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Message: 1625 - Contents - Hide Contents Date: Mon, 24 Sep 2001 20:56:44 Subject: Re: semi-periodic scales From: Paul Erlich --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:>> Only an MOS scale generated by approximate 3:2's will have this >> property (only one note changes with transpositions by a 3:2). >> Yes but I'm interested in a general measure for any scale. I'm > thinking of a function which can take a scale and an interval as > an input, and spit out the "extent" of the scale's symmetry at > that interval.Sounds good . . . for an MOS scale, it depends on how many generators make up the 3:2 . . . for example, any MIRACLE MOS will have 6 notes change when you transpose it by a 3:2.>>> My omnitetrachordality property is closely related. If >> omnitetrachordality holds, moves of a 3:2 will only change the >> notes within a single 9:8 span. >> Is definition in your paper still the most recent? >I don't use the word "omnitetrachordality" in my paper but the concept is there.
Message: 1626 - Contents - Hide Contents Date: Mon, 24 Sep 2001 03:23:18 Subject: Miracle postscript From: genewardsmith@xxxx.xxx Dave Keenan asked for primes to be expressed in the miracle fourth system; since this was an eminently reasonable querry I am reproducing the results here. If A is 5/3, flat by one cent, and B is 4/3, sharp by 2 cents, then we have the following approximations: 2 = A^(-6) B^13 3 = A^(-12) B^25 5 = A^(-11) B^25 7 = A^14 B^(-18) 11 = A^3 B^3
Message: 1627 - Contents - Hide Contents Date: Mon, 24 Sep 2001 22:06:25 Subject: Re: semi-periodic scales From: Carl Lumma>> >es but I'm interested in a general measure for any scale. I'm >> thinking of a function which can take a scale and an interval as >> an input, and spit out the "extent" of the scale's symmetry at >> that interval. >>Sounds good . . . for an MOS scale, it depends on how many >generators make up the 3:2 . . . for example, any MIRACLE MOS >will have 6 notes change when you transpose it by a 3:2.I wonder if Gene, or somebody else with any knowledge of group theory could help. Is there any notion like being 'partially closed' under a certian transformations?>> I don't use the word "omnitetrachordality" in my paper but the >> concept is there.Is there a place where you do use omnitetrachordality? -Carl
Message: 1628 - Contents - Hide Contents Date: Mon, 24 Sep 2001 06:41:01 Subject: Re: Miracle theory From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> 225:224 is also in the kernel for characteristic meantone, schismic and > "magic" scales. There's nothing special about associating it with > miracles.I dunno--it's associated with miracles in something like the way 81/80 is associated to meantone, and that is a completely defining way. But it seems you want more: 225:224, 2401:2400 and 3025:3024, however, completely define> miracle temperament.This means that h10 and h31 also completely define it; and we have h41 = h10 + h31 and h72 = h10 + 2 h31 in this miracle kernel. Add 81:80, and you have 31-equal (actually a 62 note> periodicity block). So is 31-equal the grand unified jumping jack > temperament?It would be better if it came out 31 and not 62, but you do get 2 h31 out of the first four jumping jacks. I've been planning to write a program to find more of these, and it will be interesting to see what they lead to.> I don't get this. The temperament consistent with 441- and 612- equal > divides the octave into 9 equal parts, with a generator mapping of [-2 -3 > -2]. The generator is around 49 cents, 18/441 or 25/612 octaves. It > covers the 9-limit with 37 notes, with 0.2 cents accuracy.Sounds interesting, I will check this out.> I don't see why these are better views, other than for meantone. As I > make the last case, the interval of repetition is 3, and a 7:4 is two > generators. 5:3 and 7:5 are both a single generator.I'm not sure what you mean by this--7/4 and 3 are two generators, just as 3/2 and 2 are two generators or 5/3 and 4/3 are two generators. Wherein lies the difference?
