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Message: 4675 - Contents - Hide Contents Date: Thu, 18 Apr 2002 03:54:51 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> I did some more work on the symbols in a second file: > >> Yahoo groups: /tuning- * [with cont.] > math/files/secor/notation/symbols2.bmp > > which I will put out there, once Yahoo gets over its cranky spell and > lets me upload it.I'm still waiting to see that, but in the meantime I've put up my latest versions with changes based on several of your suggestions, and one innovation. Yahoo groups: /tuning-math/files/Dave/Sagittal... * [with cont.] C2DK.bmp Yahoo groups: /tuning-math/files/Dave/Sagittal... * [with cont.] 2DK.bmp
Message: 4676 - Contents - Hide Contents Date: Thu, 18 Apr 2002 04:35:29 Subject: Re: 41 ET 11-tone diatonic From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> of course, we've discussed not only two-dimensional but also > three- and four-dimensional periodicity blocks quite a bit -- we've > focused more attention on the ones where there's only one chromatic > unison vector and all the rest are commatic, but there's no 'rule' > that says there shouldn't be more than one chromatic unison vector.How would you work it?
Message: 4677 - Contents - Hide Contents Date: Thu, 18 Apr 2002 05:05:23 Subject: Re: one from the archives From: clumma>It's been a few days, so I'm joining the resending parade; I hope >Carl still remembers what these are about! > >[1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21] > >edge connectivity = 3I remember connectivity; 3 isn't so hot for a 12-tone scale in the 7-limit, eh?>characteristic polynomial = >x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968* >x^4-8*x^3+304*x^2+96*xNever did the homework to understand these.>This tells us there are 29 consonant intervals and 22 consonant >triadsThat doesn't seem like much to write home about either. What are some good scales here in your experience, Gene? The three winners on my list were: 1/1 21/20 9/8 7/6 5/4 4/3 7/5 3/2 14/9 5/3 7/4 15/8 1/1 16/15 28/25 7/6 5/4 4/3 7/5 112/75 8/5 5/3 7/4 28/15 1/1 21/20 7/6 6/5 5/4 21/16 7/5 3/2 8/5 42/25 7/4 9/5 I ended up using the first for melodic reasons.>The scale is not epimorphic, but can be extended in various ways, >for instance to the h15-epimorphic scale /.../ >I've attached two files which show the graph for each of these >scales. Thanks! >Now I'm wondering what the story is--where does this scale >arise from?I came up with it when drawing lattices on paper some time ago, when looking for good 12-tone 7-limit tunings for my piano. -Carl
Message: 4678 - Contents - Hide Contents Date: Thu, 18 Apr 2002 05:10:16 Subject: Re: one from the archives From: clumma I wrote...> The three winners on my list were: > > 1/1 21/20 9/8 7/6 5/4 4/3 7/5 3/2 14/9 5/3 7/4 15/8 "lester" > 1/1 16/15 28/25 7/6 5/4 4/3 7/5 112/75 8/5 5/3 7/4 28/15 "prism" > 1/1 21/20 7/6 6/5 5/4 21/16 7/5 3/2 8/5 42/25 7/4 9/5 "stelhex"And this one by Wilson/Hahn: 1/1 21/20 35/32 6/5 5/4 21/16 7/5 3/2 25/16 42/25 7/4 15/8 "class" -Carl
Message: 4679 - Contents - Hide Contents Date: Thu, 18 Apr 2002 07:49:59 Subject: Euclidean reduced 9-note scales From: genewardsmith Here are the Euclidean reductions for h9 in the 5 and 7 limits: 5-limit: [1, 16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8] This is pretty much of a classic--a rhombic block. 7-limit: [1, 15/14, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 28/15] This, with steps of size 16/15 and 15/14, as well as 9/8 and 28/25, fairly cries aloud to be tempered via 225/224~1. If we do that of course both scales become the same. I looked at all of the permutations of the 72-et version of this scale, and the original form turned out to be the clear winner: [0, 7, 19, 23, 30, 42, 49, 53, 65] Here are the 5,7, and 9-limit characteristic polynomials: x^9-16*x^7-16*x^6+57*x^5+84*x^4-34*x^3-84*x^2-8*x+16 x^9-21*x^7-28*x^6+65*x^5+100*x^4-71*x^3-116*x^2+26*x+44 x^9-29*x^7-80*x^6-39*x^5+70*x^4+52*x^3-16*x^2-9*x+2 We have 40 9-limit triads here, which looks pretty good! Here is a runner-up scale: [0, 7, 19, 23, 30, 42, 49, 53, 60] x^9-16*x^7-16*x^6+57*x^5+86*x^4-27*x^3-80*x^2-18*x+6 x^9-19*x^7-20*x^6+67*x^5+76*x^4-79*x^3-84*x^2+30*x+28 x^9-27*x^7-64*x^6+3*x^5+104*x^4+37*x^3-40*x^2-14*x
