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Message: 5725 - Contents - Hide Contents Date: Thu, 12 Dec 2002 02:07:32 Subject: Relative complexity for 225/224 planar From: Gene Ward Smith Here is the ordered listing of class representatives: 4/3, 15/14, 5/4, 7/5, 6/5, 9/8, 7/6, 8/7, 9/7, 21/20, 10/9, 21/16, 25/21, 25/24, 35/32, 48/35, 60/49, 80/63, 50/49, 27/20, 36/35, 25/18, 63/50, 49/40, 35/27 ... Here is a basis for the lattice: 3/2 ~ [2.973037720, 1.272695617] 15/14 ~ [-2.808540818, 2.730573683] Here are lattice points. Plotting the middle column and labeling the points using the first column is what I have done before, in a rough and ready way. Plotting the third column and lableing by the first column, and -1 times the third column and lableing by 2 / first column, gives something which does not depend on the particular choice of coordinates, but is intrinsic to the temperament, and which seems to be a good way to look at it. 4/3 [-1, 0] [-2.973037720, -1.272695617] 15/14 [0, 1] [-2.808540818, 2.730573683] 5/4 [-1, -1] [-.164496902, -4.003269300] 7/5 [1, -1] [5.781578538, -1.457878066] 6/5 [2, 1] [3.137534622, 5.275964917] 9/8 [2, 0] [5.946075440, 2.545391234] 7/6 [-1, -2] [2.644043916, -6.733842983] 8/7 [0, 2] [-5.617081636, 5.461147366] 9/7 [2, 2] [.328993804, 8.006538600] 21/20 [2, -1] [8.754616258, -.185182449] 10/9 [-3, -1] [-6.110572342, -6.548660534] 21/16 [1, -2] [8.590119356, -4.188451749] 25/21 [-3, 0] [-8.919113160, -3.818086851] 25/24 [-3, -2] [-3.302031524, -9.279234217] 35/32 [-1, -3] [5.452584734, -9.464416666] 48/35 [2, 3] [-2.479547014, 10.73711228] 60/49 [0, 3] [-8.425622454, 8.191721049] 80/63 [-3, 1] [-11.72765398, -1.087513168] 50/49 [-2, 2] [-11.56315708, 2.915756132] 27/20 [4, 1] [9.083610062, 7.821356151] 36/35 [3, 3] [.493490706, 12.00980790] 25/18 [-4, -2] [-6.275069244, -10.55192983] 63/50 [4, 0] [11.89215088, 5.090782468] 49/40 [1, -3] [11.39866017, -6.919025432] 35/27 [-4, -3] [-3.466528426, -13.28250352]

