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Message: 5176 Date: Fri, 06 Sep 2002 02:58:38 Subject: [tuning] Re: Proposal: a high-order septimal schisma From: genewardsmith --- In tuning-math@y..., paul.hjelmstad@u... wrote: > Let p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4, then > >Why are there two "vectors?" What is p and what is q?? We need two of something to define a linear 7-limit temperament--two generators, two equal temperaments, two commas. In general two commas define something of codimension two, but in four dimensions this is the same. The above would be the two commas; so, for instance, we could define meantone using 81/80 and 126/125, and miracle using 225/224 and 1029/1024. > p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, > u1*v3-v1*u3,u1*v4-v1*u4] > >Got it. It's like some kind of dot product, with every combination of > pairs of p and q? So say it's the four-dimensional analog of the cross-product would be more correct. There's a web site which some people found useful for getting the gist of it: Grassmann Algebra Book * > Let r be the mapping to primes of an equal temperament given > by r = [u1, u2, u3, u4], and s be given by [v1, v2, v3, v4]. This > means r has u1 notes to the octave, u2 notes in the approximation of 3, and > so forth; hence [12, 19, 28, 24] would be the usual 12-equal, and [31, 49, > 72, 87] the usual 31-et. > >Got it > > The wedge now is > > r ^ s = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4, > u4*v2-u2*v4,u2*v3-v2*u3] > > OK, what is r and s (again?) Two "vals", or duals to intervals; in the above example, the 12 and 31 equal temperaments. > Whether we've computed in terms of commas or ets, the wedge product of the > linear temperament is exactly the same, up to sign. > >So signs can be reversed? Standardizing so the first non-zero entry is positive is what Graham and I have agreed on as a convention for the "wedgie", or wedge invariant of a temperament. > If the wedgie is [u1,u2,u3,u4,u5,u6] then we have commas given by > > Is this the same wedgie as above? (Based on r ^ s for example)? It is the same numerically, and so for our purposes identical; in theory you get into Poincare duality, or you might do things as in the Grassman book I gave a url for. Did you find what you wanted re the Riemann Zeta function? Can I ask what your math background is?
Message: 5182 Date: Sun, 08 Sep 2002 02:25:42 Subject: Re: A common notation for JI and ETs From: David C Keenan At 01:39 PM 6/09/2002 -0700, George Secor wrote: >[dk:] > > So, in > > other words, we should have a way of interpreting a certain set of > > single-shaft symbols (about 13 of them) as specific offsets from >12-ET > > between about 2.5 and 60 cents (an alternative to writing plus or >minus > > cents next to the notes) while preserving their (preferably lowest >product > > complexity) comma meanings. > > > > Do you want to propose a set of symbols to do that? > >Yes, and here it is! > >I have limited this to multiples of 12 through 96 and rational notation >to the 13 limit. The commas are assigned the following values: > > 5 comma: 15 cents > 7 comma: 31 cents >11 diesis: 50 cents >13 diesis: 38 cents >|( comma: 14 cents > >The value of the 13 diesis is intended to approximate 8:13, 12:13, and >9:13. It is closest to making 12:13 exact and gives approximately >equal (but opposite) error to 8:13 and 9:13. > >The value of 14 cents for the |( comma is a practical value midway >between the sizes required for the two roles it plays in the 13 limit, >as indicated in the following table: > >For 12 through 96-ET For rational notation >--------------------- --------------------- > |( not used 16 cents as 7/5 comma > and 12 cents as 13/11 comma >/| = 13 to 20 cents 15 cents (5 comma) >(| not used 19 cents (11/7 comma) >//| not used 30 cents (5+5 comma) > |) = 25 to 33 cents 31 cents (7 comma) >(|( not used 35 cents as 11/5 comma > and 31 cents as 13/7 comma > |\ not used 35 cents as 11-5 comma >/|) = 38 to 43 cents 38 cents (13 diesis) >/|\ = 50 cents 50 cents (11 diesis) >(|\ = 57 to 63 cents 62 cents (13' diesis) >/|| not used 65 cents as 11-5 comma complement >~||( not used 65 cents as 11/5 comma complement > and 69 cents as 13/7 comma complement > ||) = 67 to 65 cents 69 cents (7 comma complement) >~|| not used 70 cents (5+5 comma complement) >)|~ not used 81 cents (11/7 comma