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Message: 5700 - Contents - Hide Contents Date: Sat, 07 Dec 2002 22:07:05 Subject: For Monz: More preposterous commas From: Gene Ward Smith While they have no known use, here are some more preposterously good 5-limit ets and corresponding preposterous commas: et: 78005 comma: [-573, 237, 85] et: 1251764 commmas: [1634, 1502, -1729], [2033, 90, -937] 2^1634 3^1502 5^(-1729) achieves new heights of preposterousness!
Message: 5702 - Contents - Hide Contents 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 * [with cont.] (Wayb.)
Message: 5703 - Contents - Hide Contents Date: Sun, 08 Dec 2002 02:15:39 Subject: Re: 25/24, 49/48, 50/49 From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho<kalleaho@m...>" <kalleaho@m...> wrote:>>> How is complexity calculated and why? >> I calculated complexity using a rather complicated formula, but >Paul tells me that Graham complexity would have been better;actually the representation of the chromatic unison vector comes into it too.>evidentally you are interested only in complete tetrads. It iseasy >to compute: take the range in generator steps of the>representations of > {1,3,5,7,5/3,7/3,7/5} and multiply by the number of periods in an >octave.that's graham's complexity. but you still haven't calculated the number of complete tetrads. to do that, you also need to calculate the number of notes in the scale. first calculate the cardinality per period, by determining (from the mapping) the number of generators comprising a 49:48 if 25:24 is tempered out, a 25:24 if 49:48 is tempered out, or either (they'll be the same) if 50:49 is tempered out. then multiply by the number of periods in an octave. finally, the number of tetrads will be the graham complexity minus the number of notes in the scale -- times two (since both otonal and utonal tetrads count).
Message: 5705 - Contents - Hide Contents Date: Mon, 09 Dec 2002 20:54:58 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> 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.Precisely what I concluded over the past week.> 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.> 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. 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.)> 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. 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. 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? 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. I think that we have done the best we could in keeping a balance between precision (of notation) vs. complexity (of symbols).> 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.And it seems to have worked pretty well.> 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 seehow 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.And will probably continue to happen.>> 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.Now I'm having second thoughts about making anything that complicated -- yes //| looks like stacked 5 commas, but looks aren't everything, and once you become fluent with the notation you remember the other two roles as well, which come pretty close to two stacked 5 commas. There's a problem with having too many symbols, and if there's only a tiny difference between the commas, why bother making a distinction? Not all of our schimas are +-0.5 cents, but they're still very small. It looks like we're starting to get into the sort of precision that is going to have practical value only if the pitches are being produced electronically, in which case you *could* use additional (more complicated) symbols (or variations thereof, probably in ascii form), since they wouldn't need to be read in real time.> 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, but I'm speaking of a performance notation. If we're considering electronics, then we could adapt what we have into something that could have any number of flags or symbols (or combinations thereof) -- something that software could easily translate into the sagittal performance notation for study or transcription to acoustic instruments. But that's an entirely different ballgame.