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Message: 5702 Date: Sun, 08 Dec 2002 19:33:20 Subject: Re: A common notation for JI and ETs From: David C Keenan At 11:31 AM 25/11/2002 -0800, you wrote: >I also discuss a 494-ET JI mapping in my paper, but recent private >correspondence with Dave Keenan has turned up a problem with using 494 >for JI notation (apart from its complexity, which makes 217 look like a >walk in the park). So I am having serious thoughts about scratching >that idea. (I'll be moving that discussion onto this list so we have >it documented here.) I don't think I ever suggested that 494-ET notation should be used for JI notation, although I agree that it could be (based on the perceptual definition of just intonation), however 217-ET should be fine for that. 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. 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. 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. 494-ET originally entered this discussion because, in the single-symbol version of the notation for rational pitches we need to assign pairs of symbols as being apotome complements of each other. When we minimised the offsets of these pairs the result happened to agree with apotome complementation in 494-ET. This was reassuring because agreement with _some_ large ET seemed to guarantee a certain kind of consistency. I felt that it meant we would not get any nasty surprises somewhere down the track. Much later I suggested that we could actually notate 494-ET itself. At the time we were just pushing the notation up through the ETs to see how far it could go without too much additional complexity. I never really imagined anyone would want to notate 494-ET. But stranger things have happened. >I don't think the deadline for the article is that close, so I took >some time to look at the 7:17 comma problem. I thought the problem >might be a matter of requiring the //| symbol to represent all of the >roles ranging from ~42.0c (for the 7:17 comma) to ~43.8c (for the 5:13 >comma), but I see that there are a number of divisions above 217 in >which //| is valid for all of these: 224, 270, 282, 342, 388, and 612. >Those in which //| is not valid for all of these are 306, 311, 364, and >400, and curiously, in most of these the 7:17 comma is the same number >of degrees as the 5:13 comma, but different for the 5+5 comma. So I >think that we just ran out of luck with 494. OK. >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. A 5:13 symbol might be \(|\. which means a 13' symbol with an upside-down 5-comma flag added. > In >looking for a rational complement I find that ~)|| is very close, but >it's already the rational complement of ~|\, although it could serve >for both. > >But it could be argued that this adds too many complications in trying >to solve a problem that seems to be of concern only insofar as it >applies to 494 (and which we wouldn't be facing if you weren't >attaching so much importance to 494). I hope you understand now that it isn't 494 per se that I'm attaching importance to. But it seems the complementation should work in some large ET (inaddition to 217), which does not itself need to be fully notatable. 624-ET might be worth a look. > But I will briefly consider the >other alternative. > >I see that ~|\, which is already the (11-5)+17 comma (4352:4455, >~40.496c) and 23' comma (16384:16767, ~40.004c) is valid for both of >these plus the 7:17 comma (448:459, ~41.995c) in quite a few of the >better divisions: 94, 111, 118, 140, 171, 183, 193, 200, 217, 282, 311, >and 494. So it's valid in *both* 217 and 494, the divisions for which >we wanted a free-modulation JI option to be available. The only hitch >is that the schisma for ~|\ is 1155:1156, ~1.498c, compared to the >schisma for //|, 1700:1701, (~1.018c, the difference between the 5+5 >comma and the 7:17 comma). I think this is unacceptable. 1.5 cents is way too big. >This one's a tough call, because, although ~|\ works as the 7:17 comma >in both 217 and 494, the schisma is larger than for //|. Also, this >would require two different symbols for 8 degrees of 217 when it's used >for freely modulating JI, which would unncessarily complicate the >217-JI-17 notation, which does just fine with //|. (I really wonder >how many composers would consider using the 494 notation for JI, when >it requires so many more single-shaft symbols, 26 vs. 12 for 217, so I >am beginning to think that a 494-JI option isn't really practical. I never thought it was. >This is in addition to the potentially confusing symbol-size reversal >between 3 and 4deg494.) > >If you feel that it's necessary to have the notation validated by a >high-precision division like 494, then I would suggest using ~|() for >the exact 7:17 comma for theoretical purposes and electronic music >(with ~)|| as rational complement, if needed), and replacing ~|() with >//| for 217-based JI. I would view this as a compromise that would >keep the notation simple for the simpler 217-JI mapping. I believe >that others are going to find that 217 is complicated enough for them, >and 494 would be unthinkable. Agreed. I think the wrong-way pointing flag idea mentioned above, as representing