Message: 1629 - Contents - Hide Contents Date: Mon, 24 Sep 2001 22:23:54 Subject: Re: semi-periodic scales From: Paul Erlich --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:>>> Yes but I'm interested in a general measure for any scale. I'm >>> thinking of a function which can take a scale and an interval as >>> an input, and spit out the "extent" of the scale's symmetry at >>> that interval. >>>> Sounds good . . . for an MOS scale, it depends on how many >> generators make up the 3:2 . . . for example, any MIRACLE MOS >> will have 6 notes change when you transpose it by a 3:2. >> I wonder if Gene, or somebody else with any knowledge of group > theory could help.Help how? Aren't we done?>>> I don't use the word "omnitetrachordality" in my paper but the >>> concept is there. >> Is there a place where you do use omnitetrachordality?On the tuning list. It means the same thing as the property mentioned in my paper.
Message: 1630 - Contents - Hide Contents
Date: Mon, 24 Sep 2001 08:26:00
Subject: Block generator theorem
From: genewardsmith@xxxx.xxx
Suppose we have two vals u and v which together with two intervals e
and f define a tempered notation. We want to take e to be an interval
of equivalence, and we want conditions on the vals; the frst is that
the matrix
[u, v] = [u(e) v(e)]
[u(f) v(f)]
be unimodular; that is that it have determinant +-1, so that
[u,v]^(-1) is an integral matrix. The second is that [u, v] is to
have positive values, the third that u(e) > v(e), the fourth that
gcd(u(e), v(e)) = 1, and finally that gcd(u(e), u(f)) = 1. We want to
construct a tempered block of u(e) notes in the temperament defined
by this notation, hence we want a complete set of residues mod u(e)
from the first coordinates of our block, which is to be sorted
according to the second coordinate, and all of it to satisfy the
conditions for a semiblock.
We first shift coordinates to the notation [u, w], where
w = u(e) v - v(e) u. Since u(e) and v(e) are relatively prime, we can
without loss of generality take a generator to be of the form [n 1],
where n is relatively prime to u(e). However, not all such choices
give us a note, since the determinant of [u, w] =
u(e) (u(e)v(f) - v(e) u(f)) = +- u(e) because of the unimodularity
condition. The sublattice of actual notes is generated by
(e, f) = [u, v]^(-1) (in the u, v notation), transforming this to the
[u, w] notation by multiplying by [u, w] gives us [u, v]^(-1) [u, w],
which is
s [1 -v(e)]
[0 u(e)],
where s = det([u,v]) = +-1. This tells us that notes are of the form
[n, m u(e) - n v(e)], so that modulo u(e) they are [n, - n v(e)].
Choosing a generator of the form [n 1] therefore means solving the
congruence n = -1/v(e) (mod u(e)); since v(e) is prime to u(e), we
may solve this congruence for a unique positive n less than u(e),
giving us a generator. If we transform this generator back to the
[u, v] notation by [a, b] = [n 1] [u, w]^(-1), we get
[ n 1] [1/u(e) 0] = [p -s],
[-s u(f)/u(e) -s]
where the value of p is an integer, and s of course is +-1. Hence
[a, b] corresponds to an interval r = e^p f^(-s), so that modulo our
interval of equivalence e it is f or 1/f. The canonical example would
be where e is 2, f is 3, and the generator we get is, modulo octaves,
a fifth or a fourth.
This argument is too convoluted and I should try to improve it, but
the theorem is more widely applicable than it at first may seem.