> ... > 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.This reminds me of when I was experimenting with putting knobs on the ends of flags (in connection with the lateral confusability issue). If certain flags had a knob (or some other such feature) on the end now, then we could reduce them by a schisma by just removing the knob. Or better yet: have a version of the notation where all of the present symbols have a filled knob at the head of the arrow; to decrease the alteration by a 5' comma (as you call it below), remove the filling so that the knob becomes a small open circle. This way the 5' comma is on neither side and can therefore be used with any symbol. (Didn't you have an idea of this sort before for the 19 comma before you came up with the wavy flag?) Or perhaps the existing symbols would work without the filled knob, and an unfilled circle at the arrowhead could be accepted as a subtraction in size (suitably symbolized by a kind of emptiness). These are just some ideas that would avoid a negative flag, yet allow us to build on what we already have. Once again you've gotten me thinking about how to make a performance notation more complicated (after I had already talked myself out of it), and now I've suggested something that I have no idea how to do in ascii.> 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.That's always a possibility.> 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.> 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.That's for sure! All of our notational schismas make the original "schisma" look huge by comparison! All of this opens up so many new possibilities that suddenly it seems that we're back where we were last spring. 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. 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? --George
Message: 5706 - Contents - Hide Contents Date: Mon, 09 Dec 2002 22:56:28 Subject: Re: Even more ridiculous 5-comma list From: wallyesterpaulrus hi paul, for meantone, see this page: W. S. B. Woolhouse's 'Essay on musical interva... * [with cont.] (Wayb.) -paul --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> > Thanks. I'm starting to get it. Could you show me the actual calculations > for the second generator (rms error) for say, meantone? > > > > wallyesterpaulrus > <wallyesterpaulrus To: tuning- math@xxxxxxxxxxx.xxx > @yahoo.com> cc: (bcc:Paul G Hjelmstad/US/AMERICAS)> Subject: [tuning-math] Re: Even more ridiculous 5-comma list> 11/30/2002 01:49 > PM > Please respond to > tuning- math > > > > > > > --- In tuning-math@y..., "paulhjelmstad" <paul.hjelmstad@u...> wrote: >>> How would one use rms to get 317.079753? >> Paul >> for 5-limit temperaments, you minimize the rms error in the 3/1, the > 5/1, and the 5/3 -- in other words, all three 5-limit consonant > interval classes. > > -another paul > > > Yahoo! Groups Sponsor > > > > ADVERTISEMENT > (Embedded image moved to file: pic04186.gif) > > > (Embedded image moved to file: pic19690.gif) > > > > > To unsubscribe from this group, send an email to: > tuning-math-unsubscribe@xxxxxxxxxxx.xxx > > > > Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.
Message: 5708 - Contents - Hide Contents Date: Tue, 10 Dec 2002 07:26:42 Subject: Transforming one temperament to another From: Gene Ward Smith I've mentioned that one 1-comma (planar, for 7-limit) temperament can be transformed to another by mapping the comma. The same is true with more than one comma; one simply applies the map to the kernel. Here is an example: The Nonkleismic temperament has map and wedgie [[1, -1, 0, 1], [0, 10, 9, 7]] [10, 9, 7, -9, 17, -9] Error, complexity and badness: 3.320167 27.531739 2516.675764 Generators: [1200., 309.9514712] and a kernel generated by <126/125, 1728/1715>. Mapping this by 2-->2, 3-->8/3, 5-->32/7, 7-->32/5 sends the above kernel to <343/360, 875/864>, a temperament whose corresponding values are [[1, 4, 4, 5], [0, -10, -7, -9]] [10, 7, 9, 1, 14, -12] 21.487964 27.215263 15915.502350 [1200., 290.6218997] It's not a very good temperament, but it certainly counts as one. Probably there are better examples.
Message: 5709 - Contents - Hide Contents Date: Wed, 11 Dec 2002 19:15:40 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> 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 have had a bit more time to think about your 5'-comma proposal now, and I've read the following a little more carefully (especially the part about having the flag either decreasing or increasing), so I have some more thoughts on this.