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Message: 5025 - Contents - Hide Contents Date: Mon, 17 Jun 2002 06:24:18 Subject: Re: Seven and eleven limit comma lists From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> Here are comma lists for the 7 and 11 limits. Each comma is less >> than fifty cents, and each has the property that if the comma is >> p/q>1 in reduced form, then ln(p-q)/ln(q) >> that's my complexity heuristic -- did you arrive at it independently?I thought your complexity heuristic was (p-q)/(q ln q); in any case I came up with this and some other things similar (but not, I think identical) to your heuristic independently. If you consider this to be a complexity heuristic, it gives a complexity of 0 to any superparticular ratio, which I don't think is what you wanted to do. It works more like a weakened superparticularity condition than complexity, I think.
Message: 5026 - Contents - Hide Contents Date: Mon, 17 Jun 2002 19:17:29 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> I would therefore recommend going back to the rational >> complementation system and doing the ET's that way as well. >> Agreed. Provided we _always_ use rational complements, whether this > results in matching half-apotomes or not.In other words, you would prefer having this: 152 (76 ss.): )| |~ /| |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||\ ~||) (||~ /||\ instead of this: 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ /|| ||\ /||~ /||) /||\ even if it isn't as easy to remember. I suggest that you try some more ET's before insisting on rational complements across the board. In addition to less memorable symbol sequences, strict rational complementation will also result in some bad symbol arithmetic in instances where the complement symbols are not consistent in some ET's. I will accept some symbol arithmetic inconsistency (e.g., with ||) in 72-ET), if it isn't too disorienting, but I think that users will need all the help they can get to keep the symbols straight in the larger ET's, and too many flags and bad symbol arithmetic aren't going to help.>> I would be agreeable to doing all of the ET's (with the rational >> complementation scheme) using the symbols that we agreed on in >> message #4443. > > OK.I erroneously stated that everything that we last agreed on (using what I would call "inverse complements") would stay the same. However, there is one exception. This: 32: )| /|\ (|) (||\ /||\ (DK - inverse complements would become this: 32: )| /|\ (|) (||~ /||\ (rational complements) To this I am agreeable.> I will respond to your suggestions for the remaining ones of 6 or less > steps per apotome when I get more time. ...My time will also be rather limited for at least the next several days, so I will not be working on this a great deal. --George
Message: 5027 - Contents - Hide Contents Date: Mon, 17 Jun 2002 15:38:25 Subject: Re: Seven and eleven limit comma lists From: Gene W Smith On Mon, 17 Jun 2002 18:31:08 -0000 "emotionaljourney22" <paul@xxxxxxxxxxxxx.xxx> writes:> a more pointed reply to this is that we've already settled on a > complexity measure and an error measure, so we shouldn't regress > backwards to less accurate evaluations of these quantities.We've settled on this only for linear temperaments, actually. what we> may want to consider is a different formula for combining the two > into a badness function.I suppose the most obvious possibility is to adjust the exponent of complexity. What did you think of my comma lists just as comma lists? The condition I put on them has a certain logic to it, and I think the results were pretty good, in a finite-list kind of way.
Message: 5029 - Contents - Hide Contents Date: Tue, 18 Jun 2002 21:27:47 Subject: Yet another 5-limit comma list From: Gene W Smith This one comes from the condition ln(p-q)/ln(q) < 3/4, cents < 50. One way to think of the first condition is as follows: writing the comma p/q as 1 + a/q, if ln(a)/ln(q) < e then a < q^e. We thus have a weaker condition than requiring a to be less than a constant, by limiting its size relative to q. 250/243, 1638400/1594323, 128/125, 1594323/1562500, 1990656/1953125, 3125/3072, 20000/19683, 531441/524288, 81/80, 2048/2025, 67108864/66430125, 129140163/128000000, 78732/78125, 393216/390625, 2109375/2097152, 15625/15552, 1600000/1594323, 1224440064/1220703125, 10485760000/10460353203, 6115295232/6103515625, 32805/32768, 274877906944/274658203125, 7629394531250/7625597484987 What do people think of this as a list of 5-limit temperaments?
