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Message: 3975 - Contents - Hide Contents Date: Sun, 3 Mar 2002 19:27:00 Subject: help with ratio/vector algorithm From: monz i need help with an algorithm. within the 3-limit, i'm trying to get an Excel spreadsheet to automatically calculate the exponent of 2 when it knows the exponent of 3, so that the resulting ratio is in the usual form where n > d. my first attempt worked fairly well, but had a few systematic errors. i'll give it in a format that will look familiar to programmers: ratio r = 2^p * 3^q if q < 0 then p = int((log(3^abs(q)) / log(2)) + 0.5) else p = (int((log(3^abs(q)) / log(2)) + 1.5)) * -1 end if but it's not foolproof: sometimes the exponent of 2 is one less or one more than it should be. i've tried setting up further nested if-statements, to check if the absolute value of p is greater than that of q and adjust accordingly, but there's always an error somewhere. help! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3976 - Contents - Hide Contents Date: Sun, 3 Mar 2002 19:33:39 Subject: Re: help with ratio/vector algorithm From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, March 03, 2002 7:27 PM > Subject: [tuning-math] help with ratio/vector algorithm > > > i need help with an algorithm. > > within the 3-limit, i'm trying to get an Excel > spreadsheet to automatically calculate the exponent > of 2 when it knows the exponent of 3, so that the > resulting ratio is in the usual form where n > d. > > my first attempt worked fairly well, but had a > few systematic errors. i'll give it in a format > that will look familiar to programmers: > > ratio r = 2^p * 3^q > > if q < 0 > then p = int((log(3^abs(q)) / log(2)) + 0.5) > else p = (int((log(3^abs(q)) / log(2)) + 1.5)) * -1 > end if > > but it's not foolproof: sometimes the exponent of 2 is > one less or one more than it should be. i've tried > setting up further nested if-statements, to check if > the absolute value of p is greater than that of q and > adjust accordingly, but there's always an error somewhere.the errors are occurring because the spreadsheet has to check to see if n > d, and i can't figure out how to implement that without getting a circular reference. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3977 - Contents - Hide Contents Date: Sun, 03 Mar 2002 20:11:29 Subject: Re: listing linear temperaments From: Carl Lumma>I took a look at this problem, and it seems to me that starting from >single ets and using generators from them ought to turn up everything >of interest if we check everything which makes some sort of sense in >non-consistent cases (which are the only hard ones.) Making that >precise and putting it into effect would still make for a big search; >I would like something like Paul's heuristic to speed that up.What's the status of your heuristic, Paul?>Right now I'm writing, so I'm thinking through the whole thing anew, >and hope that will help. Cool!>> Speaking of this method, nobody ever answered this: >> >> Carl wrote...>>> More to the point, every line on this plane is a linear temperament, >>> right? So what makes low-numbered (less than 100) equal >>> temperaments cluster on some of them? >>Those are the better low-rent commas. Not all temperaments were born >equal.What makes a comma "better"?>> A map uniquely defines a linear temperament? Or do you also >> need period? >>I thought a map *had* a period. What do you mean by a map?A list of identities given in terms of generators, ie [1 4] for meantone. I suppose the rms optimum generator can be constructed from this alone, and a generator plus a period specifies a linear temerament. I really don't know where the period comes from.>A map with two vectors (period and octave) can be wedged to >get the corresponding wedgie, so it clearly defines the >temperament.Hmmm... not sure I follow. What do octaves have to do with anything? Isn't it just that some temperaments have a period that is a fraction of an octave, and so two or more periods of the temperament are used in practice to get octaves for musical reasons?>>> Finally, re the jumping jacks / ideal comma question... what's the >>> question? How are we defining "most powerful" comma? >>At this point, we aren't. Paul's discovery I suppose could be used >to do that, if you want to define it in terms of Fourier analysis!Well, I'd certainly like to hear more about this. Are you referring to the periodicity in the badness curve? You want to run a Fourier analysis on the wave and see the commas with the biggest peaks?>>> What's the relationship between a comma vanishing and a map? >>If the map maps the comma to zero, the comma vanishs for the >map--it's in the kernel of the map.You mean: if I factor a comma in terms of the identities in a map, sum the corresponding number of generators on each side of the fraction, subtract the numerator's sum from the denominator's, and get zero? So, 81/80 -> (3^4)/(5* 2^4), so if the map is [1 4], I get 4 - 4 = 0.>> Can we get a list with optimum generator, et series, commas, maps, >> periods for these (and the rest of the top 20)? Are any of the >> "Monzo's lines" temperaments in here? >>These are 7-limit temperaments; Monzo would need to graph a plane.You mean a volume? Anyway, yes, they are 7-limit -- I've been thinking about the relationship between good linear temperaments at limit n and good ones at higher limits. You just append the extra identities onto the map. Porcupine, for example, is on Monzo's chart but extends to the 7-limit. I'm interested in how the commas change... 64:63 becomes a porcupine comma in the 7-limit, for example. IOW, will we need separate top-20 lists for each limit? -Carl

