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Message: 9900 - Contents - Hide Contents Date: Mon, 09 Feb 2004 20:49:54 Subject: Re: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Graham Breed Dave Keenan wrote:> Psychologically it would seem that there is some point in the > complexity of low-error 5-limit linear temperaments where one would > rather have the planar complexity of 5-limit JI than bother with the > linear complexity of a temperament. I suggest that occurs somewhere > between the complexities of schismic and the least complex temperament > with error less than schismic. which is:5/31, 193.2 cent generator basis: (1.0, 0.16099797612742392) mapping by period and generator: [(1, 0), (4, -15), (2, 2)] mapping by steps: [(25, 6), (40, 9), (58, 14)] highest interval width: 17 complexity measure: 17 (19 for smallest MOS) highest error: 0.000068 (0.081 cents) unique Graham

Message: 9901 - Contents - Hide Contents Date: Mon, 09 Feb 2004 21:41:58 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Complexity is horizontal, error is vertical, > > Aha. >>> labels are the notes per octave of the ET. >> How can error be in notes?Sorry. I was referring to the labels on the points. i.e. each point is labelled with the n of the n-tET that it is. The error is minimax error in cents where the weighting is log_2(n*d) for the ratio n/d in lowest terms. The complexity I'm not sure about. Paul? But the point is, if you believe these are good error and complexity measures, to try to draw a simple curve that cuts off those ETs that have historically been used or recommended as approximations of 5-limit JI (you are allowed to exclude from your curve if you wish, some that may only have been used because they were multiples of 12). Or alternatively, those you would include in a catalog of "useful 5-limit ETs" or an article about the same.

Message: 9902 - Contents - Hide Contents Date: Mon, 09 Feb 2004 02:07:04 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote: >>>>> Some evidence you've actually considered it would be nice. A plot >>> would be grand. Some attempt to theoretically justify what you > two are>>> doing would be appreciated. >>>> I'm not sure what "it" is that you think we haven't considered. >> The "it" is using log(complexity) and log(error) in drawing moats, > and trying to make the moats involve straight lines if possible.That would be nice, but I just don't think we can make a straight line on a loglog chart agree with what we know about the historical (with much weight on the last few decades) use of various ETs as 5 and 7-limit approximations. But Paul, would you care to replot those ET plots as loglog and we'll have a go. I'm pretty sure we're gonna want to have almost quarter-elliptical cutoffs on a loglog plot, so it's simpler to use linear plots in that case.> Paul > thinks JI is a temperament too and should be on the chart somewhere, > which I think is an absurd thing to worry about. What's your view?I don't think that's quite what Paul means. It's not that we need to see JI plotted, and certainly not because "it's a temperament too". It's that when the error gets below say 0.3 of a cent it's as good as JI in most circumstances (I'll never forget how your persistence payed off in getting Johnny to agree to that :-) so you're not going to put up with any significant amount of extra complexity just because some temperament gets you to 0.03 cents or 0.003 cents. So any straight line on a loglog plot will be too extreme in this regard.>> And by "theoretically justify" do you mean justify purely from >> mathematical considerations? >> I mean that combining error and complexity, and not their logs, seems > a little like adding the distance to the moon in inches to the gross > national product in dollars--where's the justification that we are > talking about something comparable?That's a good point. When we move to a psychological model we are effectively considering a quantity that has been called "pain". And we are considering how to convert error in cents to pain (in ouches?) and complexity in notes to pain in ouches. Then we add pains to get the total pain. And we assume there is a certain threshold of total pain beyond which few musicians are willing to venture. Calculating pain as log of error or complexity just doesn't produce cutoffs that agree with what we've seen on this list over the years.>> I understand you're still in favour of log-flat cutoffs which can be >> written in the form > >> log(err) + k * log(complexity) < x >> Those seem to me to make more sense logically.But not psychologically.>> Paul and I have been considering those of the form >> >> err^p + k * comp^p < x >> >> which can be made to look a lot like the previous one when 0<p<0.5. >> Which is what I thought. Why then the fanatical opposition to even > thinking about it?I'm sorry it came across that way. But the fact is we had already thought about it and found it too extreme, not possible to match up with the historical data (vague though that is). Sorry we didn't spell that out.>> And we find that what works best is a value of p that's slightly > less>> than one, i.e. the cutoff functions that we construct based on our >> knowledge of which ETs have been popular historically, are somewhere >> between log and linear, but much closer to linear. >> This is based on actually looking at loglog charts?No. Paul started off giving those (or were they log-linear, I forget) but it soon became apparent that the cutoffs we wanted (based, admittedly on our somewhat intuitive distillation of what has been talked about on the tuning lists for the last decade or so, and Herman Miller's experiments) would be much less curved on a linear-linear plot.> I'd like to put this moat business on a theoretical basis which makes > sense to me, and a good way to start would be shifting to loglog. I > really don't see why that idea is so horrible. Of course if we must > use curved lines the difference between the approaches is less > important, but first can we show it is somehow better to use curved > lines?I suspect the only thing that would convince you, and fair enough, would be some kind of a survey of the tuning list. Perhaps we could list a bunch of what we think are borderline useful 5-limit ETs and ask people to say which are in and which are out based on considering both the error and the complexity. Trouble is, to have an appreciation of the complexity, it isn't enough to hear Pachelbel's Canon played in each, you have to have considered composing or playing in the temperament or building a fixed-pitch instrument for it or some such. So there would be few people qualified to participate.