Message: 4680 - Contents - Hide Contents Date: Fri, 19 Apr 2002 11:23:07 Subject: Re: My Approach Generalized Diatonicity From: Carl Lumma>Two dissonant chords with no particular rationalisation may be hard to >tell apart. But there are plenty of dissonances, like 11-limit intervals, >which have their own quality distinct from other dissonances. And miracle >tuning is specifically optimized for them. > >I'm certainly not going to reject scales because they don't have enough >unambiguous consonances. We should be collecting scales that have most >reasonable properties, and see how well they work as diatonics. That's >going to take a long time, because it means writing fairly complex music >for each.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. 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? -Carl
Message: 4682 - Contents - Hide Contents Date: Fri, 19 Apr 2002 20:08:00 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> I'll assume for the moment that you accept the addition of a (17'- 17) flag > as the best way of giving us 17', because it's the only such choice that > also gives us 41 (assuming we can only use 1600-ET schismas).Some other things I like about it is that: 1) It fills in a size gap between the 19 and 17 commas (e.g, giving 2deg494); 2) It gives 2deg311 (added to the 19-comma), should someone want to notate that division.> Now if we ignore for the moment the relative sizes of the commas, and > therefore ignore which pairs we might want to have as left and right > varieties of the same type, I get the following left-right assignment of > flags as being the one that minimises flags-on-the-same-side for as far > down the list of prime commas as possible. > > By the way, if this list has only one comma for a given prime, the reason > is that the same comma is optimal for both diatonic-based (F to B relative > to G) and chromatic-based (Eb to G# relative to G) notations. > > Symbol Left Right > for flags flags > ------------------------------ > 5 = 5 > 7 = 7 > 11 = 5 + (11-5) > 11' = 29 + 7 > 13 = 5 + 7 > 13' = 29 + (11-5) > 17 = 17 > 17' = 17 + (17'-17) > 19 = 19 > 19' = 19 + 23 > 23 = 23 > 23' = 17 + (11-5) > 29 = 29 > 31 = 19 + (11-5) > 31' = 5 + (17'-17) + 7 > or 5 + 23 + 23 > 37 = 29 + 17 > 37' = 19 + 23 + 7 > or 5 + 17 + 23 > or 5+5+19 > 41 = 17 + (17'-17) + (17'-17) > 43 = 19 + 19 + (17'-17) > 47 = 19 + 23 + 23 > or 5 + 17 + (17'-17) > ... So my new proposal for the flags is > > | Left Right > ---------+--------------- > Convex | 29 7 > Straight | 5 (11-5) > Wavy | 17 23 > Concave | 19 (17'-17)This looks very workable, and I am about 99 percent sold on it. (Just give me some more time.)> Complementary > Flag Size Size Flag > comma in steps of comma > name 217-ET name > ---------------------------- > Left > ---- > 29 6 -2 none available with same side and direction > 5 4 0 blank > 17 2 2 17 > 19 1 3 none available with same side and direction > > Right > ----- > 7 5 1 (17'-17) > (11-5) 6 0 blank > 23 3 3 23 > (17'-17) 1 5 7 Likewise.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, because I don't see any good reason. While it is nice to have everything come out exact using 1600 as a frame of reference, do you think anyone is actually going to be able to use it in a performance to produce pitches? The increments are much smaller than 1 cent, and the pitches can't be related easily to 12-ET, as Johnny Reinhard is doing. (i.e., not a subdivision, as is 1200-ET), ) So if we're trying to accommodate him with this notation, all that's really necessary is to keep the schismas small and provide the number of cents somewhere on the score, at least in a table with the symbols. --George
Message: 4686 - Contents - Hide Contents Date: Fri, 19 Apr 2002 21:19:31 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:>> I added some more to this file: >> >> Yahoo groups: /tuning- * [with cont.] >> math/files/secor/notation/symbols1.bmp >> >> I think the problem with reading them under poor conditions is a >> combination of factors -- vertical lines that are rather thin *and* >> vertical lines that are too close together, with the second factor >> being more of a problem than the first. I re-did the straight- flag >> symbols on the fourth staff using single-pixel vertical lines with >> enough space between them to make them legible at a distance. I also >> put a couple of them in combination with conventional sharps and >> flats at the upper right, for aesthetic evaluation. >> I like this better, but part of the family resemblance of the existing > symbols is that they are all ectomorphs, except for the rarely seen > double-sharp being a mesomorph. They are not endomorphs like your||| and X> symbols. Even some of my | symbols are pushing it.The symbols get fatter as the alterations become larger, which is only logical. 