complement) > ||\ = 80 to 88 cents 85 cents (5 comma complement) >/||) not used 84 cents as 7/5 comma complement > and 88 cents as 13/11 comma complement >/||\ = 100 cents 100 cents (apotome) > >This same information may be easier to digest if it is displayed >graphically: > >12: /||\ > 100 >24: /|\ /||\ > 50 100 >36: |) ||) /||\ > 33 67 100 >48: |) /|\ ||) /||\ > 25 50 75 100 >60: /| /|) (|\ ||\ /||\ > 20 40 60 80 100 >72: /| |) /|\ ||) ||\ /||\ > 17 33 50 67 83 100 >84: /| |) /|) (|\ ||) ||\ /||\ > 14 29 43 57 71 86 100 >96: /| |) /|) /|\ (|\ ||) ||\ /||\ > 13 25 38 50 63 75 88 100 > >Ratl: /| |) /|) /|\ (|\ ||) ||\ /||\ > 15 31 38 50 62 69 85 100 > //| |\ /|| ~|| > 30 35 65 70 > |( (| (|( ~||( )||~ /||) > 7/5 11/7 11/5 11/5 11/7 7/5 > 16 19 35 65 81 84 > 13/11 13/7 13/7 13/11 > 12 31 69 88 > >No attempt has been made to make the flag sizes add up so that /| + |) >= /|) exactly. In order to do that, the 7 comma must become half of >the 11 comma, or 25 cents (as long as the 11 comma and 11' comma are >equal). Also, the 5 comma becomes 14 cents and the 13 diesis 39 cents. > Then |( will be 11 cents as both the 7/5 and 13/11 comma, and (|( will >be 36 cents as both the 11/5 (or 11'-5) and 13/7 (or 13'-7) comma. >This is okay for the 5 comma and 13 diesis, but the value for the 7 >comma is about 6 cents too small to make 7/4 just, and the error for >7/6 and 9/7 is even greater; 7/5 likewise suffers. So there is not >much point in doing this. > >Does this look like it will work for what you had in mind? Yes. That looks very good. As far as it goes. There are obviously some big gaps, e.g. between 0 and 15 cents. We could use: Sym Approximate offset and Comma interpretation ------------------------------------------------ ~|( 3 cents as large 9:17, 3:17, 1:17 commas ~|~ 10 cents as 15:19, 5:19 commas /| 15 cents as 5:9, 3:5, 1:5, 1:15 commas * (| 18 cents as 7:11 comma or |( 18 cents as 5:7 and 7:15 commas big gap, nothing near 22 cents ~|) 26 cents as 17 comma + 7 comma )|) 29 cents as 7:19 comma |) 33 cents as 7:9, 3:7, 1:7 commas * |\ 35 cents as 11 comma - 5 comma or (|( 36 cents as 5:11, 11:15 comma /|) 39 cents as 9:13, 3:13, 1:13 dieses * big gap, nothing near 44 cents. Seems a bad idea to use //| for 47 cents as 5:13, 13:15 commas /|\ 49 cents as 9:11, 3:11, 1:11 dieses * (|~ 54 cents as 11:19 comma (|\ 61 cents as large 9:13, 3:13, 1:13 dieses * For the cent values, I've taken the mean of the commas listed (19 limit) and rounded to the nearest cent. The asterisks are the ones that agree with the first line of your "rational" notation. I don't think we can call it rational. It's really only for approximation, of any tuning as cent offsets from 12-ET. For example, low numbered ETs that are not multiples of 12 could be notated approximately, using these symbols to represent offsets from 12-ET. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5190 Date: Wed, 11 Sep 2002 11:15:04 Subject: Re: A common notation for JI and ETs From: David C Keenan At 01:47 PM 10/09/2002 -0700, George Secor wrote: >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4619] > > There are obviously some big gaps, e.g. between 0 and 15 cents. We >could use: > > > > Sym Approximate offset and Comma interpretation > > ------------------------------------------------ > > ~|( 3 cents as large 9:17, 3:17, 1:17 commas > > ~|~ 10 cents as 15:19, 5:19 commas > > /| 15 cents as 5:9, 3:5, 1:5, 1:15 commas * > > (| 18 cents as 7:11 comma > > or > > |( 18 cents as 5:7 and 7:15 commas > > big gap, nothing near 22 cents > > ~|) 26 cents as 17 comma + 7 comma > > )|) 29 cents as 7:19 comma > > |) 33 cents as 7:9, 3:7, 1:7 commas * > > |\ 35 cents as 11 comma - 5 comma > > or > > (|( 36 cents as 5:11, 11:15 comma > > /|) 39 cents as 9:13, 3:13, 1:13 dieses * > > big gap, nothing near 44 cents. Seems a bad idea to use > > //| for 47 cents as 5:13, 13:15 commas > > /|\ 49 cents as 9:11, 3:11, 1:11 dieses * > > (|~ 54 cents as 11:19 comma > > (|\ 61 cents as large 9:13, 3:13, 1:13 dieses * > > > > For the cent values, I've taken the mean of the commas listed (19 >limit) > > and rounded to the nearest cent. > >This looks pretty good, except for the gaps that you noted, and it is >in reasonable agreement with the latest 120 and 144 notation proposals. > There are two 132-ET proposals that both use (|~ for 5deg (45 cents), >which is in conflict with the above, but we could use /|~ for 5deg132 >instead. > >I would make /|\ 50 cents instead of 49 (rounding 49.363 up instead of >down); that way it's easier both to remember and to execute, so that, >for example, E/|\ would be the same as F\!