a pair of intervals as being so close they'll be heard as equal. If a scale has low Rothenberg stability, you need a lot of context to decide which interval class each interval belongs to. An example of such a scale is the rendering of Rast in 10-equal, which has a propriety grid 2 1 1 2 2 1 1 3 2 3 4 3 2 3 4 4 5 5 4 4 4 6 6 6 6 6 5 5 8 7 7 8 7 6 7 9 8 9 9 8 8 9 That shows the number of steps of 10-equal to each second, third, fourth and so on down the grid. Seconds and thirds can both be 2 steps, thirds and fourths can both be 4 steps, fourths and fifths can both be 5 steps, fifths and sixths can both be 6 steps and sixths and sevenths can both be 8 steps. It's only intervals of 1, 3, 7 or 9 steps that are unambiguous. If a proper scale has a small number of ambiguous intervals that may be better than none at all. Rothenberg says that an ambiguous interval resolving to an ambiguous one adds to the cadential effect. I'm dubious about this -- the tritone is used in diatonic cadences because there's only one of them, so it almost describes the scale. But tritone substitutions only work because of the ambiguity, so if you want an analog of tritone substitutions you need at least one ambiguous interval. Rothenberg efficiency relates to patterns of notes that uniquely determine the key. The more notes you can play without the key being specified, the higher the efficiency. MOS scales with an octave period always have a high efficiency because you can play all but one note and there's still ambiguity. The diatonic scale in 12-equal is particularly efficient because the interval that shares the same interval class as the generator (the tritone) is ambiguous. In 31-equal, if you assume the listener can distinguish 7:5 and 10:7, you can uniquely specify the key by playing one such interval. Scales with low efficiency either have lots of different sized intervals or a number of periods to the octave. The latter case is what Rothenberg usually seems to be thinking of by low efficiency scales. He says they're suitable for atonal type music because they don't have a clearly defined key center. That's the opposite of what efficiency is supposed to show, and only works this way because of a peculiarity of the definition. So, I'm coming to dislike efficiency. I'd rather rate scales according to the number of periods to the octave (which should be low for a diatonic), the largest number of notes you can play without establishing the key (which should be high for modulation to work, and is easier to calculate than Rothenberg efficiency) and the smallest number of notes you need to establish the key (which should be low, to give a strong sense of tonality). Carl did ask before about Consistency, the measure Rothenberg uses to rate how good a scale will be for modal transposition. I think I've worked it out now. You start with the ordered mapping rather than the matrix above. But after working out what the ordered mapping (alpha_ij) is, I don't think it matters. So, I'll keep working with the usual matrix (delta_ij). With the only example Rothenberg works through, the 12-equal diatonic, the two are the same anyway. 2 2 2 1 2 2 1 4 4 3 3 4 3 3 6 5 5 5 5 5 5 7 7 7 6 7 7 7 9 9 8 8 9 9 8 11 10 10 10 11 10 10 You then construct "consistent sets" out of columns of this matrix that don't share ambiguous intervals. In this example, any set that doesn't contain both tritones is consistent. Consistency is defined as the average of the proportion of sets with each number of columns greater than 2 which are consistent. So, for this example there are 7 consistent sets of 12 column, which will be true for all 7 note scales so we ignore this statistic. There are 20 consistent sets of 2 columns. The total number of combinations of 2 objects from 7 is 21. One of those is invalid -- the two columns that both have a tritone in them. Hence 21-1=20. 30 out of 35 sets with 3 columns are consistent. 25 out of 35 sets with 4 columns are consistent. 11 out of 21 sets with 5 columns are consistent. 2 out of 7 sets with 6 columns are consistent (the ones you get by excluding either of the columns with a tritone in it). There are no consistent sets with 7 columns. The calculation then is ((20/21) + (30/35) + (25/35) + (11/21) + (2/7))/6 = 70/126 = 5/9 = 0.556 which agrees with Rothenberg. So there you go. I prefer Lumma stability which does much the same job, and is much simpler. It also works for improper an strictly proper scales. Which means it doesn't depend on assumptions about the smallest perceivable differences between intervals and doesn't suddenly change when you hit an equal temperament. Graham
Message: 4703 - Contents - Hide Contents Date: Sat, 20 Apr 2002 18:24 +0 Subject: Re: My Approach Generalized Diatonicity From: graham@xxxxxxxxxx.xx.xx Carl Lumma wrote:> It would be fun to discuss this, but I'm not sure what it has to do with > where we started from. Do we agree that a principle thing about the > diatonic scale, that is not a common thing among scales, is the ability > to harmonize a melody inside the scale and have the harmony voice > still sound like the same melody? If we do, then we can proceed to > figure > out which properties have anything to do with it, and which don't.Maybe, I'm not sure if that's the most important property.> Regardless of if we agree about this, we can ask if there are any > other principle things that differentiate the diatonic scale from the > wide variety of possible scales. What might they be?Having a large proportion of consonant triads with a consistent pattern is one of the hardest to find. But that may be at too low a level of abstraction. It may be easier to list the rare things about tonality, to see if they can be applied to other scales. - Stable but complex melody. It has a sense of being 7 almost equal notes, but they're not so equal as to be boring. - Different melodies can work together with consonant harmony. - Key center can be established through harmony. - Melodies and chord sequences have a sense of direction. You can often predict what will come next, but still be surprised. - Notes from outside the key can be used to reinforce it. Graham
Message: 4704 - Contents - Hide Contents Date: Sat, 20 Apr 2002 12:07:34 Subject: Re: More 12-tone JI scale comparisons From: Carl Lumma>If a proper scale has a small number of ambiguous intervals that may be >better than none at all. Rothenberg says that an ambiguous interval >resolving to an ambiguous one adds to the cadential effect. I'm dubious >about this -- the tritone is used in diatonic cadences because there's >only one of them, so it almost describes the scale.That's due to it's effect as a nearly "sufficient subset" of the scale, which R. also discuses.>Scales with low efficiency either have lots of different sized intervals >or a number of periods to the octave. The latter case is what Rothenberg >usually seems to be thinking of by low efficiency scales. He says they're >suitable for atonal type music because they don't have a clearly defined >key center. That's the opposite of what efficiency is supposed to show, >and only works this way because of a peculiarity of the definition.Namely, that you only need to hear one note of the scale to determine the key... or maybe you can never determine the key. I think my solution at lumma.org/gd3.txt is elegant.>So, I'm coming to dislike efficiency. I'd rather rate scales according >to the number of periods to the octave (which should be low for a >diatonic), Agree. >the largest number of notes you can play without establishing the key >(which should be high for modulation to work,Modulation? Are you sure?>the smallest number of notes you need to establish the key (which should >be low, to give a strong sense of tonality).So you're saying you want there to be one sufficient subset which is small, and the rest maximally large? This is a tonality measure, which as I say, I'm avoiding on purpose. It could be a good one, though.>You start with the ordered mapping rather than the matrix above. But >after working out what the ordered mapping (alpha_ij) is, I don't think >it matters. So, I'll keep working with the usual matrix (delta_ij). >With the only example Rothenberg works through, the 12-equal diatonic, >the two are the same anyway.It doesn't matter, but with the ordered mapping you get results applicable to any scale that shares it, whereas the usual matrix uniquely specifies the scale in question.> 2 2 2 1 2 2 1 > 4 4 3 3 4 3 3 > 6 5 5 5 5 5 5 > 7 7 7 6 7 7 7 > 9 9 8 8 9 9 8 >11 10 10 10 11 10 10 > >You then construct "consistent sets" out of columns of this matrix that >don't share ambiguous intervals. In this example, any set that doesn't >contain both tritones is consistent. > >Consistency is defined as the average of the proportion of sets with each >number of columns greater than 2 which are consistent. > >So, for this example there are 7 consistent sets of 12 column, which will >be true for all 7 note scales so we ignore this statistic. > >There are 20 consistent sets of 2 columns. The total number of >combinations of 2 objects from 7 is 21. One of those is invalid -- the >two columns that both have a tritone in them. Hence 21-1=20. > >30 out of 35 sets with 3 columns are consistent. > >25 out of 35 sets with 4 columns are consistent. > >11 out of 21 sets with 5 columns are consistent. > >2 out of 7 sets with 6 columns are consistent (the ones you get by >excluding either of the columns with a tritone in it). > >There are no consistent sets with 7 columns. > >The calculation then is > >((20/21) + (30/35) + (25/35) + (11/21) + (2/7))/6 > >= 70/126 = 5/9 = 0.556 which agrees with Rothenberg. > > >So there you go. I prefer Lumma stability which does much the same job, >and is much simpler. It also works for improper an strictly proper >scales. Which means it doesn't depend on assumptions about the smallest >perceivable differences between intervals and doesn't suddenly change when >you hit an equal temperament.Thanks for explaining this, Graham. This rings a bell now. Yes, I agree that Lumma stability is usually better, especially when you leave the world of et subsets that R. worked in. -Carl
Message: 4705 - Contents - Hide Contents Date: Sun, 21 Apr 2002 05:20:37 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau Hi George, Is this more like what you had in mind? Yahoo groups: /tuning-math/files/Dave/SymbolsB... * [with cont.]
