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Message: 3325 - Contents - Hide Contents Date: Thu, 17 Jan 2002 08:08:12 Subject: Re: ERROR IN CARTER'S SCHOENBERG From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 17, 2002 12:50 AM > Subject: Re: [tuning-math] ERROR IN CARTER'S SCHOENBERG > >>> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> >> >> You better tell the tuning list if you believe that Partch's >> criticism of Schoenberg was based on a mistranslation!!! > >> Wow, I hadn't even *thought* of *that*! I'll have to get some > sleep and check those details tomorrow. But thanks for the > suggestion.Actually, Partch's examination is based on Schoenberg's 1927/34 theory from "Problems of Harmony", so there's no typo anywhere in that. And as I've shown, that theory results in a 12-tone PB with consistent notation for the 11- and 13-limit ratios. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3326 - Contents - Hide Contents
Date: Thu, 17 Jan 2002 17:49 +0
Subject: Re: algorithm sought
From: graham@xxxxxxxxxx.xx.xx
In-Reply-To: <a26m3j+6noq@xxxxxxx.xxx>
I've been doing a bit of research. There are two GNU implementations of
Common Lisp:
<GCL - GNU Common Lisp * [with cont.] (Wayb.)>
<CLISP Common Lisp Summary * [with cont.] (Wayb.)>
I don't actually know the differences between Common Lisp and Scheme,
hence why it would be heretical to move from one to the other.
Enumerating all 15-limit, 5-note chords is actually a 4 from 48
combinations problem, which takes 45 seconds on my machine with
unoptimized Python.
Graham
Message: 3327 - Contents - Hide Contents Date: Thu, 17 Jan 2002 20:17:32 Subject: scheme, etc., OT (was re: algorithm sought) From: clumma>> >easuring algorithms, or their compactness in different languages? >>Measuring efficiency in one particular language implementation. >Profilers are tools that inspect running code, and record how much >time is spent in each section. That means you can go straight to >the least efficient parts, and not waste effort optimising things >that aren't a problem in the first place.Sounds handy, but I wasn't referring to how much work the computer does in scheme. Obviously, at least in my implemetation, it's quite a bit more than in other languages! I was referring to how compactly the language represents algorithms to humans. Like Knuth, I think that computer programs can be useful not only for the answers they give, but for the explanations they present to humans. There's a continuum in abstraction from assembly to C to functional languages like scheme, to actual math. For me, scheme is the ideal point on this continuum -- math is too compact, C is egregious.>> Aren't there any ASSes that contain more than one instance of >> an 11-limit interval?Measuring intervals from the root is equivalent to the choosing notes method, and I thought you weren't talking about that. Just got our wires crossed, is all.>There's a GNU interpreter for Common Lisp, at least. Perhaps it's >one of the C translators.Oh no, I don't think so. Too bad scheme code doesn't compile as common lisp.>You may need a good garbage collector, but that's independent of >being compiled or interpreted.I suppose that's true.>I expect a GNU interpreter will be fine.As I say, I suspect these commercial outfits cripple their free interpreters.>RMS is a Lisp programmer, after all. Who's he?>> There's nothing NP here that I can see. >>Polynomial time is all you need if it gets complex enough. 10^50 >operations does it. That's roughly the total number of chess >games. And it's only 48!. That's true. > Carl: >> Generator? >>See <PEP 255 -- Simple Generators * [with cont.] (Wayb.)> for Simple >Generators. The code I have now uses a function to generate a huge >list, and then iterates over it. If I turned the function into a >generator, it'd only have to return one result at a time, without >the code getting much more complex. I haven't looked at this yet, >partly because I'm trying to keep compatibility with older versions >of Python.Interesting. These guys are really up to some cool stuff. To me, as someone who finally got decent at scheme, it all seems redundant. But compared to C, it sounds pretty revolutionary.>I think you can do similar, but more advanced things with the Lisp >family. The interpreter/compiler may even do them for you. You >can also write a function that takes a function as argument, and >calls it for each item.That's map. It's just simple recursion on the cdr. (define map (lambda (proc ls) (if (null? ls) '() (cons (proc (car ls) (map proc (cdr ls)))))) Then you can pass it any function that returns a list, including map! And if you pass it a function that doesn't return a list, instead of getting a bug, you get an error. What a concept. But I digress, this OT.