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Message: 8275 Date: Sat, 15 Nov 2003 00:04:27 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >>>>>>I started with six rationals and ended up with 6 > >>>>>>integers. What's the problem? > >>>>> > >>>>>Are your integers consecutive? > >>>> > >>>> No, and that's part of the def. of standard val, but what > >>>> motivates it? > > > >what does that have to do with the definition of standard val? > > Sorry, it doesn't. I forgot the definition doesn't mention > consecutive. It's just this particular case. > > Wait... is this true: 'For a scale with card k, if there is > no standard val with n=k that consistently maps the scale, the > scale is not a Constant Structure.' Carl, Drop the word "standard". There's absolutely no relationship between "standard" vals and Constant Structure. Sorry if anything I said mislead you in that direction.
Message: 8276 Date: Sat, 15 Nov 2003 05:42:38 Subject: Re: Vals? 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: > > > How does that generalise to other than the 5-limit? i.e. vectors > with > > other than 3 components? > > In the 7-limit, the wedge product of two monzos is a 6D wedge product > vector, (which is the two intervals are commas gives us on reduction > a wedgie for a temperament) How do you reduce it? Is there a direct interpretation of the coefficients of the 6D wedge-product in tuning terms, either before or after the reduction? As there is in the 3D case? > wedging it with a monzo again gives us a > val. The wedge product of two vals (I'm assuming things are set up > the way I define them) gives us, once again, a 6D wedge product > vector, (which if the two vals are et vals gives us on reduction a > wedgie for a temperament) wedging it with a val again gives us a > monzo. This has to be done carefully in terms of basis elements to > make the equivalencies work. I think this is coming a bit too fast for me yet.
Message: 8277 Date: Sat, 15 Nov 2003 19:33:14 Subject: Re: Vals? From: Graham Breed Paul Erlich wrote: > similarly, if i take the (transpose of the?) val for 12-equal: > > |12 19 28> > > and take the cross product with the val for 22-equal: > > <22 35 51| > > i get the monzo for the diaschisma, the interval that vanishes in > both tunings: > > [-11 4 2] > > again, not sure what's going on notationally, but the numbers > work . . . As Gene's said, this should be written <12 19 28] ^ <22 35 51] = <-11 4 2] meaning the wedge product of the two vals is equivalent to that monzo. Here's how you write it using my Python module: >>> from temper import Wedgie as Val >>> (Val((12,19,28))^Val((22,35,51))).complement().flatten() (-11, 4, 2) That the interval vanishes in both tunings can be expressed by the brakets (or whatever the products are called without complex numbers) equalling zero: <12 19 28 | -11 4 2> = 0 <22 35 51 | -11 4 2> = 0 to check: >>> Monzo=Val >>> int(Val((22,35,51))^~Monzo((-11,4,2))) 0 >>> int(Val((12,19,28))^~Monzo((-11,4,2))) 0 (If the module knew the difference between covariant an contravariant vectors (as one version did) you wouldn't need that ~ .) The complement (~ or .complement()) is not the same as a matrix transpose. Which way round you do the wedge product only affects the sign of the result. >>the symbol normally indicates the cross-product, which is extremely >>useful in tuning: for example, if i take the monzo for the > > diaschisma > >>[-4 4 -1> >> >>and cross it with the (transpose of the?) monzo for the syntonic > > comma > >><-11 4 2] >> >>i get the val for the et where they both vanish: >> >>[12 19 28] >> >>not sure how gene would do this notationally, probably i did >>something terrible, but without it i could not have made those >>charts . . . >>> (Monzo((-4,4,-1))^Monzo((-11,4,2))).complement().flatten() (12, 19, 28) Graham
Message: 8278 Date: Sat, 15 Nov 2003 00:05:23 Subject: Re: Vals? From: Dave Keenan Perhaps I should have said: There's absolutely no relationship between the "standardness" of the vals and Constant Structure.
