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Message: 5875 - Contents - Hide Contents Date: Tue, 07 Jan 2003 22:51:44 Subject: Re: Temperament notation From: Carl Lumma>>> >hat, exactly, do you mean by "septimal notation"? >>>> A notation with 7 nominals! >> What I would prefer to call a heptatonic notation.Of course; much better.>> And in harmonics 6-12, the aug 3rd and dim 4th don't function >> differently? //>It is interesting to contemplate that if we used a notation >with 12 nominals for 12-ET that we would be unable to observe >a distinction on the printed page. Exactly.>> Msg. #s 35809 and 41680. >>Thanks! I'll have to take a look.Promising technology for microtonalists, to be sure. For infinite flexibility we loose velocity and aftertouch, so we'd be stuck to organ-type patches if we wanted them to sound good. We also loose tactile feedback, which would probably make sight reading impossible. Notation could be replaced on these instruments by a 'follow the lights' approach, in which the whole key can light up, if you like! Graham, are you listening? Certainly exciting that technology exists to bring the holy grail of an infinitely configurable, extremely portable keyboard within reach of the consumer (indeed: cheap!). In the year between msg. 35809 and 41680, it went from trade-show demo to at least two companies providing OEM kits. Assuming there's a flexible inferface in there somewhere, a microtonal keyboard software project (perhaps two projectors would be needed for a full keyboard) might not be too difficult... -Carl

Message: 5876 - Contents - Hide Contents Date: Tue, 07 Jan 2003 02:23:54 Subject: Re: 31, 112 and 11-limit Meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> I guess I don't really understand this poptimal stuff after all. > Isn't there rather a big difference between the RMS and the max- > absolute optimum generators for meantone, at least at the 5 limit.The rms (and least-fourth power) value is 7/26-comma, and the minimax value is 1/4-comma, which I would call a small difference.

Message: 5877 - Contents - Hide Contents Date: Tue, 07 Jan 2003 23:50:29 Subject: Re: Poptimal generators From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith >> >>> "Poptimal" is short for "p-optimal". The p here is a real variable >>> p>=2, which is what analysts normally use when discussing these >> Holder>>> type normed linear spaces. >>> >>> A pair of generators [1/n, x] for a linear temperament is >>> *poptimal* if there is some p, 2 <= p <= infinity, >>>> why not go all the way to 1? MAD, or p=1, error certainly seems > most>> appropriate for dissonance curves such as vos's or secor's -- which >> are in fact even pointier at the local minima (resembling exp >> (|error|)) . . . >> Possibly because no one in the history of this endeavour has ever > before now suggested that mean-absolute error corresponds in any way > to the human perception of these things.no one in history? you've gotta be kidding me. sum or mean of absolute errors is a quite common error criterion.>> "A "poptimal" generator can lay claim to being absolutely and ideally >> perfect as a generator for a given temperament ..." >> When we're talking about human perception, as we are, it should be > obvious that nothing can be absolutely and ideally perfect for > everyone. Even a single person might prefer slightly different > generators for different purposes. To validate such a claim > of "perfection" you would at least need to produce statistics on the > opinions of many listeners.clearly dave missed the clever mockery hidden in gene's statement.

Message: 5878 - Contents - Hide Contents Date: Tue, 07 Jan 2003 04:02:38 Subject: Re: 31, 112 and 11-limit Meantone From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> The rms (and least-fourth power) value is 7/26-comma, and theminimax value is 1/4-comma, which I would call a small difference. OK. With regard to sagittal notation, 112-ET fails the following test. (b) The best fifth (approx 2:3) in the ET must be the same as the fifth calculated by applying the temperament's mapping to the best approximation of the generator (and period) in that ET. The best fifth in 112-ET is 66 steps. The best approximation of the meantone fifth generator in 112-ET is 65 steps. I agree with George ("At last!", he says) that sagittal notation of open meantone chains of up to 30 notes should be based on 31-ET notation. 31-ET passes Gene's earlier test, of being the highest denominator of a convergent for both RMS and max-absolute generators.