Message: 1631 - Contents - Hide Contents Date: Mon, 24 Sep 2001 23:15:52 Subject: Re: semi-periodic scales From: Carl Lumma>> > wonder if Gene, or somebody else with any knowledge of group >> theory could help. >>Help how? Aren't we done?I won't speak for everybody else, but I'm not done before I have a metric. Something that can return more than a boolean... as in, "such-and-such scale in x % symmetrical at the 3:2", or something. What makes sense here? I've got: "The percentage of a scale's modes containing a 3/2, in which the pattern of intervals generating scale degrees between 1/1 and 3/2 generates no pitches outside the scale when started at 3/2." Which adds a modal aspect to the metric. But I wonder if there's a raw measure of symmetry?>>> I don't use the word "omnitetrachordality" in my paper but the >>> concept is there. >>>> Is there a place where you do use omnitetrachordality? >>On the tuning list. It means the same thing as the property >mentioned in my paper.Cool. I've seen it on the lists. Just wondering if a Monzo dictionary entry, or something, had been made. -Carl
Message: 1632 - Contents - Hide Contents Date: Mon, 24 Sep 2001 11:45 +0 Subject: Re: Miracle theory From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9omkht+amc8@xxxxxxx.xxx> Gene wrote:>> 225:224 is also in the kernel for characteristic meantone, schismic > and>> "magic" scales. There's nothing special about associating it with >> miracles. >> I dunno--it's associated with miracles in something like the way > 81/80 is associated to meantone, and that is a completely defining > way. But it seems you want more:81:80 completely defines meantone because it converts the three dimensional 5-limit into a two-dimensional linear temperament. (Or 2-D into 1-D with octave equivalence.) One dimension is being lost, so you need one commatic unison vector. 225:224 is a 7-prime limit interval, and can't of itself define a linear temperament.> 225:224, 2401:2400 and 3025:3024, however, completely define >> miracle temperament. >> This means that h10 and h31 also completely define it; and we have > h41 = h10 + h31 and h72 = h10 + 2 h31 in this miracle kernel.Yes, that's right.>> I don't see why these are better views, other than for meantone. > As I>> make the last case, the interval of repetition is 3, and a 7:4 is > two>> generators. 5:3 and 7:5 are both a single generator. >> I'm not sure what you mean by this--7/4 and 3 are two generators, > just as 3/2 and 2 are two generators or 5/3 and 4/3 are two > generators. Wherein lies the difference?No, 7:4 and 3 are two intervals, but they aren't generators of this scale. The 3 is wrong anyway. I meant to say that the interval of equivalence is 1/9 of an octave, whereas you seemed to say 1/3. 5:3 and 7:5 are both equivalent to each other and to the generator in this system. 7:4 is not a generator -- it's represented by two generator steps, the same as 9:8 in meantone. The system you describe is different to the one I get by following your instructions, so I don't understand where it comes from. Graham
Message: 1633 - Contents - Hide Contents Date: Mon, 24 Sep 2001 23:24:14 Subject: Re: semi-periodic scales From: Carl Lumma I wrote...> "The percentage of a scale's modes containing a 3/2, in which the > pattern of intervals generating scale degrees between 1/1 and 3/2 > generates no pitches outside the scale when started at 3/2." > > Which adds a modal aspect to the metric... But I wonder if there's > a raw measure of symmetry?One try is to root each mode of the scale on a random pitch, then mulitply all those pitches by 3:2, then count the number of new pitches and divide by the number of notes in the scale. -Carl
Message: 1634 - Contents - Hide Contents Date: Mon, 24 Sep 2001 11:45 +0 Subject: Re: Miracle theory From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9olut9+7mhg@xxxxxxx.xxx> Gene wrote:> The miracle generator certainly does better than the meantone, which > may be why it is a miracle, but I am not convinced. The JI diatonic > scale requires 25 miracle steps from 5/3 at -13 to 9/8 at 12, whereas > the compass is 14 for the fourth-in-5/3 version. Paul was actually > worried that Jacky's Blackjacky sounded too diatonic, an issue which > would not even arise in the other system.Why is the JI diatonic being taken as the standard? Even so, your calculation only seems to cover one octave. Merely to represent two octaves with the 5:3 equivalence, you need 26 fourths. With octave equivalence, any number of octaves can be represented with the same number of miracle steps. With the 5:3 equivalence, an 11:9 is 43 fourths, as compared to 3 Secors. So a neutral third scale will come out a lot worse in your 5:3 system. I thought Jacky's piece sounded too diatonic because the retuning wasn't working.>> But the value for quarter comma meantone is the minimax for the 5- > limit.>> There doesn't seem to be an analogy here. >> I don't know what you mean by "the" minimax, nor why you could not do > a similar calculation here.The 5-limit minimax is the tuning where the largest 5-limit error (relative to JI) is as small as possible. The relevant limit for miracle is 11, and indeed Dave Keenan did this calculation a long time ago. The result is a 116.7 cent generator giving a worst error of 3.3 cents. Graham
Message: 1635 - Contents - Hide Contents Date: Mon, 24 Sep 2001 23:35:46 Subject: Re: semi-periodic scales From: Paul Erlich --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:>>> I wonder if Gene, or somebody else with any knowledge of group >>> theory could help. >>>> Help how? Aren't we done? >> I won't speak for everybody else, but I'm not done before I have > a metric. Something that can return more than a boolean... as in, > "such-and-such scale in x % symmetrical at the 3:2", or something. > > What makes sense here?How about the number of notes that stay the same, divided by the total number? So for the diatonic scale, it's 6/7 . . . for blackjack, 15/21 . . . for canasta, 25/31 . . . my decatonics would be 4/5 . . . one can easily calculate this as 1-(number of generators in 3:2)/(number of notes).