Message: 3978 - Contents - Hide Contents Date: Sun, 03 Mar 2002 21:29:26 Subject: maple for graph theory From: Carl Lumma Gene, I've upgraded to Maple 7. Where did you get the graph theory package? -Carl

Message: 3979 - Contents - Hide Contents Date: Mon, 4 Mar 2002 11:34 +00 Subject: Re: maps, uvs From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <002001c1c314$5539d420$af48620c@xxx.xxx.xxx> monz wrote:> to find the mapping to EDOs, put the unison-vectors > in vector form into a matrix, then calculate the > determinant and the inverse. if the inverse is > unimodular (= has a determinant = 1), then it gives > the mapping to EDOs, the cardinality of which (i.e., > mapping of prime-factor 2) is in the top row. see: > > Tuning Dictionary, "matrix" > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)Note that mappings to EDOs are covered in <Equal temperaments from matrix formalism * [with cont.] (Wayb.)>. That's old, but the method still stands. The generator mapping is the same idea but for octave-equivalent matrices, and common factors give you the octave division. Graham

Message: 3980 - Contents - Hide Contents Date: Mon, 04 Mar 2002 22:42:13 Subject: Re: error-free badness From: Carl Lumma>but you can always enforce GIGO. GIGO? >Enumerating all mappings requires 100 times as many.I'll take your word for it.>If a temperament doesn't include a fairly good ET with fewer than >100 notes, you could consider that to be badness in itself.Could you go over the your reasoning here? -Carl

Message: 3981 - Contents - Hide Contents Date: Mon, 4 Mar 2002 11:34 +00 Subject: Re: help with ratio/vector algorithm From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <004101c1c32d$603c6540$af48620c@xxx.xxx.xxx> monz wrote:>> ratio r = 2^p * 3^q >> >> if q < 0 >> then p = int((log(3^abs(q)) / log(2)) + 0.5) >> else p = (int((log(3^abs(q)) / log(2)) + 1.5)) * -1 >> end if >> >> but it's not foolproof: sometimes the exponent of 2 is >> one less or one more than it should be. i've tried >> setting up further nested if-statements, to check if >> the absolute value of p is greater than that of q and >> adjust accordingly, but there's always an error somewhere. > >> the errors are occurring because the spreadsheet has to > check to see if n > d, and i can't figure out how to > implement that without getting a circular reference.I think =MOD(N20, 1)-N20 does it where N20 is the magnitude of the original interval in octaves. So you have to substitute that with log(3^abs(q)) / log(2). Graham

