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Message: 2250 - Contents - Hide Contents Date: Thu, 6 Dec 2001 08:59:33 Subject: Re: The wedge invariant commas From: monz Hi Gene,> From: ideaofgod <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 05, 2001 5:08 PM > Subject: [tuning-math] Re: The wedge invariant commas > > > ... so I've messed > things up around here by introducing multilinear algebra, Baker's > theorem and what-not, as well as something I (and Pierre) saw as > relevant already, namely abelian groups (or Z-modules, as Pierre > prefers to call them), and quadratic forms in connection with > lattices.I'm having lots of trouble understanding what's been discussed on this list since you joined. But this bit of your post jumped out at me, and I thought you'd find this profitable: Mark Lindley & Ronald Turner-Smith. 1993. _Mathematical Models of Musical Scales: A New Approach_. Orpheus-Schriftenreihe zu Grundfragen der Musik vol. 66, Verlag für systematische Musikwissenschaft, Bonn-Bad Godesberg. Lindley, Mark and Ronald Turner-Smith. "An Algebraic Approach to Mathematical Models of Scales", Music Theory Online vol. 0 no. 3, June 1993. M U S I C T H E O R Y O N L I... * [with cont.] (Wayb.) Lindley/Turner-Smith view tuning systems as abelian groups. (see especially paragraph [5] of the latter article) love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2251 - Contents - Hide Contents Date: Thu, 06 Dec 2001 00:35:04 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>> An order of growth estimate shows there should be an infinite list >> for step^2, but not neccesarily for anything higher, and looking far >> out makes it clear step^3 gives a finite list. What this means, of >> course, is that in some sense step^2 is the right way to measure >> goodness. >> Yes! Only squared, not cubed. >>> Step^3 weighs the small systems more heavily, and that is >> why we see so many of them to start with. >> I believe the way to fix this is not to go to step^3 (I don't thinkthere's any human-perception-or-cognition-based justification for doing that), What human-perception-or-cognition-based justification is there for using step^2 ???> Yes. Once the deviation goes past about 20 cents it's irrelevant >how big it is, That's not true -- you're ignoring both adaptive tuning and adaptive timbring.>and a 0.1 cent deviation does not sound 10 times better than a 1.0 >cent deviation, it sounds about the same.In my own musical endeavors, this is true, but with all the strict-JI obsessed people out there, a 0.1 cent deviation may end up being 10 times more interesting than a 1.0 cent deviation.> I suggest this figure-of->demerit. > > step^2 [...]Again, what on earth does step^2 tell you about how composers and performers would rate a temperament? OK, step^2 is the number of possible dyads in the typical scale. Step^3 is the number of possible triads. Why is the former so much more "human-perception-or-cognition- based" to you than the latter? As for the other part, the dissonance measure . . . by doing it Gene's way, we're going to end up with all the most interesting temperaments for a wide variety of different ranges, from "you'll never hear a beat" to "wafso-just" to "quasi-just" to "tempered" to "needing adaptive tuning/timbring". Thus our top 30 or whatever will have much of interest to all different schools of microtonal composers.
Message: 2252 - Contents - Hide Contents
Date: Thu, 6 Dec 2001 19:03 +00
Subject: Wedge products
From: graham@xxxxxxxxxx.xx.xx
Okay, let's go right back to the beginning. I already have a tutorial
online for matrix algebra. See
<Matrix tutorial * [with cont.] (Wayb.)>. If you take the basic
equivalence there, that
5x + 3y + z
is the same as
(5 3 1)(x)
(y)
(z)
you can similarly write any row vector (a b c) as
(a b c)(e1)
(e2)
(e3)
or
a*e1 + b*e2 + c*e3
where * is a normal multiplication and ei simply means the ith element of
the basis. You can then multiply two vectors (a b) and (c d) to get
(a*e1 + b*e2) * (c*e1 + d*e2)
which comes out as
a*c*e1*e1 + a*d*e1*e2 + b*c*e2*e1 + b*d*e2*e2
The music matrices I usually talk about have a basis H, where each entry
is a number. So a normal multiplication really does work
(a b)H * (c d)H
= (a*log(2) + b*log(3)) * (c*log(2) + d*log(3))
= a*c*log(2)*log(2) + a*d*log(2)*log(3)
+ b*c*log(3)*log(2) + b*d*log(3)*log(3)
= a*c*log(2)**2 + (a*d + b*c)*log(2)*log(3) + b*d*log(3)**2
But in general you can't do that, because there isn't a rule for
multiplying elements of a basis. The wedge product is a specific rule
for multipying bases. NOW PAY ATTENTION!