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?>> Just above the "conventional accidentals" staff I also added my >> conventional sharp to the left of yours. Mine (more than yours) >> looks more like what I found in printed music, and I suspect that >> Tartini fractional sharps constructed (or written) with too- narrow >> spacing between the vertical lines (such as we have here) are what >> led to Ted Mook's observation. (And I do prefer your version.) >> Maybe, but part of the problem is just that it is hard to tell 2 identical > side-by-side things from 3 identical side-by-side things with the same > spacing. I've made my ||| and X's wider now, but not as wide as yours, and > shortened the middle tail of the 3 by 3 pixels relative to the others, so > they are not 3 identical things any more.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). 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. (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.)>>> Frankly, I think the best solution is to use two symbols side by side >>> instead of the 3 and X shaft symbols, the one nearest thenotehead being a>>> whole sharp or flat (either sagittal or standard). I thinkwe're packing so>>> much information into these accidentals that we can't afford totry to also>>> pack in the number of apotomes. I suggest we provide single symbols from >>> flat to sharp and stop there. At least you have provided the double-shaft >>> symbols so you never have to have the two accidentals pointing in opposite >>> directions. >>>> What? Did I understand this correctly? Are you considering using >> the double-shaft symbols??? > > No. >>> (Or are you suggesting that I should do >> this and forget about the ||| and X symbols?) > > Yes. >>>> The fact that in all the history of musical notation, a single symbol for >>> double-flat was never standardised, tells me that it isn't very important, >>> and we could easily get by without a single symbol for double- sharp too. >>>> I think that it's because two sharps placed together looks a little >> weird, but not two flats. >> You're just strengthening my case. I don't find that two of our symbols > side by side have the same problem as the conventional sharp symbol (making > a third phantom symbol in between). I find them to be more like the > conventional flat symbol, for which no single-symbol double has ever been > seen as necessary. >>> If I remember correctly, I think that >> double-flats are placed in contact with one another, as I put them >> above the staff (but I will need to check on this.) >> Those I have found have not been touching. I do not propose to have ours > touching either.Yes, I saw that, once I found an example.> ... I just thing this is too confusing, and the ||| and X symbols are not > required anyway. >>> I did make my new symbols (in the fourth staff) one pixel longer (at >> the tip of the arrow), which can be seen only when the note is on a >> line. I didn't think that it was wise to overlap the flag in the >> other direction, because this would make the nubs at the ends of the >> flags less visible if a staff line were to pass through them. Where >> I now have them, the nubs (actually 3x3 pixel squares) are in both >> cases immediately adjacent to staff lines. >> I don't see what's wrong with the nubs straddling the line in one case, and > being in free space in the other case. i.e. put them, not one, but 2 pixels > further out than you have them now.They bigger they get, the uglier they look. I eventually realized that the reason why they had to be so big for the straight flags is that the straight lines are of constant thickness, whereas the others get thinner at the ends, making the nubs easier to see. 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. 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.>> By the way, if you look at the concave flags in my earlier figures, >> you will see that one part of the curving flag has postive and >> another part negative slope (and I am still making them this way); a >> nub on the end of this sort of flag can be seen very easily. > > True.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.>>> ... I copied three of your symbols so I can comment >> on them. In all three of them the concave or wavy flag is >> significantly lower than the line or space for its note. >> You mean the upward pointing ones? Concave I can understand, but wavy? The > horizontally inflected part of the wavy flag is always exactly centred > relative to the center of the notehead.