Message: 3982 - Contents - Hide Contents Date: Mon, 4 Mar 2002 11:34 +00 Subject: Re: listing linear temperaments From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4.0.1.20020223185452.019836c0@xxxxx.xxx> Carl Lumma wrote:> A paper. I think it's a great idea. And, the 569 of us who don't > have a computer set up to do calculations on linear temperaments > need a list!You can get "set up" with only free software, and all 569 of you could do that if you really wanted. I've also got my ISP's computer set up to do the calculations at <Linear Temperament Finding Home * [with cont.] (Wayb.)>. That is limited, mainly because you can't choose your own badness measures, and it can only handle the odd limits. As my free time's limited, what would people prefer I do this evening? Improving the CGI, writing up the algorithms and playing with some drum loops are all options.> Graham's catalog, "The grooviest 7-limit temperaments", Monzo's lines, > and Herman Miller's "Carl's favorite page on the internet" Warped > Canons page are huge, huge, huge. But wouldn't it be cool to really > get the goat?What else do you want?> Is this still state-of-the-art-badness? I seem to remember > something about different exponents for each prime/odd (?) identity, > taken from coefficients of Diophantine equations, or some such? > Don't need details, just want to know if we need a new top 20.I don't think any badness will ever be good enough for everybody. My solutions are 1) Put the calculations online, so people can specify what they want in terms of accuracy and complexity, instead of generating as many as 20 options. 2) Allow custom badness measures. Hopefully this can be done by entering Python expressions in text boxes. I will need to check the security on this. The badness as a function of error (RMS or minimax) and complexity should be the most useful thing to customize.> ~~~~~~~~~~~~~~~~~~~~~ > (1b) The slippery six > ~~~~~~~~~~~~~~~~~~~~~ > > I wasn't reading the tuning-math very closely back then, but Gene, > your top 20 is generated by starting with some large number of > ets and then seeing what temperaments they share, sort of like a > more precise version of looking for lines on Herman Miller's / Paul's > charts, right? But you found that some temperaments only hit a > single et up to your cutoff -- those were the slippery six, right? > Do we have a general solution to this problem -- making the cutoff > really high, an entirely different method, etc.?I'm not sure about Gene, but that's how I do it. Raising the cutoff -- not insisting on consistency -- does get you more temperaments, and this is an option on the CGI. You'll still miss some because they aren't represented by a pair of nearest-prime ETs. One way round this is to consider all sane options of inconsistent temperaments. Another is to use a different method.> Speaking of this method, nobody ever answered this: > > Carl wrote...>> More to the point, every line on this plane is a linear temperament, >> right? So what makes low-numbered (less than 100) equal >> temperaments cluster on some of them? >> What makes some linear temperaments belong to more than one et, out > of ets as high as some given number? They would have to share a > common generator... Is sharing a common generator related to the > un-even distribution of the rationals on the number line (such as > makes harmonic entropy work)?I'd say ETs belong to linear temperaments rather than the other way round. And ET is a linear temperament where the large and small steps happen to make a rational number. How many you get depends on your badness cutoff.> Carl wrote...>> Intersections (by eye) >> ---------------------- >> 12 - 8 >> 31 - 5 >> 22, 37, 15, 34, 19 - 4 >> 29, 41, 75, 61, 53, 72, 23 - 3 >> Why are some of the 'best' ets (ones that have gotten so much > attention on these lists for so many different reasons, for > so long) here? Is it because we've often defined "best" as > "consistent", and where two lines cross the same tuning is being > reached two different ways (via two different maps), which > requires consistency?No, because they don't have to be consistent. But the approximations for a linear temperament must be at least as good as its best constituent ET. The complexity must be somehow related as well. I've conjectured that the complexity of an LT must be no worse than the larger number of notes of a pair of consistent ETs consistent with it.> A map uniquely defines a linear temperament? Or do you also > need period? Looking at Graham's catalog, I'm not sure how to > use maps with non-octave periods.The simple, one-dimensional maps assume an equivalence interval. All you need to do is set this other period to be that interval. Otherwise you have to give the octave specific map, as for the Bohlen-Pierce entry. Or describe it in words, as the golden non-meantone entry.> Carl wrote...>> I say the most powerful maps are the ones with the smallest >> numbers in them. Sum of abs value would work. >> Or, maybe the sum of the abs values of the max and min numbers in > the map, for a given limit (or divided by the card of the map, if > you want to compare across limits). Which is better?You have to consider 0 as well, for the unison. Then, the max-min is my complexity. I think Gene's using the RMS.> There's definitely some overlap with badness here, but by not > considering the quality of the approximations, doesn't this tell > us more about the abstract musical-theoretic properties of a > temperament?My keyboard mapping lists (.key suffix) do this.> Carl wrote...>> Finally, re the jumping jacks / ideal comma question... what's the >> question? How are we defining "most powerful" comma?One rule of thumb, which relates to Paul's heuristic, is to consider the number of consonances that make up the comma, and the size of the comma. The simpler the interval the better, and the smaller it is the more accurate a temperament can be. The optimal accuracy is the size of the comma in cents divided by its complexity in consonances. Here, the "consonances" are the list of intervals you want to optimize.> Carl wrote...>> What's the relationship between a comma vanishing and a map? > > ?The map will approximate all vanishing commas to a unison.> Paul wrote...>> i bet gene can do this in a jiffy. maybe graham too. >> and oh, we need the period as well as the generator. >> I completely agree.I can't tell from the context what you want here. I probably thought the CGI could do it originally. I can't be wiping your noses forever.>> (1) <21/20,27/25> >> (2) <8/7,15/14> >> (3) <9/8,15/14> >> (4) <25/24,49/48> >> (5) <15/14,25/24> >> (6) <21/20,25/24> >> (7) <15/14,35/32> >> (8) <7/6,16/15> >> (9) <16/15,21/20> >> Can we get a list with optimum generator, et series, commas, maps, > periods for these (and the rest of the top 20)?I've got a CGI that can do some of that. Graham

