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Message: 1375 - Contents - Hide Contents Date: Wed, 22 Aug 2001 19:43:47 Subject: Re: Interpreting Graham's matrix From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> Over to Paul: is this a general method for finding a periodicity block?This is how I've been doing it all along -- see part 3 of the _Gentle Introduction_.
Message: 1376 - Contents - Hide Contents Date: Wed, 22 Aug 2001 19:51:02 Subject: Re: Generators and unison vectors From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:>>> Here, the octave and twelfth are taking the place of the chromatic >> unison>>> vectors. So, can we call them unison vectors? >>>> A unison vector has to come out to a musical _unison_! An augmented >> unison maybe, but still a unison. >> Fokker says, in "Unison Vectors and Periodicity Blocks" > <A.D. Fokker: Unison Vectors and Periodicity Bl... * [with cont.] (Wayb.)> "All pairs of notes > differing by octaves only are considered unisons." That's not far from > "an octave is a unison vector".The octave, OK, but the twelfth?> > You can't define the 5:4 approximation by adding octaves and twelfths.Why not? We've been doing so all along. The 5:4 approximation is four twelfths minus six octaves.> > Four fifths add up to a 5:1 approximation. But you can't define a > diatonic scale by adding fifths and octaves.Why not? We've been doing so all along, right?>>> An alternative to "unison vector" would be "melodic generator". >>>> Ouch! Are you sure? The steps in the scale can't be unison vectors, >> nor can the generator of the scale be a unison vector. >> I don't know what to call them.Something else, I beseech you!> > Probably it'd be better written as > > v v v v v v v v v v v v p > > where v is a fifth and p is a Pythagorean comma. That means p is the > (chromatic) unison vector. That leaves v as "the thing you have left when > you take out all the unison vectors". That's what we need a name for. The generator? >>>> Defining an MOS in octave-equivalent terms is easy, once you >> realize that>>> the generators that pop out needn't be unison vectors in octave- >> specific >>> space. >>>> They can't be unison vectors in any space! If the generator is a >> unison vector, you never get beyond the first note of the scale and >> its chromatic alterations. >> But the generator does have a close relationship to the chromatic unison > vector, in an octave equivalent system. You're taking out everything > except the chromatic unison vector, and describing what you have left in > terms of the generator.Great! That's a valuable insight! But that doesn't make the generator a unison! Look, the unison vectors together define N equivalence classes in the infinite lattice. All that unison vectors can do is to move you around within a single equivalence class. You'll never get to the other N-1 equivalence classes using unison vectors.
Message: 1377 - Contents - Hide Contents Date: Wed, 22 Aug 2001 19:57:32 Subject: Re: Chromatic = commatic? From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> I don't think MOS is any more clearly defined than WF in this respect. So > let's assume they are identical. In which case for the things my program > spits out to be MOSes, and therefore your hypothesis to be true, they must > be the same. Or, at least, we need to redefine both to use the "period" > instead of "interval of equivalence".I must be on Saturn today. None of this makes any sense to me.> The octave-equivalent algebra doesn't distinguish the different repeating > blocks.Sure it does. The symmetrical decatonic scale (LssssLssss) has 10 notes -- two repeating blocks per octave.> The linear temperament result tells you the number of > times the period goes into the interval of equivalence (octave)Exactly! So the WF definition, where the period _is_ the interval of equivalence, won't do.> and the > mapping in terms of generators. Again, you need the octave- specific > algebra, or the metric, to get the mapping by generators *and* periods. > Effectively, the octave-equivalent algebra collapses so that the interval > of equivalence becomes the period in any given context.What do you mean? For the symmetrical decatonic, the interval of equivalence remains _two_ periods whether you're looking octave- equivalently or not.
Message: 1378 - Contents - Hide Contents Date: Wed, 22 Aug 2001 20:01:44 Subject: Re: Chromatic = commatic? From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> This is because you're expecting to get an octave-specific result. The > octave-equivalent equivalent of an equal temperament is an MOS.In what way? What do you mean by that? That certainly doesn't seem to be true.> (Or > family of MOS scales, there's nothing special about the particular numbers > of generators that give two interval sizes.)As opposed to the ones that don't??