Message: 4706 - Contents - Hide Contents Date: Sun, 21 Apr 2002 12:13:10 Subject: Re: More 12-tone JI scale comparisons From: Carl Lumma>Um, which bit of gd3?Where I set the score of the efficiency section equal to the generalized fifths section if there are only 1 or 2 unique keys.>I'd rather have one maximally large sufficient subset, and the rest small. >It's only the former that's rare, and apart from the bizarre way periodic >modes are treated I don't see another reason for efficiency being >considered.I'm afraid this would be far too specific for me.>> Thanks for explaining this, Graham. This rings a bell now. Yes, I >> agree that Lumma stability is usually better, especially when you >> leave the world of et subsets that R. worked in. >>Rothenberg worked with discrete measuring scales, not ET subsets. You >could apply all the measures to arbitrary scales if you defined a >tolerance within which intervals are considered equal. And you really >wanted to.Yes, he says that a couple of times, but he only ever calculates anything based on et subsets. -Carl
Message: 4707 - Contents - Hide Contents Date: Sun, 21 Apr 2002 12:16 +0 Subject: Re: More 12-tone JI scale comparisons From: graham@xxxxxxxxxx.xx.xx Me:>> Scales with low efficiency either have lots of different sized> intervals >or a number of periods to the octave. The latter case is > what Rothenberg >usually seems to be thinking of by low efficiency > scales. He says they're>> suitable for atonal type music because they don't have a clearly > defined>> key center. That's the opposite of what efficiency is supposed to> show, >and only works this way because of a peculiarity of the > definition. Carl: > Namely, that you only need to hear one note of the scale to determine > the key... or maybe you can never determine the key. I think my > solution > at lumma.org/gd3.txt is elegant.I suppose the idea is that once you've heard a few notes (one for an equal temperament, a number of pairs for an octatonic scale) you know all the notes in the scale, even if there isn't a key center. Um, which bit of gd3?>> the largest number of notes you can play without establishing the key >> (which should be high for modulation to work, >> Modulation? Are you sure?I don't see anything wrong there. If you want to change key according to shared notes you need a lot of notes to be shared. There may be other ways of doing modulations, but this is an obvious criterion to add. An alternative would be to modulate between pairs of diatonics, like I suggested for neutral third scales.>> the smallest number of notes you need to establish the key (which > should>> be low, to give a strong sense of tonality). >> So you're saying you want there to be one sufficient subset which is > small, and the rest maximally large? This is a tonality measure, which > as I say, I'm avoiding on purpose. It could be a good one, though.I'd rather have one maximally large sufficient subset, and the rest small. It's only the former that's rare, and apart from the bizarre way periodic modes are treated I don't see another reason for efficiency being considered.> Thanks for explaining this, Graham. This rings a bell now. Yes, I > agree that Lumma stability is usually better, especially when you > leave the world of et subsets that R. worked in.Rothenberg worked with discrete measuring scales, not ET subsets. You could apply all the measures to arbitrary scales if you defined a tolerance within which intervals are considered equal. And you really wanted to. Graham
Message: 4708 - Contents - Hide Contents Date: Sun, 21 Apr 2002 20:13:08 Subject: An amazing new kind of transformation From: Gene W Smith Since I just discovered it, perhaps "amazing" is immodest, but that is how it struck me. This is a transformation which transforms from the 64/63 planar temperament to the 126/125 planar temperament and back! It works as follows: using the approximations 64/63~1 or 126/125~1 respectively, the 7-limit music can be pulled back to a 5-limit preimage in a canonical way, since 7~64/9 in the first instance, and 7~125/18 in the second instance. We then apply a standard triad preserving mapping of order 2, namely 2-->2 3-->10/3 5-->5 While sending triads to triads, it also sends tetrads in 64/63~1 to tetrads in 126/125~1, and tetrads in 126/125~1 to tetrads in 64/63~1; for instance 1-5/4-3/2-7/4 in 64/63~1 becomes 1-5/4-5/3-10/7 in 126/25~1, and vice-versa.
Message: 4709 - Contents - Hide Contents Date: Mon, 22 Apr 2002 00:58:22 Subject: Re: A common notation for JI and ETs From: David C Keenan At 01:26 22/04/02 -0000, George Secor wrote:>In your table of symbols: > >Symbol Left Right >for flags flags >------------------------------ > >23' = 17 + (11-5) > >31' = 5 + (17'-17) + 7 > or 5 + 23 + 23 > >37 = 29 + 17 > >options can be added for the following: > >23' = 17 + (11-5) > or 29 + (17'-17) > >31' = 5 + (17'-17) + 7 > or 5 + 23 + 23 > or 7 + 7 > >37 = 29 + 17 > or 5 + 5 > >These 5+5 option for the 37-comma uses a much smaller schisma >(6553600:6554439, ~0.222 cents) than what you have. But the problem >with these three options that I have given is that none of the >schismas vanish in 1600-ET. > >Should we rethink the question of whether it is really necessary for >these schismas to vanish in 1600-ET ... ?I believe we can forget 1600-ET. It was just a handy place to look for suitably small notational schismas. I've been foolishly failing to check the size in cents of some of the more recent 1600-ET schismas. As you point out, c37 = c29 + c17 involves a largish schisma of 0.57 cents, but one alternative I gave, c37' = c5 + c5 + c19, is completely unconscionable at 1.04 cents. Also c43 = c19 + c19 + c(17'-17), 0.72 cents, which I don't consider usable either. I've now exhaustively searched all combinations of up to 3 of our flags. Here's what I end up with. Symbol Left Right Schisma for flags flags (cents) ------------------------------------- 5 = 5 0 7 = 7 0 11 = 5 + (11-5) 0 11' = 29 + 7 0.34 13 = 5 + 7 0.42 13' = 29 + (11-5) 0.08 17 = 17 0 17' = 17 + (17'-17) 0 19 = 19 0 19' = 19 + 23 0.16 23 = 23 0 23' = 17 + (11-5) 0.49 or 29 + (17'-17) 0.52 * 29 = 29 0 31 = 19 + (11-5) 0.12 31' = 29 + 5 0.03 * or 7 + 7 0.44 * or 5 + (17'-17) + 7 0.19 or 5 + 23 + 23 0.37 37 = 5 + 5 0.22 * or 29 + 17 0.57 37' = 19 + 23 + 7 0.25 or 5 + 17 + 23 0.65 41 = 5 0.26 * or 17 + (17'-17) + (17'-17) 0.51 43 = 19 + 19 + (17'-17) 0.72 [schisma too big] 47 = 17 + 7 0.45 or 19 + 29 0.42 or 19 + 23 + 23 0.02 or 5 + 17 + (17'-17) 0.21 pythagorean comma = 17 + 17 + (17'-17) 0 diaschisma = 19 + 23 0.37 [same symbol as 19'] diesis = 17 + (11-5) 0.56 [same symbol as 23'] * doesn't vanish in 1600-ET. So, in addition to c37 = c5 + c5, there are some other schismas available to us, that don't vanish in 1600-ET and are smaller than those that do. Namely: 31' = 29 + 5 0.03 cents 41 = 5 0.26 cents We should definitely stop at prime 41, since there is no way to get 43 with sufficient accuracy using our 8 existing flags. We're under the half cent otherwise. In the application you (or Erv) found for 41, would a 0.26 cent error in the 41 have rendered it useless? Why not simply reuse the 5 comma as the 41 comma? If we do that we eliminate one major reason for choosing (17'-17) as our final comma (over 17'-19 or simply 17'). No other comma symbols depend on it. But it is the only one that has good complementation rules in 217-ET. Actually, it might be better to stop at 31, since symbols with more than 2 flags (e.g. 37') are getting too difficult, for my liking. I've uploaded a new version of Yahoo groups: /tuning-math/files/Dave/SymbolsB... * [with cont.] based on the first option for each symbol, up to the prime 41, in the table above. I realised recently that some of those alternate commas (the primed ones that are intended for a diatonic-based notation) should not really be defined as they currently are, but as their apotome complements, because that's how they will be used. They are 17', 19', 23' and 25. Let's call the apotome complements of these 17", 19", 23" and 25". For diatonic-based purposes, these should be defined as 17:18, 18:19, 23:24 and 24:25 respectively, and should be assigned appropriate double-shaft symbols. The question is, can their symbols be sensibly based on the complementation rules which we derived in the context of 217-ET? Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4710 - Contents - Hide Contents Date: Mon, 22 Apr 2002 19:50:38 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: >> The symbols get fatter as the alterations become larger, which is >> only logical. > > Sure. >>> And I even put the fattest ones on a diet, and now >> none of them is wider than its height. So what is the problem? >> I'm ignoring the tails. With the standard symbols the _body_ of the symbol > is never wider than it is high. But hey, I can live with it. Okay.>> I believe that shortening the middle line makes it more difficult to >> see it, thereby making it *more* difficult to distinguish three from >> two. This is particularly true when the symbol modifies a note on a >> line and the middle line terminates at a staff line (so you see only >> two lines sticking out). >> Good point. How about making the middle one only 2 pixels shorter than the > outer ones. That will solve the latter problem.The problem is that that middle line needs to be noticed as much as the other two, so that we can see that there are three of them, and making it shorter tends to de-emphasize it.>> In fact, after looking at this again, I >> think I would be in favor of shorting all of the symbols from 17 to >> 16 pixels so that no vertical line would terminate at a staff line. >> Then I think the sagittals will look odd with sharps too, not just flats. > And it will worsen the aspect-ratio problem. I believe flats have such long > tails, precisely to give them a similar aspect ratio to sharps and naturals. >>> (This would also keep symbols modifying notes a fifth apart from >> colliding. But you made a comment below regarding how the length of >> a new symbol looks when placed beside a conventional flat, so I need >> to evaluate this further.) >> I don't see a problem with them colliding. Have you found examples of flats > doing that yet? I have.Why don't we just make all three of the arrow shafts the same length, and I'll forget about making the symbols shorter than 17 pixels.