Message: 9903 - Contents - Hide Contents Date: Mon, 09 Feb 2004 14:42:48 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>> >omplexity is horizontal, error is vertical, >> >> Aha. >>>>> labels are the notes per octave of the ET. >>>> How can error be in notes? >> Sorry. I was referring to the labels on the points. i.e. each point > is labelled with the n of the n-tET that it is. > >The error is minimax error in cents where the weighting is log_2(n*d) >for the ratio n/d in lowest terms. Ok, thanks. >But the point is, if you believe these are good error and complexity >measures, to try to draw a simple curve that cuts off those ETs that >have historically been used or recommended as approximations of >5-limit JI (you are allowed to exclude from your curve if you wish, >some that may only have been used because they were multiples of 12).I don't place much stock in this sort of game. I have no idea what ETs I'd include or exclude. -Carl

Message: 9904 - Contents - Hide Contents Date: Mon, 09 Feb 2004 02:19:29 Subject: Re: Beep isn't useless.... From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-beep.mid in Wayback Machine > > It doesn't sound as bad as I was imagining it would. It really warps the > melody and harmony, but it could have its uses. Compare with the > superpelog version, which I originally thought was beep before I figured > out the mapping. > > 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-superpelog.mid in Wayback Machine > > I still think the best use of extreme temperaments like beep and father > is for their exotic melodic and harmonic properties, and not as > approximations of JI. But the beep version doesn't seem as extreme as > the father version: > > 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-father.mid in Wayback MachineWhat qualifies this tune as a good test of a temperament's approximation of 7-limit JI? It seems the tune was not composed for 7-limit JI and you are even unsure of how it should be mapped to 7-limit JI. The Canon was ideal for 5-limit as there was no such doubt. Can't we find some "classic" 7-limit-JI piece that demonstrates a lot of different 7-limit consonances and cadences, and warp that?

Message: 9905 - Contents - Hide Contents Date: Mon, 09 Feb 2004 15:16:01 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >.. the dreaded error and complexity bounds. >>My objection was not to limits on them per se, but to acceptance >regions shaped like this (on a log-log plot). > >err >| >| (a) >|---\ >| \ >| \ >| \ (b) >| | >| | >------------ comp > >as opposed to a smooth curve that rounds off those corners marked (a) >and (b).Aha, now I understand your objection. But wait, what's stopping this from being a rectangle? Is the badness bound giving the line AB? If so, it looks like a badness cutoff alone would give a finite region...>It turns out that the simplest way to round off those corners is to >do the following on a linear-linear plot. > >err >| >| >|\ >| \ >| \ >| \ >| \ >------------ compWhy not this on a loglog plot? -C.