ei^ej = - ej^ei
According to Gene, that's the full definition of a wedge product. So, as
long as you paid attention when I told you you can ignore the rest. You
know what a wedge product is. It follows that
ei^ei = -ei^ei
ei^ei + ei^ei = 0
2*(ei^ei) = 0
ei^ei = 0
So the wedge product of (a b) and (c d) is
(a b)^(c d)
= (a*e1 + b*e2) ^ (c*e1 + d*e2)
= a*c*e1^e1 + a*d*e1^e2 + b*c*e2^e1 + b*d*e2^e2
= a*d*e1^e2 - b*c*e2^e1
= (a*d - b*c)
the same as the determinant
|a b|
|c d|
It gets more complicated in three dimensions
A^B
= (a1*e1 + a2*e2 + a3*e3)^(b1*e1 + b2*e2 + b3*e3)
= a1*b1*e1^e1 + a1*b2*e1^e2 + a1*b3*e1^e3
+ a2*b1*e2^e1 + a2*b2*e2^e2 + a2*b3*e2^e3
+ a3*b1*e3^e1 + a3*b2*e3^e2 + a3*b3*e3^e3
= a1*b2*e1^e2 + a1*b3*e1^e3
- a2*b1*e1^e2 + a2*b3*e2^e3
- a3*b1*e1^e3 - a3*b2*e2^e3
= (a1*b2 - a2*b1)*e1^e2 + (a1*b3 - a3*b1)*e1^e3
+ (a2*b3 - a3*b2)*e2^e3
which is the same as the determinant
|e2^e3 e3^e1 e1^e2|
| a1 a2 a3 |
| b1 b2 b3 |
That's analogous to the cross product of vectors, where AxB is
|x y z|
|a1 a2 a3|
|b1 b2 b3|
and, if A and B are octave-specific 5-limit unison vectors, it happens
that AxB gives the number of steps to each prime consonance. Or, if A and
B are octave-equivalent 7-limit commatic unison vectors, AxB is the
generator mapping.
I can't be bothered to work out the triple wedge product, but you'll find
it's all in terms of e1^e2^e3. So it's something like the determinant
|a1 a2 a3|
|b1 b2 b3|
|c1 c2 c3|
And the right number of wedge products in the right number of dimensions
will always give a determinant.
As determinants and cross products are already useful in dealing with
unison vectors, it shouldn't be such a surprise that wedge products in
general are also useful. I still don't know how they're useful, but Gene
assures us that they are.
What is this "wedge invariant" he keeps using, and how do you go from it
to get a list of unison vectors?
I'm not sure how to implement these things in Python. I could implement
vectors like dictionaries, so
A = {e1:a1, e2:a2, e3:a3}
B = {e1:b1, e2:b2, e3:b3}
then (A^B)[ei^ej] = A[ei]*B[ej] - A[ej]*B[ei]
so the next problem is what data type ei and ej should be. Probably
something like tuples:
A = {(1,):a1, (2,):a2, (3,):a3}
B = {(1,):b1, (2,):b2, (3,):b3}
that makes them look a bit like lists, so instead of A[1]=a1, we have
A[1,]=a1. Then, (A^B)[1, 2] = A[1,]*B[2,] + A[2,]*B[1,].
So this is making some sense, but I still haven't worked out all the
details needed to write the code.