Message: 5879 - Contents - Hide Contents Date: Tue, 07 Jan 2003 23:50:26 Subject: Re: Poptimal generators From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith >> >>> "Poptimal" is short for "p-optimal". The p here is a real variable >>> p>=2, which is what analysts normally use when discussing these >> Holder>>> type normed linear spaces. >>> >>> A pair of generators [1/n, x] for a linear temperament is >>> *poptimal* if there is some p, 2 <= p <= infinity, >>>> why not go all the way to 1? MAD, or p=1, error certainly seems > most>> appropriate for dissonance curves such as vos's or secor's -- which >> are in fact even pointier at the local minima (resembling exp >> (|error|)) . . . >> Possibly because no one in the history of this endeavour has ever > before now suggested that mean-absolute error corresponds in any way > to the human perception of these things.no one in history? you've gotta be kidding me. sum or mean of absolute errors is a quite common error criterion.>> "A "poptimal" generator can lay claim to being absolutely and ideally >> perfect as a generator for a given temperament ..." >> When we're talking about human perception, as we are, it should be > obvious that nothing can be absolutely and ideally perfect for > everyone. Even a single person might prefer slightly different > generators for different purposes. To validate such a claim > of "perfection" you would at least need to produce statistics on the > opinions of many listeners.clearly dave missed the clever mockery hidden in gene's statement.

Message: 5880 - Contents - Hide Contents Date: Tue, 07 Jan 2003 06:02:33 Subject: Nonoctave scales and linear temperaments From: Gene Ward Smith I pointed out over on PostTonality that the 88CET is closely related to the Octafifths temperament; you might say it is Octafifths without the octave reduction. If Octafifths (with a generator around 8/109) has a generator small enough for this, so does Quartaminorthirds, with a generator arounf 7/108. Moreover, if the nonoctave people have not yet tried the secor, they are missing a bet. Slender, with a generator of about 4/125 gives something perversely related to 31-et, and we might even try Porcupine, Tertiathirds or Hemikleismic. The Wendy Carlos alpha is related to an 11-limit temperament which appeared on my best-20 list; it is what you get by wedging 121/120, 126/125 and 176/175, [9,5,-3,7,-13,-30,-20,-21,-1,30], and we might call it Wendy or Alpha. Which is best? I also looked at beta and gamma, but I didn't find much. Beta can be related to a cheeseball system obtainable as h75&h94, and it is possible to regard gamma as 5/171. Maybe Graham or Dave can point out something I am missing here.

Message: 5881 - Contents - Hide Contents Date: Tue, 07 Jan 2003 23:57:22 Subject: Re: thanks manuel From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e... wrote:> Sure, I've enjoyed it too. >>> so one might either be interested in the *average* >> complexity of the intervals formed by the note in question from all >> the other notes in the scale, or, in special cases, the complexity of >> the interval formed by the note from the tonic (1/1). >> There's an idea, it could be added to the output of > "show/attribute intervals" which I didn't show you. > > Manueli think it would be good to have a graphical scale analysis tool. the dyadic one would look like the blackjack interval matrix (.jpg file to be referenced in my next messaage), and you could move the scale tones around by dragging the lines (or indicators on a separate slider, which would then move the lines accordingly). that would be cool.

Message: 5882 - Contents - Hide Contents Date: Tue, 7 Jan 2003 10:24:49 Subject: Re: thanks manuel From: manuel.op.de.coul@xxxxxxxxxxx.xxx Sure, I've enjoyed it too.> so one might either be interested in the *average* >complexity of the intervals formed by the note in question from all >the other notes in the scale, or, in special cases, the complexity of >the interval formed by the note from the tonic (1/1).There's an idea, it could be added to the output of "show/attribute intervals" which I didn't show you. Manuel

Message: 5883 - Contents - Hide Contents Date: Wed, 08 Jan 2003 09:55:44 Subject: Notating Kleismic From: Gene Ward Smith I've been pondering this, and I think there is a strong argument in favor of using 53. The top of the poptimal range for septimal kleismic and the bottom of the poptimal range for 5-limit kleismic coincide at the minimax generator of 3^(1/6), which is the same for both. This is the only generator which is poptimal in both limits, but of course 53, which has a much better fifth than 72, comes a lot closer. Moreover, kleismic is more important as a 5-limit system (where it is very strong) than as a 7-limit system, and I think we should try to use one et for all of the versions of a temperament when we can. I vote (if that is how this is done) for 53. Anyone else care to chime in?