Message: 1636 - Contents - Hide Contents Date: Mon, 24 Sep 2001 23:37:02 Subject: Re: semi-periodic scales From: Paul Erlich --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:> One try is to root each mode of the scale on a random pitch, then > mulitply all those pitches by 3:2, then count the number of > new pitches and divide by the number of notes in the scale.I don't see what the mode has to do with it. What changes when you label one note as the root, as opposed to not doing so?
Message: 1637 - Contents - Hide Contents Date: Tue, 25 Sep 2001 20:04:31 Subject: Re: semi-periodic scales From: Carl Lumma>> >hich adds a modal aspect to the metric. But I wonder if there's >> a raw measure of symmetry? >> Do you care about the relative proportions of the scale steps?Not sure what you mean... sounds like the answer would be yes... maybe an example would help make it clear. -Carl
Message: 1638 - Contents - Hide Contents Date: Tue, 25 Sep 2001 20:08:20 Subject: Re: semi-periodic scales From: Carl Lumma> Carl, how about the autocorrelation of the scale with its > transposition to the degree representing 3/2? > In Scala, look up the interval class for the nearest interval > to 3/2, do SHOW TRANSPOSE and look at the value for this > interval class.Don't really know what how autocorrelation is calculated. How many lines of code does it take? You can take your pick between posting sample code and attempting a verbal explanation.> It wouldn't be defined if the scale is not CS though, in any > case not if there's more than one i.c. for the fifth. > Or we can think further for a way to define it in that case > too, which may not be useful.What I'm after here should definitely be independent of CS. -Carl
Message: 1639 - Contents - Hide Contents Date: Tue, 25 Sep 2001 20:34:15 Subject: Re: semi-periodic scales From: Paul Erlich Carl, you and I independently came up with the same measure here, so aren't we done? That is, once you realize that mode is irrelevant.
Message: 1640 - Contents - Hide Contents Date: Tue, 25 Sep 2001 22:00:53 Subject: Re: Miracle theory From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9oo2ic+tpk6@e...> > Gene wrote:>> What would you suggest >> as a standard?> The 11-limit tonality diamond.If you look at the 11-limit diamond, it takes an average of 10.4 generator steps to get its elements using (7, 72) and 9.1 using (30, 53) within 5/3; however we need to adjust that value by 72/53, which gives 12.36; by this measure (7, 72) is better. However, where (30, 53) really shines is in generating harmonies; we have triads in a compass of 1, which adjusts to 1.358, compared to 13 for the miracle system; we have 7-chords in a compass of 4, which adjusts to 5.434, compared to 13 again, and 11-chords in a compass of 6, adjusting to 8.151, as compared to 22. The (30, 53) system is jam- packed with harmonies, which to my mind makes it more "normal"; it certainly helps make it practical.> What??? So if it's practical, presumably you have music made with it to > show us ...Save this sort of thing for Page Wizard or someone of that ilk, don't try it on me. :) You are well-equipped by knowledge and ability to evaluate this scale without needing to write a piece of music in it, and so am I. Having said that, I did spend yesterday working on getting myself to the point of actually being able to produce music, and indeed plan on writing a piece to demonstrate this scale. The relevance of that to this discussion is more or less like the relevance of reading on April 10, 1912 on that Thomas Andrews was going to be on the maiden voyage of the Titanic--essentially nil, unless there is a design flaw. You seem to have responded to a posting I canceled shortly after posting it--I didn't like my discussion of your minimax stuff, and was going to find the original and re-do it; however I looked through the archives and didn't locate it.