>> In your latest figures I notice that you are making a noticeable >> difference in width between the left and right flags, which is very >> effective with the straight flags. >> They were like that from the start. For straight and concave I have 5 > pixels wide versus 7 and for wavy I have 4 versus 5, but concaveare both 4> pixels. >>> Perhaps this will be the best > way>> to distinguish left from right. A very small nub could still be > used>> at the end of the larger of each pair of curved flags as a stylistic >> embellishment. >> I agree it would help with the lateral confusability. But from a purely > aesthetic point of view, I think I'd prefer not.So would you then be satisfied with a difference in width alone to aid in making the lateral distinction?>> With your concave flags, half of the length of the curve is >> coincident with the vertical arrow shaft, which makes it difficult > to>> tell that this was intended to be a concave curve. The portion of >> the curve with least slope is much thicker, and taken together with >> the overall lateral narrowness of the flag, it comes out looking > more>> like a blob than a curved line. >> You're absolutely right. The concaves just don't work at only 4 pixels > wide. It's interesting how the knowledge of what it's supposed to look like > can blind one to alternative interpretations. That's why it's so good to > cooperate the way we are. > > Trouble is, I just can't accept a 19 comma flag that's wider than 4 pixels > (including shaft) since it represents a barely perceptiple comma of about 3 > cents. I'd really prefer to make it only 3 pixels, but that seemstoo low res.> > How about we forget abour concave and _make_ it a (circular or semicircular > or triangular) blob? And move it up the shaft as you suggest, to center the > blob on the notehead. Too bad about the convex/concave complementarity.Why not just go with my version of the concave symbols: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/symbols2.bmp (see upper right, top staff)? The left flag is 3 pixels wide, and the right flag is 4 pixels wide, yet they are clearly identifiable. (I also threw in a complement symbol.)>> As with the concave flag, the top part of the curve is coincident >> with the arrow shaft, so it (i.e., the version on which I was >> commenting) tends to look like a smaller and lower convex flag that >> is modifying a note one staff position lower. Your latest version >> (19 April) of the wavy flag is identical to what I now have, except >> that I have made the (vertical) extremity of the flag one pixel >> shorter.This observation applied to your (wavy) flag for the 23 comma, not the one for the 17-comma.>> Why shorter? I think that the concave and wavy flags >> should be smaller than the convex and straight flags -- both in >> length and thickness. >> Yes. The wavy doesn't work at 4 pixels wide, and apparently you find it > only barely works at 5 pixels. I like your idea of making both concave and > wavy vertically shorter than the others too. And I agree that the vertical > position should be a sort of compromise between centering the flag > _including_ the part coincident with the shaft, and centering it > _excluding_ the part coincident with the shaft.I wouldn't make the vertical arrow shaft shorter, though. To the right of our convex symbols are our latest versions of the wavy flags for comparison. I made the left wavy flag 4 pixels wide, like the concave right flag, and the right wavy flag is 5 pixels wide. Both of our wL+wR symbols have flags 4 pixels wide on each side. As with the concave symbols, I also threw in a complement symbol. I also experimented with taking the curves out of the wavy symbols, making them right-angle symbols, which I put at the far right. (The left vs. right line lengths are different in both the horizontal and vertical directions to aid in telling them apart.) We already have two kinds of curved-line symbols, and substituting these for the wavy symbols would give us two kinds of straight-line symbols as well. It's not that I don't like the wavy symbols (I do like them), but I thought that this would make it easier -- both to remember and to distinguish them. (This one's your call.)>> I would further like to modify what I have for these by using >> different lateral widths (left vs. right), so I still have some work >> to do on the symbols before putting a new file out there. >> I look forward to it.I copied your symbols (unaltered) into the second staff. Below that I put my versions of the symbols for comparison. I found that when I draw *convex* flags free-hand that I tend to curve the end of the flag inward slightly to make sure that it isn't mistaken for a straight flag, and I have been doing something on this order for some time with my bitmap symbols as well. I have modified these also to reflect this, and you can let me know what you think. (I notice that the right flag of your 47.4-cent symbol has this sort of feature -- was that a mistake?) Or possibly only the left convex flag could be given this feature to further distinguish it from the convex right flag. Also, observe my 43-cent and 55-cent symbols -- the ones with two flags on the same side.>> Okay, I'll try this and let you know what I think. (But I always >> thought that the tails of conventional flats were too long anyway.) >> I believe flats have such long tails, precisely to give them a similar > aspect ratio to sharps and naturals.Yes, good point, and one reason why I'm not reluctant to discard the idea of making the symbols any shorter than 17 pixels. When I put a 5-comma-down symbol next to a flat the new symbol has a shorter stem than the flat. I don't think that this is inappropriate, inasmuch as: 1) the two symbols are in about the same proportion length-to-width; and 2) the difference in height is the same as that in the two vertical lines of a conventional sharp symbol.>> Slowly, but surely, we are making progress. >> Yes indeed. :-)And I can't imagine that anyone else has ever worked out a notation in this much detail. --George
Message: 4711 - Contents - Hide Contents Date: Mon, 22 Apr 2002 03:22:48 Subject: Bearings From: genewardsmith My new type of transformation was discovered in the course of looking at the position of the minor triad 7/6-7/5-7/4 when reduced to the 5-limit lattice by means of a 7-limit comma in which 7 appears only to a power of +-1, such as 64/63, 126/125, 225/224 or 4375/4374. In this case we can describe the position by a bearing (here in degrees on one or another side of one of the six 5-limit consonances) and distance. One way to produce scales suitable for these planar temperaments would be to produce 5-limit scales running in the general direction of the bearings below. Hence, 64/63 works well with chains of fifths, 126/125 with chains of minor thirds, 225/224 with chains of 16/15s (secors), and 4375/4374 with chains of minor thirds. I obtained the following: Bearing for 64/63 7/4 ~ 16/9 distance = 2, bearing 4/3 7/6 ~ 32/27 distance = 3, bearing 4/3 7/5 ~ 64/45 distance sqrt(7) = 2.64575, bearing 19.10661 4/3 by 8/5 Triad distance sqrt(57)/3 = 2.51661, bearing 6.58678 4/3 by 8/5 Bearing for 126/125 7/4 ~ 125/72 distance sqrt(7) = 2.64575, bearing 19.10661 5/3 by 5/4 7/6 ~ 125/108 distance 3, bearing 5/3 7/5 ~ 25/18 distance 2, bearing 5/3 Triad distance sqrt(57)/3 = 2.51661, bearing 6.58678 5/3 by 5/4 Bearing for 225/224 7/4 ~ 225/128 distance 2sqrt(3) = 3.46410, bearing 30 3/2 by 5/4 7/6 ~ 75/64 distance sqrt(7) = 2.64575, bearing 19.10661 5/4 by 3/2 7/5 ~ 45/32 distance sqrt(7), bearing 19.10661 3/2 by 5/4 Triad distance 5sqrt(3)/3 = 2.88675, bearing 30 3/2 by 5/4 Straight at 15/8 = 2/secor Bearing for 4375/4374 7/4~2187/1250 distance sqrt(37) = 6.08276, bearing 25.28500 6/5 by 3/2 7/6~729/625 distance 2sqrt(7) = 5.29150, bearing 19.10661 6/5 by 3/2 7/5~4374/3125 distance sqrt(39) = 6.24500, bearing 16.10211 6/5 by 3/2 Triad distance sqrt(309)/3 = 5.85947 bearing 20.17357 6/5 by 3/2
Message: 4712 - Contents - Hide Contents Date: Mon, 22 Apr 2002 20:01:14 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> > Why not just go with my version of the concave symbols: > > Yahoo groups: /tuning- * [with cont.] > math/files/secor/notation/symbols2.bmpSorry -- wrong file!!! This is the new one: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/SymAllSz.bmp --George
Message: 4713 - Contents - Hide Contents
Date: Tue, 23 Apr 2002 09:50:09
Subject: Comma pairs
From: genewardsmith
I searched among 7-limit intervals less than 50 cents in size and
within a radius of 10 of 1 in the symmetric 7-limit lattice (so that
the first invariant is <= 100) and came up with a list of matching
pairs. Particularly noteworthy, I think, are the following, all within
6 of the unison:
4375/4374 829440/823543 35 56 4180
589824/588245 28672/28125 33 -60 4744
33075/32768 8748/8575 33 84 4456
110592/109375 5103/5000 31 -72 6228
2048/2025 15876/15625 28 0 2304
16875/16807 300125/294912 27 -120 6676
2430/2401 84035/82944 23 -40 2164
2430/2401 51200/50421 23 -40 2164
2430/2401 31104/30625 23 -40 2164
2430/2401 9604/9375 23 -40 2164
3136/3125 3125/3087 19 0 900
3136/3125 250/243 19 0 900
3125/3087 250/243 19 0 900
4000/3969 5625/5488 19 -72 2772
1029/1024 81/80 13 0 144
2401/2400 525/512 11 8 148
1728/1715 875/864 10 -9 180
1728/1715 1029/1000 10 -9 180
875/864 1029/1000 10 -9 180
126/125 64/63 7 0 36
49/48 36/35 3 0 4
The complete cycle of 2430/2401 is noteworthy, as are the
{1728/1715, 875/864, 1029/1000} and {3136/3125, 3125/3087, 250/243}
sets. The 2401/2400 and 525/512 matchup seems worth investigating, and
of course 49/48 and 36/35 is a no-brainer.