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. Why shorter? I think that the concave and wavy flags should be smaller than the convex and straight flags -- both in length and thickness.>> 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 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.>> At the top right (under altitude considerations) I put my latest >> version of the symbols in combination with conventional sharps and >> flats, with single-symbol equivalents included (above the staff). >> I hope you show some down pointing ones next to a flat, because I think > they look strange if their tails are too much shorter than the flat's tail. > In fact I'd be in favour of making the tails of down-pointing arrows longer > than up-pointing ones.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.) Slowly, but surely, we are making progress. --George
Message: 4687 - Contents - Hide Contents Date: Fri, 19 Apr 2002 11:24:35 Subject: Re: one from the archives From: Carl Lumma>>> >he adjacency matrix of a graph is a square matrix labeled by >>> verticies; it has a "1" if they are connected, and a "0" if not >>> (counting the vertex to itself as a "0".) The characteristic >>> polynomial of this is as above, and is a graph invariant. The >>> n-2 term gives the number of edges, and the n-3 term twice the >>> number of triads. >> >> That's crazy. > >Why?What makes anything crazy? That you don't understand it? That's a pretty good def., I guess.>> Does it show tetrads? >>It doesn't show anything. To show tetrads, we would need to count >principal minors which were all 1s except along the diagonal, which >would be 0. I don't know if that can be done using the coefficients >of the characteristic polynomial. Noted. -Carl
Message: 4688 - Contents - Hide Contents Date: Fri, 19 Apr 2002 00:33:41 Subject: Re: A common notation for JI and ETs From: David C Keenan --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>>> I think there should be a strong family resemblance between the > standard>>> symbols and our new ones, or they will not be found unacceptable > on visual >>> aesthetic grounds. >>>> Yes, that is a very valid point.Of course you knew I meant to write "will be found unacceptable" or "will not be found acceptable".> I did some more work on the symbols in a second file: > >> Yahoo groups: /tuning- * [with cont.] > math/files/secor/notation/symbols2.bmp > > which I will put out there, once Yahoo gets over its cranky spell and > lets me upload it.Got it at last. Thanks.> The more I look at your symbols, the more I like their style,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.> so > (assuming that the file is out there) please follow along with me. > > The fifth staff is a synthesis of features from both of our efforts > above that. I made the sesequisharp (|||) and double-sharp (X) group > of symbols intermediate in width between what each of us had,Ok. We agree on the line-thickness and overall width of all the tails now, 5 pixels for ||, 7 pixels for both ||| and X. I hope you like the idea of shortening the middle stem of the ||| by 3 pixels so it's more like |'|. I think that having that ^ shape in the tail tends to put them psychologically in the same apotome as the X tails. 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. 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.> 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. By the way, I'm finding Elton John and Bernie Taupin's 'Goodbye Yellow Brick Road' songbook to contain examples of just about everything with regard to accidentals. My recommendation is to make the nubs 4 x 4 with the corner pixels knocked out. More round, less square. @@@@ @@@@@@@@ @@@@@@@@ @@@@> To the right of that I copied three of your symbols so I can comment > on them. In all three of them the concave or wavy flag is > significantly lower than the line or space for its note.Again, I don't understand this statement in regard to the wavy flags. But I do notice you're missing one pixel from your copy of one of my wavys, which makes it look a little bit lower. I hope the drawings in the flags.doc file will help you understand where I'm coming from on the wavy and concave flags. Unfortunately, in this conception, the concaves do not lend themselves to the addition of nubs, because they are already quite thick on the ends.