Message: 1641 - Contents - Hide Contents Date: Tue, 25 Sep 2001 22:15:04 Subject: Re: Miracle theory From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., graham@m... wrote: >> In-Reply-To: <9oo2ic+tpk6@e...> >> Gene wrote: >>>> What would you suggest >>> as a standard? >>> The 11-limit tonality diamond. >> If you look at the 11-limit diamond, it takes an average of 10.4 > generator steps to get its elements using (7, 72) and 9.1 using > (30, 53) within 5/3; however we need to adjust that value by 72/53, > which gives 12.36; by this measure (7, 72) is better. However, where > (30, 53) really shines is in generating harmonies; we have triads in > a compass of 1, which adjusts to 1.358,I don't understand the compass or the adjustment. What kind of triads are you talking about? How can you get three notes with only 1 iteration of a generator??
Message: 1642 - Contents - Hide Contents Date: Tue, 25 Sep 2001 22:19:44 Subject: Re: Miracle theory From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> However, where > (30, 53) really shines is in generating harmonies; we have triads in > a compass of 1,Oh . . . I understand this now . . . but that's cheating! Since you don't have octave-equivalence, you're quite constrained in terms of the inversions of the triads you can use . . . this fact can't go unpenalized. In particular, to make nice music you'll often want to be able to use a bass note which is the implied fundamental of the chord.
Message: 1643 - Contents - Hide Contents Date: Tue, 25 Sep 2001 23:32:25 Subject: Re: semi-periodic scales From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:>> Do you care about the relative proportions of the scale steps?> Not sure what you mean... sounds like the answer would be yes... > maybe an example would help make it clear.Suppose we look at 12-note scales of the form abababaababa, formed by iterating fifths within octaves. We have 7 a and 5 b, and if we take scale steps such that a=2, b=1, we get the 19 et, a=3, b=2 gives the 31 et, a=4 b=5 the 53 et, and a=5 b=4 the 55 et. One way of measuring self-similarity would make all of these the same, however we could also look at it in proportion to the average number of et intervals making up a scale step. In all cases a shift of a fifth interchanges an a and a b; for the 19-et that means an exchange of 2 and 1, in a sitation where the average step size is 19/12, whereas for the 55-et it is and exchange of 5 and 4 compared to an average size of 55/12; both are an exchange of one et interval so we might measure the goodness of the first by 12/19 and the second by 12/55. This is related to the approximation of the fifth of the et by 7/12; a standard measure here would measure the relative goodness of the approximation to x by the reduced fraction p/q by q^2 |x - p/q|. If we do this, we find 12^2 |7/12 - 11/19| = 12/19 and 12^2 |7/12 - 32/55| = 12/55, etc. If a scale of n steps is generated from a single cycle of a generator in an m-et, this suggests n^2 |a/n - b/m|, where b/m is the generator and a/n is what we might call the scale-step generator, as a measure of self-similarity. We can also adapt it to more than one cycle of the generator; for instance a 12-cycle in the 22 et gives us 12^2 |7/12 - 13/22| = 12/11 as a measure of self-similarity, whereas if we have two cycles of six tones a half-ocatave apart, we have two cyles each of which has a self-similarity measure of 6^2 | 1/6 - 2/11| = 6/11, suggesting the total self-similarity should be the same at 12/11.
Message: 1644 - Contents - Hide Contents Date: Tue, 25 Sep 2001 23:38:02 Subject: Re: Miracle theory From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Oh . . . I understand this now . . . but that's cheating!It would be cheating except for the fact that I adjusted by a factor of 72/53, which makes them directly comperable. Since you> don't have octave-equivalence...But I do have octave equivalence--the proposal is to run the 4/3 around inside the 5/3 for however many times we want to do that, and then run that pattern out to 2. We then use 2, not 5/3, as the interval of equivalence.