Graham
Message: 2254 - Contents - Hide Contents Date: Thu, 06 Dec 2001 00:57:16 Subject: Re: The slippery six From: paulerlich --- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> (6) [-2,4,-30,-81,42,11] ets: 46,80 >>> >>> [ 0 2] >>> [-1 4] >>> [ 2 3] >>> [-15 18] >>> >>> a = 33.01588032 / 80 (~4/3); b = 1/2 >>> measure 26079 >>>> So this _isn't_ 46+34?? >> It is, but it won't show up as the *sum* of ets, only as the > difference. This is because the 34-et map in question is > h80 - h46, and that isn't h34 in the 7-limit, since h34(7)=95 and > (h80-h46)(7)=96. Maybe adding in a list of differences would be a > good idea.Hmm . . . you keep avoiding my whining about consistency (most recently with regard to 21), and this would seem to be a good place to bring it up again. You told Graham that something like 46+34 to you would be _defined_ so that the 80 would come out right, not necessarily the individual ETs. Now you seem to be contradicting yourself. What gives?> The consequence of being 46+34 of course is that this system is a > hell of a lot better in the 5-limit than it is in the 7-limit; the > 5-limit comma I get from the wedgie is 2048/2025--the diaschisma. > Graham devotes a web page to the diaschismic temperament as a 5- limit > temperament, where it makes a lot of sense.And you brought up 80 when we were discussing ways of extending diaschismic to 11-limit, if you recall . . . probably this same mapping through the 7-limit.
Message: 2257 - Contents - Hide Contents Date: Thu, 06 Dec 2001 01:22:04 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>> Yes. Once the deviation goes past about 20 cents it's irrelevant >> how big it is, > > That's not true -- you're ignoring both adaptive tuning and adaptive > timbring.You can adaptively tune or timbre just about anything, so it seems like we _should_ ignore it.>> and a 0.1 cent deviation does not sound 10 times better than a 1.0 >> cent deviation, it sounds about the same. >> In my own musical endeavors, this is true, but with all the strict-JI > obsessed people out there, a 0.1 cent deviation may end up being 10 > times more interesting than a 1.0 cent deviation.A strict JI obsessed person will not be the slightest bit interested in linear temperaments, or at least that has been my experience. If they are at all interested then think they will be quite happy to have a 1c error rather than a 0.1c one if it lets them halve (actually divide by 10^(1/3)) the number of notes in the scale. Given that 1c is way below the typical accuracy of non-electronic instruments.>> I suggest this figure-of->demerit. >> >> step^2 [...] >> Again, what on earth does step^2 tell you about how composers and > performers would rate a temperament? OK, step^2 is the number of > possible dyads in the typical scale. Step^3 is the number of possible > triads. Why is the former so much more "human-perception-or-cognition- > based" to you than the latter?Ok. Maybe I don't have good argument for that. Try step^3 * exp((cents/k)^2)> As for the other part, the dissonance measure . . . by doing it > Gene's way, we're going to end up with all the most interesting > temperaments for a wide variety of different ranges, from "you'll > never hear a beat" to "wafso-just" to "quasi-just" to "tempered" > to "needing adaptive tuning/timbring". Thus our top 30 or whatever > will have much of interest to all different schools of microtonal > composers.I think it has some extreme cases that are of interest to no one. This can be fixed.
Message: 2258 - Contents - Hide Contents Date: Thu, 06 Dec 2001 01:33:09 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>>> Yes. Once the deviation goes past about 20 cents it's irrelevant >>> how big it is, >> >> That's not true -- you're ignoring both adaptive tuning and adaptive >> timbring. >> You can adaptively tune or timbre just about anything,Not true -- in adaptive tuning, you don't want the horizontal shifts to be too big, or you lose the melodic coherence of the scale; and in adaptive timbring, you don't want the partials to deviate too far from a harmonic series, or you'll lose the sense that each note has a definite pitch.> A strict JI obsessed person will not be the slightest bit interested > in linear temperaments, or at least that has been my experience. If > they are at all interested then think they will be quite happy to have > a 1c error rather than a 0.1c one if it lets them halve (actually > divide by 10^(1/3)) the number of notes in the scale.You don't know that for sure. But look, I myself was trying to get Gene to adopt some exponential, rather than polynomial, function of the number of notes in the scale. He resisted . . .> Given that 1c is > way below the typical accuracy of non-electronic instruments.Hey, it won't be the first time a feature of tuning that is highly removed from most musicians' possible realm of experience has gotten published!>>>> I suggest this figure-of->demerit. >>> >>> step^2 [...] >>>> Again, what on earth does step^2 tell you about how composers and >> performers would rate a temperament? OK, step^2 is the number of >> possible dyads in the typical scale. Step^3 is the number of > possible>> triads. Why is the former so much more > "human-perception-or-cognition->> based" to you than the latter? >> Ok. Maybe I don't have good argument for that. Try > > step^3 * exp((cents/k)^2)That's the _last_ conclusion I wanted you to reach!> I think it has some extreme cases that are of interest to no one. This > can be fixed.I tried to argue this point to Gene, but he seems to really like Ennealimmal. Hey, if we're getting mathematical elegance with this criterion, and all our favorite systems are showing up (I'm still waiting for double-diatonic ~26), shouldn't we be willing to pay the price of letting the guy who's doing all the work get his favorite system in too?