>But in this case it'll probably be easier to forget about >reusability, and put the functionality in the middle of the >combinations code, so that the following can work: >>>> Rejecting all chords that are already outside the limit before >>> you add notes to them should speed it up a great deal more.There may be a way to keep the combinations code generalized and still do this, by taking advantage of lazy evaluation. I can't think of how, though, at the moment.>I think a GNU implementation of some Lisp dialect has beaten >Python in benchmarks.I'm surprised. I'll look in to how much it would take to port myself over to common Lisp. -Carl
Message: 3328 - Contents - Hide Contents Date: Thu, 17 Jan 2002 20:58:59 Subject: Re: algorithm sought From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> As in, I have to tack on the ASSes. I can just use your table. > I admitted this is quite satisfactory, that I was just being > obstinate, from the start. It isn't just question of results > for me -- it's understanding. The nature of the problem is > fairly simple, and the ASSes and o- and u-tonalities should all > spin out from the same process. If you could get your ASS > method to produce Partchian tonalities...Mine should, but I'm with Graham--this problem doesn't sound that bad. Should I give it a try, and see if I'm wrong?> Actually, the picking notes instead of intervals thing is worth > trying. I'll do that. Thanks. I'm sure I've given away to > you, by stating my eventual need for inversions, that I'm going > after n-adic uniqueness.If you start saying "scheme" and "n-adic" in the same sentence I'm going toget confused.
Message: 3329 - Contents - Hide Contents Date: Thu, 17 Jan 2002 21:06:03 Subject: ERROR IN CARTER'S SCHOENBERG (Re: badly tuned remote overtones) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, January 16, 2002 4:09 PM >> Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned > remote overtones) >> >>>>> So I will give Paul Erlich the benefit of the doubt >>> and assume that Schoenberg was following tradition as >>> closely as possible in his note naming, by thinking in >>> terms of meantone, and therefore the most likely candidate >>> for the other independent UV is the 5-limit diesis >>> 128:125 = [ 7 0 -3 ] . >>>> This is not like anything I would say. Giving me the benefit of the >> doubt, huh? This reminds me of where you recently told Klaus that you >> disagree with me about sustained notes changing their >> intonation . . . if your interpretations of Schoenberg and other >> theorists are as good as your interpretation of me . . . well, >> whatever, I love you Monz, you get a big ice cream with a cherry on >> top. > >> OK, so I misunderstand things sometimes too ... like most people. > Sorry. :( > > All I'm saying is that *if* Schoenberg had any kind of meantone > conception in mind -- which I think is quite likely, given its > ubiquity in European music, right down to our current notation, > (*this* is why I'm giving a nod to you, Paul!!) -- then the > unison-vector he'll bump into, in 1/4-comma meantone, is 128:125.No way, dude. This is your "rational implications" business rearing its ugly head again -- in no way do I endorse that view.
Message: 3330 - Contents - Hide Contents Date: Thu, 17 Jan 2002 21:10:17 Subject: Re: The 'Arabic' temperament From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> The 7-tone MOS is Mohajira: > > [5, 4, 5, 4, 5, 4, 4] > [9, 7, 9, 7, 9, 7, 7]And the alteration with 5, rather than 3, homotetrachordal octave species is the Arabic Diatonic: 5, 4, 4, 5, 5, 4, 4. Whenever you find an MOS, you should always check if some alteration of the MOS gives more homotetrachordal octave species. In fact, Dave Keenan and I found a scale with three step sizes that was omnitetrachordal. Although the scale had 22 notes, I think this points out that one cannot assume that the melodically "nicest" scales will always come directly from altering an MOS in this way.
Message: 3331 - Contents - Hide Contents Date: Thu, 17 Jan 2002 21:13:49 Subject: Re: ERROR IN CARTER'S SCHOENBERG From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Actually, Partch's examination is based on Schoenberg's > 1927/34 theory from "Problems of Harmony", so there's no > typo anywhere in that. And as I've shown, that theory > results in a 12-tone PB with consistent notation for the > 11- and 13-limit ratios.Where did you show that?