Message: 9906 - Contents - Hide Contents Date: Mon, 09 Feb 2004 02:44:43 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> That would be nice, but I just don't think we can make a straight line > on a loglog chart agree with what we know about the historical (with > much weight on the last few decades) use of various ETs as 5 and > 7-limit approximations.To convince me of this, I'll need to see the plots.> But Paul, would you care to replot those ET plots as loglog and we'll > have a go.That's log of complexity and error, not relative error, or epimericity, or anything else. Stating exactly what is being plotted would be nice.> It's that when the error gets below say 0.3 of a cent it's as good as > JI in most circumstances (I'll never forget how your persistence payed > off in getting Johnny to agree to that :-) so you're not going to put > up with any significant amount of extra complexity just because some > temperament gets you to 0.03 cents or 0.003 cents. So any straight > line on a loglog plot will be too extreme in this regard.Your conclusion doesn't follow from your premises, but it is likely that a straight line would include micros as well as macros. I don't see why that is bad; I want a few of them included.> Calculating pain as log of error or complexity just doesn't produce > cutoffs that agree with what we've seen on this list over the years. Show me.The nice thing about log(error) vs log(complexity) in my mind is that we know they are related.> I'm sorry it came across that way. But the fact is we had already > thought about it and found it too extreme, not possible to match up > with the historical data (vague though that is). Sorry we didn't spell > that out.It would be nice if some attempt was made to bring the rest of us on board. I don't know what Carl or Graham think, but I have not been convinced.>> I'd like to put this moat business on a theoretical basis which makes >> sense to me, and a good way to start would be shifting to loglog. I >> really don't see why that idea is so horrible. Of course if we must >> use curved lines the difference between the approaches is less >> important, but first can we show it is somehow better to use curved >> lines? >> I suspect the only thing that would convince you, and fair enough, > would be some kind of a survey of the tuning list.Seeing the plots would be nice. It remains vaporware to me, even if Paul has them, if they aren't made available. Perhaps we could> list a bunch of what we think are borderline useful 5-limit ETs and > ask people to say which are in and which are out based on considering > both the error and the complexity.Well, hey, what do you think I keep pesting people about, and why? I've been trying to get a data set. Trouble is, to have an appreciation> of the complexity, it isn't enough to hear Pachelbel's Canon played in > each, you have to have considered composing or playing in the > temperament or building a fixed-pitch instrument for it or some such.High complexity really isn't such a big deal for some uses. JI can be said to have infinite complexity in a sense (no amount of fifths and octaves will net you a pure major third, etc) which I think shows Paul's worry about where it is on the graph is absurd, and also shows high complexity is not something we must necessarily be concerned about. The question for some uses simply is, are we getting anything out of this approximation?> So there would be few people qualified to participate.I'm working away at it.

Message: 9907 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:21:22 Subject: Re: Loglog From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I checked the files I saved of the graphs being posted, and found no > loglog examples. I then went over to tuning-files, and found one > example,You missed quite a few then, like Yahoo groups: /tuning_files/files/Erlich/gene1... * [with cont.] Yahoo groups: /tuning_files/files/Erlich/herma... * [with cont.] Yahoo groups: /tuning_files/files/Erlich/herma... * [with cont.] Yahoo groups: /tuning_files/files/Erlich/herma... * [with cont.] Yahoo groups: /tuning_files/files/Erlich/dave2... * [with cont.] Yahoo groups: /tuning_files/files/Erlich/dave4... * [with cont.] Yahoo groups: /tuning-math/files/Paul/com5monz... * [with cont.] Yahoo groups: /tuning-math/files/Paul/com5rat.gif * [with cont.]> uploaded today.Yes, for you. No comments, just general derision?> I can't tell by looking at it what the logs > are logs of, however. Clarifying this would be nice.Complexity and error -- what else could it be? Some of the graphs above are even labeled ;)> It would also be > nice if, having created all these loglog images, they were made > available to the rest of us.Yes, I've tried to be nice like this, and will continue to do so.

