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Message: 3175 - Contents - Hide Contents Date: Sat, 12 Jan 2002 04:18:04 Subject: Re: More proposed definitions From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> From: unidala <JGill99@i...> >> To: <tuning-math@y...> >> Sent: Friday, January 11, 2002 8:05 PM >> Subject: [tuning-math] Re: More proposed definitions >> >> >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>> I'd like to hear from some others besides Gene and Paul >>> ... should these math terms go directly into the >>> Tuning Dictionary, or should they live "off-campus"? >>> >>> >>> -monz >> >>>> J Gill: While it might not be the easiest task to implement, >> how about presenting it as non-esoterically as possible in >> the main definition, with links (much as Monz allready does) >> within that text which (heirarchically) enter the realms of >> complexity (deeper and deeper) if the reader is so inclined? >> >> That way the information is pre-compiled in levels of detail. > >> Hey J, thanks for your input ... and I think, thanks to your > suggestions, that I've already hit on the right way to do this: > simply include Gene's defintions "as is" as individual entries > in the Dictionary, but then have a "see also" link at the bottom > of each, which leads to an elementary tutorial webpage which > explains this brand of tuning math. > > Feedback?I don't think you'd want to do this for "pitch", "interval", etc., though . . .
Message: 3176 - Contents - Hide Contents Date: Sat, 12 Jan 2002 13:48:51 Subject: The International Linear Algebra Society From: monz I thought some folks here might appreciate this: The International Linear Algebra Society ILAS - The International Linear Algebra Society * [with cont.] (Wayb.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3178 - Contents - Hide Contents Date: Sat, 12 Jan 2002 13:54:17 Subject: Linear Algebra Toolkit From: monz Linear Algebra Toolkit Linear Algebra Toolkit * [with cont.] (Wayb.) This looks really useful. Gene, can you please show me by example how to use it? How about if we examine the Schoenberg unison-vector matrix I posted yesterday? 2 3 5 7 11 unison vectors ~cents [ 11 -4 -2 0 0] = 2048:2025 19.55256881 [ -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 (without powers of 2) = [-1 0 0 2] [-4 0 0 -4] 1 [ 2 0 -12 -4] * -- [ 1 12 0 -2] 12 -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3179 - Contents - Hide Contents Date: Sat, 12 Jan 2002 21:54:53 Subject: Wedge product definitions from mathworld From: genewardsmith Here is exterior algebra: http://mathworld.wolfram.com/ExteriorAlgebra.html * [with cont.] Here is wedge product: http://mathworld.wolfram.com/WedgeProduct.html * [with cont.] This could and should be made a lot simpler for musical purposes.
Message: 3180 - Contents - Hide Contents Date: Sat, 12 Jan 2002 06:06:04 Subject: Re: More proposed definitions From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> For those of you who are more familiar with algebra (and > specifically, multilinear algebra) than I am, I guess what > I'm really driving at with this is: are these terms mainly > quite personal to Gene and his work, or are they more > widely accepted among other mathematically-savvy tuning > theorists?The list I recently gave of other things which might go into a tuning dictionary was all standard mathematical terminology. Some else I would like to have available is that terminology adapted to the particular uses of it I've found in music--terms like "val" and "wedgie" are of that latter type. However, I see no difference in principle between this and terms such as "unique" and "consistent" which people have been coming up with. I've also comeup with rather musically specialized definitions, such as for "notation". Some of them are proposals, put out there for consideration. I'm in effect asking "what do you think of this as a mathematically precise definition ofscale?" or "what do you think of using 'tone group' in this way?" If the former, then are there terms used by> others which are equivalent to Gene's, and therefore which > also need to be included in the Dictionary as synonyms?If there are synonyms, I'd like to know.> This is really important to me, because I get a sense (and > an eyewitness account from Paul) that the last three months > or so on the tuning-math list (i.e., since Gene's arrival) > have borne some of the most comprehensive and most > widely-applicable concepts and algorithms in the history > of tuning theory, and it's high time that me and the rest > of us join the party.I'm not an authority on the history of tuning-theory, but I would presume this has been more evolutionary than revolutionary. It is clear there are certain (so far, not particularly difficult ones, so things could be much worse!) mathematical concepts which it would be very good for people to know who work in this area. Just as you would be severely limited not to know anything about logarithms or determinants, a little knowledge of groups and multilinear algebra would go a long way.