> 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. I'm guessing that you need the huge recurve to convince yourself that concave can represent larger commas than wavy? I would have agreed that, if you want the set of 3 flag types that are maximally distinct from one another, (to be used for the lowest primes) it is probably {concave, straight, convex}. However in typing those curly brackets above, I had the thought: Isn't it interesting that our character set includes brackets that correspond to some of our flags (in like pairs turned sideways). It has (-- convex, <-- straight, and {-- wavy, but _not_ concave. And of course we also have [-- convex right angle, but my feeling is that it would be hard to make those fit the style of the standard accidentals.> 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.> Perhaps the wavy flag would now be most appropriate for the > smallest intervals.I'd need to know what they all mean re commas or see a complete set, in order, for the first 12 degrees of 217-ET. I thought it made sense for apotome-complements, that wavy should be its own complement, and convex and concave should be complements, when they _have_ complements (which is only on the right).> I put a set of convex flag symbols on the 6th staff, which (like the > straight-flag symbols) combines features from both of our previous > efforts).These all look fine to me, except I'd leave off the nubs for those with two flags of the same type. I don't have a clear preference yet for nubs versus change-of-width, for indicating relative size while reducing lateral confusability. Maybe we can perfect both, then present them and ask folks to vote on them. On a single shaft I made sL 5 pixels (including the shaft) and sR 7 pixels. But when I combine the two I make them both 6 pixels wide for a total symbol width of 11. I did the same thing with xL 7 pixels (not shown) and xR 5 pixels. I made wL 4 pixels and wR 5 pixels. wR represents a smaller comma than sL, so it couldn't be more than 5. A 3 pixel wavy wouldn't work, but when they are both on the same shaft I would make them both 4 pixels. Both vL and vR would be 4 pixels because they represent smaller commas than the wavys, and I figure you just can't go narrower than 4 pixels. I want to make all the flags as narrow as is reasonable so that the double flag symbols are not getting too wide and out-of-character with the standard symbols.> At the top right (under altitude considerations) I put my latest > version of the symbols in combination with conventional sharps and > flats, with single-symbol equivalents included (above the staff).As I said before, it would be good to show some down-pointing ones with conventional flats.> Let me know what you think. Done.-- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4689 - Contents - Hide Contents Date: Fri, 19 Apr 2002 07:45:47 Subject: Re: 41 ET 11-tone diatonic From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> what do you mean, how would you work it? and btw, didn't you produce > a 9-tone scale that is a perfect example of this, having two > chromatic unison vectors?You tell me, starting with the scale and passing on to the unison vectors.
Message: 4690 - Contents - Hide Contents Date: Fri, 19 Apr 2002 10:20:23 Subject: Re: 41 ET 11-tone diatonic From: manuel.op.de.coul@xxxxxxxxxxx.xxx Yahoo wrote:>------------------------ Yahoo! Groups Sponsor ---------------------~--> >Kwick Pick opens locked car doors, >front doors, drawers, briefcases, >padlocks, and more. On sale now!Kwick Pick? They're carrying advertisements for the burglar's guild now? I'm in favour of moving to the Columbia list too. Manuel
Message: 4691 - Contents - Hide Contents Date: Fri, 19 Apr 2002 07:50:41 Subject: Re: one from the archives From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>> The adjacency matrix of a graph is a square matrix labeled by >> verticies; it has a "1" if they are connected, and a "0" if not >> (counting the vertex to itself as a "0".) The characteristic >> polynomial of this is as above, and is a graph invariant. The >> n-2 term gives the number of edges, and the n-3 term twice the >> number of triads. > > That's crazy. Why?> What happened to the n-1 term, above?Since the diagonal terms are all 0, the trace is 0 and so the trace term is 0.> Does it show tetrads?It doesn't show anything. To show tetrads, we would need to count principal minors which were all 1s except along the diagonal, which would be 0. I don't know if that can be done using the coefficients of the characteristic polynomial.