Message: 5031 - Contents - Hide Contents Date: Tue, 25 Jun 2002 08:13:26 Subject: Another approach to notating JI From: genewardsmith It occured to me that if we notate ennealimmal or hemiennealimmal, we effectively notate JI up to the 7 (or 11) limit. The only problem is, JI enthusthiasts would probably never believe in it or accept it. The idea is to take an ordinary five-line staff, and have each line and space on it correspond to one step of 9-et. We would not use the space between staves as a space for a note, so that the bottom line of a treble staff would be exactly an octave above the bottom line of a bass staff. We would then use sharp to mean 21/20, and flat 20/21. This suffices for the 7-limit; for the 11-limit, we would introduce a symbol meaning exactly half of a 9-et step; i.e., 1/18-th of an octave. We could of course use multiple sharp and flat symbols, and introduce a symbol which is a compliment to the sharp--that is, a step up and a flat down; or almost precisely 36/35, which could be useful though it isn't necessary. Any thoughts? It seems like an awfully easy way to get the job done.
Message: 5032 - Contents - Hide Contents Date: Tue, 25 Jun 2002 02:59:34 Subject: Re: Another approach to notating JI From: Gene W Smith On Tue, 25 Jun 2002 10:43 +0100 (BST) gbreed@xxx.xxxxxxxxx.xx.xx (Graham Breed) writes:> Yes, but we can still notate the same limits uniquely using miracle, > but > much more efficiently. Why would ennealimmal be an advantage?Because it is far more accurate, and hence can claim to be effecively JI. Miracle doesn't really work to notate JI, but this is so close you essentially can't tell the difference.> What's > hemiennealimmal?Like ennalimmal, only dividing the octave in 18 parts, and then using this to go to the 11-limit with extreme accuracy. [[18, 30, 44, 52, 63], [0, -2, -3, -2, -1]] would be one way to do the mapping.
Message: 5033 - Contents - Hide Contents Date: Tue, 25 Jun 2002 03:25:07 Subject: Re: Another approach to notating JI From: Gene W Smith On Tue, 25 Jun 2002 12:15:38 +0200 manuel.op.de.coul@xxxxxxxxxxx.xxx writes:> > Very nice, I will make a Scala notation for it, and call it EL72 > (EL=ennealimmal). Note names C D E F G H J A B C and ascii > accidentals:Looks good; one of the things I like about it is that you can notate with ordinary musical notation programs such as Noteworthy.> This can also be used for 27-tET. Maybe I'll make a EL99 too. > Rather than introducing an extra symbol for 36/35, I thought > that 7/ is fine too.One might find extra symbols for commas such as 126/125=1728/1715, etc. useful as well. To clarify what I said about this being "much more accurate" than miracle, we are talking about an order of magnitude difference, resulting in something which basically is indistinguishable from JI.
Message: 5034 - Contents - Hide Contents Date: Tue, 25 Jun 2002 03:33:09 Subject: Re: Another approach to notating JI From: Gene W Smith On Tue, 25 Jun 2002 12:15:38 +0200 manuel.op.de.coul@xxxxxxxxxxx.xxx writes:> > Very nice, I will make a Scala notation for it, and call it EL72 > (EL=ennealimmal). Note names C D E F G H J A B C and ascii > accidentals: > > 0: 1/1 C > 1: 16.667 cents C/ DbL > 2: 33.333 cents C7 Db\ > 3: 50.000 cents C7/ Db > 4: 66.667 cents C#\ Db/ > 5: 83.333 cents C# DL\ > 6: 100.000 cents C#/ DL > 7: 116.667 cents C#7 D\ > 8: 133.333 cents D133.333 cents is right for the 1/9-octave steps, but these intermediate steps ought to be concocted out of 21/20 = 84.467 cents, or something close to it, so this and not 83.333 is what I was thinking of as C#; this 72-et version may be good for naming things, but what I was thinking of is, as I said, something an order of magnitude more accurate--at least, if needed.