Message: 1380 - Contents - Hide Contents Date: Wed, 22 Aug 2001 21:27:19 Subject: Re: A little research...will elaborate further when time allows From: genewardsmith@xxxx.xxx --- In tuning-math@y..., BobWendell@t... wrote: This isn't what you asked for, but 311 represents all primes up through 41 with an error of less than a cent; it's a remarkable et in some ways. Your question could be answered easily with a brute-force search, however; and I might do it if you define P3 and M5 for me.
Message: 1381 - Contents - Hide Contents Date: Wed, 22 Aug 2001 22:17:57 Subject: Re: A little research...will elaborate further when time allows From: Carl> This isn't what you asked for, but 311 represents all primes up > through 41 with an error of less than a cent; it's a remarkable et > in some ways. Your question could be answered easily with a brute- > force search, however; and I might do it if you define P3 and M5 > for me.As I'm sure others will be quick to point out, if one intends to use anything more than dyadic harmony, he must also check the errors of these larger chords in the given et. This gave rise to the notion of consistentcy. See: Music (and Music Theory) * [with cont.] (Wayb.) Also, aside from looking at max errors, Paul Erlich has suggested considering an et as an an approximation. See: Question What would be most helpful in music t... * [with cont.] (Wayb.) As far as replacing cents with the units of some et, it isn't likely to happen. Ellis' cents enjoy wide use, provide good accuracy regardless of wether the target interval is just, make the octave highly divisable without fractional results, and provide a handy reference to 12-tone equal temperament. -Carl
Message: 1382 - Contents - Hide Contents Date: Wed, 22 Aug 2001 22:21:21 Subject: Re: A little research...will elaborate further when time allows From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:> I have long been interested in coming up > with an EDO that would meet the following requirements: > > 1) P5s and M3s less than 3 cents from JIWhat about m3s and M6ths?> > 2) all primes through 19 less than 5 cents from JIWhy just primes?> 3) The smallest number of scale degrees possible while meetingIf you insist on _consistency_ through the 19-limit (which I would), and maximum error less than 5 cents in the 19-limit, then 121-tET is for you. It has maximum errors of 2.18 cents in the 3-limit 2.18 cents in the 5-limit 3.07 cents in the 7-limit 4.35 cents in the 9-limit 4.35 cents in the 11-limit 4.35 cents in the 13-limit 4.35 cents in the 15-limit 4.35 cents in the 17-limit 4.35 cents in the 19-limit 111-tET would probably satisfy you, though, since its maximum errors are 0.75 cents in the 3-limit 2.88 cents in the 5-limit 4.15 cents in the 7-limit 4.15 cents in the 9-limit 4.15 cents in the 11-limit 4.15 cents in the 13-limit 4.15 cents in the 15-limit 4.15 cents in the 17-limit 5.19 cents in the 19-limit> > The purpose is to provide myself and anyone else interested with a > simple, quick and but rather clean tool for dealing conceptually with > JI and tuning systems in general.These systems (121 and 111) will not deal with adaptive JI very well.> 1200 is a huge number. The goal is > not to find a scale division so low as to be terribly practical > physically, but to serve as a finite analytical tool that reflects > the realities of intonation as accurately as possible with the least > encumberance, requiring only simple arithmetical tools for its > practical implementation in anaylzing or composing. See above. > > I just finished some research that took as a conceptual starting > point 12-EDO for the P5 as derived from the 7/12 lowest rational > approximation of a JI P5 in terms of its pitchwise portion of the > octave. Using the same principle of rational approximation of JI > pitch ratios (NOT their frequency ratios), I found the lowest > rational approximation of P5 to M3 (exact pitch ratio = 1.8171) to be > 9/5. 9 X 12 = 108-EDO. I'm sure this must be a well-known EDO > division, since it has so many compelling characteristics, but I > don't know about it from any source other than my own research of the > last two days.It isn't. As for multiples of 12, 72-tET is superior in all respects. I'm not sure what your derivation seeks to accomplish.> > It does satisfy the above conditions, except for #3But it's not even consistent beyond the 7-limit. if someone finds> a lesser division that meets conditions 1 & 2. The greatest error is > 5.0 cents for 17/16. The P5 is -2.0 cents as in 12-EDO, and the M3 is > +2.6 cents. The lowest deviation is the 16/15 M2 at -0.6 cents. The > M2+ (8/7) is 2.2 cents, so the greatest error for 7-limit is 2.6 > cents for the M3.108-tET has a maximum error of 4.53 cents in the 5-limit and 4.73 cents in the 7-limit.