Message: 5726 - Contents - Hide Contents Date: Thu, 12 Dec 2002 17:29:42 Subject: Re: A common notation for JI and ETs From: David C Keenan At 11:21 AM 11/12/2002 -0800, George Secor wrote:>> JI can of course also be notated with rational sagittal notation, not >tied to>> any ET. But such a notation must of course be limited in some way, >since>> there is an infinite number of rationals. >>And this is where the 217 mapping comes in. Since the notational >schismas all vanish in 217, all of the comma roles are usable. No >other division has this property.True, but not all of the schismas that vanish in it are small enough to be acceptable for rational notation.>> I wonder if Manuel Op de Coul could easily write a program that would >go>> through every file in the Scala archive and count the number of times >each>> rational pitch occurs and then list them in order of popularity (I >think we>> can safely omit 2/1 :-). It may be that we are worrying about the >notation>> of 17/7 when in fact we don't have a single symbol for many others >that are>> in far greater demand. >>Those are the instances that the 217 mapping is supposed to handle.Yes. But that's a choice a strict JI-ist may or may not be willing to make, so I'd prefer to say that we do not yet have a symbol for a 7:17 comma rather than tell them to use a symbol that has a 1.8 cent error relative to another use of the same symbol, namely as the 5:13 comma.>But I wonder how much help a popularity poll will be, because I can >give you an uncontrived example in which the way we are notating 11:14 >won't even be acceptable. Suppose C is 1/1 and Margo wants to notate a >22:28:33 triad on C. The notation we have for 14/11 in the two >versions is F)!!~ and Fb(|, but she wants E-something. The 217 mapping >gets her out of a pinch by letting her use E|(. Even if she tried to >notate the same triad on 11/8 or F/|\, she would want A-something for >7/4 as the third of the triad, and a 217 mapping would give her A(|). >I can imagine the gears turning in your head as you're asking, "what's >the schisma?" (If you have to ask, then you can't afford it.) This is >one of those times when we don't have enough commas, as you noted >above. (This would all become much more useful if we had 31-ET >instruments that could be used with the 217 notation.)Good point. But I don't see that it negates the desirability of those statistics. In addition to trying to notate the most popular intervals we should try to notate them _as_ the appropriate interval class(es).>> From the other side, why are we concerned with the complete 17-limit > >> diamond when we don't have unique symbols for the commas involved in >the>> 13-limit diamond. |( is used for both 5:7 comma and 11:13 comma, and >(|(>> for both 5:11 comma and 7:13 comma. 0.83 cents different. Strict JI >types>> are probably not going to accept this. At one stage we were keeping >the>> notational schismas below 0.5 cents, but they seem to have crept up >as time >> went on. >>The symbols for the 5:7 and 11:13 commas don't have to be unique in >order for the ratios in a tonality diamond to be notated uniquely.I realised that.>Even if you fall back on a 217 mapping for JI and use the 217-ET >standard symbols, you can still notate a 19-limit tonality diamond >uniquely (as letter-plus-symbol combinations). Uniqueness is lost only >if you start using multiple tonality diamonds in the same composition.Sure. But composers do. The proposed Scala archive stats would give us at least some kind of handle on that.>Do you seriously think that a composer is going to get upset because a >player missed a pitch by ~0.83 cents on account of an insufficiently >precise notation?No because they wont be able to hear it (although some will claim otherwise). But some will be upset at the _idea_ of it being possible. And we're actually talking about 1.8 cents here if a 7:17 from one note is mistaken for a 5:13 from another.> Or that a composer is going to specify two >consecutive pitches differing by 0.83 cents in a composition (or if so, >I think that they would be treated like adaptive JI)? We need to step >away from the nitty-gritty details and consider the big picture for a >moment: what is our objective, anyway? > >This is supposed to be a performance notation, and to keep the number >of symbols manageable, we have: > >1) Allowed a number of small schismas to vanish; and >2) Allowed the flags and symbols to vary in size according to the >tuning. > >Since the symbols don't indicate precise intervals; the composer must >provide some sort of indication as to how they are being used in a >composition, and we probably should have some sort of spreadsheet that >would automate this (and which would simultaneously calculate Reinhard >1200-ET notation). I'm trying to look at this in the practical way >Johnny does: with the notation you give enough of an indication to get >the player very close and you then depend on the player's ear to handle >the rest -- so if it's exact JI that is desired, let the player listen >in order to make the fine adjustments.Sure. But we're just disagreeing on how close iks close enough. Johnny gives it within 0.5 cents. All I'm saying is that 1.8 cents is too far.>I think that we have done the best we could in keeping a balance >between precision (of notation) vs. complexity (of symbols).I think so too.>>> The symbol with which we would have no problem is the one that >>> represents the 7:17 comma exactly (a zero schisma, so it would be >valid>>> everywhere that both the 17' and 7 commas are valid): the 17'+7 >comma,>>> or ~|(). It's three flags, but I tried making the symbol, and it >looks>>> nice enough (i.e., it's easy enough to identify all the flags). >>>> Might be a good idea. I don't think strict JI-ists will accept a >symbol>> that looks so obviously like a stacked pair of 5-comma symbols, as a >7:17>> comma symbol _or_ a 5:13 comma symbol. These also involve schismas > >0.8 cents. >>Now I'm having second thoughts about making anything that complicatedOK. Forget it.>> A 5:13 symbol might be \(|\. which means a 13' symbol with an >upside-down>> 5-comma flag added. >>I don't like the idea of adding more to a symbol to make it smaller in >size, Good point.>> Since it is even smaller than the 19-comma, a 5-schisma flag will >make it>> possible to fully notate ETs even larger than 494, for what that's >worth. >> Try 624. >>Everyone would ask why we didn't do 612.Sure, look at that too. I just suggested 624-ET because it's 27-limit consistent where 612 is only 11-limit, but I'm guessing you'll tell me the error in its second-best x:13 is good enough and then it's 29-limit unique or better.>All of this opens up so many new possibilities that suddenly it seems >that we're back where we were last spring.Not at all. No one is proposing to throw away any of that.> I don't know what to say >about the paper now, because I thought that most everything works out >pretty well if you don't go above 217. Absolutely. > Do you really think that a 5' >comma wouldn't be too complicated for a performance notation? Or >should it instead be incorporated into an ascii-based expanded/modified >version (for theoretical and electronic applications) of what we now >have?You're right. It would probably be too complicated. But in any case we have agreed that whenever we add new complexity to the notation it should not make the simple stuff more complicated. So it's just a matter of deciding what we have really acomplished with the notation as it stands, and what should be left to a possible future extension of the notation (possibly along the lines of the +- 5'-comma). I propose that the diaschisma and 7:17 comma be left to such a future extension. I'd prefer that they were not defined in the current XH18 article. If you felt it necessary you could still mention that it is a property of 217-ET that its 7:17 comma is the same size as its 5+5-comma. But many other ETs have similar properties for lower-limit commas and I wouldn't expect you to list all of them in this article. That could wait for a more detailed catalog of ETs and their notations. I'd like to suggest that we not have any notational schismas larger than half the 5'-comma (i.e. none larger than 0.98 cents), given that adding or subtracting this comma is a possible way of extending the notation. Is it only 7:17 that that would kill? I also propose that ETs larger than 217 be left to a possible future extension, and maybe some of the more difficult ones above 72-ET. I'd prefer to reach agreement on the limitations of the existing flags and the XH18 article, before further discussions on new flags. But I will say: Now that you've centered those right triangles, the filled ones look too much like concave flags. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5727 - Contents - Hide Contents Date: Thu, 12 Dec 2002 10:23:17 Subject: Planar plots From: Gene Ward Smith I've figured out how to get Maple to do these plots, so I am going to put up some jpg files of lattice plots of planar temperaments; the first is: Yahoo groups: /tuning-math/files/planar20plots... * [with cont.]