Message: 9908 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:22:24 Subject: Re: loglog! From: Carl Lumma>> >ttp://groups.yahoo.com/group/tuning-math/files/Paul/et5loglog.gif >> >> Ok, easy! No moat needed, at least for ETs. Just draw a >> circle around the origin and grow the radius until it would >> include something that exceeds a single bound -- a "TOP >> notes per 1200 cents" bound. For ETs at least. Choose a >> bound according to sensibilities in the 5-limit, round it >> to the nearest ten, and use it for all limits. >>That's great, Carl, but in loglog land the origin is arbitrary. Lessee,log(TOP notes per 1200 cents) log2(2)= 1 log2(1)= 0 stop! ...and if you take 1 cent as being the Reinhard cutoff, you get another zero for log(error). Poof! Instant origin. -Carl

Message: 9909 - Contents - Hide Contents Date: Mon, 09 Feb 2004 03:42:16 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> It's that when the error gets below say 0.3 of a cent it's as good > as>> JI in most circumstances (I'll never forget how your persistence > payed>> off in getting Johnny to agree to that :-) so you're not going to > put>> up with any significant amount of extra complexity just because some >> temperament gets you to 0.03 cents or 0.003 cents. So any straight >> line on a loglog plot will be too extreme in this regard. >> Your conclusion doesn't follow from your premises, but it is likely > that a straight line would include micros as well as macros. I don't > see why that is bad; I want a few of them included.Yeah but it may well include femtos as well. If we assume for the sake of argument that 0.3 c is as good as JI, and you have a 0.3 c temperament that is just barely acceptable because of its complexity. Then all the pain is caused by the complexity, not the error. Therefore a further reduction in error from 0.3 c to 0.03 c is not going to reduce the pain and therefore you will not be able to accept any more complexity.>> Calculating pain as log of error or complexity just doesn't produce >> cutoffs that agree with what we've seen on this list over the years. > > Show me.I'm hoping paul can easily replot those ET plots loglog. I don't have the data and am not confident to calculate it.> > The nice thing about log(error) vs log(complexity) in my mind is that > we know they are related.But to the musician they are experienced as two very different things.>> I'm sorry it came across that way. But the fact is we had already >> thought about it and found it too extreme, not possible to match up >> with the historical data (vague though that is). Sorry we didn't > spell >> that out. >> It would be nice if some attempt was made to bring the rest of us on > board. I don't know what Carl or Graham think, but I have not been > convinced.At one stage Carl gave some good arguments why the cutoff might be as far from loglog as err^2 + k * comp^2 < x And I went along with this until I saw the ET plots. Perhaps this still makes sense as a badness measure for ranking temperaments, but not as a cutoff for what to include in an article. But I'm not even sure if that's a coherent suggestion.>> I suspect the only thing that would convince you, and fair enough, >> would be some kind of a survey of the tuning list. >> Seeing the plots would be nice. It remains vaporware to me, even if > Paul has them, if they aren't made available. > > Perhaps we could>> list a bunch of what we think are borderline useful 5-limit ETs and >> ask people to say which are in and which are out based on > considering>> both the error and the complexity. >> Well, hey, what do you think I keep pesting people about, and why? > I've been trying to get a data set.Right. Well maybe you could put together something more formal that sets out everything we know that's relevant about all the borderline 5-limit ETs along with something to listen to and then have a web form where we just click yes, no, or don't know for each. But again, I don't know how to ensure people have thought about how painful the complexity might be, and aren't just responding to the error alone. Hmm. Now that I think about it. We seem to disagree most about the temperaments near the axes. So what we most need to agree on is how much error is acceptable and how much complexity, independently of each other. That would go a long way to nailing things down. So we could have separate surveys for error and complexity, for starters.> High complexity really isn't such a big deal for some uses. JI can be > said to have infinite complexity in a sense (no amount of fifths and > octaves will net you a pure major third, etc) which I think shows > Paul's worry about where it is on the graph is absurd, and also shows > high complexity is not something we must necessarily be concerned > about. The question for some uses simply is, are we getting anything > out of this approximation?Yes. That's a good point (about e.g. 5-limit JI having infinite complexity as a linear temperament), but obviously there's another point of view available where 5-limit JI has finite complexity as a planar temperament. Psychologically it would seem that there is some point in the complexity of low-error 5-limit linear temperaments where one would rather have the planar complexity of 5-limit JI than bother with the linear complexity of a temperament. I suggest that occurs somewhere between the complexities of schismic and the least complex temperament with error less than schismic.

Message: 9910 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:26:39 Subject: Re: A post with pending questions From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >> 9052 >> I thought this was supposed to be a guessing game to which you knew > the answer.No, I ask a lot of questions, and they're almost never rhetorical. Now, if we could only rewind a couple of years and start over . . .