Message: 5035 - Contents - Hide Contents Date: Tue, 25 Jun 2002 04:44:33 Subject: Re: Another approach to notating JI From: Gene W Smith On Tue, 25 Jun 2002 11:45 +0100 (BST) gbreed@xxx.xxxxxxxxx.xx.xx (Graham Breed) writes:> In-Reply-To: <20020625.025934.-1949755.0.genewardsmith@xxxx.xxx> > Gene W Smith wrote: > It doesn't matter how accurate the temperament is, as long as the > notation > can uniquely identify each just interval.Ennealimmal is accurate enough that it doesn't need to do this.> I've added it to the catalog. It's accurate but complex. On a 9 > note > staff, you need an accidental to get the 18 notes and 3 more to get > all > the intervals, giving 8 different combinations.My system only needs a single accidental.>Hemiennealimmal notation doesn't seem to > offer any > advantages unless you're using the temperament.The point is, you do use temperament--you temper out 2401/2400 and 4375/4374, and nobody misses them.
Message: 5038 - Contents - Hide Contents Date: Tue, 25 Jun 2002 16:29:29 Subject: Re: Another approach to notating JI From: Gene W Smith On Tue, 25 Jun 2002 13:25 +0100 (BST) gbreed@xxx.xxxxxxxxx.xx.xx (Graham Breed) writes:> Gene:>> Ennealimmal is accurate enough that it doesn't need to do this. >> If you make the colossal assumption that your performers can > accurately > reproduce such a temperament without using a just reference.Why worry about a just preference? We won't know how this works as a performance system unless it is tried, but it does locate the JI intervals, and I doubt very much adding goofy accidentals for 2401/2400 and 4375/4374, making it a "true" JI system, would help any.>> My system only needs a single accidental. >> If you can get an 11-limit otonality with 9 nominals and 1 > accidental, > that is good. I'll believe it when I see it.18 nominals and 1 accidental for the 11-limit; 9 for the 7-limit.> Then you're notating a temperament. We don't know what commas will > be > missed because we can't get a straight answer out of the JI > proponents.The answers we seem to be getting from those who are willing to answer tell us that the difference between ennealimmal and JI is not perceptible in practice.
Message: 5039 - Contents - Hide Contents Date: Wed, 26 Jun 2002 00:46:00 Subject: Fw: tuning-math choice From: Gene W Smith Personally, I was hoping to wait a bit and see which group emerged victorious. --------- Forwarded message ---------- From: Carl Lumma <carl@xxxxx.xxx> To: tuning-math@xxxxxxxxx.xxx Date: Tue, 25 Jun 2002 09:05:44 -0700 Subject: tuning-math choice Message-ID: <5.0.2.1.2.20020625085721.00b08478@xxxxx.xxx> I suggest it would now be appropriate to kill one of the lists. Gene, since I'm not a member of the yahoo list, maybe you could post this over there. If we decide to kill freelists, I think I'll just unsubscribe everybody and let the archives sit. Or maybe subscribe myself to yahoo briefly and forward everything there (only the messages that started here) and then nuke it completely. If we decide to kill yahoo, I can send a 'follow this link to complete your subscription' message to all members of the yahoo group. Then, we can let the archives sit over there, or perhaps there's a way to extract them -- or perhaps somebody has kept every message in their mail client (I have 17 Feb 2002 through early June). -Carl ____________________________________________________________ To learn how to configure this list via e-mail (subscribe, unsubscribe, etc.), send a message to listar@xxxxxxxxx.xxx with the subject line "info tuning-math". Or visit the website: < Welcome to FreeLists - Free, No-hassle Mailing... * [with cont.] (Wayb.) > .