Message: 5728 - Contents - Hide Contents Date: Thu, 12 Dec 2002 08:48:37 Subject: Re: Planar plots From: monz ----- Original Message ----- From: <genewardsmith@xxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Thursday, December 12, 2002 2:23 AM Subject: [tuning-math] Planar plots> I've figured out how to get Maple to do these plots, > so I am going to put up some jpg files of lattice plots > of planar temperaments; the first is: > > > Yahoo groups: /tuning-math/files/planar20plots... * [with cont.]that link is wrong ... it's Yahoo groups: /tuning-math/files/planar plots/... * [with cont.] how about including, on the diagram itself, a URL to a post explaining what it shows? -monz

Message: 5729 - Contents - Hide Contents Date: Fri, 13 Dec 2002 20:08:36 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> At 12:44 PM 12/12/2002 -0800, George Secor wrote:>> So I'll leave the 7:17 comma out of the list of commas in Table 1. >> I'll also leave a listing of the ratios of 17 out of Table 2 and just >> give the notation for 17/16 and 32/17, as I did with the higher primes. >> The ratios of 17 were taking up a lot of space anyway, and a complete >> 15-limit listing (plus the odd harmonics and subharmonics up to 29) is >> still pretty impressive. >> Yes. 15-limit is impressive enough, and along with the higher harmonics, > should keep most people happy.After unsuccessfully trying to figure out how I would redo Table 2 without the ratios of 17 (problems with the layout), I changed my mind and left both Tables 1 and 2 as is, except that I added a bold asterisk in both tables to identify the 7:17 comma role for //| as "controversial" (a word I used in several places). I decided that it would be appropriate to give the notation for the 17-limit consonances, inasmuch as they are fully compatible with a 217 JI mapping and inasmuch as there is only one comma role that would be open to question. In connection with this I stated that "it is possible that this symbol may not be acceptable to some JI practitioners for this comma role, in which case another symbol apart from those given in Table 1 would be used." (As to what that symbol would be, see below.)>>>> ... I think that we have done the best we could in keeping a balance >>>> between precision (of notation) vs. complexity (of symbols). >>>>>> I think so too. >>>> And as we introduce more complexity we can get more precision. Onward! >> OK. But not for the XH18 article, right?Right! That will have only 18 symbols and no ETs above 217. And I deleted the 99-ET symbol set from Table 4. I also added a sentence in one place that reads, "At the present time work on some of the advanced features of the notation is still in process, and this is expected to include even larger divisions of the octave and possibilities for notating an even greater variety of rational intervals." Just so that it is clear that the notation is a work in progress.>>>>>> The symbol with which we would have no problem is the one that >>>>>> represents the 7:17 comma exactly (a zero schisma, so itwould be valid>>>>>> everywhere that both the 17' and 7 commas are valid): the 17'+7 comma, >>>>>> or ~|(). It's three flags, but I tried making the symbol,and it looks>>>>>> nice enough (i.e., it's easy enough to identify all the flags). >>>>>>>>>> Might be a good idea. I don't think strict JI-ists willaccept a symbol>>>>> that looks so obviously like a stacked pair of 5-commasymbols, as a 7:17>>>>> comma symbol _or_ a 5:13 comma symbol. These also involveschismas >0.8 cents.>>>> >>>> Now I'm having second thoughts about making anything that complicated >>>>>> OK. Forget it. >>>> But we can't forget it if we don't have anything else for the 7:17 >> comma. With the new 5'-comma symbols we could use ~|\` -- with ~|\ as >> the 23' comma -- which is almost exact (~0.036c schisma), and with ~|\ >> as the (11-5)+17 comma the schisma is ~0.455c. A three-flag symbol >> would then be permitted if one of the flags is a 5' comma. >> (Unfortunately, this isn't valid in 494 or most other ET's where it >> might be useful, so perhaps we'll want the 17'+7 comma after all. I >> will discuss combining the 5' comma with a straight right flag below.) >> I prefer ~|() to ~|\`Since having thought more about it, I agree with you. There are too many places where ~|\` won't work.. So that one's settled, then, and we'll allow a three-flag symbol if there's a very good reason for having it. This might also become the standard symbol for 17deg494 instead of ~|\.>>> You're right. It would probably be too complicated. But in anycase we have>>> agreed that whenever we add new complexity to the notation it should not >>> make the simple stuff more complicated. So it's just a matter of deciding >>> what we have really acomplished with the notation as it stands, and what >>> should be left to a possible future extension of the notation (possibly >>> along the lines of the +- 5'-comma). >>>> Now that I have had a chance to play around with some symbols for the >> 5' comma, I just may change my mind. It seems to be easy enough to >> understand, as long as the symbols are legible. >> Unfortunately I think that the four existing flag types are so well > distributed around the space of possible flag types that anything else is > bound to look too much like one of them.An even more severe problem that you raise below is that my triangular flags aren't smaller than concave flags. I have yet another idea, which I'll give below.>>> ... >>> I'd prefer to reach agreement on the limitations of theexisting flags and>>> the XH18 article, before further discussions on new flags. >>> >>> But I will say: Now that you've centered those right triangles, the filled >>> ones look too much like concave flags. >>>> I also concluded that they're all too hard to read -- the triangles are >> too small. I made them a little larger and discarded the filled ones. >> I put these in the same file with the previous ones, so we can make a >> comparison with what I had: >> >> Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Schisma.gif >> >> I also threw in a few of the flag combinations that I suggested for >> these ETs: >> >> 99: /|' /| //|' //| //|` ~|| ~||` ||\ ||\` /||\ >> 140 (70 ss.): |` /|' /| /|` (|( /|) ?|? (|\ ~||( ||\' ||\ >> ||\` /||\' /||\ >> >> I don't know how quickly these could eventually be read, but I think >> the meanings are clear enough. >> I think they suffer from the problem that the size of the modification is > visually _way_ out of proportion with the size (and direction) of the > alteration in pitch.The idea with the open triangles is that they're hollow like balloons and would be smaller if you let all of the air out. :-) Just a thought. But at least we have something now with which to compare other proposals.> What if we either deleted or thickened the part of the shaft that aligns > with the notehead. Or instead of thickening we could extend it beyond the > usual tip.The idea of not having the indication on either the left of right side is a good one in that it would probably allow us to alter any existing symbol by the small 5' comma in either direction. We'll have to try these sorts of things to see how they look. Another possibility that I wonder if I should mention: since we haven't yet published anything yet, we're still free to use a concave flag for the 5' comma and come up with another shape of flag for the present concave-flag commas. (Or something intermediate in size between wavy and straight to replace the wavy flag and reassign the wavy flags to replace the concave ones, etc., etc.) Or we could still keep )| for the 19 comma and just reassign |( for the 5' comma. I tried mirroring a concave flag vertically (to look like a small convex flag) to see how well it would serve as a 5'-comma decrease; and I think it would fly. I realize that this may seem a bit inconvenient, now that we've gotten used to the symbols that we have (so much so that I wondered if I should even suggest it, because I like the look of the concave flags for their present uses better than triangles, for example), but it's now or never. --George