Message: 8279 Date: Sat, 15 Nov 2003 06:33:26 Subject: Re: Vals? From: monz hi Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > > In the 7-limit, the wedge product of two monzos is a > 6D wedge product vector, (which is the two intervals are > commas gives us on reduction a wedgie for a temperament) > wedging it with a monzo again gives us a val. The wedge > product of two vals (I'm assuming things are set up > the way I define them) gives us, once again, a 6D wedge > product vector, (which if the two vals are et vals gives > us on reduction a wedgie for a temperament) wedging it > with a val again gives us a monzo. This has to be done > carefully in terms of basis elements to make the > equivalencies work. i sure wish i knew what the hell this was all about. especially since my name is being used as a term all thru it. you guys (Gene, paul, Dave) lost me on this long ago. but it sure seems interesting. -monz
Message: 8281 Date: Sat, 15 Nov 2003 00:32:02 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > > > wrote: > > > > > > > If we are told that the mapping is for a tET then _which_ tET > it is > > > > for can be read straight out of the mapping, as the coefficient > for > > > > the prime 2 (the first coefficient). And the generator is > simply one > > > > step of that tET. > > > > > > just wondering why you keep saying "tET" -- 'If we are told that > the > > > mapping is for a tone equal temperament then . . .' ?? > > > > I agree it's awkward. Carl objected so vehemently to EDO and I > wanted > > to reserve ET for the most general term (including EDOs ED3s cETs). > > Perhaps this would be a misuse of ET. Do we have some other term for > > the most general category of 1D temperaments, i.e. any single > > generator temperament whether or not it is an integer fraction of > any > > ratio? I guess "1D-temperament" will do. > > > > > actually, > and < fit together and create a X (as in times) ! > > > > Oops. Well we could interpret that as the matrix-product as opposed > to > > the scalar-product (dot-product), but I don't know of any meaning > for > > that in tuning. > > the symbol normally indicates the cross-product, which is extremely > useful in tuning: for example, if i take the monzo for the diaschisma > > [-4 4 -1> > > and cross it with the (transpose of the?) monzo for the syntonic comma > > <-11 4 2] Should have been [-11 4 2> > i get the val for the et where they both vanish: > > [12 19 28] Now you could write <12 19 28] That's magic! I never knew that! But of course if someone ever said it before I wouldn't have understood it since I didn't have a clue what a val was. So [-4 4 -1> (x) [-11 4 2> = <12 19 28] Where (x) is a rather poor ASCII version of the cross-product operator. Is there a standard ASCII version of that. And while we're at it how about an ASCII version of the matrix transpose operator. "^T" ? It's obviously bad having letters in operator symbols since they invite confusion with variables. For non-math types: The cross-product of vectors <a1 a2 a3} and <b1 b2 b3] is [a2b3-a3u2 a3b1-a1b3 a1b2-a2b1>
Message: 8282 Date: Sat, 15 Nov 2003 07:16:21 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > i sure wish i knew what the hell this was all about. > especially since my name is being used as a term all thru it. > > you guys (Gene, paul, Dave) lost me on this long ago. > but it sure seems interesting. I think I'll eventually be able to explain it in a way you can understand it. But it wouldn't do to try until I'm sure I've actually got it all sorted out myself. Paul could probably do the job too.
Message: 8284 Date: Sat, 15 Nov 2003 00:34:56 Subject: Re: Vals? From: Dave Keenan I should probably have made it clearer by writing: The cross-product of vectors <a1 a2 a3} and <b1 b2 b3] is [a2*b3-a3*b2 a3*b1-a1*b3 a1*b2-a2*b1> How does that generalise to other than the 5-limit? i.e. vectors with other than 3 components?
Message: 8286 Date: Sat, 15 Nov 2003 01:05:29 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > similarly, if i take the (transpose of the?) val for 12-equal: > > |12 19 28> > > and take the cross product with the val for 22-equal: > > <22 35 51| > > i get the monzo for the diaschisma, the interval that vanishes in > both tunings: > > [-11 4 2] I never knew this either! Although your use of notation sucks, as you suggested it might. I'd write <12 19 28] (x) <22 35 51] = [-11 4 2> I don't think the bra and ket notation was particularly designed to help us with what can be crossed with what and what the result is. Although we can see that ]< and >[ are both cross products while ][ (which can be relaced by | is the dot product and >< is the matrix product.