Here is the complete list:
250047/250000 256000/250047 45 -324 29160
4375/4374 829440/823543 35 56 4180
2401/2400 525/512 11 8 148
52734375/52706752 2579890176/2573571875 82 -945 164196
52734375/52706752 1313046875/1289945088 82 -945 164196
420175/419904 5898240/5764801 47 80 8356
420175/419904 390625/381024 47 80 8356
2460375/2458624 3623878656/3603000625 85 -864 150480
2460375/2458624 15752961/15625000 85 -864 150480
2460375/2458624 679477248/669921875 85 -864 150480
703125/702464 843750/823543 49 -252 25992
48828125/48771072 67528125/67108864 83 440 69844
48828125/48771072 34034175/33554432 83 440 69844
13841287201/13824000000 14155776000/13841287201 99 648 107892
65625/65536 2240/2187 38 35 2524
95703125/95551488 1019215872/1008840175 73 -540 96264
95703125/95551488 17578125/17210368 73 -540 96264
134217728/133984375 725594112/720600125 97 0 69696
134217728/133984375 177147/175616 97 0 69696
134217728/133984375 5764801/5668704 97 0 69696
2100875/2097152 1071875/1048576 49 0 14400
9191328125/9172942848 257298363/256000000 85 -1764 299880
283115520/282475249 15116544/14706125 73 180 37224
2579890176/2573571875 1313046875/1289945088 82 -945 164196
3276800000/3268642167 41006250/40353607 93 -1440 270544
3276800000/3268642167 3268642167/3200000000 93 -1440 270544
589824/588245 28672/28125 33 -60 4744
26873856/26796875 144120025/143327232 63 -336 44836
26873856/26796875 321489/312500 63 -336 44836
4096000/4084101 273375/268912 54 -525 58300
4096000/4084101 2151296/2109375 54 -525 58300
5120/5103 117649/116640 37 -36 2664
5120/5103 3645/3584 37 -36 2664
3136/3125 3125/3087 19 0 900
3136/3125 250/243 19 0 900
40500000/40353607 458752/455625 67 -216 51444
49009212/48828125 367653125/362797056 91 0 108900
49009212/48828125 1977326743/1944000000 91 0 108900
49009212/48828125 48828125/47775744 91 0 108900
49009212/48828125 268435456/262609375 91 0 108900
10976/10935 83349/81920 42 -105 12916
118098/117649 9834496/9765625 76 0 57600
118098/117649 9765625/9529569 76 0 57600
235298/234375 15625/15309 45 -84 9160
235298/234375 786432/765625 45 -84 9160
16875/16807 300125/294912 27 -120 6676
823543/820125 66706983/65536000 69 -672 94672
823543/820125 1404928/1366875 69 -672 94672
823543/820125 160000000/155649627 69 -672 94672
19683/19600 157464/153125 57 180 17896
282475249/281250000 413343/409600 93 -180 41224
282475249/281250000 847425747/838860800 93 -180 41224
321489/320000 37748736/37515625 60 -384 52480
321489/320000 75497472/73530625 60 -384 52480
1029/1024 81/80 13 0 144
1500625/1492992 50625/50176 36 -192 14848
158203125/157351936 161414428/158203125 97 -1296 251424
144120025/143327232 321489/312500 63 -336 44836
3623878656/3603000625 15752961/15625000 85 -864 150480
3623878656/3603000625 679477248/669921875 85 -864 150480
2109375/2097152 282475249/279936000 79 0 44100
2109375/2097152 286654464/282475249 79 0 44100
37748736/37515625 75497472/73530625 60 -384 52480
67528125/67108864 34034175/33554432 83 440 69844
2097152/2083725 9765625/9680832 69 300 37000
2097152/2083725 4194304/4134375 69 300 37000
393216/390625 4194304/4117715 57 0 3136
393216/390625 839808/823543 57 0 3136
725594112/720600125 177147/175616 97 0 69696
725594112/720600125 5764801/5668704 97 0 69696
9834496/9765625 9765625/9529569 76 0 57600
34420736/34171875 18345885696/18015003125 87 -1680 292996
7558272/7503125 1063125/1048576 71 200 44500
3200000/3176523 5359375/5308416 43 -360 31284
1728/1715 875/864 10 -9 180
1728/1715 1029/1000 10 -9 180
78732/78125 40353607/40000000 67 0 15876
78732/78125 20588575/20155392 67 0 15876
4000/3969 5625/5488 19 -72 2772
126/125 64/63 7 0 36
1250000/1240029 200120949/195312500 98 -385 158404
15752961/15625000 679477248/669921875 85 -864 150480
1071875/1062882 2125764/2100875 97 720 114336
117649/116640 3645/3584 37 -36 2664
177147/175616 5764801/5668704 97 0 69696
9765625/9680832 4194304/4134375 69 300 37000
40353607/40000000 20588575/20155392 67 0 15876
282475249/279936000 286654464/282475249 79 0 44100
413343/409600 847425747/838860800 93 -180 41224
390625/387072 214375/209952 45 96 10960
33075/32768 8748/8575 33 84 4456
1058841/1048576 6561/6400 52 0 9216
1058841/1048576 401408/390625 52 0 9216
514714375/509607936 191102976/187578125 63 -840 106324
1019215872/1008840175 17578125/17210368 73 -540 96264
52428800/51883209 19140625/18874368 68 -256 34816
10616832/10504375 153664/151875 41 -400 32800
65536/64827 78125/76832 37 0 7056
110592/109375 5103/5000 31 -72 6228
2048/2025 15876/15625 28 0 2304
5832000/5764801 3176523/3125000 55 -144 23940
5832000/5764801 28824005/28311552 55 -144 23940
2500000/2470629 234375/229376 50 -49 4900
2500000/2470629 559872/546875 50 -49 4900
9882516/9765625 3828125/3779136 77 140 25096
9882516/9765625 9765625/9633792 77 140 25096
9882516/9765625 535815/524288 77 140 25096
2430/2401 84035/82944 23 -40 2164
2430/2401 51200/50421 23 -40 2164
2430/2401 31104/30625 23 -40 2164
2430/2401 9604/9375 23 -40 2164
11529602/11390625 16000000/15752961 76 -1152 177408
3125/3087 250/243 19 0 900
20253807/20000000 262144000/257298363 97 -1260 268776
20253807/20000000 7529536/7381125 97 -1260 268776
875/864 1029/1000 10 -9 180
3828125/3779136 9765625/9633792 77 140 25096
3828125/3779136 535815/524288 77 140 25096
84035/82944 51200/50421 23 -40 2164
84035/82944 31104/30625 23 -40 2164
84035/82944 9604/9375 23 -40 2164
51883209/51200000 53747712/52521875 59 -400 55300
367653125/362797056 1977326743/1944000000 91 0 108900
367653125/362797056 48828125/47775744 91 0 108900
367653125/362797056 268435456/262609375 91 0 108900
11390625/11239424 1838265625/1811939328 81 -972 169128
9765625/9633792 535815/524288 77 140 25096
49545216/48828125 48828125/48009024 90 231 60820
40960000/40353607 20000/19683 61 0 32400
40960000/40353607 1990656/1953125 61 0 32400
40960000/40353607 19683/19208 61 0 32400
2985984/2941225 765625/746496 40 -144 11520
51200/50421 31104/30625 23 -40 2164
51200/50421 9604/9375 23 -40 2164
31104/30625 9604/9375 23 -40 2164
20000/19683 1990656/1953125 61 0 32400
20000/19683 19683/19208 61 0 32400
41006250/40353607 3268642167/3200000000 93 -1440 270544
5859375/5764801 13671875/13436928 75 -144 26500
5859375/5764801 327680/321489 75 -144 26500
5859375/5764801 100352/98415 75 -144 26500
3176523/3125000 28824005/28311552 55 -144 23940
273375/268912 2151296/2109375 54 -525 58300
1977326743/1944000000 48828125/47775744 91 0 108900
1977326743/1944000000 268435456/262609375 91 0 108900
13671875/13436928 327680/321489 75 -144 26500
13671875/13436928 100352/98415 75 -144 26500
66706983/65536000 1404928/1366875 69 -672 94672
66706983/65536000 160000000/155649627 69 -672 94672
4194304/4117715 839808/823543 57 0 3136
262144000/257298363 7529536/7381125 97 -1260 268776
1990656/1953125 19683/19208 61 0 32400
327680/321489 100352/98415 75 -144 26500
120000/117649 12005/11664 27 24 1204
15625/15309 786432/765625 45 -84 9160
49/48 36/35 3 0 4
234375/229376 559872/546875 50 -49 4900
48828125/47775744 268435456/262609375 91 0 108900
4689453125/4586471424 257298363/250000000 91 -1512 281268
5159780352/5044200875 40353607/39366000 90 -729 131220
89915392/87890625 7247757312/7061881225 97 -540 111816
89915392/87890625 52734375/51380224 97 -540 111816
5898240/5764801 390625/381024 47 80 8356
5904900/5764801 7061881225/6879707136 92 -640 138496
6561/6400 401408/390625 52 0 9216
40353607/39321600 131072/127575 61 108 15048
7247757312/7061881225 52734375/51380224 97 -540 111816
1404928/1366875 160000000/155649627 69 -672 94672