Message: 5730 - Contents - Hide Contents Date: Fri, 13 Dec 2002 20:10:12 Subject: Re: Relative complexity and scale construction From: Carl Lumma>> >s I say, I'm fine by that. But you didn't answer if your method >> keeps the 'minimum-notes' aspect of Graham complexity. >>I don't know what you are asking for--it's two dimensional.Graham complexity tells me the minimum number of notes of the temperament I need to play all the identities in question. Does relative complexity?>>> You geometers might want to look at it geometrically--take the 3D >>> lattice of 7-limit JI classes, rotate it so that things separated >>> by the comma in question are stacked--in other words, the comma is >>> perpendicular to your "eye"--and then project down onto a plane, >>> making the comma vanish. Since we now have a number of 7-limit >>> classes coinciding, we pick the one of smallest tenney height >>> within the ocatave to label the point. >>>> This gets you a block, but not a temperament. >>It does no such thing. We are just projecting a 3D lattice down >to a 2D lattice; no blocks appear.By hiding commatic versions behind the pitches visible on the plane, you're applying unison vector(s), but not tempering. Perhaps you don't have enough uvs to close a block, but you're certainly on you're way. No? -Carl

Message: 5731 - Contents - Hide Contents Date: Fri, 13 Dec 2002 22:47:40 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> Graham complexity tells me the minimum number of notes of the > temperament I need to play all the identities in question. > Does relative complexity?What you want would be geometric complexity, and it doesn't. However a measure of the convex hull of the identities would certainly be possible.>> By hiding commatic versions behind the pitches visible on the> plane, you're applying unison vector(s), but not tempering.I'm showing relationships between tempered classes; it is analogous to the 5-limit JI lattice.> Perhaps you don't have enough uvs to close a block, but you're > certainly on you're way. No?If you think 5-limit JI is on the way to being a block.

Message: 5732 - Contents - Hide Contents Date: Fri, 13 Dec 2002 01:30:16 Subject: Re: Relative complexity and scale construction From: Carl Lumma>> >n the case linear temperaments, the it is equivalent to measuring >> the taxicab distance of the target intervals on a lattice 'defined >> by the temperament' (the chain of generators). Is it accurate to >> say that you intend to extend this second aspect of Graham >> complexity to planar temperaments, and not the first? If so, then >> I only need to understand how you build the lattice, and how you >> measure distance on it... >>Perhaps I should choose a better name than "relative complexity"-- >what about "relative distance" using the "relative metric"? In any >case, for linear temperaments, you can consider it taxicab if you >like, but it is all occurring along a single line and I am >considering that to be Euclidean distance.As I say, I'm fine by that. But you didn't answer if your method keeps the 'minimum-notes' aspect of Graham complexity.>> ...sounds like you measure distance with a fancy Euclidean metric, >> which I'm happy to accept as such. However, I'd like to have a >> picture of how the lattice you're measuring the distance on looks. >>The lattice is *defined* by the distance. If you look at the plots >of what I have up so far (126/125, 225/224, 1728/1715) it should >be obvious how closely the lattice is connected with the planar >temperament, so I'm not quite sure what the problem is.Unfortunately, it isn't obvious to me.>> Perhaps the lattice you just posted will help, but it doesn't >> look 'special' to me (or to monz, apparently) yet... >>Why not? Have you tried relating the lattice to the temperaments >which correspond to the lattice?What are the temperaments? I'm looking at "126/125", but I'm not sure what that means. It means you've tempered out one out of three unison vectors in a 7-limit block? What are the other two?>You geometers might want to look at it geometrically--take the 3D >lattice of 7-limit JI classes, rotate it so that things separated >by the comma in question are stacked--in other words, the comma is >perpendicular to your "eye"--and then project down onto a plane, >making the comma vanish. Since we now have a number of 7-limit >classes coinciding, we pick the one of smallest tenney height >within the ocatave to label the point.This gets you a block, but not a temperament. -Carl