Message: 5885 - Contents - Hide Contents Date: Wed, 8 Jan 2003 12:28:29 Subject: Re: thanks manuel From: manuel.op.de.coul@xxxxxxxxxxx.xxx Paul wrote:>i think it would be good to have a graphical scale analysis tool.Do you mean with a colour representation of attribute values, like in your gif picture? There are some other graphical analysis things in the to-do list already, don't think I'll get to it soon. Manuel

Message: 5886 - Contents - Hide Contents Date: Wed, 08 Jan 2003 21:15:20 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> In case anyone has already looked at the .bmp for my latest suggestion > regarding symbolising the 5'-comma (5-schisma) up and down: > > It was riddled with vertical alignment errors so I've had anothergo at it.> > See Yahoo groups: /tuning- * [with cont.] math/files/Dave/5Schismas.bmp > -- Dave Keenan > Brisbane, AustraliaI've looked at it and thought about it for a couple of days. Good points: 1) The +-5' symbols can easily be used in conjunction with any existing symbol. 2) They clearly indicate (by vertical position) the line or space of the note being modified. 3) There is also a helpful indication (a difference in vertical position in addition to slope) of whether the 5' is plus or minus. Comment on point 3: The slope is most meaningful if the new symbols are placed to the left, as in the upper staff (which you also favor). Problems: 1) The +-5' symbols are detached from the others, so are too easy to overlook (particularly if this is the only thing modifying a natural note). 2) Since they are detached from the others, we technically have two new modifying symbols used together, so the double-symbol version of the notation might now become a triple-symbol version -- something to think about. Since looking at this I also tried something else, which I have added to this file (on the third staff): Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Schisma.gif Note: If you don't see 4 staves in the figure, then click on the refresh button on your browser to ensure that you're looking at the latest version of the file. I tried small arrowheads to indicate the 5' down and up symbols. In the 3rd staff I attached them to the point of an existing sagittal symbol; for the up-arrow I removed the pixel at the end of the shaft to clarify the symbol. The big advantage here is that we would avoid having detached symbol elements. In the 4th staff (up to the first double bar) I placed the arrows to the left of existing sagittal symbols, but they could just as easily be placed to the right, or on either side, depending on where they would look or fit best. Wherever you put them, I think that these small arrowheads are easier to see than those tiny slanted lines, and they give a better indication of direction of alteration. While I was writing this I got a couple of other ideas that use 5' flags, so I quickly added them on the fourth staff. I lowered the short straight -5' flag to the same vertical position that we seem to be agreeing on to see how that would look and made 3 symbols that way. Then, after the next double bar, I used the small arrowheads as right flags and tried some symbols that way. (The 5:7 comma is also there for comparison.) After I looked at them for a little while, I decided to move the 5' flags one pixel to the right, so that they are almost, but not quite touching the rest of the sagittal symbol (to avoid confusion with a concave right flag). I think that this last group is my preference in that: 1) The 5' flags are clear and logical; 2) The 5' symbol elements aren't off by themselves, therefore don't get overlooked; 3) Their vertical positions are well placed; 4) They aren't larger than concave flags. These are just a bunch of ideas that I'm tossing out there. Let me know what you think. --George

Message: 5887 - Contents - Hide Contents Date: Wed, 08 Jan 2003 00:02:14 Subject: blackjack interval matrix as promised From: wallyesterpaulrus Yahoo groups: /tuning/files/perlich/secor2.gif * [with cont.]

Message: 5888 - Contents - Hide Contents Date: Wed, 08 Jan 2003 00:08:32 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> The problem with this is that it makes the question of what the >> consonances of the temperament are murky--we can't simply use >> everything in the odd limit for some odd n. Of course, we could >> simply create a set of intervals and then temper, or use n-limit >> including evens. >> Perhaps I'm not seeing it, but I don't think we need to change > our concept of limit.we certainly would, and could use "integer limit" as gene suggests, or use product limit (tenney).