Message: 4692 - Contents - Hide Contents Date: Fri, 19 Apr 2002 11:40 +0 Subject: Re: My Approach Generalized Diatonicity From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4.2.2.20020418115051.01e58740@xxxxx.xxx> Me:>> That looks like the heart of the issue. I find it much easier to tell >> two dissonances apart than a dissonance from a consonance. Carl:> You do? Have you got that backward? Anyway, that isn't the question. > The question is: if I randomly play you either 6:5 or 5:4, harmonically, > and ask you to identify them, would you perform better than if I had > used 11:9 and 9:7?Um, yes, wrong way round. I don't have very good relative pitch at all. The real test would be 9:7 and 11:8, because they're close enough to be confused. The easiest way of telling intervals apart is by how consonant they are, so I should easily be able to differentiate 5:4 from either 11:9 or 9:7 if they're tuned well enough. I'm sure I can hear the in-tune-ness of 11:8 as well, in the right circumstances. So an interval class containing both 9- and 11-limit "consonances" should have more audible variety than one with only 5-limit consonances.>> With my schismic keyboard setup I found it remarkably difficult to >> distinguish 5:4 and 6:5 what with them both being well tuned >> consonances and not far apart. >> Really? Maybe I just find consonances easier to recognize because > I've trained myself to do it... maybe it isn't innate.Really, I think it's your training at work here. Fifths and thirds are easy to tell a part, of course, because of the change in consonance. Thirds can also be distinguished in 12- or 31-equal because major thirds are better tuned. I was surprised at how the nature of a minor third changes in 19-equal, and how similar major and minor triads (to be specific, rather than thirds) become when the two thirds are well tuned, and slightly closer in pitch than they would be in JI. Two dissonant chords with no particular rationalisation may be hard to tell apart. But there are plenty of dissonances, like 11-limit intervals, which have their own quality distinct from other dissonances. And miracle tuning is specifically optimized for them. I'm certainly not going to reject scales because they don't have enough unambiguous consonances. We should be collecting scales that have most reasonable properties, and see how well they work as diatonics. That's going to take a long time, because it means writing fairly complex music for each. Graham
Message: 4693 - Contents - Hide Contents Date: Fri, 19 Apr 2002 11:40 +0 Subject: Re: scala stability logic From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a9n8mm+abjt@xxxxxxx.xxx> emotionaljourney22 wrote:> i, for one, am opposed to defining an "extra-sensory" chromatic, as i > complained before in reference to balzano and clough. i'm glad to > hear mark gould is with me on this -- sorry if i implied otherwise, > mark.I'm not suggesting anything "extra-sensory" either. If your target listener senses the 612 cent interval as belonging to a distinct interval class, rather than being a tuning of a "tritone" of around 600 cents, we'll consider the Pythagorean diatonic independent of the 12 note chromatic. If they can hear a 612 cent interval as being wider than a 588 cent interval, when they're at unrelated pitches, we can even call the Pythagorean diatonic improper. Graham
Message: 4694 - Contents - Hide Contents Date: Fri, 19 Apr 2002 11:40 +0 Subject: Re: scala stability logic From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4.2.2.20020418114756.01ece7b0@xxxxx.xxx> Carl Lumma wrote:>> For Rothenberg stability and efficiency to work properly, you need to >> define each diatonic on a chromatic. >> ? What terminology is this?Mark Gould defines diatonics, pentatonics and chromatics. I'm removing the distinction between diatonics and pentatonics. Treating the chromatic as an equal temperament, even if it's tuned differently, is needed for Rothenberg's conclusions about the ambiguity of the tritone to be valid in meantone. Graham