/. Maybe we should round them all to the nearest 5 cents. Most are already within 1 cent. This has the side-effect of making the gap near 22 cents vanish. By the way, my apologies. Just when you start using the terminology such as "7/5 comma" that I suggested earlier, I decided that it was better to use the colon-based interval notation rather than slash-based pitch notation to refer to the commas.I figure we really are referring to intervals, not pitches. And I prefer to give them with no factors of 2 rather than in first octave form here, since then the two odd numbers involved can be read directly. What do you think? >I think we will have to use //| to fill the gap between /|) and /|\. >In looking for candidates for this position, I rejected (|( as the >17/11 and 19/11 commas, and then I came across //| as the 19/13 comma, >which seems to validate its use as the 13/5 and 15/13 commas. I don't understand "validate" here. In rational terms (i.e. relative to strict Pythagorean) the 13:19 comma (38:39) is 44.97 cents while the 1:25 comma (6400:6561) is 43.01 cents. That's a 1.96 cent schisma, far greater than any notational schisma we've accepted before. We can't use //| for the 13:19 comma anywhere. > There >are two problems with this: > >1) It's not intuitive relative to the /| comma. > >So what? The actual size of the //| comma is around 43 cents. We're >just not using it as the 5+5 comma. But we agreed we shouldn't use //| (at least for notating ETs) unless it _is_ the 5+5 comma. I don't see the lack of a 45 cent symbol in the 12-relative notation as a good enough reason to do something that is _so_ counterintuitive. >2) It's not compatible with the 108-ET notation: > >108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ > 11 22 33 44 56 67 78 89 100 > >We could eliminate the 108 incompatibility by doing 108 this way: > >108b: /| |( |) /|) (|\ ||) )||) ||\ /||\ > 11 22 33 44 56 67 78 89 100 > >This essentially writes off the ratios of 11 in this division. In a >previous message I said that I thought that it was not very good to do >something like this, but I don't see any other choice for what is >admittedly not a very good division (not even 9-limit consistent). > >However, after writing the above I found another way. The ~|\ symbol >could be interpreted as the 19'/11 (or 11'-19' comma, 297:304,~40.330 >cents) comma (the difference between 19/11 and 27/16), for which the >required alteration in 12-ET is ~46 cents. This would then allow us to >have //| available as the 5+5 comma and to keep the 108 notation as in >version a, above, if we wish. Again, interpreting ~|\ as an 11:19 comma would involve a schisma of more than 2 cents relative to (19'-19)+5 and 23+5. Unacceptable. >To fill the gap around 22 cents, I found ~|~ as the 23/17 (or 17+23) >comma, which is ~23.3 cents in 12-ET, but we have already used this as >the 19/5 comma. I then found that /|~ can approximate the 23/22 comma >(as 24057:23552, ~36.729 cents, the difference between the apotome and >23/22); this symbol represents almost exactly 23 cents in 12-ET Apart from the fact that I'd rather not to go beyond 19-limit unless I really have to; using /|~ as the 11:23 comma (24057:23552) of 36.73 cents is out of the question since as the (11-5)+17 comma (4352:4455) /|~ is 40.5 cents (all relative to strict Pythagorean), a schisma of more than 3 cents. Anyway, as I said above, the gap near 22 cents disappears if we round to nearest 5 cents. >I agree that "rational" is not the right term. How about "relational" or >"12-relational"? We could abbreviate this as "12-R" notation. I suggest "12-relative", which can of course also be abbreviated 12-R. > > For > > example, low numbered ETs that are not multiples of 12 could be >notated > > approximately, using these symbols to represent offsets from 12-ET. > >I can see the value of this for notating approximations of rational >intervals relative to 12-ET, but I am a bit skeptical about how well >this would work for notating ETs other than multiples of 12. It's as >if we're trying to shoehorn everything into a 12-ET framework to make >it more convenient (at least