> 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.Yes, this is good. I looked at some of the graphics I tried on my own earlier this year to notate small commas, and I found a couple of arrow types that might work. I'll be referring to the figure in this file: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Schisma.gif The new flag I am proposing is a small right triangle with one leg coinciding with part of the arrow shaft. For a 5'-reduction the base of the triangle is near the tip of the arrow, and for a 5-increase the apex of the triangle is near the arrow tip. I tried both filled and hollow triangles without arriving at a preference, but if we wish to make it easier to distinguish between the 5' reduction and increase, then I suggest that we use a hollow triangle for a reduction and a filled one for an increase. (I put asterisks under the ones in the diagram that would correspond to these.) Of course, the 5'-down symbol that I put there would not be used -- we would vertically mirror the 5'-up symbol, but I made it just so it is easier to see how it looks (without anything else).> 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.We shall see.> 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.) --George
Message: 5710 - Contents - Hide Contents Date: Wed, 11 Dec 2002 01:56:00 Subject: Re: Even more ridiculous 5-comma list From: wallyesterpaulrus rms error is the whole focus of the woolhouse calculation. it minimizes the rms error. complexity is a function of the mapping, and badness is a function of both complexity and rms error. --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> > Thanks. Worked through Woolhouse/Monzo's derivation of 7/26-meantone. Now I> see how the generators are calculated. Now, how are (the other) rms, > complexity and badness calculated? > > Paul > > > > "wallyesterpaulrus > <wallyesterpaulrus@ To: tuning- math@xxxxxxxxxxx.xxx > yahoo.com>" cc: (bcc:Paul G Hjelmstad/US/AMERICAS)> <wallyesterpaulrus Subject: [tuning-math] Re: Even more ridiculous 5-comma list> > 12/09/2002 04:56 PM > Please respond to > tuning- math > > > > > > > hi paul, > > for meantone, see this page: > > W. S. B. Woolhouse's 'Essay on musical interva... * [with cont.] (Wayb.) > > -paul > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > <paul.hjelmstad@u...> wrote: >>>> Thanks. I'm starting to get it. Could you show me the actual > calculations>> for the second generator (rms error) for say, meantone? >> >> >> > >> > wallyesterpaulrus >>>
Message: 5711 - Contents - Hide Contents 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: 5712 - Contents - Hide Contents Date: Wed, 11 Dec 2002 03:03:01 Subject: Relative complexity and scale construction From: Gene Ward Smith I have a nifty new concept, "relative complexity", which should be useful for constructing scales in non-linear regular temperaments. If w is the wedgie of a temperament, and q is an interval, then define the relative complexity of q with respect to w as the (geometric) complexity of w^q. If the kernel of w is generated by <c1, ..., ck> then this is the complexity of c1^c2^...^ck^q. In the case where w is a linear temperament, the relative complexity of q is proportional to the number of generator steps for q. Hence, scale construction using relative complexity can be thought of as a generalization of scales constructed by contiguous generators, such as MOS. For an example, consider the 126/125 "Starling" planar temperament. If we order equivalence classes of intervals 1 <= q < sqrt(2) by relative complexity, we get 1, 6/5, 4/3, 5/4, 7/5, 10/9, 21/20, 15/14, 8/7, 7/6, 16/15, 125/112, 80/63, ... Here 6/5, for instance, represents the class {(6/5)*(125/125)^n} of intervals; I choose the representative of smallest height in each case. Now we may use these intervals in the above order to construct scales; I give the JI version, then the version in 336-et (using the mapping h336+v3-v5 = [336, 532, 781, 943] which totally nails Starling.) [1] [0] [336] [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] [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] [1, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3] [0, 87, 109, 140, 162, 174, 196, 227, 249] [87, 22, 31, 22, 12, 22, 31, 22, 87] [1, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5] [0, 87, 109, 140, 162, 174, 196, 227, 249, 283] [87, 22, 31, 22, 12, 22, 31, 22, 34, 53] [1, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5] [0, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283] [53, 34, 22, 31, 22, 12, 22, 31, 22, 34, 53] [1, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5, 40/21] [0, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283, 314] [53, 34, 22, 31, 22, 12, 22, 31, 22, 34, 31, 22] [1, 21/20, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5, 40/21] [0, 22, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283, 314] [22, 31, 34, 22, 31, 22, 12, 22, 31, 22, 34, 31, 22] [1, 21/20, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5, 28/15, 40/21] [0, 22, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283, 302, 314] [22, 31, 34, 22, 31, 22, 12, 22, 31, 22, 34, 19, 