Message: 3332 - Contents - Hide Contents Date: Thu, 17 Jan 2002 14:12:38 Subject: Re: ERROR IN CARTER'S SCHOENBERG From: monz ----- Original Message ----- From: paulerlich <paul@xxxxxxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Thursday, January 17, 2002 1:13 PM Subject: [tuning-math] Re: ERROR IN CARTER'S SCHOENBERG> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> Actually, Partch's examination is based on Schoenberg's >> 1927/34 theory from "Problems of Harmony", so there's no >> typo anywhere in that. And as I've shown, that theory >> results in a 12-tone PB with consistent notation for the >> 11- and 13-limit ratios. >> Where did you show that?I'll have to find it tonight when I have time. It's in one of the first posts I sent in this discussion, probably on Christmas day. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3333 - Contents - Hide Contents Date: Thu, 17 Jan 2002 22:32:54 Subject: Re: scheme, etc., OT (was re: algorithm sought) From: clumma>> >here's a GNU interpreter for Common Lisp, at least. Perhaps it's >> one of the C translators. >>Oh no, I don't think so.Well, that's wrong as of the first list item on the page. :) Just goes to show how low-level C is, that they can get the kind of performance they claim by translating to it. -Carl
Message: 3334 - Contents - Hide Contents Date: Thu, 17 Jan 2002 00:00:19 Subject: [tuning] Re: badly tuned remote overtones From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Right, but . . . did he apply any of them explicitly within any of >> the 'constructions of unison vectors' you gleaned from him? >> Otherwise, you're just "assuming the answer". > >> Well, on p 176 of _Harmonielehre_ (p 155 of the Carter translation), > Schoenberg illustrates the "Circle of 5ths", and explicitly notates > the equivalences Cb=B, Gb=F#, and Db=C# for the major keys, and > ab=g#, eb=g#, and bb=a# for the minor keys that are +5, +6, > and +7 "5ths" (respectively) from the origin C-major/a-minor.OK, but isn't this separate from the 'constructions of unison vectors' in which Schoenberg tries to arrive at a 13-limit justification of 12-tET? I mean, this is traditional harmony, which Schoenberg loves to explain, but is trying to break away from in his own music, no?> > To my mind, the 3-limit (linear, 1-D) is both historically and > conceptually more basic than 5-limit (planar, 2-D). The > notational difference between a "sharp" and what later became > its enharmonically equivalent "flat", ocurred first in Pythagorean > tuning. And so, along this line of reasoning, the Pythagorean > comma is historically and conceptually a more basic enharmonicity > than any of the 5-limit examples. However, as implied above, I > will also grant the possibility that Schoenberg may have intended > the diesis as a unison-vector, and will examine that case below > as well.Schoenberg sees the diatonic scale not as an essentially 3-limit entity, as I do, but as an essentially 5-limit entity. So why would the chromatic scale fall back to 3-limit in his thinking? Doesn't seem to make sense.> > > Also, I understand Gene's "notation" a little better now. > So, taking this particular matrix as an example, > > > 2 3 5 7 11 unison-vector ~cents > > [ -2 2 1 0 -1 ] = 45:44 38.90577323 > [-19 12 0 0 0 ] = 531441:524288 23.46001038 > [ -5 1 0 0 1 ] = 33:32 53.27294323 > [ 6 -2 0 -1 0 ] = 64:63 27.2640918 > [ -4 4 -1 0 0 ] = 81:80 21.5062896 > > > inverse > > [ 12 -7 12 0 12 ] > [ 19 -11 19 0 19 ] > [ 28 -16 28 0 27 ] > [ 34 -20 34 -1 34 ] > [ 41 -24 42 0 41 ] > Adjoint?