Message: 5735 - Contents - Hide Contents Date: Fri, 13 Dec 2002 06:53:38 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> As I say, I'm fine by that. But you didn't answer if your method > keeps the 'minimum-notes' aspect of Graham complexity.I don't know what you are asking for--it's two dimensional.>> You geometers might want to look at it geometrically--take the 3D >> lattice of 7-limit JI classes, rotate it so that things separated >> by the comma in question are stacked--in other words, the comma is >> perpendicular to your "eye"--and then project down onto a plane, >> making the comma vanish. Since we now have a number of 7-limit >> classes coinciding, we pick the one of smallest tenney height >> within the ocatave to label the point. >> This gets you a block, but not a temperament.It does no such thing. We are just projecting a 3D lattice down to a 2D lattice; no blocks appear.

Message: 5736 - Contents - Hide Contents Date: Fri, 13 Dec 2002 08:43:57 Subject: Planar temperament plots From: Gene Ward Smith I thought I should standardize these to the extent of making the smallest step size run along the vertical axis, so I am uploading some replacement versions.

Message: 5737 - Contents - Hide Contents Date: Fri, 13 Dec 2002 20:56:13 Subject: Re: A common notation for JI and ETs From: David C Keenan At 12:44 PM 12/12/2002 -0800, George Secor wrote:>So I'll leave the 7:17 comma out of the list of commas in Table 1. >I'll also leave a listing of the ratios of 17 out of Table 2 and just >give the notation for 17/16 and 32/17, as I did with the higher primes. > The ratios of 17 were taking up a lot of space anyway, and a complete >15-limit listing (plus the odd harmonics and subharmonics up to 29) is >still pretty impressive.Yes. 15-limit is impressive enough, and along with the higher harmonics, should keep most people happy.>On an instrument of flexible pitch the player might have no idea of the >harmonic function of the tone until it was played, so it would probably >be read as a 5+5 comma. If the 5+5 comma were played exactly, then >fine-tuning by ear would require ~0.83 cents up (if it was supposed to >be a 5:13 comma) or ~1.02 cents down (if it was supposed to be 7:17). >Is there experimental evidence to support the notion that pitches can >be initiated this accurately?I doubt it very much.> Even on the microtonal valved-brass >instrument designs that I've sketched out, errors caused by addition of >valves (with a compensating mechanism for the 4:5 valve) on the order >of 2 to 5 cents are commonplace, so I think players will be depending >on their hearing to adjust the pitch subsequent to the attack (on >longer notes) for reasons apart from the notation. Sure. >So I feel that even a 217 mapping for JI should be close enough for all >practical purposes. > >But for the JI purists and theoreticians we'll still have to come up >with a more complicated option. :-)As you guessed, my concern here is not based on experimental evidence or practical purposes, but ideology and politics. I suspect it is important that no one can claim that the rational notation is based on any particular ET, or at least not one as small as 217.>> Sure. But we're just disagreeing on how close iks close enough. >Johnny>> gives it within 0.5 cents. All I'm saying is that 1.8 cents is too >far. >>The difference isn't as much as you're making it out to be, because >you're not comparing the same things. The figure you give for Johnny >is a max *pitch* deviation, while the figure you're giving for the 7:17 >vs. the 5:13 comma is an interval, a *difference* between *two* >pitches. Johnny's notation is one for 1200-ET, and it can approximate >any *pitch* of any tuning to within half a cent, but any *interval* (a >difference of two pitches) only to within one cent.I'm not sure I follow this, but even so, it doesn't really matter since I still I think it's too big.>>> I think that we have done the best we could in keeping a balance >>> between precision (of notation) vs. complexity (of symbols). >>>> I think so too. >>And as we introduce more complexity we can get more precision. Onward!OK. But not for the XH18 article, right?>>>>> The symbol with which we would have no problem is the one that >>>>> represents the 7:17 comma exactly (a zero schisma, so it would >be valid>>>>> everywhere that both the 17' and 7 commas are valid): the 17'+7 >comma,>>>>> or ~|(). It's three flags, but I tried making the symbol, and >it looks>>>>> nice enough (i.e., it's easy enough to identify all the flags). >>>>>>>> Might be a good idea. I don't think strict JI-ists will accept a >symbol>>>> that looks so obviously like a stacked pair of 5-comma symbols,>as a 7:17>>>> comma symbol _or_ a 5:13 comma symbol. These also involve >schismas > >>> 0.8 cents. >>>>>> Now I'm having second thoughts about making anything that >complicated >>>> OK. Forget it. >>But we can't forget it if we don't have anything else for the 7:17 >comma. With the new 5'-comma symbols we could use ~|\` -- with ~|\ as >the 23' comma -- which is almost exact (~0.036c schisma), and with ~|\ >as the (11-5)+17 comma the schisma is ~0.455c. A three-flag symbol >would then be permitted if one of the flags is a 5' comma. >(Unfortunately, this isn't valid in 494 or most other ET's where it >might be useful, so perhaps we'll want the 17'+7 comma after all. I >will discuss combining the 5' comma with a straight right flag below.)I prefer ~|() to ~|\`>>>> A 5:13 symbol might be \(|\. which means a 13' symbol with an >upside-down>>>> 5-comma flag added. >>>>>> I don't like the idea of adding more to a symbol to make it smaller >in >>> size, >> >> Good point. >>But I'll make an exception for the 5'd flag, since it's so small.Yeah. Forget \(|\. Silly idea.>> You're right. It would probably be too complicated. But in any case >we have>> agreed that whenever we add new complexity to the notation it should >not>> make the simple stuff more complicated. So it's just a matter of >deciding>> what we have really acomplished with the notation as it stands, and >what>> should be left to a possible future extension of the notation >(possibly>> along the lines of the +- 5'-comma). >>Now that I have had a chance to play around with some symbols for the >5' comma, I just may change my mind. It seems to be easy enough to >understand, as long as the symbols are legible.Unfortunately I think that the four existing flag types are so well distributed around the space of possible flag types that anything else is bound to look too much like one of them.>Yes, and omitting the ratios of 17 would make Table 2 a lot more >readable and less cluttered. A 13 limit is my own personal minimum >desired requirement in a JI or near-JI tonal system, so this would not >disappoint me if I were someone who was reading the article for the >first time. And wherever I've used primes above 13, they have always >been in conjunction with 1/1 as one of the natural notes, and we've >also got that covered. So, personally speaking, I'm pretty happy with >what we can present in the article up to the point. OK. Great!>> I'd like to suggest that we not have any notational schismas larger >than>> half the 5'-comma (i.e. none larger than 0.98 cents), given that >adding or>> subtracting this comma is a possible way of extending the notation. >Is it>> only 7:17 that that would kill? >>That's the only one (sort of). Otherwise, the worst case we have is >with |(: >7-5 comma (5103:5120, ~5.758c) >11-13 comma (351:352, ~4.925c) >17'-17 comma (288:289, ~6.001c) >The extremes are just over a cent, but we don't advocate the 17'-17 >comma for notating a JI consonance as we would with the 7:17 comma.Yeah. Those are fine.>> I also propose that ETs larger than 217 be left to a possible future >> extension, and maybe some of the more difficult ones above 72-ET. >>Yes. To that end I believe that I should delete 99-ET from Table 4, >but I think that all of the others I listed are straightforward enough. OK.>> I'd prefer to reach agreement on the limitations of the existing >flags and>> the XH18 article, before further discussions on new flags. >> >> But I will say: Now that you've centered those right triangles, the >filled>> ones look too much like concave flags. >>I also concluded that they're all too hard to read -- the triangles are >too small. I made them a little larger and discarded the filled ones. >I put these in the same file with the previous ones, so we can make a >comparison with what I had: > >Yahoo groups: /tuning-math/files/secor/notatio... * [with cont.] > >I also threw in a few of the flag combinations that I suggested for >these ETs: > >99: /|' /| //|' //| //|` ~|| ~||` ||\ ||\` /||\ >140 (70 ss.): |` /|' /| /|` (|( /|) ?|? (|\ ~||( ||\' ||\ >||\` /||\' /||\ > >I don't know how quickly these could eventually be read, but I think >the meanings are clear enough.I think they suffer from the problem that the size of the modification is visually _way_ out of proportion with the size (and direction) of the alteration in pitch. What if we either deleted or thickened the part of the shaft that aligns with the notehead. Or instead of thickening we could extend it beyond the usual tip. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5739 - Contents - Hide Contents Date: Sat, 14 Dec 2002 11:10:37 Subject: Pauline tempered Duodene From: Gene Ward Smith I was looking at my 225/224 lattice jpeg, and it occurred to me that the Duodene would be a good candidate for tempering by this. I may have done this before when I was tempering various things in this way, but in any case it adds to the 6 major and 6 minor triads of the Duodene 2 major and 2 minor tetrads. The tetrads alone are enough to cover every note. The count of tetrads, as expected, corresponds to what this souped-up Graham complexity business gives. Now to upgrade it to something really workable!