Message: 5889 - Contents - Hide Contents Date: Wed, 08 Jan 2003 00:31:52 Subject: Re: Poptimal generators From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" > <d.keenan@u...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus >> >>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith >>> >>>> "Poptimal" is short for "p-optimal". The p here is a real variable >>>> p>=2, which is what analysts normally use when discussing these >>> Holder>>>> type normed linear spaces. >>>> >>>> A pair of generators [1/n, x] for a linear temperament is >>>> *poptimal* if there is some p, 2 <= p <= infinity, >>>>>> why not go all the way to 1? MAD, or p=1, error certainly seems >> most>>> appropriate for dissonance curves such as vos's or secor's -- which >>> are in fact even pointier at the local minima (resembling exp >>> (|error|)) . . . >>>> Possibly because no one in the history of this endeavour has ever >> before now suggested that mean-absolute error corresponds in any way >> to the human perception of these things. >> no one in history? you've gotta be kidding me. sum or mean of absolute > errors is a quite common error criterion.By "this endeavour" I meant specifically the mathematical modelling of perceptual optimality of generators for musical temperaments, not mathematic or statistics in general. I also only said "possibly". I'm happy to be corrected.>>> "A "poptimal" generator can lay claim to being absolutely and ideally >>> perfect as a generator for a given temperament ..." >>>> When we're talking about human perception, as we are, it should be >> obvious that nothing can be absolutely and ideally perfect for >> everyone. Even a single person might prefer slightly different >> generators for different purposes. To validate such a claim >> of "perfection" you would at least need to produce statistics on the >> opinions of many listeners. >> clearly dave missed the clever mockery hidden in gene's statement.Sorry. I must have missed some previous discussion that would have made it clear that irony was intended. A smiley or winky after it wouldn't have gone astray. I just read it and thought, hey this is the sort of talk that gets the non-math folk pissed at us math folk. Thanks Gene, for being kind enough to ignore my patronising pedantry.

Message: 5890 - Contents - Hide Contents Date: Wed, 08 Jan 2003 01:15:21 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >erhaps I'm not seeing it, but I don't think we need to change >> our concept of limit. >>we certainly would, and could use "integer limit" as gene >suggests, or use product limit (tenney).Maybe so, but I don't see why. I'm suggesting we think only of the map, and let it do the walking. We get to pick what goes in the map. Picking 2, 3, 5, 7 and calling it "7-limit" seems fine to me.>> If you weight the error right, you shouldn't have to weight >> the complexity. >>I'll return to this after I've had my breakfast coffee. :)Sorry, all I should have said is, * is communitive. So it's really... Sum ( raw-error(i) * graham-complexity(i) * weighting-factor(i) ) I'll wager a coke this eliminates the need for an averaging function over the intervals of the limit. If so, it would approximate traditional badness, and this could be checked. -Carl

Message: 5891 - Contents - Hide Contents Date: Wed, 08 Jan 2003 01:17:59 Subject: 8-limit meantone From: Gene Ward Smith If we look at all-number optimization, we can consider septimal meantone inthe 7, 8, 9 and 10 limits, for all the different p values. Since anything other than p=2 or p=infinity is asking for a headache, Idid those for the 8-limit. The minimax calculation simply gave us our old friend, 1/4-comma meantone. For rms I got stretched octaves: 2~1200.416 cents 3~1897.321 cents. If we take the ratio, we get .580551, which still doesn't quite do it for us.

Message: 5892 - Contents - Hide Contents Date: Wed, 08 Jan 2003 01:22:48 Subject: Re: Poptimal generators From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> Sorry. I must have missed some previous discussion that would have > made it clear that irony was intended. A smiley or winky after it > wouldn't have gone astray."Irony" is too strong, but a knowing grin would do excellently.

Message: 5893 - Contents - Hide Contents Date: Wed, 08 Jan 2003 01:25:53 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> Sum ( raw-error(i) * graham-complexity(i) * weighting-factor(i) ) > > I'll wager a coke this eliminates the need for an averaging > function over the intervals of the limit. If so, it would > approximate traditional badness, and this could be checked.I've had my coffee, but still can't see the advantage of this system. We want to have a measure of absolute error independent of complexity in any case, do we not?