Message: 2260 - Contents - Hide Contents Date: Thu, 06 Dec 2001 01:56:20 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Personally I'd feel much better if everyone could somehow agree what > was the overall most sensible measure regardless of the results!Fat chance :)> In Gene's case, I would hope that it would be some elegant internal > consistency that ties the whole deal together. I'd personally settle > for that even if the results were a tad exotic.I feel the same way.> Of course it might help if I understood it all a bit better too! I > feel like I'm getting there though, I just wish Gene were a little bit > more generous with the narrative--either that or someone else besides > him were saying the same things slightly differently... that helps me > sometimes too.I think he's the only one who understands abstract algebra around here, so in a lot of cases, that isn't really possible, unfortunately . . . of course, I should study up on it, but I should also make more music, and get more sleep, and . . .
Message: 2262 - Contents - Hide Contents Date: Thu, 06 Dec 2001 02:45:38 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> You don't know that for sure. But look, I myself was trying to get > Gene to adopt some exponential, rather than polynomial, function of > the number of notes in the scale. He resisted . . .You wanted to have exponential growth for the "step" factor, and Dave for the "cents" factor, which have opposite tendencies; Dave seems to want to filter the very things out on the low end that you wanted included. If we added an exponential growth to "cents", I would suggest trying k sinh (cents/k) for various k.>> Given that 1c is >> way below the typical accuracy of non-electronic instruments. >> Hey, it won't be the first time a feature of tuning that is highly > removed from most musicians' possible realm of experience has gotten > published!It seems to me it is quite relevant to the strict JI school of thought. I got roasted for mentioning Partch in such a connection, but it's hard to see what theoretical objection he could raise to 45 notes of ennealimmal in the 7-limit.> I tried to argue this point to Gene, but he seems to really like > Ennealimmal. Hey, if we're getting mathematical elegance with this > criterion, and all our favorite systems are showing up (I'm still > waiting for double-diatonic ~26), shouldn't we be willing to pay the > price of letting the guy who's doing all the work get his favorite > system in too?I think the only way you will get rid of Ennealimmal is to have an upper-end cut-off, and you said you wanted none. Sorry, you are stuck with it, and it has nothing to do with my liking it really. I've never even tried it!
Message: 2263 - Contents - Hide Contents Date: Thu, 06 Dec 2001 02:53:30 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> In Gene's case, I would hope that it would be some elegant internal > consistency that ties the whole deal together. I'd personally settle > for that even if the results were a tad exotic.Elegant internal consistency suggests to me steps^2 cents as a measure, but that would need an upper cut-off. We do it for ets, however, so I don't see that as a bif deal myself.> Of course it might help if I understood it all a bit better too! I > feel like I'm getting there though, I just wish Gene were a little bit > more generous with the narrative--either that or someone else besides > him were saying the same things slightly differently... that helps me > sometimes too.I'm hoping Paul will absorb it all and start coming out with his own interpretations, but I can't get him to compute a wedge product. :)
Message: 2264 - Contents - Hide Contents Date: Thu, 06 Dec 2001 02:59:36 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> You wanted to have exponential growth for the "step" factor, and Dave > for the "cents" factor,I think you misunderstood Dave -- he wanted the *goodness* for the cents factor to be a Gaussian.