Message: 4695 - Contents - Hide Contents Date: Fri, 19 Apr 2002 11:07:02 Subject: Re: one from the archives From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> [1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21] >> >> edge connectivity = 3 >> characteristic polynomial = >> x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968* >> x^4-8*x^3+304*x^2+96*xLet's compare this to some other possibilities: [1, 21/20, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] This is a sentimental favorite--my first scale. x^12-27*x^10-38*x^9+168*x^8+366*x^7-206*x^6-950*x^5-474*x^4+400*x^3+437*x^2+130*x+12 connectivity = 2 Not quite up to your numbers, but a good extension of JI diatonic. [1, 15/14, 9/8, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3, 7/4, 15/8] x^12-26*x^10-36*x^9+156*x^8+334*x^7-176*x^6-768*x^5-266*x^4+370*x^3+190*x^2-36*x-15 2 Another verion of the above. [1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 28/15] This is Euclidean reduced, vertex centered. x^12-27*x^10-38*x^9+168*x^8+368*x^7-172*x^6-792*x^5-232*x^4+368*x^3+96*x^2-64*x 2 [1, 15/14, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] This is Euclidean reduced, tetrad (shallow hole) centered. x^12-28*x^10-42*x^9+175*x^8+430*x^7-70*x^6-812*x^5-396*x^4+374*x^3+302*x^2+ 32*x-3 2 [1, 15/14, 8/7, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 7/4, 15/8] This is Euclidean reduced, hexany (deep hole) centered. Finally something to beat your numbers; I think your scale is actually quite good. This has 30 intervals and 25 triads. x^12-30*x^10-50*x^9+189*x^8+506*x^7-103*x^6-1118*x^5-487*x^4+772*x^3+508*x^2-128*x-96 3 [1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] A modified version of the shallow hole reduction. x^12-30*x^10-48*x^9+193*x^8+498*x^7-96*x^6-1096*x^5-735*x^4+238*x^3+278*x^2-8*x-27 2 [1, 21/20, 35/32, 7/6, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] Another modified shallow hole reduction. x^12-29*x^10-44*x^9+183*x^8+432*x^7-148*x^6-924*x^5-444*x^4+276*x^3+177*x^2-22*x-15 2
Message: 4696 - Contents - Hide Contents Date: Fri, 19 Apr 2002 12:07:41 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau See my latest attempt at the variable-width nub-free style. I've included a symbol for every one of our prime commas up to 47, and shown the purely sagittal double-symbol notation between sharp and double-sharp. Yahoo groups: /tuning-math/files/Dave/SymbolsD... * [with cont.]
Message: 4697 - Contents - Hide Contents Date: Sat, 20 Apr 2002 01:28:16 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> | Left Right >> ---------+--------------- >> Convex | 29 7 >> Straight | 5 (11-5) >> Wavy | 17 23 >> Concave | 19 (17'-17) >> This looks very workable, and I am about 99 percent sold on it. > (Just give me some more time.)Sure. We want to be sure we've explored every option thoroughly.> 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, because I don't see any good > reason.It doesn't have to be 1600-ET. It doesn't even have to be an ET. It might be a linear or planar or whatever temperament. But I feel it is highly desirable to know that the schismas we are using do not somewhere add up to something considerably more than 0.5 cents. i.e. I want to know what maximum error (over all the intervals in our highest odd limit) is implied by our choice of notational schismas. If we don't know what temperament it is based on, we may happen to have two near 0.5 cent schismas that "pull in opposite directions".> While it is nice to have everything come out exact using > 1600 as a frame of reference, do you think anyone is actually going > to be able to use it in a performance to produce pitches?Not at all. But it is significant that (in the simplest example) our single symbols for 13 and 35 are identical. We are presenting the composer with a choice. Either use a pair of separate symbols for 35, or accept that the performer will read it as a 13 diesis (or the corresponding number of cents) and introduce a certain error. We're trying to keep that error below 0.5 cents, although I think we've already got one of 0.6 c.> The > increments are much smaller than 1 cent, and the pitches can't be > related easily to 12-ET, as Johnny Reinhard is doing. (i.e., not a > subdivision, as is 1200-ET), )Although it may be convenient that its divisions are exactly 3/4 of a cent.