Message: 4714 - Contents - Hide Contents Date: Tue, 23 Apr 2002 18:45:13 Subject: Re: A common notation for JI and ETs From: gdsecor In light of your recent difficulties with recognizable concave symbols (and possibly the wavy ones), I'll do my best to respond to some of the things in this message (#4117): --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: > [dk:] > I've just uploaded a MsWord document containing drawings that show how I > conceive of these flags in a resolution-independent manner, so as to > produce that style. You will see how the style is designed to be compatible > with the conventional symbols, in particular the conventional sharp and > flat symbols which the sagittals will most often have to appear next to. > > Yahoo groups: /tuning-math/files/Dave/Flags.doc * [with cont.] > > I suggest you print it and take a hiliter pen and colour in the parts that > actually make up the flags and stems. I couldn't figure any easyway to do> that in Word. I'm sure you'll figure out what needs colouring. Then turn > the second page upside down and hold each flag in turn, beside the standard > flat and then the standard sharp. > > Notice that the prototype convex and concave flags are exact 180 degree > rotations of each other, and wavy is an exact 180 degree rotation of > itself. This was partly intended to help with flag complementation in 217-ET.This is somewhat different than what I had envisioned. In your plan each curve changes direction by 90 degrees, whereas, in drawing the symbols freehand, I found that it was most natural to exceed 90 degrees for the convex symbols and even to exceed 180 degrees for the concave symbols. I found that, in order for the narrower concave symbols to be recognizable, I had to curve the line more sharply the closer I approached the end of the flag. Regarding my comments yesterday about wavy vs. right-angle (i.e., straightened-wavy) symbols: << I also experimented with taking the curves out of the wavy symbols, making them right-angle symbols, which I put at the far right. (The left vs. right line lengths are different in both the horizontal and vertical directions to aid in telling them apart.) We already have two kinds of curved-line symbols, and substituting these for the wavy symbols would give us two kinds of straight-line symbols as well. It's not that I don't like the wavy symbols (I do like them), but I thought that this would make it easier -- both to remember and to distinguish them. (This one's your call.) >> After trying out a few things with right-angle symbols, I would have to say that I'm in favor of the wavy flags. Also, regarding this comment: << (I notice that the right flag of your 47.4-cent symbol has this sort of feature -- was that a mistake?) >> I didn't notice until later that it is actually a 3-flag symbol with a combination of convex and wavy flags on the right side, which would indicate that the combination as you have it isn't recognizable.> Ok. We agree on the line-thickness and overall width of all the tails now, > 5 pixels for ||, 7 pixels for both ||| and X. Yes. > We also agree on how far the tail projects away from the centreline of the > corresponding notehead. That's 11 pixels not including the pixel that's > _on_ the centreline. That's the same as a sharp or natural, but two pixels > shorter than a flat. These agreements are good. Yes. > But we still don't agree on the height of the X's. Your X's are not > constant. They vary according to what flags they have on them, and are > often not laterally symmetrical. My X's are all the same height as they are > wide (7 pixels) and are laterally symmetrical. They just meet the concave > flags, but for other flag types they are extended by two parallel lines at > the same spacing as the outer two of the |'|. If nothing else, it certainly > simplifies symbol construction, not to have to design a new X tail for > every possible combination of flags. And if we get into using more than one > flag on the same side (e.g. for 25) with these X tails, I figure we're > gonna need those parallel sides.I tried this out myself and ended up with exactly what you have for these. I don't like the way the x's look with the straight and convex flags for a couple of reasons: 1) The x appears too remote or detached from the flag(s), and 2) The x would seem to be indicating an alteration to a note on a line or space two steps away from the note actually being altered, which would tend to be confusing. (The Sims square root symbol also has this problem.) When I draw the X-symbols freehand, I imagine that I am constructing the diagonals of a trapezoid having 3 right angles. The size and shape of the symbol is the same as the corresponding one with 3 arrow shafts, and the four corners of the trapezoid are determined by the two ends of the outside shafts and the points of intersection of those shafts with a flag. This is the way I would construct the X's for a scalable font.>> while >> the semisharp (|) and sharp groups (||) are either the same as or >> very close to your symbols. The biggest problem I had was with the >> nubs (which I made rather large and ugly) still tending to get lost >> in the staff lines. I tried one symbol (in the middle of the staff) >> with a triangular nub, which looks a little neater, I think. >> Yes it looks neater, but I fear it is out of character with the standard > accidentals. I even think that maybe _any_ nubs are out-of- character. Of > course we have the precedent of the double-sharp symbol, but I tend to > think of _it_ as being out-of-character with the other 3 standard symbols. > I suspect it is more often seen as the unpitched notehead than as an > accidental.I hope, then, that by agreeing on appropriate distinctions in size between left and right flags we can eliminate them entirely.>> I propose >> using instead the concave style of flag that I described before, for >> which I prepared a set of symbols on the 7th staff. (Note that the >> nubs don't get lost, even though they are quite small.) >> I'm not averse to a slight recurve on the concaves, but I'm afraid I find > some of those in symbols2.bmp, so extreme in this regard, that they are > quite ambiguous in their direction. With a mental switch akin to the Necker > cube illusion, I can see them as either a recurved concave pointing upwards > or a kind of wavy pointing down. Apart from any nub, I don't think that > they should go more than one pixel back in the "wrong" direction. Those at > the extreme lower left of the page look ok.You would have to think of the end of the flag as pointing and not the curve of the flag. Putting a nub on the end might even help in that regard. But I can't see how I can get away from using a rise of several pixels to indicate the line or space of the note being altered. The part of the curve coincident with the arrow shaft just isn't going to accomplish that.> I'm guessing that you need the huge recurve to convince yourself that > concave can represent larger commas than wavy?No, forget about that. I want to keep the concave flags for the two smallest alterations. The convex-to-concave conversion in the complementary symbols is a concept that I wouldn't want to discard.>> I like your wavy flag, but I would propose waving it a little > higher,>> as I did in the symbols just to the right of yours (back on the 5th >> staff); I seemed to be getting a better result with a thinner flag, >> which would also serve to avoid confusion of the wavy with the > convex >> flag. >> It seems to me that you have increased the possibility of confusion of wavy > with convex, by waving it higher.We'll just have to evaluate our latest efforts to see if that's a problem, but I don't think it is. --George
Message: 4715 - Contents - Hide Contents Date: Tue, 23 Apr 2002 18:57:32 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> > When I draw the X-symbols freehand, I imagine that I am constructing > the diagonals of a trapezoid having 3 right angles.Oops! Make that "the diagonals of a trapezoid having three of its sides at right angles (i.e., having two right angles) with the vertical sides parallel." (Come to think of it, don't all trapezoids have two right angles?) --George
Message: 4716 - Contents - Hide Contents
Date: Tue, 23 Apr 2002 05:56:28
Subject: Yet another new kind of transformation
From: genewardsmith
I did a search of possible transformations among the commas in the list
25/24, 28/27, 36/35, 49/48, 50/49, 64/63, 81/80, 2048/2025, 245/243,
126/125, 4000/3969, 1728/1715, 1029/1024, 225/224, 3136/3125,
5120/5103, 6144/6125, 2401/2400, 4375/4374. There is nothing special
about this list, it's just one I had handy. I checked to see which
pairs had the same full octahedral group invariants, where for 3^x 5^y
7^z the invariants I used were
Degree 2 x^2+y^2+z^2+x*y+x*z+y*z
Degree 4 y*x^2*z+x*y*z^2+x*y^2*z
Degree 6
y^4*z^2+y^4*x^2+2*y^3*z^3+2*y^3*z^2*x+2*y^3*z*x^2+2*y^3*x^3+y^2*z^4+2*y^2*z^3*x+4*y^2*z^2*x^2+2*y^2*z*x^3+y^2*x^4+2*y*z^3*x^2+2*y*z^2*x^3+z^4*x^2+2*z^3*x^3+z^2*x^4
This gave me the following possibilites, of pairs of commas which had
the same values for all three invariants:
25/24 36/35 3 0 4
25/24 49/48 3 0 4
28/27 64/63 7 0 36
28/27 126/125 7 0 36
36/35 49/48 3 0 4
64/63 126/125 7 0 36
81/80 1029/1024 13 0 144
The last one was unexpected and particularly intriging; on checking
it, I found it associated to the 7-limit transformation (of order 4)
given by 2->2, 3->7/2, 5->14/3, 7->28/5. The orbit of 81/80 under this
transformation is 81/80->1029/512->2401/2000->378/625->81/80.