Message: 3984 - Contents - Hide Contents Date: Mon, 4 Mar 2002 12:18 +00 Subject: Re: error-free badness From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <200203040803.g2483r130861@xxxxx.xxxxxx.xxx> Carl Lumma wrote:> Anywho, does anyone else agree on the utility of this measure? > If the approximations are acceptable, this is very much like > badness. What I'm thinking is, instead of searching comma space > or generator space, why not search map space? For values of > Dave's measure that are reasonable, there aren't that many > possibilities, and each one specifies its own optimum generator. > You find it, calculate the errors, rank them by badness, and > you're done.If you mean the octave-equivalent map, you first need an algorithm to get a full description of the temperament from that. It's more difficult than you might think, but shouldn't be as difficult as Paul and Gene are suggesting. You might need to guess generators within a half-period. You can assume the number of periods to the equivalence interval is the GCD of the map. That won't work with torsion, but you can always enforce GIGO. The map, period and example generator are enough to fully define a temperament. In some cases different generators give different octave-specific mappings that are equally good. So pick one. All you have to do is try enough generators that you'll know you got the best one. Now, the problem with this method is that it is complex to calculate. Take a calculation that would give Miracle -- all 11-limit temperaments with a complexity no greater than 22. Relative to the lowest-mapped prime, each prime can take any value between 0 and 22. There are 4 entries in the map, one of which is fixed. So the number of options is 22*22*22 or 22**3. For the total number, each prime could be the lowest-mapped, which will give slightly less than 4*22**3 options. That is 42,592 different temperaments which have to be considered. Certainly possible, but this wasn't a problem anyway. My method of taking pairs of ETs only needs to consider the best 20 ETs to get miracle and a load of more complex temperaments. That means it only has to consider 400 linear temperaments. Enumerating all mappings requires 100 times as many. On top of that, it's The general formula, for n dimensions, is n*complexity**(n-1). My top 17-limit temperament has a complexity of 66. To get this by guessing maps you'd need to consider 6*66**5 = 7,513,995,456 different options, which is huge. My program gets good results (may miss some) from only those 400 linear temperaments. Okay, the inconsistent list does give one with a complexity of 34. To get that, you'd need 6*34**5 = 272,612,544 which is better than before. But it must be less efficient than a modified version of my program that considers all versions of inconsistent temperaments. An alternative approach which is certainly worth trying (and Gene may have tried it) is to take all mappings for the ETs you're looking at. That should be quadratic time, same as the ET-pair algorithm, and you can probably get away with looking at fewer ETs. If a temperament doesn't include a fairly good ET with fewer than 100 notes, you could consider that to be badness in itself. That's partly the logic behind my algorithm. One of the good things about miracle is that it unifies three 11-limit consistent ETs. If an LT doesn't even cover two almost-consistent ETs, I probably won't be interested in it. Graham

Message: 3985 - Contents - Hide Contents Date: Mon, 04 Mar 2002 23:07:45 Subject: Re: listing linear temperaments From: Carl Lumma>You can get "set up" with only free software, and all 569 of you could do >that if you really wanted. I've also got my ISP's computer set up to do >the calculations at <Linear Temperament Finding Home * [with cont.] (Wayb.)>. That is limited, >mainly because you can't choose your own badness measures, and it can >only handle the odd limits.Rock! I hadn't looked at these for a while. Some nice-to-haves: () document it () allow input of identities, not just odd limit () return the name of the temperament, if known () return all fields for each temperament (ie "not unique" or "unique")>As my free time's limited, what would people prefer I do this evening? >Improving the CGI, writing up the algorithms and playing with some drum >loops are all options.Hope you enjoyed what you picked. My biggest complaint is too many worthy options. But it's better than not enough! Really shows how spoiled I am.>> Graham's catalog, "The grooviest 7-limit temperaments", Monzo's lines, >> and Herman Miller's "Carl's favorite page on the internet" Warped >> Canons page are huge, huge, huge. But wouldn't it be cool to really >> get the goat? >>What else do you want?Slightly improved cgi stuff (see above) and/or a list. My ideal list would have: () Top 20 temperaments, by Gene's favorite badness measure, in each odd limit from 5 to 17. () Show a name, map, rms optimum generator, rms error, simplest commas, and complexity for each. () Make sure the names hold for a given LT if it makes it into the top 20 of higher and higher limits. () Uniqueness level.>> Is this still state-of-the-art-badness? I seem to remember >> something about different exponents for each prime/odd (?) identity, >> taken from coefficients of Diophantine equations, or some such? >> Don't need details, just want to know if we need a new top 20. >>I don't think any badness will ever be good enough for everybody. My >solutions are > >1) Put the calculations online, so people can specify what they want in >terms of accuracy and complexity, instead of generating as many as 20 >options.It would still be nice to report the top 20 to people who aren't going to learn to use the script.>2) Allow custom badness measures. Hopefully this can be done by entering >Python expressions in text boxes. I will need to check the security on >this. The badness as a function of error (RMS or minimax) and complexity >should be the most useful thing to customize.This is a good idea, but it would be the last thing I would spend time implementing. IMO rms is always better than minimax, and complexity should be kept separate from badness, and for badness I'll happy to trust Gene!>I'm not sure about Gene, but that's how I do it. Raising the cutoff -- >not insisting on consistency -- does get you more temperaments, and this >is an option on the CGI. You'll still miss some because they aren't >represented by a pair of nearest-prime ETs.Nearest-prime? Anyway, all you have to do is show why a temperament with low badness is bound to appear in multiple ets, as you claim above, and I'll be happy.>> What makes some linear temperaments belong to more than one et, out >> of ets as high as some given number? They would have to share a >> common generator... Is sharing a common generator related to the >> un-even distribution of the rationals on the number line (such as >> makes harmonic entropy work)? >>I'd say ETs belong to linear temperaments rather than the other way >round.Okay, fine. But I don't see how this answers the question.>> Carl wrote...>>> I say the most powerful maps are the ones with the smallest >>> numbers in them. Sum of abs value would work. >>>> Or, maybe the sum of the abs values of the max and min numbers in >> the map, for a given limit (or divided by the card of the map, if >> you want to compare across limits). Which is better? >>You have to consider 0 as well, for the unison. Then, the max-min >is my complexity.Okay, great! We agree.>I think Gene's using the RMS.Wow. z'that true, Gene?>>> Finally, re the jumping jacks / ideal comma question... what's the >>> question? How are we defining "most powerful" comma? >>One rule of thumb, which relates to Paul's heuristic, is to consider the >number of consonances that make up the comma, and the size of the comma. >The simpler the interval the better, and the smaller it is the more >accurate a temperament can be. The optimal accuracy is the size of the >comma in cents divided by its complexity in consonances. Makes sense. >The map will approximate all vanishing commas to a unison.Gene showed me this. Cool.>>> i bet gene can do this in a jiffy. maybe graham too. >>> and oh, we need the period as well as the generator. >>>> I completely agree. >>I can't tell from the context what you want here.More details on Monzo's chart. But Monzo's chart is just a list of LTs that we eyeballed from Paul's graph. But if what you say is right about LTs sharing ETs and badness, the eyeballing might not be so bad afterall. But still, we shouldn't bother with it any more. We should create a badness-ranked list!>I probably thought the CGI could do it originally. I can't be >wiping your noses forever.I think your cgi can do most of it. -Carl