Message: 3335 - Contents - Hide Contents Date: Thu, 17 Jan 2002 22:34:45 Subject: Re: algorithm sought From: clumma>> >s in, I have to tack on the ASSes. I can just use your table. >> I admitted this is quite satisfactory, that I was just being >> obstinate, from the start. It isn't just question of results >> for me -- it's understanding. The nature of the problem is >> fairly simple, and the ASSes and o- and u-tonalities should all >> spin out from the same process. If you could get your ASS >> method to produce Partchian tonalities... >>Mine should, but I'm with Graham--this problem doesn't sound >that bad. Should I give it a try, and see if I'm wrong?It depends on how long you're willing to wait to find out. It could take me a long time indeed to be able to tell you.>> Actually, the picking notes instead of intervals thing is worth >> trying. I'll do that. Thanks. I'm sure I've given away to >> you, by stating my eventual need for inversions, that I'm going >> after n-adic uniqueness. >> If you start saying "scheme" and "n-adic" in the same sentence >I'm going to get confused. ? -Carl
Message: 3336 - Contents - Hide Contents Date: Thu, 17 Jan 2002 00:01:17 Subject: Re: algorithm sought From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Cool! One thing it will allow us to do is subtract out the > natural JI chords and leave only magic chords in your lists > on the main list.How's this as a method: using the standard o-limit metric, take everything in a radius of 1 of the unison, which should give you the o-limit diamond. Now take all subsets of size k, find the centroid by averaging the coordinates (which should be in the prime-power basis, so that in the 9-limit 5/3 would be 9^(-1/2) * 5^1 * 7^0 = [-1/2, 1, 0], for instance) and test if everything is within a radius of 1/2 of the centroid, in which case put it on your list. For larger values of o, this would be faster than simply testing for pairwise consonance.
Message: 3337 - Contents - Hide Contents Date: Thu, 17 Jan 2002 22:52:42 Subject: No sevens From: genewardsmith Here is a list of "no sevens" ets up to 500, with a "no sevens" badness measure. Of course corresponding "no sevens" linear temperaments can be treated just like 7-limit linear temperaments. The no-sevens version of Arabic would come from combining 24 and 31. 1 .874469118 2 .6628051687 3 .8991089882 4 .7970714166 6 .8907979318 7 .4849216657 8 .8491505000 9 .7640348589 15 .5562849116 16 .9056313011 22 .6677836946 24 .9023455495 31 .8249555856 65 .6413351099 72 .7439328269 87 .5140313008 152 .8880115141 183 .9262407104 270 .8140628426 342 .8784426512 407 .6748936949 494 .5791226647
Message: 3338 - Contents - Hide Contents Date: Thu, 17 Jan 2002 00:02:10 Subject: Re: Iterated CPS From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> If S is a set of octave-equivalence classes, we can denote by CPS(S) the set of ratios of these classes, CPS already has another meaning -- combination product set. What you're looking at here might be called the "diamond".
Message: 3339 - Contents - Hide Contents Date: Thu, 17 Jan 2002 23:06:08 Subject: Re: No sevens From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Here is a list of "no sevens" ets up to 500, with a "no sevens" > badness measure.Meaning the nine intervals 1:3 1:5 3:5 1:9 5:9 1:11 3:11 5:11 9:11 are equally weighted?
Message: 3340 - Contents - Hide Contents Date: Fri, 18 Jan 2002 00:10:06 Subject: Re: metric visualization From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> I wrote, >>> In any case, if I directly use the formula you use to evaluate ETs > on >> Searching Small Intervals * [with cont.] (Wayb.): >> >> f3 + log(3)/log(5)*f5 + log(3)/log(5)*f(5/3) >> >> intepret f3, f5, and f(5/3) as absolute values, and minimize this >> sum, I do get 1/4-comma meantone. But if I take the MAX instead of >> the sum of these three terms, I get a fifth of 697.1160 cents.A more exact calculation gives me 697.1383 cents . . .