12, 22] [1, 21/20, 15/14, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5, 28/15, 40/21] [0, 22, 34, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283, 302, 314] [22, 12, 19, 34, 22, 31, 22, 12, 22, 31, 22, 34, 19, 12, 22] [1, 21/20, 15/14, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 7/4, 9/5, 28/15, 40/21] [0, 22, 34, 53, 87, 109, 140, 162, 174, 196, 227, 249, 271, 283, 302, 314] [22, 12, 19, 34, 22, 31, 22, 12, 22, 31, 22, 22, 12, 19, 12, 22] [1, 21/20, 15/14, 10/9, 8/7, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 7/4, 9/5, 28/15, 40/21] [0, 22, 34, 53, 65, 87, 109, 140, 162, 174, 196, 227, 249, 271, 283, 302, 314] [22, 12, 19, 12, 22, 22, 31, 22, 12, 22, 31, 22, 22, 12, 19, 12, 22] [1, 21/20, 15/14, 10/9, 8/7, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 9/5, 28/15, 40/21] [0, 22, 34, 53, 65, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 283, 302, 314] [22, 12, 19, 12, 22, 22, 31, 22, 12, 22, 31, 22, 12, 10, 12, 19, 12, 22] [1, 21/20, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 9/5, 28/15, 40/21] [0, 22, 34, 53, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 283, 302, 314] [22, 12, 19, 12, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 12, 19, 12, 22] [1, 21/20, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 9/5, 28/15, 15/8, 40/21] [0, 22, 34, 53, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 283, 302, 305, 314] [22, 12, 19, 12, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 12, 19, 3, 9, 22] [1, 21/20, 16/15, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 9/5, 28/15, 15/8, 40/21] [0, 22, 31, 34, 53, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 283, 302, 305, 314] [22, 9, 3, 19, 12, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 12, 19, 3, 9, 22] [1, 21/20, 16/15, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 224/125, 9/5, 28/15, 15/8, 40/21] [0, 22, 31, 34, 53, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 280, 283, 302, 305, 314] [22, 9, 3, 19, 12, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 9, 3, 19, 3, 9, 22] [1, 21/20, 16/15, 15/14, 10/9, 125/112, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 224/125, 9/5, 28/15, 15/8, 40/21] [0, 22, 31, 34, 53, 56, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 280, 283, 302, 305, 314] [22, 9, 3, 19, 3, 9, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 9, 3, 19, 3, 9, 22] [1, 21/20, 16/15, 15/14, 10/9, 125/112, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 63/40, 8/5, 5/3, 12/7, 7/4, 224/125, 9/5, 28/15, 15/8, 40/21] [0, 22, 31, 34, 53, 56, 65, 75, 87, 109, 140, 162, 174, 196, 218, 227, 249, 261, 271, 280, 283, 302, 305, 314] [22, 9, 3, 19, 3, 9, 10, 12, 22, 31, 22, 12, 22, 22, 9, 22, 12, 10, 9, 3, 19, 3, 9, 22] [1, 21/20, 16/15, 15/14, 10/9, 125/112, 8/7, 7/6, 6/5, 5/4, 80/63, 4/3, 7/5, 10/7, 3/2, 63/40, 8/5, 5/3, 12/7, 7/4, 224/125, 9/5, 28/15, 15/8, 40/21] [0, 22, 31, 34, 53, 56, 65, 75, 87, 109, 118, 140, 162, 174, 196, 218, 227, 249, 261, 271, 280, 283, 302, 305, 314] [22, 9, 3, 19, 3, 9, 10, 12, 22, 9, 22, 22, 12, 22, 22, 9, 22, 12, 10, 9, 3, 19, 3, 9, 22] These scales vary in their degree of regularity, but Carl might find the 8 or 13 note scales acceptable, for example.
Message: 5713 - Contents - Hide Contents Date: Wed, 11 Dec 2002 20:44:13 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> >> From: <gdsecor@y...> >> 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! > > -monzOkay, once we settle on the symbols. I've redone the file already, because I realized that the small arrows weren't centered on the lines or spaces of the notes that they would be modifying: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Schisma.gif --George
Message: 5714 - Contents - Hide Contents Date: Wed, 11 Dec 2002 05:42:07 Subject: Planar temperament lattice diagrams From: Gene Ward Smith The octave equivalence classes of a planar temperament system can be reduced to a plane (hence the name.) Relative complexity gives us a Euclidean metric on that plane, so it is an (honest to goodness, genuine mathematician-style) lattice. If Paul or Monz felt ambitious, lattice diagrams drawn in such a way that this distance was respected might prove illuminating. To do it for Starling, one might take 6/5 represented by [2.851474223, 0] 3/2 represented by [1.349572175, 4.477746219] These can of course be rescaled or rotated, but the relative lengths and the angle between them should be kept the same. In Starling, since 126/125~1, we have (6/5)^(-3) 3 ~ 7/4. Hence every 7-limit octave reduced note class can be expressed in terms of the two vectors above.