Message: 9911 - Contents - Hide Contents Date: Mon, 09 Feb 2004 22:22:35 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >og-flat is natural, in a way. And it should be one of the easier >> concepts around here to explain to musicians. >>I don't recall even Dave understanding its derivation, let along any >full-time musicians.I recall that Dave rejected the idea of a critical exponent. But I didn't understand it until I coded it. But anyway, it's no big deal. At the level I'd imagine this stuff being explained to a full-time musician, it wouldn't be any harder to explain than a moat.>> So far, you and Dave >> have not done any kind of job explaining "moats", >>I thought we had, over and over again.Can you give a definition? Is it a )) shaped region on a log-linear plot, or...?>> or why we should >> want to add instead of multiply to get badness. >>Why should we want to multiply instead of add?Gene multiplies logs, and you and Dave are adding them. Or so I thought... -Carl

Message: 9912 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:30:37 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>>> A bit more concavity still and we include >>> >>> 45. Blackwood >>>> Following what Dave did for the 5-limit and ET cases, I found that an >> exponent of 2/3 produces the desired moat, for example when >> >> err^(2/3)/6.3+complexity^(2/3)/9.35 < 1. >> I prefer to put the scaling constants inside the exponentiation like this > > (err/15.8)^(2/3) + (complexity/28.6)^(2/3) < 1. > > Then you can see at a glance what maximum error and complexity are > allowed by this cutoff. For similar reasons I prefer to show the chart > with both axes starting from zero. >>> Please look at the resulting graph: >> >> Yahoo groups: /tuning_files/files/Erlich/7lin2... * [with cont.] >> >> The temperaments in thie graph are identified by their ranking >> according to the badness measure implied above: >> >> 1. Huygens meantone >> 2. Pajara >> 3. Magic >> 4. Semisixths >> 5. Dominant Seventh >> 6. Tripletone >> 7. Negri >> 8. Hemifourths >> 9. Kleismic/Hanson >> 10. Superpythagorean >> 11. Injera >> 12. Miracle >> 13. Biporky >> 14. Orwell >> 15. Diminished >> 16. Schismic >> 17. Augmented >> 18. 1/12 oct. period, 25 cent generator (we discussed this years ago) >> 19. Flattone >> 20. Blackwood >> 21. Supermajor seconds >> 22. Nonkleismic >> 23. Porcupine >> This looks reasonable to me as a cutoff, although maybe still too > many, but making a badness measure out of it may be going too far.The badness measure is only "implied" by the above; as you know, I don't favor using actual badness measures.

Message: 9913 - Contents - Hide Contents Date: Mon, 09 Feb 2004 03:56:58 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Hmm. Now that I think about it. We seem to disagree most about the > temperaments near the axes. So what we most need to agree on is how > much error is acceptable and how much complexity, independently of > each other. That would go a long way to nailing things down.Those are the dreaded error and complexity bounds.

Message: 9914 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:34:44 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:>> Paul >> thinks JI is a temperament tooriiight . . .> and should be on the chart somewhere,Of course not -- it has infinite complexity according to any of the charts, since it exceeds the dimension being considered.>> And by "theoretically justify" do you mean justify purely from >> mathematical considerations? >> I mean that combining error and complexity, and not their logs, seems > a little like adding the distance to the moon in inches to the gross > national product in dollars--where's the justification that we are > talking about something comparable?What do I get if I add the log of the distance to the moon in inches to the log of the gross national product in dollars??> Which is what I thought. Why then the fanatical opposition to even > thinking about it?I've been thinking about it and only it for years. The only fanaticism I've seen is the opposition to thinking about something different (and far more practical).

Message: 9915 - Contents - Hide Contents Date: Mon, 09 Feb 2004 04:01:47 Subject: Re: Beep isn't useless.... From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> In any case, the point is that a tuning that seemed useless, after a bit > of experimentation, turned out to have some attractive features after all. >Sure. I only ever said it was useless _as_an_approximation_of_JI_, in particular 5-limit.