Message: 1383 - Contents - Hide Contents Date: Wed, 22 Aug 2001 22:24:40 Subject: Re: A little research...will elaborate further when time allows From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., BobWendell@t... wrote: > > This isn't what you asked for, but 311 represents all primes up > through 41 with an error of less than a centNot only that, it's actually _consistent_ through the 41-limit. We noticed this amazing fact a few years ago -- it's a huge spike in the consistency-limit as a function of increasing ET cardinality. Gene, you still haven't brought up anything about the zetafunction. Could it help explain why 311 is so special?
Message: 1384 - Contents - Hide Contents Date: Wed, 22 Aug 2001 23:38:06 Subject: Re: A little research...will elaborate further when time allows From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Not only that, it's actually _consistent_ through the 41-limit.That's right--this is what really made my eyes open wide when I noticed it some time ago. We> noticed this amazing fact a few years ago -- it's a huge spike in the > consistency-limit as a function of increasing ET cardinality. Gene, > you still haven't brought up anything about the zetafunction. Could > it help explain why 311 is so special?It can express it, but it can't really explain it any better than the charts I've been looking at by you and Paul Hahn (which was like meeting old friends!) I promise to explain it sometime, but in fact there are lots of things still locked away in my head unpublished (because there didn't seem to be any place to publish.) I'm still trying to figure out your hypothesis. :(
Message: 1385 - Contents - Hide Contents Date: Wed, 22 Aug 2001 23:45:44 Subject: Re: A little research...will elaborate further when time allows From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Carl" <carl@l...> wrote:> As far as replacing cents with the units of some et, it isn't > likely to happen. Ellis' cents enjoy wide use, provide good > accuracy regardless of wether the target interval is just, make > the octave highly divisable without fractional results, and > provide a handy reference to 12-tone equal temperament.I've often used the 612 system for my own use, but I don't expect anyone else to adopt it. I like it partly because I have some of it memorized.
Message: 1386 - Contents - Hide Contents Date: Thu, 23 Aug 2001 12:22 +0 Subject: Re: Chromatic = commatic? From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9m1338+rgjv@xxxxxxx.xxx> Paul wrote:>> This is because you're expecting to get an octave-specific result. > The>> octave-equivalent equivalent of an equal temperament is an MOS. >> In what way? What do you mean by that? That certainly doesn't seem to > be true.There both one dimensional sets of points.>> (Or >> family of MOS scales, there's nothing special about the particular > numbers>> of generators that give two interval sizes.) >> As opposed to the ones that don't??Not for these purposes. Graham
Message: 1387 - Contents - Hide Contents Date: Thu, 23 Aug 2001 12:22 +0 Subject: Re: Chromatic = commatic? From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9m12rc+9cqs@xxxxxxx.xxx> Paul wrote:>> The octave-equivalent algebra doesn't distinguish the different > repeating >> blocks. >> Sure it does. The symmetrical decatonic scale (LssssLssss) has 10 > notes -- two repeating blocks per octave.But how can you say "per octave" in a system that doesn't recognize octaves?>> The linear temperament result tells you the number of >> times the period goes into the interval of equivalence (octave) >> Exactly! So the WF definition, where the period _is_ the interval of > equivalence, won't do.Only if their definition of "interval of equivalence" is the same as yours. Whatever the definition, I don't expect they intended to exclude this case.>> and the >> mapping in terms of generators. Again, you need the octave- > specific>> algebra, or the metric, to get the mapping by generators *and* > periods.>> Effectively, the octave-equivalent algebra collapses so that the > interval>> of equivalence becomes the period in any given context. >> What do you mean? For the symmetrical decatonic, the interval of > equivalence remains _two_ periods whether you're looking octave- > equivalently or not.It doesn't look that way to me. That scale can be defined as a periodicity block by [-1 2 0] [ 0 2 -2] [-2 0 -1] The adjoint is [-2 2 -4] [ 4 1 -2] [ 4 -4 -2] The left hand column defines the generator mapping, which we can write as [ 1] -2*[-2] [-2] The minus sign isn't important. The 2 tells us the new interval of equivalence is half the old one. The [ 1] [-2] [-2] gives us the mapping by the generator modulo this new interval of equivalence. 16:15 is [-1 -1 0] times [ 1] [-2] [-2] or 1 generator. 3:2 is [1 0 0] or 1 generator. Where does the algebra tell us these two intervals are not equivalent? You could define it by giving each vector a "spin", but I still don't see how to derive that from the octave equivalent algebra. Graham