Message: 3986 - Contents - Hide Contents Date: Mon, 4 Mar 2002 18:15:32 Subject: Re: HEWM notation calculator From: monz re: The Proxomitron Reveals... * [with cont.] (Wayb.) here's a great example of how the JI version of HEWM differs from the 72edo version (the 72edo simplifies things): say i want to find the notation for the ratio 42:25, assuming that C = 1/1. the first thing to do on the spreadsheet is to make sure that there's a zero next to "C" in the column of cyan cells near the top. this defines "C" as 1/1. then put in the target ratio. whether using the method of inputting "comma" exponents or inputting the notation directly, the result will be the same: 2 3 5 7 11 target ratio: [ 1 1 -2 1 0] 42/25 898.2 HEWM: B bb - [ 15 -9 0 0 0] 32768/19683 882.4 ------------------- [-14 10 -2 1 0] 413343/409600 15.7 < - [ -6 2 0 1 0] 63/64 -27.3 ------------------- [ -8 8 -2 0 0] 6561/6400 43.0 + - [ -4 4 -1 0 0] 81/80 21.5 ------------------- [ -4 4 -1 0 0] 81/80 21.5 + - [ -4 4 -1 0 0] 81/80 21.5 ------------------- [ 0 0 0 0 0] 1/1 0.0 so the JI HEWM notation for 42:25 is Bbb<++. this is simplified in 72edo, because in 72edo, < = -- and > = ++, so we have Bbb<>, and the <> cancel each other out, leaving us with just Bbb. in the JI version of HEWM, > and ++ are not the same. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3987 - Contents - Hide Contents Date: Mon, 04 Mar 2002 23:12:36 Subject: some output from Graham's cgi From: Carl Lumma Unison vector input ------------------->unison vectors >· 36:35 >· 50:49 >calculated >unison vectors >· 36:35 >· 50:49Why do you say "calculated" and then list them again?>0/1, 1886.3 cent generatorA bug? Do we always want the "generator" to be the smaller generator, and the "period" the larger? Anyway, it would be nice to see the period. Are these optimum generators?>basis: >(0.25, 1.57192809489)Never heard of it.>mapping by period and generator: >[(4, 0), (0, 1), (3, 1), (5, 1)]I'm up to speed on this.>mapping by steps: >[(4, 0), (-1, 1), (2, 1), (4, 1)]? Carried out how far?>highest interval width: 1 ? >complexity measure: 4 (8 for smallest MOS)From the map above isn't it 5?>highest error: 0.027608 (33.129 cents)Why not give rms as well as or instead of this? List parameters input --------------------- """ 9 odd limit, or consonant harmonics 15 highest complexity 10.0 worstError (cents) 100 number of equal temperaments to consider 10 number of results to return 0.5 ET goodness cutoff (0.5 is Erlich consistency) _ Figure of Demerit (optional) """>9-limit >ETs considered 5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70 72 /.../ >13/41, 380.4 cent generator >mapping by period and generator: >[(1, 0), (0, 5), (2, 1), (-1, 12)] >complexity measure: 12 (13 for smallest MOS) >highest error: 0.004942 (5.930 cents) >unique >13/31, 503.6 cent generator >mapping by period and generator: >[(1, 0), (2, -1), (4, -4), (7, -10)] >complexity measure: 10 (12 for smallest MOS) >highest error: 0.009198 (11.038 cents) >17/46, 443.4 cent generator >mapping by period and generator: >[(1, 0), (-1, 7), (-1, 9), (-2, 13)] >complexity measure: 14 (19 for smallest MOS) >highest error: 0.007304 (8.765 cents) >unique >11/27, 489.6 cent generator >mapping by period and generator: >[(1, 0), (2, -1), (6, -9), (2, 2)] >mapping by steps: >[(22, 5), (35, 8), (51, 12), (62, 14)] >complexity measure: 11 (12 for smallest MOS) >highest error: 0.014090 (16.908 cents) >6/31, 232.3 cent generator >mapping by period and generator: >[(1, 0), (1, 3), (0, 12), (3, -1)] >complexity measure: 13 (16 for smallest MOS) >highest error: 0.009322 (11.186 cents) >3/13, 90.8 cent generator >mapping by period and generator: >[(3, 0), (5, -1), (7, 0), (8, 2)] >complexity measure: 12 (15 for smallest MOS) >highest error: 0.012155 (14.586 cents) >19/45, 506.3 cent generator >mapping by period and generator: >[(1, 0), (2, -1), (4, -4), (-1, 9)] >complexity measure: 13 (19 for smallest MOS) >highest error: 0.013826 (16.591 cents) Very cool. -Carl