Message: 3341 - Contents - Hide Contents Date: Fri, 18 Jan 2002 09:19:59 Subject: The 41 rank three consonantly-generated subgroups of the 11-limit group From: genewardsmith By "consonantly generated" I mean each of the generators is an 11-limit consonance, and by "rank 3" that there are three independent generators. It turns out that there are 41 of them, each of which could, in theory, be treated as something to locate linear temperaments in. Some, however, are clearly more significant than others. [3,5,7] [3,5,11] [3,7,11] [5,7,9] [5,7,11] [5,9,11] [7,9,11] [5,7,11/9] [3,7/5,11] [7/5,9,11] [5/3,7,11] [3,7,11/5] [7,9,11/5] [5,7/3,11] [9/5,7,11] [5,9/7,11] [5,7,11/3] [5,9,11/7] [3,5,11/7] [5,7/3,11/9] [5,9/7,11/3] [5/3,9/7,11] [5/3,7,11/9] [5,9/7,11/7] [7/3,9/5,11] [9/5,7,11/5] [5,7/3,11/3] [7/5,9/5,11] [5/3,7/3,11] [9,7/5,11/5] [5/3,7,11/3] [3,7/5,11/5] [9/5,7,11/3] [5/3,7/3,11/3] [7/5,9/5,11/5] [5/3,7/3,11/9] [7/3,9/5,11/3] [5/3,9/7,11/3] [7/5,9/5,11/3] [5/3,9/7,11/7] [9/5,7/3,11/5]
Message: 3342 - Contents - Hide Contents Date: Fri, 18 Jan 2002 01:11:51 Subject: Re: No sevens From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Meaning the nine intervals > > 1:3 > 1:5 > 3:5 > 1:9 > 5:9 > 1:11 > 3:11 > 5:11 > 9:11 > > are equally weighted?It probably should mean that, but I forgot to put in the nines. It's been a busy day and I was distracted.
Message: 3343 - Contents - Hide Contents Date: Fri, 18 Jan 2002 02:08:33 Subject: Re: ERROR IN CARTER'S SCHOENBERG (Re: badly tuned remote overtones) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 17, 2002 1:06 PM > Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned remote overtones) > >>> All I'm saying is that *if* Schoenberg had any kind of meantone >> conception in mind -- which I think is quite likely, given its >> ubiquity in European music, right down to our current notation, >> (*this* is why I'm giving a nod to you, Paul!!) -- then the >> unison-vector he'll bump into, in 1/4-comma meantone, is 128:125. >> No way, dude. This is your "rational implications" business rearing > its ugly head again -- in no way do I endorse that view.I think you misunderstand me, Paul. I just mean that there's probably a good chance that at least some of the time, Schoenberg thought of the "Circle of 5ths" in a meantone rather than a Pythagorean sense. This reference to you is only meant to credit you for opening my eyes to the strong meantone basis behind a good portion of the "common-practice" European musical tradition. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3344 - Contents - Hide Contents Date: Fri, 18 Jan 2002 02:52:50 Subject: Schoenberg's 1927/34 "Problems of Harmony" theory From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 17, 2002 1:13 PM > Subject: [tuning-math] Re: ERROR IN CARTER'S SCHOENBERG > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> Actually, Partch's examination is based on Schoenberg's >> 1927/34 theory from "Problems of Harmony", so there's no >> typo anywhere in that. And as I've shown, that theory >> results in a 12-tone PB with consistent notation for the >> 11- and 13-limit ratios. >> Where did you show that? Yahoo groups: /tuning-math/message/2159 * [with cont.] > Message 2159 > From: "monz" <joemonz@y...