Hence a piece in meantone, as a 7-limit planar temperament, can be
sent to something in the 1029/1024 temperament. The comma, instead of
converting into another comma and then being tempered out, converts to
1029/512, which tempers to 1/2. Hence, 7-limit harmony *is* preserved!
We now have two new, albeit related, kinds of transformations.
We might also note the larger groups of transformations arising from
the {28/27, 64/63, 126/125} and {25/24, 36/25, 49/48} sets of commas.
Message: 4717 - Contents - Hide Contents Date: Tue, 23 Apr 2002 18:47:10 Subject: Re: A common notation for JI and ETs From: gdsecor In light of your recent difficulties with recognizable concave symbols (and possibly the wavy ones), I'll do my best to respond to some of the things in this message (#4117): --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: > [dk:] > I've just uploaded a MsWord document containing drawings that show how I > conceive of these flags in a resolution-independent manner, so as to > produce that style. You will see how the style is designed to be compatible > with the conventional symbols, in particular the conventional sharp and > flat symbols which the sagittals will most often have to appear next to. > > Yahoo groups: /tuning-math/files/Dave/Flags.doc * [with cont.] > > I suggest you print it and take a hiliter pen and colour in the parts that > actually make up the flags and stems. I couldn't figure any easyway to do> that in Word. I'm sure you'll figure out what needs colouring. Then turn > the second page upside down and hold each flag in turn, beside the standard > flat and then the standard sharp. > > Notice that the prototype convex and concave flags are exact 180 degree > rotations of each other, and wavy is an exact 180 degree rotation of > itself. This was partly intended to help with flag complementation in 217-ET.This is somewhat different than what I had envisioned. In your plan each curve changes direction by 90 degrees, whereas, in drawing the symbols freehand, I found that it was most natural to exceed 90 degrees for the convex symbols and even to exceed 180 degrees for the concave symbols. I found that, in order for the narrower concave symbols to be recognizable, I had to curve the line more sharply the closer I approached the end of the flag. Regarding my comments yesterday about wavy vs. right-angle (i.e., straightened-wavy) symbols: << I also experimented with taking the curves out of the wavy symbols, making them right-angle symbols, which I put at the far right. (The left vs. right line lengths are different in both the horizontal and vertical directions to aid in telling them apart.) We already have two kinds of curved-line symbols, and substituting these for the wavy symbols would give us two kinds of straight-line symbols as well. It's not that I don't like the wavy symbols (I do like them), but I thought that this would make it easier -- both to remember and to distinguish them. (This one's your call.) >> After trying out a few things with right-angle symbols, I would have to say that I'm in favor of the wavy flags. Also, regarding this comment: << (I notice that the right flag of your 47.4-cent symbol has this sort of feature -- was that a mistake?) >> I didn't notice until later that it is actually a 3-flag symbol with a combination of convex and wavy flags on the right side, which would indicate that the combination as you have it isn't recognizable.> Ok. We agree on the line-thickness and overall width of all the tails now, > 5 pixels for ||, 7 pixels for both ||| and X. Yes. > We also agree on how far the tail projects away from the centreline of the > corresponding notehead. That's 11 pixels not including the pixel that's > _on_ the centreline. That's the same as a sharp or natural, but two pixels > shorter than a flat. These agreements are good. Yes. > But we still don't agree on the height of the X's. Your X's are not > constant. They vary according to what flags they have on them, and are > often not laterally symmetrical. My X's are all the same height as they are > wide (7 pixels) and are laterally symmetrical. They just meet the concave > flags, but for other flag types they are extended by two parallel lines at > the same spacing as the outer two of the |'|. If nothing else, it certainly > simplifies symbol construction, not to have to design a new X tail for > every possible combination of flags. And if we get into using more than one > flag on the same side (e.g. for 25) with these X tails, I figure we're > gonna need those parallel sides.I tried this out myself and ended up with exactly what you have for these. I don't like the way the x's look with the straight and convex flags for a couple of reasons: 1) The x appears too remote or detached from the flag(s), and 2) The x would seem to be indicating an alteration to a note on a line or space two steps away from the note actually being altered, which would tend to be confusing. (The Sims square root symbol also has this problem.) When I draw the X-symbols freehand, I imagine that I am constructing the diagonals of a trapezoid having 3 right angles. The size and shape of the symbol is the same as the corresponding one with 3 arrow shafts, and the four corners of the trapezoid are determined by the two ends of the outside shafts and the points of intersection of those shafts with a flag. This is the way I would construct the X's for a scalable font.>> while >> the semisharp (|) and sharp groups (||) are either the same as or >> very close to your symbols. The biggest problem I had was with the >> nubs (which I made rather large and ugly) still tending to get lost >> in the staff lines. I tried one symbol (in the middle of the staff) >> with a triangular nub, which looks a little neater, I think. >> Yes it looks neater, but I fear it is out of character with the standard > accidentals. I even think that maybe _any_ nubs are out-of- character. Of > course we have the precedent of the double-sharp symbol, but I tend to > think of _it_ as being out-of-character with the other 3 standard symbols. > I suspect it is more often seen as the unpitched notehead than as an > accidental.I hope, then, that by agreeing on appropriate distinctions in size between left and right flags we can eliminate them entirely.>> I propose >> using instead the concave style of flag that I described before, for >> which I prepared a set of symbols on the 7th staff. (Note that the >> nubs don't get lost, even though they are quite small.) >> I'm not averse to a slight recurve on the concaves, but I'm afraid I find > some of those in symbols2.bmp, so extreme in this regard, that they are > quite ambiguous in their direction. With a mental switch akin to the Necker > cube illusion, I can see them as either a recurved concave pointing upwards > or a kind of wavy pointing down. Apart from any nub, I don't think that > they should go more than one pixel back in the "wrong" direction. Those at > the extreme lower left of the page look ok.You would have to think of the end of the flag as pointing and not the curve of the flag. Putting a nub on the end might even help in that regard. But I can't see how I can get away from using a rise of several pixels to indicate the line or space of the note being altered. The part of the curve coincident with the arrow shaft just isn't going to accomplish that.> I'm guessing that you need the huge recurve to convince yourself that > concave can represent larger commas than wavy?No, forget about that. I want to keep the concave flags for the two smallest alterations. The convex-to-concave conversion in the complementary symbols is a concept that I wouldn't want to discard.>> I like your wavy flag, but I would propose waving it a little higher, >> as I did in the symbols just to the right of yours (back on the 5th >> staff); I seemed to be getting a better result with a thinner flag, >> which would also serve to avoid confusion of the wavy with the convex >> flag. >> It seems to me that you have increased the possibility of confusion of wavy > with convex, by waving it higher.We'll just have to evaluate our latest efforts to see if that's a problem, but I don't think it is. --George
Message: 4719 - Contents - Hide Contents Date: Tue, 23 Apr 2002 19:30:45 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >>>> When I draw the X-symbols freehand, I imagine that I am > constructing>> the diagonals of a trapezoid having 3 right angles. >> Oops! Make that "the diagonals of a trapezoid having three of its > sides at right angles (i.e., having two right angles) with the > vertical sides parallel." (Come to think of it, don't all trapezoids > have two right angles?)No, they don't have to have any right angles, but they always have two parallel sides. (It only took 3 messages to get it right; I must be having a bad trapezoid day!) --George
Message: 4720 - Contents - Hide Contents Date: Tue, 23 Apr 2002 19:42:56 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >>>>>> When I draw the X-symbols freehand, I imagine that I am >> constructing>>> the diagonals of a trapezoid having 3 right angles. >>>> Oops! Make that "the diagonals of a trapezoid having three of its >> sides at right angles (i.e., having two right angles) with the >> vertical sides parallel." (Come to think of it, don't all > trapezoids>> have two right angles?) >> No, they don't have to have any right angles, but they always have > two parallel sides. (It only took 3 messages to get it right; I must > be having a bad trapezoid day!) > > --GeorgeOr you could say I fell into that trapezoid! -gs