Message: 9916 - Contents - Hide Contents Date: Mon, 09 Feb 2004 22:20:11 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >o you mean TOP error for an equal >> temperament, >>Of course that's what he means.For 7-limit ets, how do you decide which comma to use? -Carl

Message: 9917 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:35:23 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> I wrote:>> This looks reasonable to me as a cutoff, although maybe still too >> many, ... >

Message: 9918 - Contents - Hide Contents Date: Mon, 09 Feb 2004 04:22:51 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> Hmm. Now that I think about it. We seem to disagree most about the >> temperaments near the axes. So what we most need to agree on is how >> much error is acceptable and how much complexity, independently of >> each other. That would go a long way to nailing things down. >> Those are the dreaded error and complexity bounds.Yes. But with a different purpose in mind. They are only intended to form part of the boundary as _points_at_the_axes_. My objection was not to limits on them per se, but to acceptance regions shaped like this (on a log-log plot). err | | (a) |---\ | \ | \ | \ (b) | | | | ------------ comp as opposed to a smooth curve that rounds off those corners marked (a) and (b). If you have those corners (a) and (b) you then have to explain what is special about, not only the max complexity and max error, but also the complexity at (a) and the error at (b). It turns out that the simplest way to round off those corners is to do the following on a linear-linear plot. err | | |\ | \ | \ | \ | \ ------------ comp

Message: 9919 - Contents - Hide Contents Date: Mon, 09 Feb 2004 22:23:17 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>>>> >y objection was not to limits on them per se, but to acceptance >>>>> regions shaped like this (on a log-log plot). >>>>> >>>>> err >>>>> | >>>>> | (a) >>>>> |---\ >>>>> | \ >>>>> | \ >>>>> | \ (b) >>>>> | | >>>>> | | >>>>> ------------ comp >>>>> >>>>> as opposed to a smooth curve that rounds off those corners marked >>>>> (a) and (b). >>>>>>>> Aha, now I understand your objection. But wait, what's stopping >>>> this from being a rectangle? Is the badness bound giving the >>>> line AB? >>> >>> Yes. >>>>>>> If so, it looks like a badness cutoff alone would give a >>>> finite region... >>>>>> No, because the zero-error line is infinitely far away on a loglog >>> plot. >>>> Can you illustrate this? >>How can I illustrate infinity? >>> It looks like the zero-error line is >> three dashes away on the above loglog plot. :) >>Since you're smiliing, I'll assume you "got it".No, I was just cracking wise. :( -Carl

Message: 9920 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:41:36 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >>> wrote: >>>>>>> Some evidence you've actually considered it would be nice. A plot >>>> would be grand. Some attempt to theoretically justify what you >> two are>>>> doing would be appreciated. >>>>>> I'm not sure what "it" is that you think we haven't considered. >>>> The "it" is using log(complexity) and log(error) in drawing moats, >> and trying to make the moats involve straight lines if possible. >> That would be nice, but I just don't think we can make a straight line > on a loglog chart agree with what we know about the historical (with > much weight on the last few decades) use of various ETs as 5 and > 7-limit approximations.The problem is that a straight line on the loglog chart will *never* cross the zero-error line! Never!!> But Paul, would you care to replot those ET plots as loglog and we'll > have a go.Yes, when I have time.> I'm pretty sure we're gonna want to have almost quarter-elliptical > cutoffs on a loglog plot,You're forgetting that the zero-error line is infinitely far away on the loglog plots.>>> And by "theoretically justify" do you mean justify purely from >>> mathematical considerations? >>>> I mean that combining error and complexity, and not their logs, seems >> a little like adding the distance to the moon in inches to the gross >> national product in dollars--where's the justification that we are >> talking about something comparable? >> That's a good point. When we move to a psychological model we are > effectively considering a quantity that has been called "pain". And we > are considering how to convert error in cents to pain (in ouches?) and > complexity in notes to pain in ouches. Then we add pains to get the > total pain. And we assume there is a certain threshold of total pain > beyond which few musicians are willing to venture. > > Calculating pain as log of error or complexity just doesn't produce > cutoffsOr pain values!>that agree with what we've seen on this list over the years.>>> And we find that what works best is a value of p that's slightly >> less>>> than one, i.e. the cutoff functions that we construct based on our >>> knowledge of which ETs have been popular historically, are somewhere >>> between log and linear, but much closer to linear. >>>> This is based on actually looking at loglog charts? >> No. Paul started off giving those (or were they log-linear, I forget)Oh crap, I must apologize to Gene. They were log-linear, weren't they? SO SORRY, GENE!