Message: 5040 - Contents - Hide Contents Date: Wed, 26 Jun 2002 09:23:14 Subject: Calculating errors From: kalleaho Hi! Do you people use least squares method or sum of absolute values of errors (or perhaps something else) when calculating the optimal values for generators? I know least squares is used in statistics but to me it seems that absolute values would be much more appropriate in tuning calculations. Fokker used least squares. Paul Erlich too I suppose. What is the idea behind using them and not absolute values or something else? Thanks, Kalle
Message: 5041 - Contents - Hide Contents Date: Wed, 26 Jun 2002 03:37:25 Subject: Re: Another approach to notating JI From: Gene W Smith On Wed, 26 Jun 2002 10:54 +0100 (BST) gbreed@xxx.xxxxxxxxx.xx.xx (Graham Breed) writes:> In-Reply-To: <20020625.162929.-1949755.4.genewardsmith@xxxx.xxx> > I said reference, not preference. If you want them to locate JI > intervals, surely it's easier to teach them JI intervals than 9 > equally > spaced notes where they have to add or subtract exactly 17.6 cents > to get > the desired intervals.How do they locate these intervals? With a ratio? A collection of accidentals? It seems to me that you can hardly complain that my system is too complicated, since any JI system will have the problem of wishing to get things very exactly in tune, and this method at least is not confusing the issue with commas which would not be relevant in performance practice. We could take a JI score, at least up to the 11-limit, and notate it in a fairly straightforward way.> I don't believe you can notate 7-limit intervals with one accidental > the > way I was counting them. Show an example of 4:5:6:7 if you think it > can > be done.A B### E## G##
Message: 5042 - Contents - Hide Contents Date: Wed, 26 Jun 2002 12:07 +0 Subject: Re: Calculating errors From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <afc162+lumk@xxxxxxx.xxx> kalleaho wrote:> Do you people use least squares method or sum of absolute values of > errors (or perhaps something else) when calculating the optimal > values for generators?I use either unweighted least squares, or the worst absolute error.> I know least squares is used in statistics but to me it seems that > absolute values would be much more appropriate in tuning > calculations. Fokker used least squares. Paul Erlich too I suppose. > What is the idea behind using them and not absolute values or > something else?I think the worst error is the most appropriate, because it tells you how bad the temperament can get. But I use least squares in the automated calculations because it's faster to calculate. See <How to find linear temperaments * [with cont.] (Wayb.)>. When evaluating all pairs of 10 equal temperaments, a slight improvement in the optimisation calculation does make a difference. There's a danger I might hit the limit where my ISP kills the CGI process (so many CPU seconds). A more advanced calculation would weight the most important intervals more highly. For example, ignore the most complex ones because they're not so likely to be played. Then you can find temperaments that work well in a subset of the limit you're interested in. Graham
Message: 5043 - Contents - Hide Contents Date: Wed, 26 Jun 2002 05:38:58 Subject: Re: Another approach to notating JI From: Gene W Smith On Wed, 26 Jun 2002 12:58 +0100 (BST) gbreed@xxx.xxxxxxxxx.xx.xx (Graham Breed) writes:> In-Reply-To: <20020626.033725.-1949755.7.genewardsmith@xxxx.xxx> > Yes, it has the same problems as any other JI system. It isn't "an > awfully easy way to get the job done" as you originally said. If > you're > score uses Partch's 43 note scale (which is the best historical > precedent) > you only need to train the performers to produce those 43 notes by > my > calculations, if you're using a JI notation.And then, that's all they can do. They can't even modulate worth a damn.> A slightly less obvious way of getting this accuracy is to use a > miracle > generator of 116.59 cents and a schisma of around 2.2 cents. That > can > give you the 11-limit to around 0.3 cents.If you really use miracle an schismic at the same time as temperaments, you get the 41-et, so this won't be a logical system. If you want to add symbols, it seems to me better to start out with something consistent. Instead of a double sharp symbol, we could add two sharps up and a step down, for a symbol covering 49/48 and 50/49. Then two steps up and three flats down give us a symbol covering 245/243, 126/125, 4000/3969 and 1728/1715 which we can use in place of a special three sharps/three flats symbol. Putting these together would give us a five sharps/five flats symbol covering 875/864, 81/80, 3125/3087 and 2430/2401. Putting this together with the three sharps symbol gives us eight sharps up and five steps down, covering 1029/1024, 225/224, 19683/19600 and 16875/16807--miracle commas. Setting this comma to unison puts you into 72-et, which you would now be notating.