Message: 2265 - Contents - Hide Contents Date: Thu, 06 Dec 2001 03:00:40 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: >>> In Gene's case, I would hope that it would be some elegant internal >> consistency that ties the whole deal together. I'd personally settle >> for that even if the results were a tad exotic. >> Elegant internal consistency suggests to me steps^2 cents as a > measure, but that would need an upper cut-off. We do it for ets, > however, so I don't see that as a bif deal myself. Who's we? >>> Of course it might help if I understood it all a bit better too! I >> feel like I'm getting there though, I just wish Gene were a little > bit>> more generous with the narrative--either that or someone else > besides>> him were saying the same things slightly differently... that helps > me >> sometimes too. >> I'm hoping Paul will absorb it all and start coming out with his own > interpretations, but I can't get him to compute a wedge product. :)I'll take a look at it again when I get a chance.
Message: 2266 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:00:40 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>>> Well, I think Gene is saying that step^2 cents is clearly the > right>>> measure of "remarkability". >>>> Huh? "Remarkability" sounds like a kind of goodness. Step^2 * cents > is>> obviously a form of badness. >> Right, but it's the _objective_ kind. Not the kind that has anything > to do with any particular musician's desiderata.Paul! You seem to have ignored the most of the rest of my message. What the heck is _objective_ about deciding that a doubling of the number of generators is twice as bad as a doubling of the error. It's completely arbitrary.> It's the only > measure that doesn't favor a certain range of acceptable values for > error or for complexity. It only favors the best examples within each > range.What _objective_ reason is there, to choose it over gens^3 * cents or gens^2.3785 * cents?
Message: 2267 - Contents - Hide Contents Date: Fri, 07 Dec 2001 08:03:05 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>>> Huh? Obviously any badness metric _must_ slope down towards (0,0) >> on>>> the (cents,gens) plain. >>>> The badness metric does, but the results don't. The results have a >> similar distribution everywhere on the plane, but only when gens^2 >> cents is the badness metric. >> You're not making any sense. The results are all just discrete points > in the badness surface with respect to gens and cents, so they have > exactly the same slope. The results have a similar distribution of > what? Everywhere on what plane?I see Gene is, at this very moment, doing a good job explaining these issues to you; meanwhile, my brain is toast.
Message: 2268 - Contents - Hide Contents Date: Fri, 07 Dec 2001 20:47:31 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> So ... What is n? What is a 7-limit et? How does one use n^(4/3) to > get a list of them? How would one check to see whether the list > favours high or low n."n" is how many steps to the octave, or in other words what 2 is mapped to. By a "7-limit et" I mean something which maps 7-limit intervals to numbers of steps in a consistent way. Since we are looking for the best, we can safely restrict these to what we get by rounding n*log2(3), n*log2(5) and n*log2(7) to the nearest integer, and defining the n-et as the map one gets from this. Let's call this map "h"; for the 12-et, h(2)=12, h(3)=19, h(5)=28 and h(7)=34; this entails that h(5/3) = h(5)-h(3) = 9, h(7/3)=15 and h(7/5)=6. I can now measure the relative badness of "h" by taking the sum, or maximum, or rms, of the differences of |h(3)-n*log2(3)|, |h(5)-n*log2(5)|, |h(7)-n*log2(7)|, |h(5/3)-n*log2(5/3)|, |h(7/3)-n*log2(7/3)| and |h(7/5)-n*log2(7/5)|. This measure of badness is flat in the sense that the density is the same everywhere, so that we would be selecting about the same number of ets in a range around 12 as we would in a range around 1200. I don't really want this sort of "flatness", so I use the theory of Diophantine approximation to tell we that if I multiply this badness by the cube root of n, so that the density falls off at a rate of n^(-1/3), I will still get an infinite list of ets, but if I make it fall off faster I probably won't. I can use either the maximum of the above numbers, or the sum, or the rms, and the same conclusion holds; in fact, I can look at the 9-limit instead of the 7-limit and the same conclusion holds. If I look at the maximum, and multiply by 1200 so we are looking at units of n^(4/3) cents, I get the following list of ets which come out as less than 1000, for n going