Message: 1389 - Contents - Hide Contents Date: Thu, 23 Aug 2001 12:22 +0 Subject: Re: Generators and unison vectors From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9m12f6+qhkq@xxxxxxx.xxx> Paul wrote:>> You can't define the 5:4 approximation by adding octaves and > twelfths. >> Why not? We've been doing so all along. The 5:4 approximation is four > twelfths minus six octaves.Do you remember, back when you were at school, a distinction being made between "adding" and "subtracting"?>> Probably it'd be better written as >> >> v v v v v v v v v v v v p >> >> where v is a fifth and p is a Pythagorean comma. That means p is > the>> (chromatic) unison vector. That leaves v as "the thing you have > left when>> you take out all the unison vectors". That's what we need a name > for. > > The generator?Yes, isn't that where I started? In octave-specific terms, when you take out 25:24 and 81:80 you're left with a generator equivalent to 16:15. In octave-invariant terms, when you take out 81:80 you're left with a generator of a fifth. In octave-specific terms, when you take out only 81:80 you have a choice of generators. They could be 2:1 and 3:2 or 16:15 and 25:24. Taking 2:1 and 25:24 as a pair is different, because the generators and unison vectors don't form a unitary matrix. So perhaps that's why we call one an "interval of equivalence" and the other a "chromatic unison vector". Graham
Message: 1391 - Contents - Hide Contents Date: Thu, 23 Aug 2001 12:22 +0 Subject: Re: Interpreting Graham's matrix From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9m121j+2t5n@xxxxxxx.xxx> Paul wrote:> This is how I've been doing it all along -- see part 3 of the _Gentle > Introduction_.Oh yes. That's <A gentle introduction to Fokker periodicity bl... * [with cont.] (Wayb.)>. The differences are that you take all elements between 0 and 1 instead of some between -0.5 and 0.5, sometimes, and you don't order them by the generator. As the PB repeats about the half octave, that means each generator occurs twice. Actually, no, it's more serious than that. You (Paul) use octave-equivalent vectors, and so the generator isn't there, or rather it's a different generator (MOS rather than ET). Would it work in general to take the generator to increase in steps until you hit the period, and find the only values for the unison vectors that work for each generator? And would this make all the scale steps members of a unitary matrix? I note that the left hand column of Paul's would cover "spin" as well. Well, this leads to a new interpretation of the matrix. Let's take the canonical example [ 4 -1 -1]-1 [ 7 5 3] [-3 -1 -2] = [11 8 5] [-4 4 -1] [16 12 7] The vectors are written with the most important at the top. That means the left hand column is the first approximation to the scale -- an equal temperament. The first two columns define the second approximation, a scale with two steps, or a linear temperament. And all three columns define the scale exactly -- or define the prime vectors in terms of these three intervals. Now, replace the melodic generator with the octave. [ 1 0 0]-1 1[ 7 0 0] [-3 -1 2] = -[11 1 2] [-4 4 -1] 7[16 4 1] The octave is still a generator here. The algebra doesn't know anything about "interval of equivalence". The left hand column is still the first approximation, but it works a bit better if we set the generator (the octave) to be exact. The other columns are correction factors which still work if the octave is exact. Hence this system can work in octave-equivalent space. The things that make this system different to the one before is that it isn't unitary, and only one column of the inverse depends on the first generator. It's the second criterion that allows us to draw the non-arbitrary distinction between "interval of equivalence" and "unison vector", and so throw away the former. The non-unitariness is because the original 16:15 generator can only be derived from the vectors we have now as [4 -1 -1] = ([1 0 0] - 5*[-3 -1 2] - 3*[-4 4 -1])/7 (I'd like a magic way of deducing this vector from the other three.) Each interval can be defined in terms of a fraction of the octave, a fraction of a chromatic semitone, and a fraction of a syntonic comma. As the first is clearly the largest, the others can be thought of as small corrections. Unlike the original system, each new approximation makes a small alteration to the previous one, instead of replacing it. The periodicity block works like that. You start out by taking all the notes of the first approximation. Then you add a correction between +/- 0.5 (or 0 and 1 which amounts to the same thing) for each other dimension to get a just interval. All but the largest correction can be taken out to leave a linear temperament. The connection between that first correction and the MOS generator can be seen when you think of practical tuning. Say you tune a keyboard to meantone with no deviation for D. The mistuning of A will be equal and opposite to that of G, relative to 12-equal. Those for E and C will be twice as much, and so on. The correction factors for the second approximation follow the generator. Graham