Message: 5715 - Contents - Hide Contents 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: 5716 - Contents - Hide Contents Date: Wed, 11 Dec 2002 08:52:15 Subject: Relative complexity for From: Gene Ward Smith This is one of the best 11-limit planar temperaments; it is of course related to Miracle, but we can with some advantage use 190-et instead of 72-et for it. Here is a list of octave equivalence class representatives between 1 and sqrt(2), in order of ascending relative complexity: 8/7, 21/16, 4/3, 21/20, 12/11, 6/5, 5/4, 7/6, 11/8, 7/5, 14/11, 49/48, 11/9, 10/9, ... Here are vectors representing 7 and 5/3, which can be used as a basis for the 11-limit equivalence classes when reduced mod 385/384 and 441/440. Plotting the classes when so reduced gives the lattice corresponding to this temperament. 7 ~ [2.921751858, -4.929080722] 5/3 ~ [20.89259427, 3.844797215]
Message: 5717 - Contents - Hide Contents Date: Wed, 11 Dec 2002 22:29:36 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> 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.I imagine that we could abbreviate the new flag names as 5'd and 5'i for 5-prime-decrease and 5-prime-increase. With these new flags we could do things like this: Use the diaschisma /|' as 4deg270, 5deg311, 6deg364, 6deg388, 6deg400, and 8deg494 (Of course, this is pretty obvious.) Make the 13:19 comma (38:39) //|` instead of |~) -- you'll love the tiny schisma for this one! This could then be used for 15deg388, 15deg400, 19deg494, and 23deg612 Make 15deg311 (/|` taking (/| as the 31' comma 99: /|' /| //|' //| //|` ~|| ~||` ||\ ||\` /||\ 140 (70 ss.): |` /|' /| /|` (|( /|) ?|? (|\ ~||( ||\' ||\ ||\` /||\' /||\ instead of those arbitrary high-prime commas, assuming that the |' and |` flags will not be easily confused and that some way will be found to depict ||\ with the 5' flags (might we want to put a 5' flags on *either* side?) -- I don't know whether it would be better to do 7deg140 as //|` or /|\` etc., etc. --George
Message: 5718 - Contents - Hide Contents Date: Wed, 11 Dec 2002 11:18:27 Subject: Relative complexity for From: Gene Ward Smith This is another of the best 11-limit planar temperaments, and even more closely allied to Miracle. The complexity ordered list of intervals for this begans: 5/4, 15/14, 4/3, 7/6, 9/7, 8/7, 6/5, 7/5, 11/8, 12/11, 9/8, 33/32, 15/11, 11/9, 25/24, 21/16 ... In terms of the [310, 491, 870, 1072] val this is: 99, 30, 129, 69, 112, 60, 82, 151, 142, 39, 52, 13, 138, 90, 17, 121 ... We may use 5/4 and 15/14 as a basis for the lattice; the corresponding vectors being: 5/4 ~ [10.16577203, 1.308331524] 15/14 ~ [-2.638975604, 10.01118040]
Message: 5719 - Contents - Hide Contents Date: Wed, 11 Dec 2002 23:48:19 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> [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.> Thirteen is more notes than I'm looking for at the moment.Twelve, then. :)> The 8-note scale seems reasonably well-approximated in 12-et...Reasonable for whom and what purpose? I wouldnt' use it.> 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.> 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.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.
Message: 5720 - Contents - Hide Contents 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
Message: 5721 - Contents - Hide Contents Date: Thu, 12 Dec 2002 20:42:31 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> At 11:21 AM 11/12/2002 -0800, George Secor wrote:>> ... 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 smallenough to be> acceptable for rational notation.Okay. This will be particularly pertinent now that we are trying to incorporate the 5' comma into the 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 :-).And we can also omit any ratios containing only powers of 2 and 3.>>> It may be that we are worrying about the notation >>> of 17/7 when in fact we don't have a single symbol for manyothers 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.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.>> 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 wouldgive 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.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? 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. 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. :-)>> 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.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 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!>>>> 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.)>>> 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.>> ... 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).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.> 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.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.> 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.> 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.> 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- * [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. --George
Message: 5723 - Contents - Hide Contents Date: Thu, 12 Dec 2002 23:24:14 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> 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...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.> ...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.> 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?
Message: 5724 - Contents - Hide Contents Date: Thu, 12 Dec 2002 23:48:34 Subject: Geometry and relative distance From: Gene Ward Smith 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.
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