Message: 3181 - Contents - Hide Contents Date: Sat, 12 Jan 2002 22:07:22 Subject: Re: Linear Algebra Toolkit From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> 2 3 5 7 11 unison vectors ~cents > > [ 11 -4 -2 0 0] = 2048:2025 19.55256881 > [ -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 21.5062896I went to the "finding the null space" module, and told it to give me four rows and five columns, and then inputted the above. It tells me the solution is [12/41] [19/41] [28/41] [34/41] [1 ] Multiplying through by 41, I find that the corresponding val is [12] [19] [28] [34] [41]
Message: 3184 - Contents - Hide Contents Date: Sat, 12 Jan 2002 08:04:41 Subject: Re: More proposed definitions From: genewardsmith --- In tuning-math@y..., "jonszanto" <jonszanto@y...> wrote:> That said, it's value will be greatly enhanced by the degree which is > serves it's main purpose: to be an online resource for the > definitions of *tuning* terminology.What about tuning terminology of a mathematical nature?
Message: 3186 - Contents - Hide Contents Date: Sat, 12 Jan 2002 01:30:32 Subject: dict/genemath.htm From: monz Here's my opening salvo in the war against tuning-math ignorance and illiteracy: Definitions of tuning terms: Gene's tuning mat... * [with cont.] (Wayb.) I left the post headers in to make it easy to track revisions, such as the one for "scale". Let me know what parts should be excised, expanded, etc. There's no link to it from anywhere yet. I'll wait for word from Gene or the consensus here as to when to announce it publicly on the big list. If right now is the right time, say so; if not, I have lots of patience. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3188 - Contents - Hide Contents Date: Sat, 12 Jan 2002 09:47:08 Subject: Re: dict/genemath.htm From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> I left the post headers in to make it easy to track > revisions, such as the one for "scale". Let me know > what parts should be excised, expanded, etc.One general rule is that when an entry has been corrected, you should always use the corrected version; so for instance, my revised definition of "scale" is the one I would want. Also, the definitions of algebraic number etc.really have nothing to do with what I'm up to; I put them out because I thought you wanted them for your regular tuning dictionary. Finally, do you really want it to be just a Gene Smith dictionary? I just spent some time trying to discover what Lumma Stability was, and failing. That's the sort of thing I like to find in your dictionary, and I don't really care if it is or isn't in general use, so long as it is in use in these precincts--and it is! I would suggest, if you want to divide things into a general usage dictionary and one of specialist terms, including ones local to this community (so far, at least) that could be done.
Message: 3189 - Contents - Hide Contents Date: Sat, 12 Jan 2002 06:01:04 Subject: Re: For Joe--proposed definitions From: Pierre Lamothe I think it would be good not only to separate math definitions but also to avoid using superficial blending of tuning notions with math concepts in terms like Tone group and Rational tone group. Gene wrote: Tone group A set of positive real numbers closed under multiplication and inversion (so that if x is in the set, so is 1/x), and regarded as a set of intervals or pitches. How could you regard an infinite group as a set of intervals or pitches? Ifa such regard implies the restriction to the audible pitches, or the intervals between distinct audible pitches, this mathematical object loose its group structure, unless to be atempered monogenic group. It is important to understand, from the modelizing viewpoint, that any group generated by more than one rational number, say 2 and 3, would have an infinite number of tones in the octave. The group structure may not modelize the operative structureon a finite set of rational intervals. I consider this group as a simple mathematical object, useful as tool but inadequate as model. That don't prevent to plunge a finite set of tones in this group when needed but to extend the mathematical properties of the group structure as it would represent the proper operative structure on the given tones. In my opinion, any attempt to understand a categorical perception of the tones in a musical context without reference to an underlying proper invariant operative structure, confined to illusion. What distinguishes a perception from a sensation, or an impression, is that the conscious part is significative (has a musical sense for the auditor) while the automated ability part to recognize a significative intention was learnedmainly from the habitual physionomy of the transformations. An isolated experience focusing on sensation is a regressive perception and attempts to reconstruct an authentic musical experience with such isolated acoustic impressions appears to me well superficial. Gene wrote: Canonical val For any positive integer n, the canonical val hn is the val such that hn(p) = round(n * log2(p)), where p is prime and where "round" means round to the nearest integer. The restriction of hn to a particular rational tone group is also denoted hn. It is important to understand that Canonical val don't mean unique pertinent val in musical sense. For instance, the canonical val h5(3) = 8 h5(5) = 12 corresponds to the chinese system (gammier 1) 1 4 2 4 2 0 3 1 . . 3 1 4 while the equally valid japanese system (gammier 8) . 3 1 4 4 2 0 3 1 . 