> > Date: Tue Dec 25, 2001 6:44 pm > Subject: lattices of Schoenberg's rational implications > > > ... > > 1934 _Problems of Harmony_ 13-limit system > > (-2 0 0 -1 1 ) = 104:99 > ( 2 0 -1 0 1 ) = 117:112 > (-2 0 -1 0 0 ) = 64:63 > ( 4 -1 0 0 0 ) = 81:80 > ( 2 1 0 -1 0 ) = 45:44 > > Determinant = 12If I include 2:1 , I get matrix [ 1 0 0 0 0 ] = 2:1 [ -2 0 0 -1 1 ] = 104:99 [ 2 0 -1 0 1 ] = 117:112 [ -2 0 -1 0 0 ] = 64:63 [ 4 -1 0 0 0 ] = 81:80 [ 2 1 0 -1 0 ] = 45:44 adjoint: [ 12 0 0 0 0 0 ] [ 19 -1 1 -1 1 1 ] [ 28 -4 4 -4 -8 4 ] [ 34 2 -2 -10 -2 -2 ] [ 42 -6 6 -6 -6 -6 ] [ 44 4 8 -8 -4 -4 ] OK, I see that the first column-vector gives a typical 12-EDO mapping. Interestingly, now the 11th harmonic is mapped to 42 degrees of 12-EDO -- if "C" is n^0, this is "F#", the opposite of how Schoenberg mapped it in 1911 (as "F"), and indeed this is exactly how Schoenberg now notates 11 in "Problems of Harmony". And the "new" 13th harmonic is mapped to the 44th degree ("Ab"), which again is how Schoenberg notates it. As for the other column-vectors: I can see that all of them map 3 = 1 generator, the "5th", typical of both meantone and Pythagorean. Columns 2, 3, 4, and 6 map 5 = 4 generators, also typical of meantone, and the 5th column maps 5 = -8 generators, typical of a Pythagorean-based schismic temperament. Columns 2, 3, 5, and 6 map 7 to -2 generators, the "minor 7th", the closest approximation in Pythagorean. Column 4 maps 7 = 10 generators, the "augmented 6th", which is a typical meantone mapping. Columns 2, 3, and 4 map 11 = 6 generators, the "tritone" or "augmented 4th", a meantone-like approximation. Columns 5 and 6 map 11 = -6 generators, the Pythagorean "diminished 5th", again only an approximation. Columns 2, 5, and 6 map 13 = -4 generators, the "minor 6th", a meantone-like approximation, and columns 3 and 4 map 13 = 8 generators, the "augmented 5th", a Pythagorean approximation. Feedback appreciated. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3345 - Contents - Hide Contents Date: Fri, 18 Jan 2002 07:14:02 Subject: Re: algorithm sought From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> If you start saying "scheme" and "n-adic" in the same sentence >> I'm going to get confused. > > ? http://mathworld.wolfram.com/Scheme.html * [with cont.] http://mathworld.wolfram.com/p-adicNumber.html * [with cont.]
Message: 3346 - Contents - Hide Contents Date: Fri, 18 Jan 2002 21:04:40 Subject: ERROR IN CARTER'S SCHOENBERG (Re: badly tuned remote overtones) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > I think you misunderstand me, Paul. I just mean that there's > probably a good chance that at least some of the time, Schoenberg > thought of the "Circle of 5ths" in a meantone rather than a > Pythagorean sense.I doubt it. For Schoenberg, the circle of 5ths closes after 12 fifths -- which is closer to being true in Pythagorean than in most meantones.> This reference to you is only meant to credit you for opening > my eyes to the strong meantone basis behind a good portion of > the "common-practice" European musical tradition.OK -- but you're confusing two completely unrelated facts -- that 128:125 is just in 1/4-comma meantone, and that 128:125 is one of the simplest unison vectors for defining a 12-tone periodicity block.
Message: 3347 - Contents - Hide Contents Date: Fri, 18 Jan 2002 21:06:43 Subject: Carl's Mathworld-Complete Correspondence Theorem From: clumma>>> >f you start saying "scheme" and "n-adic" in the same sentence >>> I'm going to get confused. >> >> ? > >http://mathworld.wolfram.com/Scheme.html * [with cont.] >http://mathworld.wolfram.com/p-adicNumber.html * [with cont.]MCCT- For all pairs of terms [t_i, t_(i+1)], from finite alphabet [gamma], it is possible to select a pair of terms [m_i, m_(i+1)] from the Mathworld website which are identical. -C.