> So if we're trying to accommodate him > with this notation, all that's really necessary is to keep the > schismas small and provide the number of cents somewhere on the > score, at least in a table with the symbols.That's right. But "keep the schismas small" means also the effective _combination_ schismas for all the intervals between pairs of odd numbers, not just the odd numbers themselves. Actually, I'm not sure if I know what I'm talking about here. At least with 1600-ET we knew where we stood. The thing may be to find another ET above 1000 or some other system that accomodates all the schismas we want. Gene was looking at these for us and he found 1600 was the best 31 limit unique ET of all those less than or equal to it in size, but hasn't gone to higher primes yet. He may have lost interest. There are 11 odd primes up to 37, if we have 11 independent schismas, and express them as prime-exponent vectors and take the determinant of the resulting square matrix, I understand we'll get the cardinality of the corresponding ET. However I'm pretty sure we don't have 11 independent schismas and I get a little hazy here about how to find generator mappings. I'm hoping Graham Breed or Gene Smith can help us here. I think we just need to give the vector for every schisma we're interested in. Then ask them to tell us the maximum error implied by various sets of these.
Message: 4698 - Contents - Hide Contents Date: Sat, 20 Apr 2002 09:07:46 Subject: More 12-tone JI scale comparisons From: genewardsmith 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. I also put in "Wille's k value" to get a start on some of the measures I don't understand; this seemed to be a good place to start since it makes no sense to me at all. Does anyone have a clue? Class [1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8] triads 26 intervals 31 connectivity 3 improper CS lumma .043920 k 437 Stelhex [1, 21/20, 7/6, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 9/5] triads 26 intervals 30 connectivity 3 improper lumma .081284 k 787 Euchex [1, 15/14, 8/7, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 7/4, 15/8] triads 25 intervals 30 connectivity 3 strictly proper CS lumma .253235 k 787 Prism [1, 16/15, 28/25, 7/6, 5/4, 4/3, 7/5, 112/75, 8/5, 5/3, 7/4, 28/15] triads 24 intervals 30 connectivity 3 strictly proper CS lumma .440966 k 262 Tet-a [1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] triads 24 intervals 30 connectivity 2 improper CS lumma .321977 k 262 Tet-b [1, 21/20, 35/32, 7/6, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] triads 22 intervals 29 connectivity 2 improper CS lumma .355766 k 262 Lumma [1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21] triads 22 intervals 29 connectivity 3 improper lumma .081284 k 262 Euctetrad [1, 15/14, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] triads 21 intervals 28 connectivity 2 improper CS lumma .166834 k 1837 Gene [1, 21/20, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] triads 19 intervals 27 connectivity 2 strictly proper CS lumma .437710 k 112 Eucvert [1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 28/15] triads 19 intervals 27 connectivity 2 strictly proper CS lumma .262832 k 367 Lester [1, 21/20, 9/8, 7/6, 5/4, 4/3, 7/5, 3/2, 14/9, 5/3, 7/4, 15/8] triads 18 intervals 26 connectivity 2 strictly proper CS lumma .490032 k 337 Gene-a [1, 15/14, 9/8, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3, 7/4, 15/8] triads 18 intervals 26 connectivity 2 strictly proper CS lumma .333977 k 787
Message: 4699 - Contents - Hide Contents Date: Sat, 20 Apr 2002 03:18:50 Subject: Re: A common notation for JI and ETs From: David C Keenan --- 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.> 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.> 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.> 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 concave are 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.> 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 seems too 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.> 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. 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 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.> 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.> Slowly, but surely, we are making progress.Yes indeed. :-) -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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