Message: 4723 - Contents - Hide Contents Date: Tue, 23 Apr 2002 16:51:20 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >The problem is that that middle line needs to be noticed as much as >the other two, so that we can see that there are three of them, and >making it shorter tends to de-emphasize it.I think we need a third (and fourth and fifth ...) opinion on this one. From a performer who sight reads.>Why don't we just make all three of the arrow shafts the same length, >and I'll forget about making the symbols shorter than 17 pixels.I have found our cooperation on this notation to be remarkable ego-less, with both of us concerned only with what will be best for the end-user, and not concerned with "getting our own way". But we've always given reasons for rejecting the other's proposal, so as to avoid any hurt feelings. I feel that any compromises we have made so far, e.g in lengths, widths, thicknesses or curvatures, have been made because we believe the best option most likely lies in between our two extremes. Now I may be reading it wrong, but the above seems to be suggesting a trade-off of two completely unrelated things, purely on the basis, "you let me have my way on this and I'll let you have your way on that". If we can't agree on something, I'd prefer to seek other opinions, rather than engage in such a tradeoff.>So would you then be satisfied with a difference in width alone to >aid in making the lateral distinction? Yes. >Why not just go with my version of the concave symbols: > >Yahoo groups: /tuning- * [with cont.] >math/files/secor/notation/SymAllSz.bmpAs I wrote in Yahoo groups: /tuning-math/message/4117 * [with cont.] I'm not averse to a slight recurve on the concaves, but I'm afraid I find your current proposals so extreme in this regard, that they are quite ambiguous in their direction. With a mental switch akin to the Necker cube illusion, I can see them as either a recurved concave pointing upwards or a kind of wavy pointing down. Apart from any nub, I don't think that they should go more than one pixel back in the "wrong" direction. Those at the extreme lower left of Symbols2.bmp look ok.>(see upper right, top staff)? The left flag is 3 pixels wide, and >the right flag is 4 pixels wide, yet they are clearly identifiable. >(I also threw in a complement symbol.)These are 4 and 5 pixels wide by my reckoning (including the part coincident with the shaft). One must define a flag as including a part coincident with the shaft so one knows what it will look like when it is sharing a || or X shaft with another flag. But I did overstate the case when I said that 4 pixels wide doesn't work. We now have agreement that the concaves on a single shaft should both be 4 pixels wide. Maybe I went too far in reducing the height of the wavy and concave to 7 pixels (including shortening the shaft at the pointy end). I see that your concaves are 9 pixels high, and your wavys are 10 pixels high, only one pixel shorter than the straight and concaves. In fact when it isn't used with the left wavy, your right wavy is the full 11 pixels in height and 6 pixels wide. I think these lead to too many symbols whose apparent visual size is too far out of keeping with their size in cents. I find that: 9.4 14.7 20.1 all look bigger than 21.5 27.5 and 30.6 look bigger than 31.8 I am proposing something between yours and mine. See Yahoo groups: /tuning-math/files/Dave/SymbolsB... * [with cont.] I don't think we actually need any lateral distinction between the two concaves because in rational tunings the (17'-17) flag will never occur on its own, and I don't think any ETs of interest below 217-ET will need to use both 19 and (17'-17). What do you think? But it wouldn't hurt if they were distinct. The biggest problem (for me) is trying to make the 19 comma (left concave) look as small as it really is without it disappearing. If its width was in proportion to the width of the 5 comma flag, you wouldn't see it for the shaft! If we look at areas and ignore the part coincident with the shaft, the 5 comma flag is 4 pixels by 11 pixels. The 19 comma flag would have to fit in a rectangle 7 pixels in area. In my 22-Apr proposal I've allowed those 7 pixels to blow out to 12, 3 wide by 4 high (excluding shaft). Yours is 18 pixels, 3 wide by 6 high. If we simply count black pixels (excluding shaft) we find that the 5 comma flag has 15, which means the 19 comma flag should only have 2.4, which we might generously round up to 3 black pixels. Mine has 7, yours has 9. So my (17'-17) (right concave) flag is about the right size, but my 19 flag is about double the size it should be. I can live with double, but I'n not sure I can handle triple. Now you probably think I'm being too literal with this representative size stuff, but the problems occur when you have the 19 flag combined with another flag and the result looks much bigger than some single flag that it should be much smaller than. In particular 9.4 looking bigger than 21.5, 20.1 looking bigger than 27.3. I suppose we can have a 19 comma flag that is lrger when used alone than when combined with others, but I'd prefer not.>I wouldn't make the vertical arrow shaft shorter, though. OK. >To the right of our convex symbols are our latest versions of the >wavy flags for comparison. I made the left wavy flag 4 pixels wide, >like the concave right flag, and the right wavy flag is 5 pixels >wide. Both of our wL+wR symbols have flags 4 pixels wide on each >side. >As with the concave symbols, I also threw in a complement >symbol.Ah, but what exactly are they complements _of_? I assume it was an oversight that left the wavy side of the 36.0 symbol unmodified.>I also experimented with taking the curves out of the wavy symbols, >making them right-angle symbols, which I put at the far right. (The >left vs. right line lengths are different in both the horizontal and >vertical directions to aid in telling them apart.) We already have >two kinds of curved-line symbols, and substituting these for the wavy >symbols would give us two kinds of straight-line symbols as well. >It's not that I don't like the wavy symbols (I do like them), but I >thought that this would make it easier -- both to remember and to >distinguish them. (This one's your call.)You're right about them being more distinct, but the aesthetics are the killer. Given more resolution, I'd go for something in between the existing wavys and these right-angle ones, but not these totally sharp corners.>I copied your symbols (unaltered) into the second staff. Below that >I put my versions of the symbols for comparison. > >I found that when I draw *convex* flags free-hand that I tend to >curve the end of the flag inward slightly to make sure that it isn't >mistaken for a straight flag, and I have been doing something on this >order for some time with my bitmap symbols as well. I have modified >these also to reflect this, and you can let me know what you think.I think they look good, aesthetically speaking. The trouble is it makes the down versions look too much like flats and backward flats. Also, you decreased the size difference between the 7 flag and the 29 flag by adding curvature on the outside of the 7 flag and the inside of the 29 flag. I find the fact that the convex flags start off at right-angles to the shaft and end parallel to the shaft, sufficient to make them distinct from straight flags, without tending towards flats.>(I notice that the right flag of your 47.4-cent symbol has this sort >of feature -- was that a mistake?)That's 37' = 19 + 23 + 7 = vL + wR + xR, so what you saw resulted from mindlessly overlaying wR and xR. Being 37', my heart wasn't in it. I've had a better go at it now, based on what you did for 25 and 31'.>Or possibly only the left convex >flag could be given this feature to further distinguish it from the >convex right flag.That would at least retain the full 2 pixel difference in width between XL and xR, but still has the problem of looking too much like a backwards flat. There is a way to make the convex more distinct from straight without taking them closer to flats. We make them closer to being right-angles, i.e. reduce the radius of the corner. I've shown comparisons with straight flags and flats at top right of my latest bitmap.>Also, observe my 43-cent and 55-cent symbols -- the ones with two >flags on the same side.Yes. I wasn't very happy with mine. I like yours better, but I've modified them very slightly. Tell me what you think. Notice that it's OK for 31' down to look like a backwards flat, because it _is_ a half-flat.>Yes, good point, and one reason why I'm not reluctant to discard the >idea of making the symbols any shorter than 17 pixels. When I put a >5-comma-down symbol next to a flat the new symbol has a shorter stem >than the flat. I don't think that this is inappropriate, inasmuch as: > >1) the two symbols are in about the same proportion length-to-width; >and > >2) the difference in height is the same as that in the two vertical >lines of a conventional sharp symbol.Good points. OK. I'll forget the idea of giving down arrows longer shafts than up arrows. Are we agreed then that all sagittals should be 17 pixels high?>And I can't imagine that anyone else has ever worked out a notation >in this much detail. Me neither.It will of course be rejected out of hand by others, for reasons we haven't even considered. :-) -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4724 - Contents - Hide Contents Date: Tue, 23 Apr 2002 20:03:44 Subject: Re: A common notation for JI and ETs From: David C Keenan Ho George, Sorry about my previous message in this thread. It should have been posted 16 hours ago, but I managed to email it to myself (replying to my own forward of your message from the Yahoo website) and only discovered my oversight the next morning (Australian time). So I sent it as soon as I discovered it. I see now, that you had already addressed some of the points in it. First a correction. I wrote: "We now have agreement that the concaves on a single shaft should both be 4 pixels wide." That was wrong. At that stage we only had agreement that vL should be 4 pixels wide. But since then I've also agreed with you that vR can be 5 pixels wide, as shown in the latest SymbolsBySize.bmp (which incidentally was sitting there for those 16 hours but you had no way of knowing it). At 00:21 24/04/02 -0000, you wrote:>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >In light of your recent difficulties with recognizable concave >symbols (and possibly the wavy ones), I'll do my best to respond to >some of the things in this message (#4117): > >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >> [dk:] >> I've just uploaded a MsWord document containing drawings that show >how I>> conceive of these flags in a resolution-independent manner, so as to >> produce that style. You will see how the style is designed to be >compatible>> with the conventional symbols, in particular the conventional sharp >and>> flat symbols which the sagittals will most often have to appear >next to. >> >> Yahoo groups: /tuning-math/files/Dave/Flags.doc * [with cont.] >> >> I suggest you print it and take a hiliter pen and colour in the >parts that>> actually make up the flags and stems. I couldn't figure any easy>way to do>> that in Word. I'm sure you'll figure out what needs colouring. Then >turn>> the second page upside down and hold each flag in turn, beside the >standard>> flat and then the standard sharp. >> >> Notice that the prototype convex and concave flags are exact 180 >degree>> rotations of each other, and wavy is an exact 180 degree rotation of >> itself. This was partly intended to help with flag complementation >in 217-ET. >>This is somewhat different than what I had envisioned. In your plan >each curve changes direction by 90 degrees, whereas, in drawing the >symbols freehand, I found that it was most natural to exceed 90 >degrees for the convex symbolsI have of course addressed the problems I see with that, in my previous message and shown it in SymbolsBySize.bmp. I expect the confusion-with-flats problem didn't occur to you because you have, in the past, only been interested in a totally sagittal notation, and had been drawing them significantly smaller than flats.