Message: 5044 - Contents - Hide Contents Date: Wed, 26 Jun 2002 07:05:52 Subject: Re: Another approach to notating JI From: Gene W Smith On Wed, 26 Jun 2002 12:58 +0100 (BST) gbreed@xxx.xxxxxxxxx.xx.xx (Graham Breed) writes:> A slightly less obvious way of getting this accuracy is to use a > miracle > generator of 116.59 cents and a schisma of around 2.2 cents. That > can > give you the 11-limit to around 0.3 cents.One way to do this would be via the 494-et, where 48/494 was one symbol, and 1/494 was another. The 494-et also convers enneadecal (with map [[19,30,44,54],[0,1,1,3]]) and hemienneadecal (with map [[38,60,88,106,131],[0,1,1,3,2]]) where we can play the same kind of game, only with 19 notes instead of 9. It's a microtemperament also, and gets things quite accurately, but I don't see any advantage to it.
Message: 5045 - Contents - Hide Contents Date: Wed, 26 Jun 2002 15:24 +0 Subject: Re: Another approach to notating JI From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <20020626.053859.-1949755.8.genewardsmith@xxxx.xxx> Me:>> Yes, it has the same problems as any other JI system. It isn't "an >> awfully easy way to get the job done" as you originally said. If >> you're >> score uses Partch's 43 note scale (which is the best historical >> precedent) >> you only need to train the performers to produce those 43 notes by >> my >> calculations, if you're using a JI notation. Gene:> And then, that's all they can do. They can't even modulate worth a damn.But that's what they *want" to do. If they wanted to do something else they could have learned that instead. The system already contains room for modulation. You can do a fair bit more without needing 234 notes. If you are doing that much modulation, it won't be a typical JI composition, judging by what people are saying in another place. Me:>> A slightly less obvious way of getting this accuracy is to use a >> miracle >> generator of 116.59 cents and a schisma of around 2.2 cents. That >> can >> give you the 11-limit to around 0.3 cents. Gene:> If you really use miracle an schismic at the same time as temperaments, > you get the 41-et, so this won't be a logical system.That isn't at all what I suggested, so it's hardly a relevant objection.> If you want to add > symbols, it seems to me better to start out with something consistent.The system's perfectly consistent, before and after adding the schisma. It starts as mapping by period: [1, 1, 3, 3, 2] mapping by generator: [0, 6, -7, -2, 15] and I've added mapping by schisma: [0, 1, 1, 1, 1] to get a planar temperament.> Instead of a double sharp symbol, we could add two sharps up and a step > down, for a symbol covering 49/48 and 50/49. Then two steps up and three > flats down give us a symbol covering 245/243, 126/125, 4000/3969 and > 1728/1715 which we can use in place of a special three sharps/three > flats > symbol. Putting these together would give us a five sharps/five flats > symbol covering 875/864, 81/80, 3125/3087 and 2430/2401. Putting this > together with the three sharps symbol gives us eight sharps up and five > steps down, covering 1029/1024, 225/224, 19683/19600 and > 16875/16807--miracle commas. Setting this comma to unison puts you into > 72-et, which you would now be notating.Where's this starting from? 49:48 and 50:49 are quommas. So the "double sharp symbol" is two steps of 72-equal? Double sharps would usually be 12 steps. With my modified system, I think 50:49 is the "true quomma" and 49:48 is a schisma wider. Your second symbol is for 3 steps of 72-equal? I think your commas map as follows to my schismas: 5 schismas =~ 19683:19600 4 schismas =~ 1029:1024 3 schismas =~ 225:224 2 schismas =~ 16875:16807 so they're all distinguished, but not in the right order (19683:19600 is the second smallest). The single schisma miracle comma 2401:2400 is around the same size. Ah, "sharps" and "steps" seem to come from your hemiennialimmal notation. So a 72 note hemiennialimmal tuning with a 7.82 cent "comma" shift? It'd certainly be worth a try if we had a group of eager volunteers to try this stuff out on. Graham