from 1 to 2000: 1 884.3587134 2 839.4327178 4 647.3739047 5 876.4669184 9 920.6653451 10 955.6795096 12 910.1603254 15 994.0402775 31 580.7780905 41 892.0787789 72 892.7193923 99 716.7738001 171 384.2612749 270 615.9368489 342 968.2768986 441 685.5766666 1578 989.4999106 This list just keeps on going, so I cut it off at 2000. I might look at it, and decide that it doesn't have some important ets on it, such as 19,22 and 27; I decide to put those on, not really caring about any other range, by raising the ante to 1200; I then get the following additions: 3 1154.683345 6 1068.957518 19 1087.886603 22 1078.033523 27 1108.589256 68 1090.046322 130 1182.191130 140 1091.565279 202 1143.628876 612 1061.222492 1547 1190.434242 My decision to add 19,22, and 27 leads me to add 3 and 6 at the low end, and 68 and so forth at the high end. It tells me that if I'm interested in 27 in the range around 31, I should also be interested in 68 in the range around 72, in 140 and 202 around 171, 612 around 441, and 1547 near 1578. That's the sort of "flatness" Paul was talking about; it doesn't favor one range over another.> But no matter what you come up with I can't see how you can get past > the fact that gens and cents are fundamentally incomensurable > quantities, so somewhere there has to be a parameter that says how bad > they are relative to each other."n" and cents are incommeasurable also, and n^(4/3) is only right for the 7 and 9 limits, and wrong for everything else, so I don't think this is the issue if we adopt this point of view. Why not> use k*gens + cents. e.g. if badness was simply gens + cents and you > listed everything with badness not more than 30 then you don't need > any additional cutoffs. You automatically eliminate anything with gens>> 30 or cents > 30 (actually cents > 29 because gens can't go below > 1).Gens^3 cents also automatically cuts things off, but I rather like the idea of keeping it "flat" in the above sense and doing the cutting off deliberately, it seems more objective.
Message: 2269 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:23:45 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Paul! You seem to have ignored the most of the rest of my message.Not at all.>> It's the only >> measure that doesn't favor a certain range of acceptable values for >> error or for complexity. It only favors the best examples within > each >> range. >> What _objective_ reason is there, to choose it over gens^3 * cents or > gens^2.3785 * cents?Because those measures give an overall "slope" to the results, in analogy to what the Farey series seeding does to harmonic entropy.
Message: 2270 - Contents - Hide Contents Date: Fri, 07 Dec 2001 08:06:28 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>>>> We could search (16/15)^a (25/24)^b (81/80)^c to start out > with, >>> and>>>> go to something more extreme if wanted. >>>>>> More extreme? I'm not getting this. >>>> (78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives the >> 5-limit, but is better for finding much smaller commas, to take a >> more or less random example. >> Once a, b, and c are big enough, the original choice of commas will > do little to induce any tendency of smallness or largeness in the > result, correct?(78732/78125)^53 (32805/32768)^(-84) (2109375/2097152)^65 = 2 I wouldn't search that far myself.
Message: 2271 - Contents - Hide Contents Date: Fri, 7 Dec 2001 21:14 +00 Subject: Re: Wedge products From: graham@xxxxxxxxxx.xx.xx genewardsmith@xxxx.xxx (genewardsmith) wrote:> First you order the basis so that a wedge product taken from two ets > or two unison vectors will correspond: > > Yahoo groups: /tuning-math/message/1553 * [with cont.]I think I've done that. All comes down to bubblesort -- one in the eye for those who say it's of no practical use!> Then you put the wedge product into a standard form, by > > (1) Dividing through by the gcd of the coefficients, andOkay, that's easy enough.> (2) Changing sign if need be, so that the 5-limit comma (or unison) > 2^w[6] * 3^(-w[2])*5^w[1] where w is the wedgie, is greater than 1. > If it equals 1, go on to the next invariant comma, which leaves out > 5, and if that is 1 also to the one which leaves out 3. See > > Yahoo groups: /tuning-math/message/1555 * [with cont.] > > for the invariant commas. The result of this standardization is the > wedge invariant, or wedgie, which uniquely determins the temperament.I'll need to study that a bit more.