Message: 5894 - Contents - Hide Contents Date: Wed, 08 Jan 2003 01:41:08 Subject: Re: 8-limit meantone From: Carl Lumma>If we take the ratio, we get .580551, which still doesn't >quite do it for us.What ratio is that? Blackwood's r? -Carl

Message: 5895 - Contents - Hide Contents Date: Wed, 08 Jan 2003 01:58:10 Subject: Re: 8-limit meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> If we take the ratio, we get .580551, which still doesn't >> quite do it for us. >> What ratio is that? Blackwood's r?It's the ratio between the approximation for 3 and the approximation for 2,minus one; or in other words the log base the approximation for two of theapproximation for 3/2. This is relevant since in the end we want to relateit all back to a division of the octave.

Message: 5896 - Contents - Hide Contents Date: Thu, 9 Jan 2003 13:29:40 Subject: Re: thanks manuel From: manuel.op.de.coul@xxxxxxxxxxx.xxx>to start with, it would be good enough to simply have all the >consonant intervals (say within a given odd limit) show up as >diagonal lines -- the rest of the chart can be all white or all black >for now . . . the point is you could visually tweak the scale with an >eye toward approximating this consonance here and that consonance >there . . . donīt know of a better way to achieve this goal than an >applet like this!Ok I understand. It probably won't be much work to expand the triad player to do this. Manuel

Message: 5898 - Contents - Hide Contents Date: Thu, 09 Jan 2003 00:04:23 Subject: Re: Poptimal generators From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" >> >>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus >>> >>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith >>>> >>>>> "Poptimal" is short for "p-optimal". The p here is a real variable >>>>> p>=2, which is what analysts normally use when discussing these >>>> Holder>>>>> type normed linear spaces. >>>>> >>>>> A pair of generators [1/n, x] for a linear temperament is >>>>> *poptimal* if there is some p, 2 <= p <= infinity, >>>>>>>> why not go all the way to 1? MAD, or p=1, error certainly seems >>> most>>>> appropriate for dissonance curves such as vos's or secor's -- which >>>> are in fact even pointier at the local minima (resembling exp >>>> (|error|)) . . . >>>>>> Possibly because no one in the history of this endeavour has ever >>> before now suggested that mean-absolute error corresponds in any way >>> to the human perception of these things. >>>> no one in history? you've gotta be kidding me. sum or mean of absolute >> errors is a quite common error criterion. >> By "this endeavour" I meant specifically the mathematical modelling of > perceptual optimality of generators for musical temperaments, not > mathematic or statistics in general. I also only said "possibly". I'm > happy to be corrected.consider yourself corrected :)>>>> "A "poptimal" generator can lay claim to being absolutely and ideally >>>> perfect as a generator for a given temperament ..." >>>>>> When we're talking about human perception, as we are, it should be >>> obvious that nothing can be absolutely and ideally perfect for >>> everyone. Even a single person might prefer slightly different >>> generators for different purposes. To validate such a claim >>> of "perfection" you would at least need to produce statistics on the >>> opinions of many listeners. >>>> clearly dave missed the clever mockery hidden in gene's statement. >> Sorry. I must have missed some previous discussion that would have > made it clear that irony was intended.nope. itīs just that an infinite number of tunings, a continuous range of generators, can be considered poptimal for a given temperament -- thus "absolutely and ideally perfect" cannot be taken seriously, and was obviously poking fun of similar claims by people like lucy. paul (stuck in mannheim right now due to trains not running on time)

Message: 5899 - Contents - Hide Contents Date: Thu, 09 Jan 2003 17:54:46 Subject: Re: Minimax generator From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> > Thanks. How would one calculate the rms generator for say, Kleismic?Kleismic tells us that 3 ~ k^6 5 ~ 2 k^5 6/5 ~ k where k is the generator. If we take log base 2 of both sides and call x=log2(k), we get u1 = 6*x-log2(3) ~ 0 u2 = 5*x+1-log2(5) ~ 0 u3 = x-log2(6/5) ~ 0. The least squares solution to this is found by finding the minimum of the polynomial function F(x) = u1^2 + u2^2 + u3^2 which we may easily do by taking the derivative and setting F'(x)=0. If you have a program which solves linear programming problems, you can setup the standard minimization problem with objective function y and constraint set {y>=u1, y>=u2, y>=u3, y>=-u1, y>=-u2, y>=-u3, x>=0, y>=0} and solve for the minimax generator x.

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