a subtraction, might be the way to deal with notating the diaschima and 5-diesis. I think we should have a short straight right flag for the 5-schisma (32768:32805), (2^15:3^8*5), 1.95 cents, which I will symbolise for now as |` (or !, when pointing down). This would give us a two-flag symbol for the Pythagorean comma, /|`. When this new flag is flipped upside down, but stays at the same end of the shaft, which I'll symbolise for now as |' (or !. when pointing down), it would give us /|' for the diaschisma 2025:2048 and //|' for the 5-diesis 125:128. Maybe this new flag and/or this new subtraction idea will open up other symbol possibilities for the dual-prime commas we're currently having trouble with, so that they will not require notational schismas any greater than 0.5 cent. 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. Given the specialised meanings we've given to "comma" and "schisma" in this discussion, it might make sense for us to refer to 32768:32805 as the 5'-comma rather than the 5-schisma. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5711 Date: Wed, 11 Dec 2002 12:18:19 Subject: Re: A common notation for JI and ETs From: monz > From: <gdsecor@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 11, 2002 11:15 AM > Subject: [tuning-math] Re: A common notation for JI and ETs > > > --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> > wrote: > > > > 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. > > And 612 will be a must-do. (Monz would want to see that.) Gene and i both are big fans of 612edo. yes, please, draw up an example musical illustration for this! -monz
Message: 5715 Date: Wed, 11 Dec 2002 22:25:41 Subject: Re: Relative complexity and scale construction From: Carl Lumma > [1, 5/3] > [0, 249] > [249, 87] > > [1, 6/5, 5/3] > [0, 87, 249] > [87, 162, 87] > > [1, 6/5, 3/2, 5/3] > [0, 87, 196, 249] > [87, 109, 53, 87] > > [1, 6/5, 4/3, 3/2, 5/3] > [0, 87, 140, 196, 249] > [87, 53, 56, 53, 87] > > [1, 6/5, 4/3, 3/2, 8/5, 5/3] > [0, 87, 140, 196, 227, 249] > [87, 53, 56, 31, 22, 87] > > [1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3] > [0, 87, 109, 140, 196, 227, 249] > [87, 22, 31, 56, 31, 22, 87] Sorry to be dense, but where are the planar temperaments (the JI versions, or the 336-et versions, or...)? How can there be a 5-limit planar temp.? > [1, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3] > [0, 87, 109, 140, 174, 196, 227, 249] > [87, 22, 31, 34, 22, 31, 22, 87] // > These scales vary in their degree of regularity, but Carl > might find the 8 or 13 note scales acceptable, for example. Thirteen is more notes than I'm looking for at the moment. The 8-note scale seems reasonably well-approximated in 12-et... And I think I prefer the 7-note version... Anyway, I have no idea what you're up to here, other than finding a way to extend our notion of complexity to non-linear temperaments... of course I see why our current definition cannot work, and I roughly get the idea behind geometric complexity, but I'm afraid you've lost me with the relative version. -Carl
Message: 5720 Date: Thu, 12 Dec 2002 19:49:57 Subject: Re: Relative complexity and scale construction From: Carl Lumma >>>[1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3] >>>[0, 87, 109, 140, 196, 227, 249] >>>[87, 22, 31, 56, 31, 22, 87] >> >>Sorry to be dense, but where are the planar temperaments (the >>JI versions, or the 336-et versions, or...)? How can there be >>a 5-limit planar temp.? > >The 5-limit intervals are merely representing intervals of the >temperament; the tuning in 336-et is given on the next line. Ok. >>Thirteen is more notes than I'm looking for at the moment. > >Twelve, then. :) Ten's my limit. >>Anyway, I have no idea what you're up to here, other than >>finding a way to extend our notion of complexity to non-linear >>temperaments... > >The idea is that this gives the classes of the planar temperament >in terms of a lattice, where distance is measured in a way >connected to the approximation(s) of the temperament, which it >already is or can be in a linear temperament or JI. This fills >in the gap between them. Graham complexity measures the complexity of a temperament based on the minimum number of *notes* of the temperament you'd need to get *all* of a target set of intervals. This aspect is what attracts me to Graham complexity, and it seems it *is* extensible to planar and higher temperaments. . . In 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... >I defined a notion of distance on ocatave equivalence classes of >p-limit JI. I then used this to define a notion of distance on >wedge products of classes, getting geometric complexity. Measuring >the complexity after wedging in the interval gives us *relative* >complexity for that interval with respect to the temperament in >question. In the case of linear temperaments, this reduces to >number of generator steps, and in the case of JI, to my original >weighted Euclidean distance measure. ...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. Perhaps the lattice you just posted will help, but it doesn't look 'special' to me (or to monz, apparently) yet... -Carl
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