Message: 1645 - Contents - Hide Contents Date: Tue, 25 Sep 2001 23:38:52 Subject: Re: semi-periodic scales From: Carl Lumma>Carl, you and I independently came up with the same measure here, >so aren't we done? That is, once you realize that mode is >irrelevant.Yeah, we may be done. Just so long as nobody comes along and says that the effect of a transposition on the entire scale doesn't nec. represent the effects of transpositions on all of its subsets. Also, I'm interested to find out how the autocorr. thing works in Scala. And maybe gene has a group-theoretical way of looking at it. -Carl
Message: 1646 - Contents - Hide Contents Date: Tue, 25 Sep 2001 00:11:34 Subject: Re: semi-periodic scales From: Carl Lumma>> >ne try is to root each mode of the scale on a random pitch, then >> mulitply all those pitches by 3:2, then count the number of >> new pitches and divide by the number of notes in the scale. >>I don't see what the mode has to do with it. What changes when you >label one note as the root, as opposed to not doing so?Quite possibly nothing... I'm cautious because I'm already thinking through a layer of abstraction -- specifically, we don't actually care about pitches moving (as we do for transpositional coherence), we care about how much of an interval pattern is destroyed when we _aren't_ allowed any new pitches. I'm trying to think if this is really equivalent to counting the new pitches we'd need. I think it is, but a lot of what I've thought has historically failed when tested against "unusual" (improper, non-just) scales... Assuming counting the changing pitches works, then we need to ask if looking at a single mode, or even all the modes, is equivalent to what we really care about. Namely, every possible interval pattern in the scale is generated, transposed by a fifth, the number of changing notes is divided by the length of the pattern in each case, and then the whole lot of results is averaged. -Carl
Message: 1647 - Contents - Hide Contents Date: Tue, 25 Sep 2001 05:26:45 Subject: Re: semi-periodic scales From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:> Which adds a modal aspect to the metric. But I wonder if there's > a raw measure of symmetry?Do you care about the relative proportions of the scale steps?
Message: 1648 - Contents - Hide Contents Date: Tue, 25 Sep 2001 11:11 +0 Subject: Re: Miracle theory From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9oo0rt+s4tp@xxxxxxx.xxx> Gene wrote:>> No, 7:4 and 3 are two intervals, but they aren't generators of this > scale. >> They are the generators of a system of scales I mentioned, so I need > a context for "this scale"--which scale? >>> temper.Temperament(441,612,temper.primes[:-2])43/117, 49.0 cent generator basis: (0.1111111111111111, 0.040838783185694151) mapping by period and generator: [(9, 0), (15, -2), (22, -3), (26, -2)] mapping by steps: [(612, 441), (970, 699), (1421, 1024), (1718, 1238)] unison vectors: [[-5, -1, -2, 4], [-1, -7, 4, 1]] highest interval width: 3 complexity measure: 27 (45 for smallest MOS) highest error: 0.000170 (0.204 cents) unique So which scale do you mean? Graham
Message: 1649 - Contents - Hide Contents Date: Tue, 25 Sep 2001 11:11 +0 Subject: Re: Miracle theory From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9oo2ic+tpk6@xxxxxxx.xxx> Gene wrote:>> Why is the JI diatonic being taken as the standard? >> Mostly because of the Blackjacky discussion. What would you suggest > as a standard?The 11-limit tonality diamond. Or the 7-limit diamond, because miracle temperament works well with that too.> Even so, your>> calculation only seems to cover one octave. Merely to represent > two>> octaves with the 5:3 equivalence, you need 26 fourths. >> That is why I suggested beginning the second 5/3 repetion, taking it > out to an octave, and then repeating that scale within an octave. I > think the result is entirely practical.What??? So if it's practical, presumably you have music made with it to show us ...> I take it you are using Paul's definition of 5-limit? The way I use > the word, this doesn't mean anything, but you can for instance look at > 3^a 5^b where a^2+ab+b^2 < 2, which I recall you doing. I don't see > why the result cannot be used for either scale.It can be, provided yours repeats at the octave. But that isn't a calculation I've seen you doing.> However, if you add 2 to the mix, you could consider 2^a 3^b 5^c with > a^2 + b^2 + c^2 + ab + ac + bc < 2, and get a closely related linear > programming exercise; the previous one being what one gets if we fix > 2 to its exact value. One could certainly fix 5/3 instead. Similar > comments apply to a least squares opitimization.Don't know. This is looking complicated. The usual rule is that the numerator shouldn't exceed a particular integer limit. In this case, limits from 7 to 12 would be appropriate. You can apply that to any generator and period you like, but you haven't applied it here, so you don't have anything to show that your method is of value. Graham
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