1 4 2 corresponds to the non canonical variant h5(5) = 11 -------------------------------------------------------------------------------- Some remarks now about the proposed definition for the tuning notion of scale. Gene wrote first: Scale A discrete set of real numbers, containing 1, and such that the distance between sucessive elements of the scale is bounded both below and above by positive real numbers. The least upper bound of the intervals between successive elements of the scale is the maximum scale step, and the greatest lower bound is the minimum scale step. The element of the scale obtained by counting up n scale steps is the nth degree, by counting down is the –nth degree; 1 is the 0th degree. and corrected with: Scale A discrete set of real numbers, containing 0, and regarded as defining tones in a logarithmic measure, such as cents or octaves, and such that the distance between sucessive elements of the scale is bounded both below and above by positive real numbers. The least upper bound of the intervals between successive elements of the scale is the maximum scale step, and the greatest lower bound is the minimum scale step. The element of the scale obtained by counting up n scale steps is the nth degree, by counting down is the –nth degree; 0 is the 0th degree. The set of positive real numbers which are the tones so represented is also regarded as the scale. Gene, is there the norm of elegance you had in head when you criticized my work at Christmas day? I can understand why Manuel said there is no need for a mathematical scale definition. However I would say that at first level, in the reduced minimal sense he uses, the scale has essentially a mathematical definition as discrete subset on an ordered set Besides, I can understand the procrastination of Monz to attack this basic definition. Intuitive as he is, I would believe that he feels its inherent difficulties. The next level implies the interval notion which requires much more than anordered set as starting structure. If a - b - c - d - ... is a such discrete ordered set, the successive steps, we would denote a:b - b:c - c:d - ... have sense only if there exist a composition law on that set permitting to define also an order on the intervals. So with the < relation (resp. > or =), depending of the law type (multiplicative or additive), we have a:b < c:d <=> bc < ad a:b < c:d <=> b+c < a+d It's there a first element that is missing. I don't have intention to explain all what is implied in a such definition, so Gene would have only to add his name. I leave him to formulate adequate proposition, reserving my comments. I would signal only an error for the moment: changing the neutral element from 1 to 0 means implicitely using now the additive composition law, so the term positive has no more sense. Pierre [This message contained attachments]
Message: 3190 - Contents - Hide Contents Date: Sat, 12 Jan 2002 03:19:52 Subject: Re: dict/genemath.htm From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, January 12, 2002 1:47 AM > Subject: [tuning-math] Re: dict/genemath.htm > > > One general rule is that when an entry has been corrected, > you should always use the corrected version; so for instance, > my revised definition of "scale" is the one I would want.OK, thanks ... I wasn't sure if the second definition of "scale" was supposed to be a correction or a variant. I've removed the original definition and the parenthetical comment, and left only the corrected version. Definitions of tuning terms: Gene's tuning mat... * [with cont.] (Wayb.)> Also, the definitions of algebraic number etc. really have > nothing to do with what I'm up to; I put them out because > I thought you wanted them for your regular tuning dictionary.Sure, I want *anything* that has anything to do with tuning. There's already a definition of "algebraic number" in there, by Paul: Definitions of tuning terms: algebraic number,... * [with cont.] (Wayb.)> Finally, do you really want it to be just a Gene Smith > dictionary?No, of course not. I just thought of that as a quick-and-dirty way to slap a name on it. Anything and everything in that webpage is open to all criticism and subject to change, right down to the title.> I just spent some time trying to discover what Lumma Stability > was, and failing. That's the sort of thing I like to find in > your dictionary, and I don't really care if it is or isn't in > general use, so long as it is in use in these precincts--and it is!"Lumma Stability" is *exactly* the kind of thing this Dictionary needs in it, because this is the first place many people will go to find out what it is! Carl, how about it?> I would suggest, if you want to divide things into a general > usage dictionary and one of specialist terms, including ones > local to this community (so far, at least) that could be done.Well, that's the idea that I had for the time being ... to just put all these highly technical definitions on a single page. The primary reason *I* wanted them segregated from the rest of the Dictionary is because *I* don't understand them yet!! Once I grok them, the plan is to flesh out each definition with lots of graphics, tables, etc., to make them easier for the novice to understand -- then, they'll be split up into separate definitions. For now, I think it's better to leave all the related terminology together on one page, and to keep adding amendments, criticisms, etc., as needed until a consensus is reached that something is ready for public airing and announcement on the big list. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3191 - Contents - Hide Contents Date: Sat, 12 Jan 2002 11:28 +0 Subject: Re: More proposed definitions From: graham@xxxxxxxxxx.xx.xx monz wrote:> Well ... right, Paul, this is exactly what's on my mind. > Terms like "pitch" and "interval", which are common currency > among all musicians, certainly belong, but most of Gene's > terms obviously lean much more towards pure mathematics.But those terms should have their common definitions, not Gene's mathematically precise ones. If Gene wants to express musical concepts in mathematical terms that's fine. But it shouldn't be in a general tuning dictionary. If Gene wants space to write his own dictionary, he can have it at microtonal.co.uk. If Joe wants to include these definitions, I suggest he puts a link at the bottom of the general definition saying something like "see Gene Ward Smith's mathematical definition".> For those of you who are more familiar with algebra (and > specifically, multilinear algebra) than I am, I guess what > I'm really driving at with this is: are these terms mainly > quite personal to Gene and his work, or are they more > widely accepted among other mathematically-savvy tuning > theorists? If the former, then are there terms used by > others which are equivalent to Gene's, and therefore which > also need to be included in the Dictionary as synonyms?You don't need to worry on that account. The problem I see is that your dictionary would turn into a dictionary of multilinear algebra, instead of tuning theory. If that's what you want, I have some comments: Please make the two separate. I know from experience that people are bad at filtering out mathematical detail they don't need to know. If they encounter such detail in you dictionary, they might give up on the whole thing. Be prepared for the mathematical dictionary to become the most popular thing on your site. My page introducing matrices (without mathematical precision, BTW) has this status. Make sure you aren't duplicating effort. If a mathematical dictionary already exists, aimed at the right audience, contribute and link to that instead.> This is really important to me, because I get a sense (and > an eyewitness account from Paul) that the last three months > or so on the tuning-math list (i.e., since Gene's arrival) > have borne some of the most comprehensive and most > widely-applicable concepts and algorithms in the history > of tuning theory, and it's high time that me and the rest > of us join the party.A simple (not mathematically precise) introduction to groups and rings may be useful. Also something about wedge products, because they aren't that prominent in linear algebra (certainly don't come up in web searches) but aren't that difficult to understand and do simplify some of what we've been talking about. Really, Gene seems to be aiming at either creating a new branch of mathematics relating to tuning theory, or defining aspects of tuning theory in such a way that they become isomorphic to an existing branch of mathematics. Either way, it's quite exciting. But explaining it all, with full mathematical precision, to a reader without a mathematical background is going to be a huge undertaking. I suggest two new dictionaries -- one containing mathematically precise definitions for tuning theory, either of new or recycled terms, with links to and from Monzo's main dictionary. And another that gives non-mathematicians enough of an idea what the mathematical terms mean that they can follow what's going on. Graham
Message: 3192 - Contents - Hide Contents Date: Sat, 12 Jan 2002 11:28 +0 Subject: Re: updated "positive" and "negative" definitions From: graham@xxxxxxxxxx.xx.xx monz wrote:> But I'm confused about one thing: on the "negative" page, > I have 53- and 65-EDO listed as negative temperaments, > and they do indeed have negatively-tempered "5ths", but > they both have p = +1. ?????Hasn't this been answered? It seems to be the first question coming round again. 53-EDO has fifths larger than 700 cents, but smaller than a 3:2 ratio. One use for the terminology that I haven't seen mentioned is that it tells you whether pitch will rise or fall as you go up one column of the Bosanquet keyboard. The 12-centricity is because Bosanquet's keyboard did have 12 columns. Wilson's layout can have 7 or 5 columns if you hold it the right way up, hence he generalized the terminology to be 5- or 7-centric as well. I have other stuff about this at <Notating and keyboarding musical scales * [with cont.] (Wayb.)>. That's somewhat out of date, but mostly holds. The & notation, for which I'm using an & again, is the most successful. Graham
Message: 3193 - Contents - Hide Contents Date: Sat, 12 Jan 2002 03:40:54 Subject: Re: For Joe--proposed definitions From: monz Hello Pierre, and thanks for the detailed comments on Gene's definitions. I will leave it to the more mathematically knowledgeable to argue on the differences ... I don't really understand them yet. But I have a few comments below ...> From: Pierre Lamothe <plamothe@xxx.xx> > To: Tuning-math <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, January 12, 2002 3:01 AM > Subject: [tuning-math] Re: For Joe--proposed definitions > > > Besides, I can understand the procrastination of Monz > to attack this basic definition. Intuitive as he is, > I would believe that he feels its inherent difficulties.I wouldn't characterize what I said as an "attack", just an alternative perspective. My main problem is not really with Gene's definition -- it's with the fact that *I* have not yet written a definition for this all-important tuning word, one that would have lots of meaning for musicians. Certainly, composers and improvisers are often guided by mathematical properties of scales, as when they make "punning" games with the harmonies. But my feeling is that when musicians are concerned mainly with *playing* music, they're guided more by the visceral sensory perception of the sound itself than by the numerical aspects of their pitch-ordering and other relationships. The fact that it's hard to talk about a musical scale without invoking the concept of a mathematical scale is what's caused me to hesitate so long in writing a definition, and I think *that* kind of definition really should be in place alongside Gene's. In simple terms, the mathematical defintion is great, but it's not enough.