Message: 5740 - Contents - Hide Contents Date: Sat, 14 Dec 2002 12:04:22 Subject: Re: Relative complexity and scale construction From: Graham Breed Gene Ward Smith wrote something which Mozilla has decided to delete along with the start of my message. So I'll have to write that bit again and then look for another program that can authenticate with an SMTP server. Okay, it was something about the general rule for parallelograms. I decided to simplify that by only looking at rectangles. The complexity of a planar temperament is the pair (i, j) where i and j are the numbers of each generator needed for a complete chord. You can then describe a rectangular scale as (I, J) where the number of notes in the scale is I*J. The general rule for the number of complete otonal chords in the scale is (I-i)*(J-j). Example: 5-limit JI (planar scale not temperament) with generators of a fifth and major third. Take the rectangle A E B F# F C G D The complexity is (1, 1). The size of the scale is (2, 4). That gives us (2-1)*(4-1)=3 major chords. They are F, C and G. Example: 4-limit JI with generators of a chromatic semitone and minor third. Take the rectangle B# D# F# A B D F Ab The complexity is (1, 2) because you need a chromatic semitone and two minor thirds to make up a fifth. The size of the scale is (2, 4) again. So there are (2-1)*(4-2)=2 major chords. They are B and D. The problem here is that the same tuning gives different complexities for different choices of generators. That's presumably what Gene was thinking of when he suggested parallelograms rather than rectangles. I can write the first scale using the second mapping as F# B D E G A C F Which is a parallelogram. So different choices of generator will always give one? You can say that the real complexity is the smallest one you get by choosing different pairs of generators. That pushes the problem a step back -- how do you find the simplest generators? Well, you probably want to do that anyway if you're working with planar temperaments. To have one number to compare planar temperaments, multiplying the two components of the complexity is the simplest option, as Gene suggested somwhere else. But we can be a bit cleverer. Take a temperament with complexity (i, j) with i<j. The simplest scales involving complete chords are with (i+1, j+n) notes, where n>=1. The number of (otonal) complete chords is (i+1-i)*(j+n-j)=n. The number of notes is (i+1)*(j+n) = ij + j + ni + n = j(i+1) + n(i+1). As n gets large, you can ignore j and make i the complexity. But if you're using that many notes you can probably get an equivalent approximation with fewer notes by using a linear temperament. And that brings us to the really difficult problem. It's already there in comparing equal with linear temperaments. The logical way of extending my complexity measure to equal temperaments gives them all a complexity of zero. So there's no reason why you should ever prefer a linear temperament over an equal temperament. Hmm. Okay, say you're using meantone with 5-limit harmony. The minimax optimum is quarter comma. For most of us, that's equivalent to 31-equal. So if you're using more than 31 notes, you can simplify the scale with no real disadvantage by making it equally tempered. So, if you're that conveniently simple composer who's only interested in the approximation to JI and the number of notes in the scale, the only reason for using 5-limit meantone is if you can get by with fewer than 31 notes. Then it's either simpler or appreciably more accurate than any equal temperament. When you hit 34 notes, you can get better harmony with an equal temperament, although it behaves differently, so I'm not sure if that's relevant. It is sort of because the most efficient equal temperaments up to this point are meantones. It looks to me that the best way of comparing the complexity of equal and linear temperaments is stating the number of notes in the smallest equal temperament that does an equivalent job. So 31 is a special number for 5-limit meantone because the error in 31-equal is about the same as the lowest possible error in meantone. You could always objectify this -- say they're equivalent when the optimum error is at least some proportion of the equal tempered error. You can also say that, for a given target error, a certain equal temperament will do the job and there's no point in looking at linear temperaments that require more notes. And in practical terms, if you have some idea of how many notes you can deal with and how accurate you want the tuning, you can then compare the number of chords that linear or equal temperaments will give you. Although I can't see a way of listing equal and linear temperaments together in a general and satisfactory way, you could place them on a graph of accuracy against number of notes. At any point, you show the temperament (of whatever kind) that gives the most complete chords. In fact, if anybody knows how to draw such graphs, I'd quite like to see a few. Adding planar temperaments doesn't really change much. The biggest problem is deciding what the "typical" scale with a given number of notes is. But you can always go for the simplest one, as I show above. So you can then add planar temperaments to that graph of size against accuracy. They'll appear for high accuracies with a large but not absurdly large number of notes. You can also say the number of notes beyond which a planar temperament is equivalent to a linear temperament that's consistent with it. Then you can look at spatial temperaments, and good luck to you ;) Graham