Message: 3988 - Contents - Hide Contents Date: Mon, 04 Mar 2002 03:24:23 Subject: Re: listing linear temperaments From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>> step^3 cent> Is this still state-of-the-art-badness? I seem to remember > something about different exponents for each prime/odd (?) identity, > taken from coefficients of Diophantine equations, or some such? > Don't need details, just want to know if we need a new top 20.No, but no consensus was reached on what we should be looking for. I like a step^2 measure of some kind, because of the log-flat business.> Gene, you initially stopped after listing 20. Did you ever list > the requested next few needed to uncover meantone? Are you still > happy with your list?No, it's not been done. I could simply go ahead and do things my way.> ~~~~~~~~~~~~~~~~~~~~~ > (1b) The slippery six > ~~~~~~~~~~~~~~~~~~~~~ > > I wasn't reading the tuning-math very closely back then, but Gene, > your top 20 is generated by starting with some large number of > ets and then seeing what temperaments they share, sort of like a > more precise version of looking for lines on Herman Miller's / Paul's > charts, right? But you found that some temperaments only hit a > single et up to your cutoff -- those were the slippery six, right? > Do we have a general solution to this problem -- making the cutoff > really high, an entirely different method, etc.?I took a look at this problem, and it seems to me that starting from single ets and using generators from them ought to turn up everything of interest if we check everything which makes some sort of sense in non-consistent cases (which are the only hard ones.) Making that precise and putting it into effect would still make for a big search; I would like something like Paul's heuristic to speed that up. Right now I'm writing, so I'm thinking through the whole thing anew, and hope that will help.> Speaking of this method, nobody ever answered this: > > Carl wrote...>> More to the point, every line on this plane is a linear temperament, >> right? So what makes low-numbered (less than 100) equal >> temperaments cluster on some of them?Those are the better low-rent commas. Not all temperaments were born equal.> What makes some linear temperaments belong to more than one et, out > of ets as high as some given number?Having common kernel elements.>> 12 - 8 >> 31 - 5 >> 22, 37, 15, 34, 19 - 4 >> 29, 41, 75, 61, 53, 72, 23 - 3 >> Why are some of the 'best' ets (ones that have gotten so much > attention on these lists for so many different reasons, for > so long) here?The best ets have the best temperaments, generally speaking.> A map uniquely defines a linear temperament? Or do you also > need period?I thought a map *had* a period. What do you mean by a map? A map with two vectors (period and octave) can be wedged to get the corresponding wedgie, so it clearly defines the temperament.>> Finally, re the jumping jacks / ideal comma question... what's the >> question? How are we defining "most powerful" comma?At this point, we aren't. Paul's discovery I suppose could be used to do that, if you want to define it in terms of Fourier analysis!>> What's the relationship between a comma vanishing and a map?If the map maps the comma to zero, the comma vanishs for the map--it's in the kernel of the map.> Can we get a list with optimum generator, et series, commas, maps, > periods for these (and the rest of the top 20)? Are any of the > "Monzo's lines" temperaments in here?These are 7-limit temperaments; Monzo would need to graph a plane.