Message: 5046 - Contents - Hide Contents Date: Wed, 26 Jun 2002 16:57 +0 Subject: Re: Another approach to notating JI From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <20020626.070552.-1949755.9.genewardsmith@xxxx.xxx> Gene W Smith wrote:> One way to do this would be via the 494-et, where 48/494 was one symbol, > and 1/494 was another. > The 494-et also convers enneadecal (with map [[19,30,44,54],[0,1,1,3]]) > and hemienneadecal > (with map [[38,60,88,106,131],[0,1,1,3,2]]) where we can play the same > kind of game, only with 19 notes instead of 9. > It's a microtemperament also, and gets things quite accurately, but I > don't see any advantage to it.494-equal's nice to know about, well done. I don't see the point about the enneadecals. Do you want them catalogued? Graham
Message: 5048 - Contents - Hide Contents Date: Wed, 26 Jun 2002 20:48:13 Subject: Re: Another approach to notating JI From: Gene W Smith On Wed, 26 Jun 2002 15:24 +0100 (BST) graham@xxxxxxxxxx.xx.xx writes:> mapping by period: > [1, 1, 3, 3, 2] > > mapping by generator: > [0, 6, -7, -2, 15] > > and I've added > > mapping by schisma: > [0, 1, 1, 1, 1] > > to get a planar temperament.This is indeed a microtemperament, closely allied to the 494-et. I was suggesting 48/494 and 1/494, and the rms generators for this temperament are 48.000/494 and .971/494, so it seems we may as well simply use 494 and be done with it for this one.
Message: 5049 - Contents - Hide Contents Date: Wed, 26 Jun 2002 20:02:09 Subject: Re: Another approach to notating JI From: Gene W Smith On Wed, 26 Jun 2002 15:24 +0100 (BST) graham@xxxxxxxxxx.xx.xx writes: If> you are doing that much modulation, it won't be a typical JI > composition, > judging by what people are saying in another place.I modulate quite a bit in electronic JI. People don't do more of it because they can't, I suspect.> Where's this starting from? 49:48 and 50:49 are quommas. So the > "double > sharp symbol" is two steps of 72-equal?It comes from 2401/2400 being a comma. The sharp is 21/20, and a step is 27/25, so a double sharp symbol represents (21/20)^2 / (27/25) = 49/48. It is two 72-et steps, five 171-et steps, and 18 612-et steps.> Your second symbol is for 3 steps of 72-equal?No, for one step. A triple flat symbol is (27/25)^2 / (21/20)^3 = 1728/1715, which is also 126/125, etc. The symbols are additive, and would come in a Fibbonaci set: {1, 2, 3, 5, 8}. I think your commas> map as > follows to my schismas: > 5 schismas =~ 19683:19600 > 4 schismas =~ 1029:1024 > 3 schismas =~ 225:224 > 2 schismas =~ 16875:16807These are all the same (being eight sharps) in my system, so the two systems seem to be very different.> Ah, "sharps" and "steps" seem to come from your hemiennialimmal > notation. > So a 72 note hemiennialimmal tuning with a 7.82 cent "comma" shift?That's one way to look at it.> It'd > certainly be worth a try if we had a group of eager volunteers to > try this > stuff out on.Always the rub. I suppose I could notate some Partch, claim it was JI, and see the fur fly over on the main group.
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