initially) for the performer. In the >process we end up with ET notations that are considerably more >complicated than those that we have already worked out, and I would >hate to see musicians become so dependent on relating microtonal >intervals to 12 that they would be unable to think of them in any other >way. And the worst case scenario would be refusal of performers to >change to the simpler, more precise ET and rational notations because >they had gotten accustomed to the 12-R notation and would not want to >change. > >Why don't we try to do something like 19 or 22-ET in 12-R notation and >see what it looks like before we go any farther with this? Probably no need to bother. I think I can see that it will be complete garbage. >I am >beginning to think that this is getting more complicated than I would >like it to be. Perhaps we should just keep it simple by restricting >the 12-R notation to the 13-limit symbols and let that serve as a >gentle introduction to the sagittal symbols. Musicians could then >learn how the same (now-familiar) symbols are used outside a 12-ET >framework. If players of flexible-pitch 12-ET instruments need some >sort of aid in remembering how many cents away from 12-ET the notes >are, then numbers can be placed above the notes in their parts. >(Hopefully this could eventually be done automatically with a >computer.) For players of fixed-pitch (i.e., retuned) or specially >built instruments, the symbols should suffice. Excellent idea. Although I still think we should give the full list of 5-cent-resolution 19-limit symbols somewhere, accompanied by some commentary to the effect that "We don't really want you to use the sagittal symbols in this way, but we suspected some of you would try anyway, because you haven't yet escaped your 12-equal dependence, so we at least wanted to make sure that it is standardised and agrees as much as possible with the rest of the system. We must warn you that if you get stuck in this cul de sac, you will be missing out on the full generality and precision of the sagittal notation. We'd probably prefer you used cents written near the noteheads." On rereading, this sounds rather patronising. But perhaps we can instead make it humorous when worked into the mythology your daughter is working on. After going thru your proposals above, I decided it was more important to have a 45 cent symbol other than //|, than a 55 cent symbol, so I reassigned (|~ for 45 cents based on its 23-limit interpretation (which you proposed above, as agreeing with your proposed 132-ET notation). But then I threw in the 31' comma symbol, with two flags on one side, for 55 cents in case anyone needs it. Here's my current 12-R proposal, where 12-R notation is for approximate notation of pitches relative to 12-equal in a manner consistent with the general sagittal notation. ~|( 5 cents as large 9:17, 3:17, 1:17 commas ~|~ 10 cents as 15:19, 5:19 commas /| 15 cents as 5:9, 3:5, 1:5, 1:15 commas |( 20 cents as 5:7 and 7:15 commas ~|) 25 cents as 17 comma + 7 comma )|) 30 cents as 7:19 comma |) 33 cents as 7:9, 3:7, 1:7 commas **** not rounded to nearest 5 **** (|( 35 cents as 5:11, 11:15 commas /|) 40 cents as 9:13, 3:13, 1:13 dieses (|~ 45 cents as 7:11' comma + 1:23 comma /|\ 50 cents as 9:11, 3:11, 1:11 dieses (/| 55 cents as 1:31' comma (|\ 60 cents as large 9:13, 3:13, 1:13 dieses Each symbol covers a +-2.5 cent range of pitches. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5196 Date: Wed, 11 Sep 2002 23:54:10 Subject: Re: [tuning] Re: Proposal: a high-order septimal schisma From: monz hi Gene, From: "genewardsmith" <genewardsmith@xxxx.xxx> To: <tuning@xxxxxxxxxxx.xxx> Sent: Wednesday, September 04, 2002 11:27 AM Subject: [tuning] Re: Proposal: a high-order septimal schisma > --- In tuning@y..., paul.hjelmstad@u... wrote: > > > > Thanks. Hope this doesn't sound stupid, but could you tell me the > > significance of each number in the "wedge invariant"? (Being really literal > > please) Are they the powers of 2,3,5,7 or something? > > It's hardly stupid, and in fact it's complicated enough I suggest further discussion should take place on tuning-math, and not here. There *are* commas of various kinds hidden in it, for which the numbers are exponents, and as we just saw, if the first number is "1" then 5 and 7 can be expressed in terms of 2 and 3; however it really comes from