Message: 5741 - Contents - Hide Contents Date: Sat, 14 Dec 2002 13:16:43 Subject: Re: Relative complexity and scale construction From: Kalle Aho Graham wrote:> Okay, say you're using meantone with 5-limit harmony. The minimax > optimum is quarter comma. For most of us, that's equivalent to > 31-equal. So if you're using more than 31 notes, you can simplify the > scale with no real disadvantage by making it equally tempered. > > So, if you're that conveniently simple composer who's only interested in > the approximation to JI and the number of notes in the scale, the only > reason for using 5-limit meantone is if you can get by with fewer than > 31 notes. Then it's either simpler or appreciably more accurate than > any equal temperament. When you hit 34 notes, you can get better > harmony with an equal temperament, although it behaves differently, so > I'm not sure if that's relevant. It is sort of because the most > efficient equal temperaments up to this point are meantones. > > It looks to me that the best way of comparing the complexity of equal > and linear temperaments is stating the number of notes in the smallest > equal temperament that does an equivalent job. So 31 is a special > number for 5-limit meantone because the error in 31-equal is about the > same as the lowest possible error in meantone. You could always > objectify this -- say they're equivalent when the optimum error is at > least some proportion of the equal tempered error. You can also say > that, for a given target error, a certain equal temperament will do the > job and there's no point in looking at linear temperaments that require > more notes. Hi Graham!I'd say you overestimate the utility of 31-equal as a 5-limit meantone system. If you measure the 5-limit errors of meantone equal temperaments as a proportion of the scale division it's actually 19- equal that comes out as the best. But 31-equal is the winner in the 7- limit. Kalle

Message: 5742 - Contents - Hide Contents Date: Sat, 14 Dec 2002 00:51:30 Subject: Re: Relative complexity and scale construction From: Carl Lumma>> >raham complexity tells me the minimum number of notes of the >> temperament I need to play all the identities in question. >> Does relative complexity? >>What you want would be geometric complexity,Why do I want that?>and it doesn't.You mean "and relative complexity doesn't"?>However a measure of the convex hull of the identities would >certainly be possible.Sounds like a good idea.>> By hiding commatic versions behind the pitches visible on the >> plane, you're applying unison vector(s), but not tempering. >>I'm showing relationships between tempered classes; it is >analogous to the 5-limit JI lattice. >>> Perhaps you don't have enough uvs to close a block, but you're >> certainly on you're way. No? >>If you think 5-limit JI is on the way to being a block.Maybe Paul can shed some light on this, when he's feeling better. -Carl