>and even to exceed 180 degrees for the >concave symbols. I found that, in order for the narrower concave >symbols to be recognizable, I had to curve the line more sharply the >closer I approached the end of the flag.That may be OK for the res-independent description, but unfortunately it's impossible to indicate it at the resolution we're using, while still keeping the concaves as small as they need to be, or sufficiently distinct from wavy's pointing the other way. And of course it spoils the visual complementarity of convex and concave, although the fact that the concaves need to be so much smaller than the convex already does that to some degree. I guess their visual complementarity could still be seen as a flip about the diagonal, rather than a rotation of 180 degrees, however that doesn't work for the wavy. A diagonal flip of wavy isn't the same kind of wavy, but something new. Anyway, visual complementarity cues are nowhere as important as distinctness from other symbols, and not just other accidental symbols. There's also the quaver-rest symbol to consider. I've just added comparisons for that, to Yahoo groups: /tuning-math/files/Dave/SymbolsB... * [with cont.]>After trying out a few things with right-angle symbols, I would have >to say that I'm in favor of the wavy flags. Good. >I didn't notice until later that it is actually a 3-flag symbol with >a combination of convex and wavy flags on the right side, which would >indicate that the combination as you have it isn't recognizable.No. It was garbage. If you haven't already seen my new c37' symbol, it would be interesting if you would have a go at one yourself and then see how similar they might be.>> But we still don't agree on the height of the X's. Your X's are not >> constant. They vary according to what flags they have on them, and >are>> often not laterally symmetrical. My X's are all the same height as >they are>> wide (7 pixels) and are laterally symmetrical. They just meet the >concave>> flags, but for other flag types they are extended by two parallel >lines at>> the same spacing as the outer two of the |'|. If nothing else, it >certainly>> simplifies symbol construction, not to have to design a new X tail >for>> every possible combination of flags. And if we get into using more >than one>> flag on the same side (e.g. for 25) with these X tails, I figure >we're>> gonna need those parallel sides. >>I tried this out myself and ended up with exactly what you have for >these. I don't like the way the x's look with the straight and >convex flags for a couple of reasons: > >1) The x appears too remote or detached from the flag(s), and > >2) The x would seem to be indicating an alteration to a note on a >line or space two steps away from the note actually being altered, >which would tend to be confusing. (The Sims square root symbol also >has this problem.)All good points. I really don't like the |||'s or X's anyway, for reasons I've given before, and that I don't feel you've addressed. I'm just sort of going along for the ride on those, assuming you're going to have them anyway, and trying to make the best of it.>When I draw the X-symbols freehand, I imagine that I am constructing >the diagonals of a trapezoid having 3 right angles. The size and >shape of the symbol is the same as the corresponding one with 3 arrow >shafts, and the four corners of the trapezoid are determined by the >two ends of the outside shafts and the points of intersection of >those shafts with a flag. This is the way I would construct the X's >for a scalable font.I understood what you meant, despite the "3" right angles. I was more disappointed to find that the following 3 messages from you and 2 from Paul, in this thread, were only trapezoid trivia. :-) Here's some more. The U.S. definitions of trapezoid and trapezium are exactly swapped relative to the British/Australian definitions. In the OED and Macquarie dictionaries, a trapezium has only one pair of sides parallel, while a trapezoid has none. Websters has it the other way 'round. There's no requirement for any right-angles anywhere. At least it's good to know Paul's reading the thread. I've been wondering whether no-one else was contributing because (a) they think we're doing such a wonderful job without them, or (b) they have no interest whatsoever in the topic, and think we're a couple of looneys? Hey, I've become so obsessed about this notation that I was lying in bed this morning thinking how my various sleeping postures could be read as various sagittal symbols. I was imagining children being taught them kinaesthetically. Sagittal aerobic workout videos by Jane Fonda! :-)>I hope, then, that by agreeing on appropriate distinctions in size >between left and right flags we can eliminate them entirely. Yes.>> I'm not averse to a slight recurve on the concaves, but I'm afraid >I find>> some of those in symbols2.bmp, so extreme in this regard, that they >are>> quite ambiguous in their direction. With a mental switch akin to >the Necker>> cube illusion, I can see them as either a recurved concave pointing >upwards>> or a kind of wavy pointing down. Apart from any nub, I don't think >that>> they should go more than one pixel back in the "wrong" direction. >Those at>> the extreme lower left of the page look ok. >>You would have to think of the end of the flag as pointing and not >the curve of the flag.Well then why shouldn't I think of the end of the wavy flag as pointing, and we're back where we started. Same problem. Of course we're talking "at a distance in poor light" here.>Putting a nub on the end might even help in >that regard.A nub on which one. The wavy or the concave? I just don't think there is room in a concave symbol, (even at _twice_ the size it should be relative to other symbols), for nubs or curlicues.> But I can't see how I can get away from using a rise of >several pixels to indicate the line or space of the note being >altered. The part of the curve coincident with the arrow shaft just >isn't going to accomplish that.So how about my latest attempt? Unfortunately it probably looks the most like a quaver-rest of any of them.>I want to keep the concave flags for the two >smallest alterations. The convex-to-concave conversion in the >complementary symbols is a concept that I wouldn't want to discard.OK. I like that too. But don't forget that it only works on the right hand side. Those complementation rules that work for 217-ET, do not work for rational tunings. I've been investigating it in some depth. It seems it is not possible to assign consistent values to the various flags when sitting atop a double shaft, so as to get apotome complements of all of 17, 17', 19, 19', 23, 23', while maintaining the same ordering of flag-combinations in the second half apotome as in the first. I'm at a loss what to do about this, except for one off-the-wall suggestion, which is to run the order of flag-combinations in the _reverse_ direction in the second half apotome. This of course means that /||\ would be the 11' comma symbol and we'd have to find a new symbol for sharp. The apotome complement of the natural would of course be a natural with two tails (and no antenna), a rhombus on stilts. And of course the same thing flipped vertically for a flat. The advantage of this would be that a symbol and its complement would always have exactly the same flags. The disadvantage is that, having worked so hard to get the single shaft symbols to be size-representative, we'd have to try to make them work in exactly the _opposite_ way when used with two shafts. A two-shaft single concave would have to look the biggest and a two-shaft double-straight /||\, the smallest. I hope to post more about the possible compromises with the original /||\ = # scheme, in future.>> It seems to me that you have increased the possibility of confusion >of wavy>> with convex, by waving it higher. >>We'll just have to evaluate our latest efforts to see if that's a >problem, but I don't think it is.No. It isn't a problem now. Actually I don't think it's a problem if a reader interprets the concave symbols as short straight flags that start part way down the shaft, and similarly if the wavys look like short convex flags that start part way down the shaft. Their shortness and their part-way-down-the-shaft-ness are quite sufficient to distinguish them. What does it matter if they don't actually look like concave and wavy. The important thing is that they are distinct from the others and each other. We could even call them short-straight and short-convex (or just short curved). Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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