multilinear algebra, and is not such a good thing to discuss here. There's quite a lot in the tuning-math archives about it. > > Let p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4, then > > p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, > u1*v3-v1*u3,u1*v4-v1*u4] > > Let r be the mapping to primes of an equal temperament given > by r = [u1, u2, u3, u4], and s be given by [v1, v2, v3, v4]. This > means r has u1 notes to the octave, u2 notes in the approximation of 3, and so forth; hence [12, 19, 28, 24] would be the usual 12-equal, and [31, 49, 72, 87] the usual 31-et. The wedge now is > > r ^ s = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4, > u4*v2-u2*v4,u2*v3-v2*u3] > > Whether we've computed in terms of commas or ets, the wedge product of the linear temperament is exactly the same, up to sign. > > If the wedgie is [u1,u2,u3,u4,u5,u6] then we have commas given by > > 2^u6 3^(-u2) 5^u1 > 2^u5 3^u3 7^(-u1) > 2^u4 5^(-u3) 7^u2 > 3^u4 5^u5 7^u6 at last, i finally understand how you're calculating wedgies! but that last bit has me a little confused. from the example meantone wedgie [1,4,10,12,-13,4] : > From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, September 09, 2002 10:50 PM > Subject: [tuning-math] [tuning] Re: Proposal: a high-order septimal schisma > > <snip> > > Wedge of two intervals: > > > p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, > > > u1*v3-v1*u3,u1*v4-v1*u4] > > For example p = 126/125 and q=81/80, then p = 2^1 3^2 5^(-3) 7^1, > so in vector form it is [1,2,-3,1]. Similarly, > q=2^(-4) 3^4 5^(-1) 7^0, which in vector form is [-4,4,-1,0]. > Wedging the two gives the wedgie for meantone, but 126/125 ^ 225/224, for example, will work also. i calculated these commas [ 4 -4 1] = 80 / 81 [-13 10 -1] = 59049 / 57344 [ 12 -10 4] = 9834496 / 9765625 [ 12 -13 4] = 1275989841 / 1220703125 OK, so the syntonic comma (81/80) is there ... but what happened to
Message: 5197 Date: Wed, 11 Sep 2002 23:59:19 Subject: commas from wedgies (was: Proposal: a high-order septimal schisma) From: monz (sorry ... the previous version of this post got away from me too soon; ignore it.) hi Gene, > From: "genewardsmith" <genewardsmith@xxxx.xxx> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Wednesday, September 04, 2002 11:27 AM > Subject: [tuning] Re: Proposal: a high-order septimal schisma > > > <snip> > > Let p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4, then > > p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, > u1*v3-v1*u3,u1*v4-v1*u4] > > Let r be the mapping to primes of an equal temperament given > by r = [u1, u2, u3, u4], and s be given by [v1, v2, v3, v4]. This > means r has u1 notes to the octave, u2 notes in the approximation > of 3, and so forth; hence [12, 19, 28, 24] would be the usual > 12-equal, and [31, 49, 72, 87] the usual 31-et. The wedge now is > > r ^ s = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4, > u4*v2-u2*v4,u2*v3-v2*u3] > > Whether we've computed in terms of commas or ets, the wedge product > of the linear temperament is exactly the same, up to sign. > > If the wedgie is [u1,u2,u3,u4,u5,u6] then we have commas given by > > 2^u6 3^(-u2) 5^u1 > 2^u5 3^u3 7^(-u1) > 2^u4 5^(-u3) 7^u2 > 3^u4 5^u5 7^u6 at last, i finally understand how you're calculating wedgies! but that last bit has me a little confused. from the example meantone wedgie [1,4,10,12,-13,4] which you gave here: > From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, September 09, 2002 10:50 PM > Subject: [tuning-math] [tuning] Re: Proposal: a high-order septimal schisma > > <snip> > > Wedge of two intervals: > > > > p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, > > > u1*v3-v1*u3,u1*v4-v1*u4] > > For example p = 126/125 and q=81/80, then p = 2^1 3^2 5^(-3) 7^1, > so in vector form it is [1,2,-3,1]. Similarly, > q=2^(-4) 3^4 5^(-1) 7^0, which in vector form is [-4,4,-1,0]. > Wedging the two gives the wedgie for meantone, but 126/125 ^ 225/224, > for example, will work also. i calculated these commas [ 4 -4 1] = 80 / 81 [-13 10 -1] = 59049 / 57344 [ 12 -10 4] = 9834496 / 9765625 [ 12 -13 4] = 1275989841 / 1220703125 OK, so the syntonic comma (81/80) is there ... but what happened to 126/125 and 225/224? why are they not in this list, and why are the other ones there? -monz "all roads lead to n^0"
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