Message: 5743 - Contents - Hide Contents Date: Sat, 14 Dec 2002 14:10:35 Subject: Re: Relative complexity and scale construction From: Graham Breed Kalle Aho wrote:>I'd say you overestimate the utility of 31-equal as a 5-limit >meantone system. If you measure the 5-limit errors of meantone equal >temperaments as a proportion of the scale division it's actually 19- >equal that comes out as the best. But 31-equal is the winner in the 7- >limit. >I was talking about the absolute error, not this "utility" measure. 31-equal gets 5-limit harmony to within 5.96 cents. Quarter comma meantone improves on that slightly with 5.38 cents. That means the best possible worst error with meantone is only 90% of the worst error with 31-equal. 19-equal has a worst error of 7.37 cents, and you can lose 27% of that by tuning to quarter comma meantone. Somewhere between the two, you can say you don't care and 31-equal and quarter comma meantone are really the same. If the last 10% (or 0.6 cents) is still important, you can set a narrower margin. But at some point it'll be reached. Graham

Message: 5744 - Contents - Hide Contents Date: Sat, 14 Dec 2002 01:22:38 Subject: Re: New file uploaded to tuning-math From: wallyesterpaulrus recently, manuel mentioned that the dude with the cule lattices had settled on a basic structure where one axis was 3/2 while the other was 27/20. what temperament would this lattice correspond to? --- In tuning-math@xxxxxxxxxxx.xxxx tuning-math@xxxxxxxxxxx.xxx wrote:> > Hello, > > This email message is a notification to let you know that > a file has been uploaded to the Files area of the tuning-math > group. > > File : /planar plots/p1728.jpg > Uploaded by : genewardsmith <genewardsmith@j...> > Description : 1728/1715 (George) planar lattice > > You can access this file at the URL > > Yahoo groups: /tuning-math/files/planar% * [with cont.] 20plots/p1728.jpg > > To learn more about file sharing for your group, please visit > > Yahoo! - Groups Files * [with cont.] (Wayb.) > > Regards, > > genewardsmith <genewardsmith@j...>

Message: 5745 - Contents - Hide Contents Date: Sat, 14 Dec 2002 01:25:47 Subject: Re: Geometry and relative distance From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> You geometers might want to look at it geometrically--take the 3D >lattice of 7-limit JI classes, rotate it so that things separated by >the comma in question are stacked--in other words, the comma is >perpendicular to your "eye"--this was the construction i used in proving the Hypothesis . . .>and then project down onto a plane, making the comma vanish.ah! there it is! just shows ya how much easier it is to visualize two dimensions than three!

Message: 5746 - Contents - Hide Contents Date: Sat, 14 Dec 2002 01:28:27 Subject: Re: New file uploaded to tuning-math From: wallyesterpaulrus this one looks "monzian". by which i mean his lattice angles, not "monzismic". uh-oh -- now i've really asked for it! --- In tuning-math@xxxxxxxxxxx.xxxx tuning-math@xxxxxxxxxxx.xxx wrote:> > Hello, > > This email message is a notification to let you know that > a file has been uploaded to the Files area of the tuning-math > group. > > File : /planar plots/p64.jpg > Uploaded by : genewardsmith <genewardsmith@j...> > Description : 64/63 (Septimus) planar lattice > > You can access this file at the URL > > Yahoo groups: /tuning-math/files/planar% * [with cont.] 20plots/p64.jpg > > To learn more about file sharing for your group, please visit > > Yahoo! - Groups Files * [with cont.] (Wayb.) > > Regards, > > genewardsmith <genewardsmith@j...>

Message: 5747 - Contents - Hide Contents Date: Sat, 14 Dec 2002 01:32:59 Subject: Re: Relative complexity and scale construction From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Graham complexity tells me the minimum number of notes of the >>> temperament I need to play all the identities in question. >>> Does relative complexity? >>>> What you want would be geometric complexity, >> Why do I want that? >>> and it doesn't. >> You mean "and relative complexity doesn't"? >>> However a measure of the convex hull of the identities would >> certainly be possible. >> Sounds like a good idea. >>>> By hiding commatic versions behind the pitches visible on the >>> plane, you're applying unison vector(s), but not tempering. >>>> I'm showing relationships between tempered classes; it is >> analogous to the 5-limit JI lattice. >>>>> Perhaps you don't have enough uvs to close a block, but you're >>> certainly on you're way. No? >>>> If you think 5-limit JI is on the way to being a block. >> Maybe Paul can shed some light on this, when he's feeling > better. > > -Carl i can't.now answer the question, carl!

Message: 5748 - Contents - Hide Contents Date: Sat, 14 Dec 2002 02:39:23 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> Graham complexity tells me the minimum number of notes of the > temperament I need to play all the identities in question. > Does relative complexity?Graham complexity along both generators, and thier product, might be what we need. We could attempt to minimize the product.

Message: 5749 - Contents - Hide Contents Date: Sat, 14 Dec 2002 03:25:37 Subject: Cule lattices From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> recently, manuel mentioned that the dude with the cule lattices had > settled on a basic structure where one axis was 3/2 while the other > was 27/20. what temperament would this lattice correspond to?(1) What in the world is a cule lattice? Did you mean cute, but with a stuffed-up nose? (2) I imagine it's just 5-limit JI, though it is possible to cook up goofy temperaments such as 4782969/4480000 to go with it.

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