Message: 3989 - Contents - Hide Contents Date: Mon, 4 Mar 2002 17:55:29 Subject: HEWM notation calculator (was: help with ratio/vector algorithm) From: monz ----- Original Message ----- From: <graham@xxxxxxxxxx.xx.xx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Monday, March 04, 2002 3:34 AM Subject: [tuning-math] Re: help with ratio/vector algorithm> In-Reply-To: <004101c1c32d$603c6540$af48620c@xxx.xxx.xxx> > monz wrote: >>>> ratio r = 2^p * 3^q >>> >>> if q < 0 >>> then p = int((log(3^abs(q)) / log(2)) + 0.5) >>> else p = (int((log(3^abs(q)) / log(2)) + 1.5)) * -1 >>> end if >>> >>> but it's not foolproof: sometimes the exponent of 2 is >>> one less or one more than it should be. i've tried >>> setting up further nested if-statements, to check if >>> the absolute value of p is greater than that of q and >>> adjust accordingly, but there's always an error somewhere. >> >>>> the errors are occurring because the spreadsheet has to >> check to see if n > d, and i can't figure out how to >> implement that without getting a circular reference. >> I think =MOD(N20, 1)-N20 does it where N20 is the magnitude of the > original interval in octaves. So you have to substitute that with > log(3^abs(q)) / log(2).thanks, Graham! (duh... of course i have to use MOD!) i've implemented this in an Excel spreadsheet that assists in finding the HEWM notation of a ratio: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) color LEGEND: cyan = data to be input by user magenta = constants, used in calculations orange = results of calculations on user data at the top are a bunch of constants that i use in the calculations. below those... on the left side, the user inputs the exponents of the target ratio first, and then the exponent of 3 which gets near it. the spreadsheet subtracts that, and indicates the letter-name and any sharp or flat. (there's still a bug here -- i don't know how to generate the negative numbers when using MOD.) then the the user enters the exponents of the notational "commas" from the magenta table above until all exponents are zero. eventually, i want to be able to have the spreadsheet determine all of the accidental symbols from the exponents entered by the user... but this is not implemented fully yet. on the right side, the user inputs the exponents of the target ratio, then directly enters the notational symbols which eventually reduce all exponents to zero. the only thing to be aware of here is that an apostrophe (') must precede the minus and plus signs when entering them. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3990 - Contents - Hide Contents Date: Mon, 4 Mar 2002 22:39:15 Subject: CGI update From: Graham Breed See <Linear Temperament Finder * [with cont.] (Wayb.)>. It does more stuff now. Graham

Message: 3991 - Contents - Hide Contents Date: Mon, 04 Mar 2002 00:04:28 Subject: error-free badness From: Carl Lumma>> > say the most powerful maps are the ones with the smallest >> numbers in them. Sum of abs value would work. >>Or, maybe the sum of the abs values of the max and min numbers in >the map, for a given limit ...This latter measure is just the length of the chain of generators needed to play the mapped chord, and has been suggested and used by Dave Keenan several times.>Which is better? Probably Dave's. >(or divided by the card of the map, if >you want to compare across limits).I remember thinking something was wrong with this, but now I can't think what it was. It would be nice if we could flatten this measure across limits. Anywho, does anyone else agree on the utility of this measure? If the approximations are acceptable, this is very much like badness. What I'm thinking is, instead of searching comma space or generator space, why not search map space? For values of Dave's measure that are reasonable, there aren't that many possibilities, and each one specifies its own optimum generator. You find it, calculate the errors, rank them by badness, and you're done. -Carl

Message: 3992 - Contents - Hide Contents Date: Mon, 04 Mar 2002 08:05:46 Subject: Re: listing linear temperaments From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> What makes a comma "better"?For 5-limit commas, simply the badness score of the associated temperament quantifies it.>>> A map uniquely defines a linear temperament? Or do you also >>> need period? >>>> I thought a map *had* a period. What do you mean by a map? >> A list of identities given in terms of generators, ie [1 4] > for meantone.That maps one generator, except it should really be [0 1 4]. You need another map for the octave, which is the other generator.> What do octaves have to do with anything? Isn't it just that > some temperaments have a period that is a fraction of an octave, > and so two or more periods of the temperament are used in > practice to get octaves for musical reasons?The period is one of the generators, if you decide to have one generator proportional to octaves.>> If the map maps the comma to zero, the comma vanishs for the >> map--it's in the kernel of the map. >> You mean: if I factor a comma in terms of the identities in a map, > sum the corresponding number of generators on each side of the > fraction, subtract the numerator's sum from the denominator's, and > get zero? So, 81/80 -> (3^4)/(5* 2^4), so if the map is [1 4], > I get 4 - 4 = 0.Pretty much; however you should really take the whole map: you have the period part as well, [1,1,0]; so 81/80 -> -4*1 + 4*0 -1*0 = 0. IOW, will we need separate top-20 lists for each> limit?For each finitely generated group of intervals we are interested in, which covers a lot more ground, I fear.

Message: 3993 - Contents - Hide Contents Date: Mon, 04 Mar 2002 08:08:59 Subject: Re: maple for graph theory From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Gene, > > I've upgraded to Maple 7. Where did you get the graph theory > package?I presume it is still the networks package, available via with(networks). SJSU supposedly has Maple 7, but it doesn't seem to work, so I'm only up to 5 still.