Message: 3348 - Contents - Hide Contents Date: Fri, 18 Jan 2002 21:48 +0 Subject: Re: scheme, etc., OT (was re: algorithm sought) From: graham@xxxxxxxxxx.xx.xx carl wrote:> Sounds handy, but I wasn't referring to how much work the computer > does in scheme. Obviously, at least in my implemetation, it's > quite a bit more than in other languages! I was referring to how > compactly the language represents algorithms to humans. Like Knuth, > I think that computer programs can be useful not only for the > answers they give, but for the explanations they present to humans. > There's a continuum in abstraction from assembly to C to functional > languages like scheme, to actual math. For me, scheme is the ideal > point on this continuum -- math is too compact, C is egregious.Well, this is something I'd prefer to leave subjective. Everybody has their own preferences with languages. Also, a lot depends on what library functions your allowed to use as "gifts". Some of the stuff with periodicity blocks is simple iff you have a library for handling and inverting matrices. There's a sense in which a brute force algorithm is less satisfactory than another kind. Comparing execution time between different programs using the same language implementation can be a good way of indicating this. Where is Haskell on your continuum?> Measuring intervals from the root is equivalent to the choosing > notes method, and I thought you weren't talking about that. Just > got our wires crossed, is all.Yes. Well, I've gone ahead and implemented this. Code is at <# Temperament finding library -- definitions * [with cont.] (Wayb.)> if I remember. The relevant stuff is in the "Tonality diamonds" section, with the combinations routine in "Utilities". CGI may follow. Here are all the 9-limit tetrads:>> for chord in temper.limit9.allConsonantChords(4): print chord[(0, 0, 0), (1, 0, 0), (-1, 0, 0), (-1, 1, 0)] [(0, 0, 0), (1, 0, 0), (-1, 0, 0), (1, -1, 0)] [(0, 0, 0), (1, 0, 0), (-1, 0, 0), (-1, 0, 1)] [(0, 0, 0), (1, 0, 0), (-1, 0, 0), (1, 0, -1)] [(0, 0, 0), (1, 0, 0), (0, 1, 0), (-1, 1, 0)] [(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)] [(0, 0, 0), (1, 0, 0), (0, 1, 0), (2, 0, 0)] [(0, 0, 0), (1, 0, 0), (-1, 1, 0), (-1, 0, 1)] [(0, 0, 0), (1, 0, 0), (1, -1, 0), (1, 0, -1)] [(0, 0, 0), (1, 0, 0), (1, -1, 0), (2, -1, 0)] [(0, 0, 0), (1, 0, 0), (0, 0, 1), (-1, 0, 1)] [(0, 0, 0), (1, 0, 0), (0, 0, 1), (2, 0, 0)] [(0, 0, 0), (1, 0, 0), (1, 0, -1), (2, 0, -1)] [(0, 0, 0), (1, 0, 0), (2, 0, 0), (2, -1, 0)] [(0, 0, 0), (1, 0, 0), (2, 0, 0), (2, 0, -1)] [(0, 0, 0), (1, 0, 0), (2, -1, 0), (2, 0, -1)] [(0, 0, 0), (-1, 0, 0), (0, -1, 0), (1, -1, 0)] [(0, 0, 0), (-1, 0, 0), (0, -1, 0), (0, 0, -1)] [(0, 0, 0), (-1, 0, 0), (0, -1, 0), (-2, 0, 0)] [(0, 0, 0), (-1, 0, 0), (-1, 1, 0), (-1, 0, 1)] [(0, 0, 0), (-1, 0, 0), (-1, 1, 0), (-2, 1, 0)] [(0, 0, 0), (-1, 0, 0), (1, -1, 0), (1, 0, -1)] [(0, 0, 0), (-1, 0, 0), (0, 0, -1), (1, 0, -1)] [(0, 0, 0), (-1, 0, 0), (0, 0, -1), (-2, 0, 0)] [(0, 0, 0), (-1, 0, 0), (-1, 0, 1), (-2, 0, 1)] [(0, 0, 0), (-1, 0, 0), (-2, 0, 0), (-2, 1, 0)] [(0, 0, 0), (-1, 0, 0), (-2, 0, 0), (-2, 0, 1)] [(0, 0, 0), (-1, 0, 0), (-2, 1, 0), (-2, 0, 1)] [(0, 0, 0), (0, 1, 0), (-1, 1, 0), (0, 1, -1)] [(0, 0, 0), (0, 1, 0), (-1, 1, 0), (-2, 1, 0)] [(0, 0, 0), (0, 1, 0), (0, 0, 1), (2, 0, 0)] [(0, 0, 0), (0, 1, 0), (0, 1, -1), (-2, 1, 0)] [(0, 0, 0), (0, -1, 0), (1, -1, 0), (0, -1, 1)] [(0, 0, 0), (0, -1, 0), (1, -1, 0), (2, -1, 0)] [(0, 0, 