> I don't have intention to explain all what is implied > in a such definition, so Gene would have only to add his > name. I leave him to formulate adequate proposition, > reserving my comments.Pierre, you are welcome to contribute anything you would like to the Tuning Dictionary, and whatever you submit will be credited with your name. BTW, I've put your definitions of "chordoid" and "gammier" into the Dictionary exactly as you posted them. Internet Express - Error 404 * [with cont.] (Wayb.) Definitions of tuning terms: Pierre Lamothe cr... * [with cont.] (Wayb.) (They're the same, just linked under different names.) But I would like to give those pages the same look as all my other webpages. Do you have any objection? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3194 - Contents - Hide Contents Date: Sat, 12 Jan 2002 03:43:25 Subject: Re: updated "positive" and "negative" definitions From: monz> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, January 12, 2002 3:28 AM > Subject: [tuning-math] Re: updated "positive" and "negative" definitions > > > monz wrote: >>> But I'm confused about one thing: on the "negative" page, >> I have 53- and 65-EDO listed as negative temperaments, >> and they do indeed have negatively-tempered "5ths", but >> they both have p = +1. ????? >> Hasn't this been answered? It seems to be the first question coming round > again. 53-EDO has fifths larger than 700 cents, but smaller than a 3:2 > ratio.Duh! Thanks, Graham. That's pretty obvious, isn't it? So I was really confused over this 12-EDO/Pythagorean question.> One use for the terminology that I haven't seen mentioned is that it tells > you whether pitch will rise or fall as you go up one column of the > Bosanquet keyboard. The 12-centricity is because Bosanquet's keyboard did > have 12 columns. Wilson's layout can have 7 or 5 columns if you hold it > the right way up, hence he generalized the terminology to be 5- or > 7-centric as well. > > I have other stuff about this at > <Notating and keyboarding musical scales * [with cont.] (Wayb.)>. That's somewhat out of date, > but mostly holds. The & notation, for which I'm using an & again, is the > most successful. Cool, thanks! -monz _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3195 - Contents - Hide Contents Date: Sat, 12 Jan 2002 06:55:52 Subject: Re: For Joe--proposed definitions From: Pierre Lamothe Hello Monz, But I would like to give those pages the same look as all my other webpages. Do you have any objection? For sure, I have no objection. What is important is the sense. Besides, I would prefer also you would correct eventually my English. Pierre [This message contained attachments]
Message: 3196 - Contents - Hide Contents Date: Sat, 12 Jan 2002 05:09:44 Subject: Re: badly tuned remote overtones From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, January 11, 2002 1:32 PM > Subject: [tuning-math] Re: badly tuned remote overtones > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> How does this compare with the other 12-tone periodicity-block >> you calculated for Schoenberg? Can you please give a listing >> of the pitches inside *this* PB? >> The set of pitches inside any non-torsional 12-tone periodicity block > with all the unison vectors tempered out is simply 12-tET.Right ... but I'm looking for the basic rational implications in the basic periodicity-block, so how does one find this? By intentionally leaving out one or more unison-vectors? Confused, -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3197 - Contents - Hide Contents Date: Sat, 12 Jan 2002 07:29:41 Subject: Re: For Joe--proposed definitions From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, January 12, 2002 3:40 AM > Subject: Re: [tuning-math] Re: For Joe--proposed definitions > > > BTW, I've put your definitions of "chordoid" and "gammier" > into the Dictionary exactly as you posted them. > Internet Express - Error 404 * [with cont.] (Wayb.) > Definitions of tuning terms: Pierre Lamothe cr... * [with cont.] (Wayb.) > > (They're the same, just linked under different names.)I decided that rather than separate these into two separate definitions, they should stay together as Pierre posted them. So the index for both terms points to "gammier.htm"; "chordoid.htm" is now a dead link. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3198 - Contents - Hide Contents Date: Sat, 12 Jan 2002 07:32:13 Subject: graphics added to "consistency" and "unique" definitions From: monz Hey Paul, I plotted the data for ETs up to 72, from Consistency limits of equal temperaments * [with cont.] (Wayb.) in my Dictionary definitions for "consistency" and "unique": Definitions of tuning terms: consistent, (c) 1... * [with cont.] (Wayb.) Definitions of tuning terms: unique, (c) 1998 ... * [with cont.] (Wayb.) Have I interpreted the data correctly? Are these graphs useful? I think they are very much so. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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