Message: 1392 - Contents - Hide Contents Date: Thu, 23 Aug 2001 00:22:05 Subject: Re: Scales From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., genewardsmith@j... wrote:>> Let's see if I understand the situation. In a p-limit system of >> harmony G, containing n primes, we select n-1 unison vectors and > the>> prime 2, which is a sort of special unison vector. This defines a >> kernel K and so congruence classes on G, and a homomorphic image H > =>> G/K which under the best circumstances is cyclic but which in any >> case is a finite abelian group of order h.> What differentiates the "best circumstances" and just "any case" here?What differentiates them is that in the best case we get a cyclic group. What differentiates cyclic groups is having only one generator. If we remove 2 from the generators of the kernel, defining a new kernel K', the "best case" is still cyclic but now of infinite order, corresponding to an et. It will fail to be cyclic if and only if there are torsion elements in G, meaning elements not in the kernel K' some power of which are in K'.> You mean an equal temperament? Sorry. :)
Message: 1394 - Contents - Hide Contents Date: Thu, 23 Aug 2001 00:23:04 Subject: Re: A little research...will elaborate further when time allows From: Carl> Thanks, Carl! Not trying to replace cents, but simply trying to > offer myself and others a simplified option to complement the use > of cents.Sure. And these large ets are interesting in their own right. -Carl
Message: 1396 - Contents - Hide Contents Date: Thu, 23 Aug 2001 01:19:19 Subject: Re: Tetrachordal alterations From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., David C Keenan <D.KEENAN@U...> wrote: > In other words, the tetrachordal scale should be a periodicity block > (not necessarily of the Fokker parallelogram type).Ok. That makes sense.>> Yes. But I wasn't considering _omni_tetrachordality. That looks too > hard>> for now. Do any popular or historical scales have it? >> The most popular ones do -- the diatonic and pentatonic.Oh yeah. Silly me.>> Here's what they look like melodically (in steps of 72-tET). > Vertical bar>> "|" is used to show tetrachords. >> >> 7 7 7 9|5 7|7 7 7 9 >> 7 7 9 7|5 7|7 7 9 7 7 7 7 9|7 5|7 7 7 9 >> 7 9 7 7|5 7|7 9 7 7 7 7 9 7|7 5|7 7 9 7 >> 9 7 7 7|5 7|9 7 7 7 7 9 7 7|7 5|7 9 7 7 >> 9 7 7 7|7 5|9 7 7 7 >> I might not call these tetrachordal. According to my paper, the > disjunction is supposed to contain some pattern of scale steps found > with the tetrachords themselves. But in this case, the 5/72 interval > isn't found in the tetrachords. > > Perhaps we can call these "weakly tetrachordal".Hmm. So in the case where the disjunction consists of a single step, you insist that that step (always an approx 8:9) appears in the tetrachords, before calling a scale tetrachordal? Isn't this just an additional desirable property because it tends to make the scale tetrachordal in more rotations. "Weakly tetrachordal" is still tetrachordal, right? Wouldn't it be better if we simply said "tetrachordal in n rotations". Of course the minimum for n is 3 (other than 0). And when n is the number of notes in the scale it is omnitetrachordal. I'll abbreviate both tetrachord and tetrachordal as "Tc" in future I'm getting tired of typing it. I suppose one could have the disjunction containing only intervals from the Tc but still only have 3 Tc rotations. But is the important thing about this the minimising of different step sizes, rather than having the steps in the disjunction the same as in the Tc?>> But I don't see any sense in which any of these tetrachordal scales > are an>> "alteration" of the 10 note MOS. >> Each note should either agree with the corresponding note in the MOS > or be a chromatic unison vector different.Right. So my alteredness metric was wrong (based on width on the chains) it should instead be melodically based. e.g. How many notes differ from the MOS, and by how many cents.> I see the periodicity block concept as underlying both.Yes. I get it now. -- Dave Keenan
Message: 1397 - Contents - Hide Contents Date: Thu, 23 Aug 2001 18:15:49 Subject: Re: Tetrachordal alterations From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:>> I might not call these tetrachordal. According to my paper, the >> disjunction is supposed to contain some pattern of scale steps found >> with the tetrachords themselves. But in this case, the 5/72 interval >> isn't found in the tetrachords. >> >> Perhaps we can call these "weakly tetrachordal". >> Hmm. So in the case where the disjunction consists of a single step, > you insist that that step (always an approx 8:9) appears in the > tetrachords, before calling a scale tetrachordal? Right. > > Isn't this just an additional desirable property because it tends to > make the scale tetrachordal in more rotations. No. > > "Weakly tetrachordal" is still tetrachordal, right? Weakly. > Wouldn't it be > better if we simply said "tetrachordal in n rotations". Of course the > minimum for n is 3 (other than 0). And when n is the number of notes > in the scale it is omnitetrachordal.This can all still be "weakly" or "strongly".> > I'll abbreviate both tetrachord and tetrachordal as "Tc" in future I'm > getting tired of typing it. > > I suppose one could have the disjunction containing only intervals > from the Tc but still only have 3 Tc rotations.Yes, like Mohajira.> But is the important > thing about this the minimising of different step sizes, rather than > having the steps in the disjunction the same as in the Tc?I think the disjuction should smoothly connect the tetrachords, rather than sounding "different".