>> I'm not sure how to impleme> nt these things in Python. > > The above should do for the 7-limit; in general is another matter.Oh, it needs to be done in general. And it's almost working now. I still need to get my wedge invariants to look like yours. Also to divide through by common factors, but I've done that before. To check:>>> wedge.wedgeProduct(h12,h22){(2, 3): 2, (0, 1): 2, (1, 3): -12, (0, 3): -4, (0, 2): -4, (1, 2): -11} That's numbering from 0 as the 2-direction, 1 as the 5-direction, etc. Then for the commas>>> for i in range(4): print wedge.interval( reduce( wedge.wedgeProduct,((zeros[:i]+(1,)+zeros[i+1:]), h12, h22))) [0, -2, -12, 11] [2, 0, 4, -4] [12, -4, 0, -2] [-11, 4, 2, 0] which looks right. I'll look at it some more tomorrow. I'm still don't know how to do the generator mapping. Here's the library code: def wedgeProduct(a, b): result = {} for base1, value1 in makewedgable(a).items(): for base2, value2 in makewedgable(b).items(): value = value1*value2 for element in base1: if element in base2: break else: base, value = wedgeEquivalent(base1+base2, value) result[base] = result.get(base, 0)+value return result def interval(wedgie): result = [] bases = wedgie.keys() for i in range(len(wedgie)): for base in bases: if i not in base: if i%2: result.append(-wedgie[base]) else: result.append(wedgie[base]) return result def wedgeEquivalent(base, value): workingBase = list(base) for i in range(len(base)): for j in range(i,0,-1): if workingBase[j]<workingBase[j-1]: workingBase[j-1:j+1] = [ workingBase[j], workingBase[j-1]] value = -value return tuple(workingBase), value def addWedges(a, b): x = makewedgable(a) y = makewedgable(b) result = {} for element in x.keys()+y.keys(): result[element]=0 for key in x.keys(): result[key] = result[key] + x[key] for key in y.keys(): result[key] = result[key] + y[key] return result def makewedgable(thing): if isinstance(thing, type({})): return thing else: result = {} for i in range(len(thing)): result[i,]=thing[i] return result
Message: 2272 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:34:59 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> The solutions represent? >> I take the 5-limit comma defined by the temperament, and then find > another comma 2^p 3^q 5^r 7 such that the wedgie of this and the 5- > limit comma is the correct wedgie, that means these two commas define > the temperament.This should be 2^p 3^q 5^r 7^s where s is gcd(a,b,c), and the 5-limit comma is 2^a 3^b 5^c.
Message: 2273 - Contents - Hide Contents Date: Fri, 07 Dec 2001 08:20:04 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>>>> (78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives > the>>> 5-limit, but is better for finding much smaller commas, to take a >>> more or less random example. >>>> Once a, b, and c are big enough, the original choice of commas will >> do little to induce any tendency of smallness or largeness in the >> result, correct? >> (78732/78125)^53 (32805/32768)^(-84) (2109375/2097152)^65 = 2 > > I wouldn't search that far myself.How do you know you wouldn't be missing any good ones?
Message: 2274 - Contents - Hide Contents Date: Fri, 7 Dec 2001 21:14 +00 Subject: Re: More lists From: graham@xxxxxxxxxx.xx.xx Dave Keenan wrote:> I note that Graham is using maximum width and (optimised) maximum > error where Gene is using rms width and (optimised) rms error. It will > be interesting to see if this alone makes much difference to the > rankings. I doubt it.I've implemented RMS error now. It's actually faster than the minimax, so I've made it the default. I've uploaded new copies of the .txt and .gauss files. There are also other changes to the code to make it more efficient. As it stands, the ET matching is broken. I've fixed that, but not uploaded. You could implement the RMS width easily enough, but I expect it'll slow down execution, so you can do it on your own time.> So I see that while the gaussian with std error of 17 cents seems to > do the right thing in eliminating temperaments with tiny errors but > huge numbers of generators, it is too hard on those with larger > errors. Notice that Ennealimmal is still in the 7-limit list (about > number 22). The problem is that Paultone isn't there at all! It has > 17.5 c error with 6 gens per tetrad.I'm dividing the 17 cents by 3 in this case, to give a figure more like what you asked for.> Those lists don't contain any temperament with errors greater than 10 > cents. The 5-limit 163 cent neutral second temperament has the largest > at 9.8 cents, with 5 generators per triad. > > So I have to agree with Paul that > badness = num_gens^2 / gaussian(error/17c) > doesn't work.It works fine. You asked for errors of around 6 cents, so why should you expect errors greater than 10 cents? Graham
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