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Message: 8275 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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: 8280 - Contents - Hide Contents Date: Sat, 15 Nov 2003 19:51:12 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:>> exponent of a particular >> prime number >> Now I'm really confused. I thought these were called monzos, not vals.The mapping which for any rational number gives the p-exponent of that rational number is called a padic valuation, and is the basis for the vals in the same way that prime numbers are the basis for monzos.
Message: 8281 - Contents - Hide Contents 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 - Contents - Hide Contents 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: 8283 - Contents - Hide Contents Date: Sat, 15 Nov 2003 19:53:03 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> So the wedge product is a generalisation of the 3D Cartesian product > or cross product. Awesome! There are really some light-bulbs coming on > in my head today. :-) Thanks Gene.You've got it. It also defines the determinant, come to that.
Message: 8284 - Contents - Hide Contents 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: 8285 - Contents - Hide Contents Date: Sat, 15 Nov 2003 19:54:27 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> The square brackets alook more enclosing, and can't be mistaken for a one.I'm happy either way. Paul? What does a physics major think?
Message: 8286 - Contents - Hide Contents 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: 8287 - Contents - Hide Contents Date: Sat, 15 Nov 2003 20:00:06 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 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) >> How do you reduce it?Take out any common factor, and make the first nonzero entry a one. The exception is for commas, which are standardized by making the comma greater than one. The point is to standardize so that there is one and only one wedgie for a given temperament, of any dimension.> Is there a direct interpretation of the coefficients of the 6D > wedge-product in tuning terms, either before or after the reduction?The first three are simply the mapping to generators, times a common factor of the number of periods per octave. The last three can be interpreted also, but less interestingly.> As there is in the 3D case?The 3D case is simply 5-limit monzos and vals; I don't get your question.
Message: 8288 - Contents - Hide Contents 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: 8289 - Contents - Hide Contents Date: Sat, 15 Nov 2003 20:04:47 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> As Gene's said, this should be written > > <12 19 28] ^ <22 35 51] = <-11 4 2]I think you mean <12 19 28] ^ <22 35 51] = [-11 4 2>
Message: 8290 - Contents - Hide Contents 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 - Contents - Hide Contents 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: 8292 - Contents - Hide Contents Date: Sat, 15 Nov 2003 03:34:25 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> But if I've made a serious mistake I really need to know: > What do the integers [the val's coefficients] represent in tuning > terms? What are they counting?They can be counting different things; steps of an equal temperament, number of octaves, number of generators, exponent of a particular prime number being the obvious examples, but not the only ones.
Message: 8293 - Contents - Hide Contents Date: Sat, 15 Nov 2003 03:42:05 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> so how can i tell which one is covariant and which one is > contravariant?Which one do you regard as the vectors you start from (contravariant vector) and which as linear functions on the space of such vectors (covariant vector?) Obviously, in our case the monzos are the objects, and the vals are the mappings, and not the other way around. However, we *can* consider linear mappings of vals, which can be identifified via unique isomorphim with monzos. Anyway, we have this: monzo = ket = contravariant val = bra = covariant
Message: 8294 - Contents - Hide Contents Date: Sat, 15 Nov 2003 03:46:39 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Ok, now we're on the right track, but I'm still not grokking >>> you. 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?Forget standard vals. The point of it is that if you leave gaps, you have no interpretation for what to put in the gap, which fits into your chord.
Message: 8295 - Contents - Hide Contents Date: Sat, 15 Nov 2003 03:48:59 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> 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.Not 1D. These are 0-dimensional temperaments, I'm afraid.
Message: 8296 - Contents - Hide Contents Date: Sat, 15 Nov 2003 03:51:05 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> not sure how gene would do this notationally, probably i did > something terrible, but without it i could not have made those > charts . . .It's a special case of the wedge product, and I'd notate it that way.
Message: 8297 - Contents - Hide Contents Date: Sat, 15 Nov 2003 03:52:24 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Without an musical example (referenced or constructed), I'm skeptical.What do you mean by an example?
Message: 8298 - Contents - Hide Contents 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
Message: 8299 - Contents - Hide Contents Date: Sat, 15 Nov 2003 03:53:23 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> What happens if we change the bases to odd numbers.You can't. It's not a basis.
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