Message: 8288 Date: Sat, 15 Nov 2003 01:07:53 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > > > > let me just repeat dave and say that this has *nothing* to do > with > > > the definition of vals -- it's a separate question that you can > > > safely ignore if you want to understand vals. > > > > I was quite aware of that. > > you should be, because as i said, i was just repeating you! > > > I was merely trying to answer Carl's questions. > > me too! Sorry. I read it as "just let me repeat, dave". i.e. I thought you were talking to me. Duh.
Message: 8290 Date: Sat, 15 Nov 2003 01:33:27 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > When we go beyond 1D > > temperaments we have prime-mappings which are matrices (one row per > > generator) and we multiply that by the transpose of a ratio's > > prime-exponent-vector (monzo) to get a vector giving the count of > each > > generator. > > can you show an example? obviously i'm plenty confused as to how to > correctly notate these things . . . Me too, since I want it to be generalisable to matrices, and it seems Gene doesn't care about that. Here's a 5-limit mapping matrix for meantone (call it "M") in one possible notation. <1 2 4] <0 -1 -4] or on one line <1 2 2; 0 -1 -4] The first row related the primes to the octave generator, the second row relates them to the fourth generator. In Gene's terminology, each row is a val. And let "a" be a prime-exponent-vector for some ratio, say 5/3 [0 -1 1> By treating M as a single matrix instead of a pair of vectors (vals) we can just use software that has matrix operations (even Excel) and write M*a (in that would be Excel {=MMULT(M,a)}). However, the fine details are that "a" has to be a column vector for this to work, and the result will be a column vector. If we want them to be rows we have to write (M*aT)T where T is the transpose operator. In Excel {=TRANSPOSE(MMULT(M,TRANSPOSE(a)))} The result is <2, -3> meaning 2 octaves up and 3 fourth-generators down.
Message: 8291 Date: Sat, 15 Nov 2003 20:10:13 Subject: Re: Vals? From: Graham Breed monz wrote: > i sure wish i knew what the hell this was all about. > especially since my name is being used as a term all thru it. > > you guys (Gene, paul, Dave) lost me on this long ago. > but it sure seems interesting. We've established that the wedge product of two monzos corresponds to the temperament in which they vanish. So, with Gene's notation, a comma and diaschisma give 12-equal. |-4 4 -1> ^ |-11 4 2> = <12 19 28| Well, on top of that, you can temper out 36:35, or |2 2 -1 -1> |-4 4 -1> ^ |-11 4 2> ^ |2 2 -1 -1> = <12 19 28 34| Which, to check with my Python module: >>> from temper import Wedgie as Monzo >>> syntonic = Monzo((-4,4,-1)) >>> diaschisma = Monzo((-11,4,2)) >>> septimal = Monzo((2,2,-1,-1)) >>> (syntonic^diaschisma^septimal).complement().flatten() (12, 19, 28, 34) If you only temper out two commas, you get a linear temperament. |-4 4 -1> ^ |2 2 -1 -1> = 7-limit meantone. I don't know how to write linear temperaments as bras, but there are some things you can show. For example, an octave equivalent mapping is like tempering out the octave. >>> octave = Monzo((1,0,0)) >>> (syntonic^septimal^octave).complement().flatten() (0, 1, 4, -2) which means |-4 4 -1> ^ |2 2 -1 -1> ^ |1 0 0> = <0 1 4 -2| and (1 4 -2) is the octave-equivalent mapping for this particular version of meantone, where C-Bb approximates 4:7, rather than C-A#. For the more accurate one, you can do |-4 4 -1> ^ |1 2 -3 1> ^ |1 0 0> = <0 1 4 10| Graham
Message: 8298 Date: Sat, 15 Nov 2003 16:47:21 Subject: Re: Vals? From: Carl Lumma >But yeah. What do others think? Square brackets or vertical >bars (pipes)? I don't care, but I think we should standardize. -Carl
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