Message: 3994 - Contents - Hide Contents Date: Mon, 04 Mar 2002 00:28:15 Subject: Re: listing linear temperaments From: Carl Lumma>> >hat makes a comma "better"? >>For 5-limit commas, simply the badness score of the associated >temperament quantifies it.We've defined good commas as those that result in good temperaments. What is it about a comma that results in a better temperaments than another? All the lines on Monz's chart could have just two dots. The badness of the associated temperaments would still quantify the situation. But instead, some lines have many dots.>> A list of identities given in terms of generators, ie [1 4] >> for meantone. >>That maps one generator, except it should really be [0 1 4]. You >need another map for the octave, which is the other generator.Okay, thanks. So is the map for the octave in meantone [1 0 -2]?>>> If the map maps the comma to zero, the comma vanishs for the >>> map--it's in the kernel of the map. >>>> You mean: if I factor a comma in terms of the identities in a map, >> sum the corresponding number of generators on each side of the >> fraction, subtract the numerator's sum from the denominator's, and >> get zero? So, 81/80 -> (3^4)/(5* 2^4), so if the map is [1 4], >> I get 4 - 4 = 0. >>Pretty much; however you should really take the whole map: you have >the period part as well, [1,1,0];What's this!? Two 1's?>so 81/80 -> -4*1 + 4*0 -1*0 = 0. ?>> IOW, will we need separate top-20 lists for each limit? >>For each finitely generated group of intervals we are interested in, >which covers a lot more ground, I fear.If we're potentially interested in all the subsets, we make software which accepts queries. For being interested in on paper, odd limits through 19 are quite enough. -Carl

Message: 3995 - Contents - Hide Contents Date: Mon, 04 Mar 2002 08:31:22 Subject: Re: error-free badness From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Anywho, does anyone else agree on the utility of this measure? > If the approximations are acceptable, this is very much like > badness. What I'm thinking is, instead of searching comma space > or generator space, why not search map space? For values of > Dave's measure that are reasonable, there aren't that many > possibilities, and each one specifies its own optimum generator. > You find it, calculate the errors, rank them by badness, and > you're done.How do you speed up the error-calculation part? I've been thinking about working directly from wedgies; one way would involve using even limits--that is, for example the 10-limit, only without octave equivalence. Any comments on that?

Message: 3996 - Contents - Hide Contents Date: Mon, 04 Mar 2002 08:36:05 Subject: Re: listing linear temperaments From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>> Pretty much; however you should really take the whole map: you have >> the period part as well, [1,1,0]; >> What's this!? Two 1's? >>> so 81/80 -> -4*1 + 4*0 -1*0 = 0.2 ~ (3/2)^0 2^1 3 ~ (3/2)^1 2^1 5 ~ (3/2)^4 2^0 So the map is [0 1] [1 1] [4 0]

Message: 3997 - Contents - Hide Contents Date: Mon, 04 Mar 2002 00:47:54 Subject: Re: error-free badness From: Carl Lumma>> >nywho, does anyone else agree on the utility of this measure? >> If the approximations are acceptable, this is very much like >> badness. What I'm thinking is, instead of searching comma space >> or generator space, why not search map space? For values of >> Dave's measure that are reasonable, there aren't that many >> possibilities, and each one specifies its own optimum generator. >> You find it, calculate the errors, rank them by badness, and >> you're done. >>How do you speed up the error-calculation part?I imagine the error-calc would be hard to get below a certain obvious footprint. So I think we need to get the number of things to check down. If we search by generators, assume one of them is a 2:1, and desire a tenth of a cent accuracy, there are 12,000 temperaments to check. I think there are fewer 7-limit maps spanning 20 generators or less.>I've been thinking about working directly from wedgiesCool. Not that I know what a wedgie is, or how to do a wedge product. Yet.>one way would involve using even limits--that is, for example the >10-limit, only without octave equivalence. Any comments on that?Well, it wouldn't be my preferred way, but it could be okay. -Carl

Message: 3998 - Contents - Hide Contents Date: Mon, 04 Mar 2002 00:49:30 Subject: Re: listing linear temperaments From: Carl Lumma>>> >retty much; however you should really take the whole map: you have >>> the period part as well, [1,1,0]; >>>> What's this!? Two 1's? >>>>> so 81/80 -> -4*1 + 4*0 -1*0 = 0. >

Message: 3999 - Contents - Hide Contents Date: Mon, 04 Mar 2002 01:02:46 Subject: Re: maple for graph theory From: Carl Lumma>I presume it is still the networks package, available via >with(networks). SJSU supposedly has Maple 7, but it doesn't >seem to work, so I'm only up to 5 still.Ah. I was under the impression this was an external plugin. Just read up on packages. What doesn't work? I seem to be able to load the package... -C.

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