0), (0, -1, 0), (0, 0, -1), (-2, 0, 0)] [(0, 0, 0), (0, -1, 0), (0, -1, 1), (2, -1, 0)] [(0, 0, 0), (-1, 1, 0), (0, 1, -1), (-2, 1, 0)] [(0, 0, 0), (1, -1, 0), (0, -1, 1), (2, -1, 0)] [(0, 0, 0), (0, 0, 1), (-1, 0, 1), (0, -1, 1)] [(0, 0, 0), (0, 0, 1), (-1, 0, 1), (-2, 0, 1)] [(0, 0, 0), (0, 0, 1), (0, -1, 1), (-2, 0, 1)] [(0, 0, 0), (0, 0, -1), (1, 0, -1), (0, 1, -1)] [(0, 0, 0), (0, 0, -1), (1, 0, -1), (2, 0, -1)] [(0, 0, 0), (0, 0, -1), (0, 1, -1), (2, 0, -1)] [(0, 0, 0), (-1, 0, 1), (0, -1, 1), (-2, 0, 1)] [(0, 0, 0), (1, 0, -1), (0, 1, -1), (2, 0, -1)] [(0, 0, 0), (2, 0, 0), (2, -1, 0), (2, 0, -1)] [(0, 0, 0), (-2, 0, 0), (-2, 1, 0), (-2, 0, 1)] The output's rather clumsy -- it's octave-equivalent vectors in no particular order. Whatever, I did spot at least one ASS in there. 15-limit tetrads take a bit of time to work out, but no much. Here's how to count them:>>> len(temper.limit15.allConsonantChords(4)) 612There are 590 15-limit pentads, and it takes two minutes to count them all on this machine. Here's how I did that:>>> import time >>> stamp=time.time();len(temper.limit15.allConsonantChords(5) );time.time()-stamp 590 126.549999952If all is correct, that should include no ASSes. The execution time may well be too great for CGI at my site. Although I have now cut it in half with some superficial changes. I wonder what I did right. Graham
Message: 3349 - Contents - Hide Contents Date: Fri, 18 Jan 2002 22:11:37 Subject: Re: scheme, etc., OT (was re: algorithm sought) From: clumma>> >here's a continuum in abstraction from assembly to C to functional >> languages like scheme, to actual math. For me, scheme is the ideal >> point on this continuum -- math is too compact, C is egregious. >>Well, this is something I'd prefer to leave subjective. Everybody >has their own preferences with languages.Very true. I was just stating mine.>Also, a lot depends on what library functions your allowed to use >as "gifts".Very true, and this should certainly be taken into account. Actually, it's nothing but library functions, all the way down; in a sense, this is my argument. As a stock language, scheme comes with very little more than a UTM. Math comes with the biggest library of all.>There's a sense in which a brute force algorithm is >less satisfactory than another kind. Right. >Where is Haskell on your continuum?Near scheme, closer to math.>Yes. Well, I've gone ahead and implemented this. Code is at ><# Temperament finding library -- definitions * [with cont.] (Wayb.)> if I remember. The relevant >stuff is in the "Tonality diamonds" section, with the >combinations routine in "Utilities". CGI may follow. Suh-weet! >The output's rather clumsy -- it's octave-equivalent vectors >in no particular order. D'oh! >Whatever, I did spot at least one ASS in there. Cool. >15-limit tetrads take a bit of time to work out, but no much. >Here's how to count them: > >>>> len(temper.limit15.allConsonantChords(4)) > 612 > > There are 590 15-limit pentads, and it takes two minutes to > count them all on this machine. Here's how I did that: > >>>> import time >>>> stamp=time.time();len(temper.limit15.allConsonantChords(5) > );time.time()-stamp > 590 > 126.549999952 > > If all is correct, that should include no ASSes. The execution > time may well be too great for CGI at my site. Although I have > now cut it in half with some superficial changes. I wonder what > I did right. Thanks, Graham! -C.
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