Message: 9921 - Contents - Hide Contents Date: Mon, 09 Feb 2004 22:28:39 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>Thus it's great for a paper for mathematicians. Not for musicians.The *contents* of the list is what's great for musicians, not how it was generated.>>> Log-flat badness with cutoffs >>>> The cutoffs are of course completely arbitrary, but can be easily >> justified and explained in the context of a paper. >>But there are *three* of them!...still trying to understand why the rectangle doesn't enclose a finite number of temperaments...>>> i.e. How should we decide what cutoffs to use on error, complexity >>> and log-flat badness? >>>> You can tweak them to satisfy your sensibilities as best as >> possible, same as you're tweaking the moat to factor infinity >>To factor infinity??Sorry, it just means "a lot" here. Like "Kill factor infinity!".>> to satisfy your >> sensibilities as best as possible. >>But there's less to tweak -- we just find the thickest moat than >encloses the systems in the same ballpark as the ones we know we >definitely want to include. This seems a lot less arbitrary than >tweaking *three* parameters to satisfy one's sensibilities as best >as possible.With moats it seems you're pretty-much able to hand pick the list, which is more arbitrary (in the above sense) than not being able to. By thoughts are that in the 5-limit, we might reasonably have a chance of guessing a good list. But beyond that, I would cry Judas if anyone here claimed they could hand-pick anything. So, my question to you is: can a 5-limit moat be extrapolated upwards nicely? -Carl

Message: 9922 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:42:15 Subject: Re: Beep isn't useless.... From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote: >> 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-beep.mid in Wayback Machine >>>> It doesn't sound as bad as I was imagining it would. It really warps the >> melody and harmony, but it could have its uses. Compare with the >> superpelog version, which I originally thought was beep before I figured >> out the mapping. >> >> 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-superpelog.mid in Wayback Machine >> >> I still think the best use of extreme temperaments like beep and father >> is for their exotic melodic and harmonic properties, and not as >> approximations of JI. But the beep version doesn't seem as extreme as >> the father version: >> >> 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-father.mid in Wayback Machine >> What qualifies this tune as a good test of a temperament's > approximation of 7-limit JI? It seems the tune was not composed for > 7-limit JI and you are even unsure of how it should be mapped to > 7-limit JI. > > The Canon was ideal for 5-limit as there was no such doubt. > > Can't we find some "classic" 7-limit-JI piece that demonstrates a lot > of different 7-limit consonances and cadences, and warp that?Cadences are too tied to particular scales and temperaments.

Message: 9923 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:45:38 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The nice thing about log(error) vs log(complexity) in my mind is that > we know they are related.Related how? Via log-flat badness? Meanwhile, error and complexity are related?> Seeing the plots would be nice. It remains vaporware to me, even if > Paul has them, if they aren't made available.OK, I'll do more of them when I have a chance, but ultimately, I don't think I want to force any musician to think about what log (error) means or what log(complexity) means.> High complexity really isn't such a big deal for some uses. JI can be > said to have infinite complexity in a sense (no amount of fifths and > octaves will net you a pure major third, etc) which I think shows > Paul's worry about where it is on the graph is absurd,No, it shows the bullshit you're putting into my mouth is absurd, as I agreed in a recent post.

Message: 9924 - Contents - Hide Contents Date: Mon, 09 Feb 2004 07:52:53 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> At one stage Carl gave some good arguments why the cutoff might be >> as far from loglog as >> >> err^2 + k * comp^2 < x >> Yes, I think I did say that, in multiplicitive form. >>> And I went along with this until I saw the ET plots. >> Ok, can you recommend a plot to look at, and what you saw that > changed your mind? None of the plots I've seen have been labeled > nor made any sense to me.Those Paul gave in Yahoo groups: /tuning-math/message/9202 * [with cont.] Particularly the 5-limit one, which I assume most people have the greatest feel for. Complexity is horizontal, error is vertical, labels are the notes per octave of the ET.>> Perhaps this >> still makes sense as a badness measure for ranking temperaments, but >> not as a cutoff for what to include in an article. But I'm not even >> sure if that's a coherent suggestion. > > Which suggestion?That something might make a good badness measure for ranking temps but not be good for determining a cutoff. I'd like to retract that now.

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