Message: 1398 - Contents - Hide Contents Date: Thu, 23 Aug 2001 01:55:36 Subject: Re: A counterexample to the hypothesis? From: Dave Keenan --- In tuning-math@y..., genewardsmith@j... wrote:> I have the idea that what we are talking about is some sort of > equidistribution property, but I can't seem to pin down the property.It has a rigorous mathematical definition which I find astonishing because it is so simple and it doesn't directly say anything about step sizes. It's essentially in 404 Not Found * [with cont.] Search for http://depts.washington.edu/pnm/CLAMPITT.pdf in Wayback Machine and I've implemented it in an Excel spreadsheet at http://uq.net.au/~zzdkeena/Music/MyhillCalculator.xls - Type Ok * [with cont.] (Wayb.) Take a generator g in cents (typically a fourth), an interval of equivalence e in cents (typically an octave), and a whole number i (typically 1, 2, 3). Then we define the period (or interval of repetition) p = e/i. Then a scale generated by iteratively adding that generator, modulo that period (doing the same in all i periods), will be MOS iff the number of notes per period n is the denominator of either a convergent or a semiconvergent of g/p. Convergents and semiconvergents are defined in relation to the continued-fraction expansion of p/g. See Clampitt's paper above if you need the details. The cardinality c of the scale is n * i. Musically, we are really only interested in those with cardinalities between about 5 and 99. Furthermore, if n (the number of notes per period) is a denominator of a convergent (not a semiconvergent), the scale is not only MOS, but is "strictly-proper", which we don't need to go into now, but generally makes it musically more interesting. -- Dave Keenan
Message: 1399 - Contents - Hide Contents Date: Thu, 23 Aug 2001 18:20:10 Subject: Re: Now I think "the hypothesis" is true :) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >>> What makes the apparent relationship between MOS and >> tempered periodicity blocks hard to fathom ... >> Well, tastes differ. I certainly hope your definition is correct,It's almost correct. There can be an interval of repetition which is 1/P of the interval of equivalence, with P an integer.> because: > > (1) It makes sense > > (2) It certainly seems like a condition we would want a scale with > one generator in the octave to satisfy > > (3) It explains why Paul's hypothesis is true. > > For the last, note if for instance we want a q-step scale generated > by interating the p-th step, where p/q is approximately log_2(3/2), > so that fifths are represented by p steps in the q-division, then if > x is log_2 of our approximation for the fifth, we want p/q to be a > semiconvergent of x. > > Now, if memory serves, if |x - p/q| < 1/q^2 then p/q is a > semiconvergent for x, and the converse is "almost" true. The details > don't much matter, unless you wanted to write this up for > publication, because it is clear that if x is anything in a certain > interval around p/q, then p/q will be a semiconvergent for it. If p/q > is a good enough approximation for log_2(3/2), say one that satisfies > |log_2(3/2) - p/q| < 1/(2*q^2), then anything close to log_2(3/2) > will be close enough to p/q so that p/q must be a semiconvergent for > it. Since the approximation x ought to be better than that given by > p/q, we would expect to find (and will find, if we wrote this up with > the details worked out as if for publication) that it gives a MOS.Lost me there. Did you see my "proof" of the hypothesis?
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