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Message: 6851 - Contents - Hide Contents Date: Wed, 04 Jun 2003 02:05:41 Subject: Re: Interval Database Experiences From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Porres" <decuritiba@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> There's no difficulty in doing that; you don't need a table. >> Oh really, how can I do that?Let <a[0], a[1], a[2], ...> be such that the a[i] are integers, and a[i]>0 for i>0. For any finite subsequence, we define a corresponding fraction as follows: let p[-2]=0, p[-1]=1, q[-2]=1, q[-1]=0, and for i>=0, define p[i] and q[i] recursively by p[i] = a[i] p[i-1] + p[i-2] q[i] = a[i] q[i-1] + q[i-2] Then we define <a[0], a[1], ... a[n]> = p[n]/q[n] to be the *nth convergent* of the continued fraction <a[0], a[1], a[2], ..., a[n]> Now suppose we have a value c given in cents, we convert it to a frequency ratio f by setting f = 2^(c/1200). If "floor(x)" is the largest integer less than or equal to x, we set f[0]=f and then a[i] = floor(f[i]) f[i+1] = 1/(f[i] - a[i]) then for increasing n <a[0], ..., a[n]> rapidly approaches f, and gives excellent approximations to it.
Message: 6852 - Contents - Hide Contents Date: Wed, 04 Jun 2003 17:22:03 Subject: Re: ...continued... From: Carl Lumma>> >an someone give the vals for 5-limit meantone? >>Various choices are possible: > >(1) Eytan, consisting of h12 = [12, 19, 28] and h7 = [7, 11, 16] > >(2) Octave and fifth, consisting of oct = [1, 1, 0] and >fif = [0, 1, 4] > >(3) Nominals and accidentals, consisting of h7 = [7, 11, 16] and >h5 = [5, 8, 12] > >(4) Very meantone, h31 = [31, 49, 72] and h19 = [19, 30, 44] > >And so forth. I suggest you try sticking each of these pairs into >my Maple routine "a5val" and seeing what comes forth.If I enter "a5val [19, 30, 44] [31, 49, 72];", it just spits back the input.>If we have linearity, then if 9 is as consonant as 5, 3 will have to >be twice as consonant. Why? -Carl
Message: 6854 - Contents - Hide Contents Date: Wed, 04 Jun 2003 20:49:08 Subject: Re: ...continued... From: Carl Lumma>>> >nd so forth. I suggest you try sticking each of these pairs >>> into my Maple routine "a5val" and seeing what comes forth. >>>> If I enter "a5val [19, 30, 44] [31, 49, 72];", it just spits back >> the input. >>a5val is a Maple function, so you need to enter > >a5val([19, 30, 44], [31, 49, 72]);Ah, I'd tried that without the comma between the two lists. Great. So they all give the comma for meantone. And I'm beginning to understand vals.>>>>>> Finally, note that I'm still confused about prime- vs. odd-limit >>>>>> as regards pi(p)-1. Obviously I'm assuming prime-limit here, >>>>>> but should pi(p) be changed to ceiling(p/2)? That is, how many >>>>>> commas does a 9-limit linear temperament require? Paul? >>>>>>>>>> ... linear independence. So the 5-limit (2-3-5) requires one >>>>> comma. So does the 2-3-7 limit. And so would a system composed >>>>> of octaves, fifths, and 7:5 tritones, although it uses 4 prime >>>>> numbers. >>>>>>>> Ok, but what about stuff like (2-3-5-9) where we don't have linear >>>> independence but wish to consider 9 as consonant as 3 or 5? How >>>> does visualization in terms of blocks work on a lattice with a >>>> 9-axis? >>>>>> If we have linearity, then if 9 is as consonant as 5, 3 will have >>> to be twice as consonant. >> >> Why? >>Because if a is the approximation to 3, a^2 will be the approximation >to 9. Then the error log(a^2/9) will be twice that of log(a/3).I'm not talking about error, but the assumption brought to the table, namely that ratios of 9 are no less consonant than ratios of 3 in a 9-limit tuning, so that whatever errors exist should not be weighted differently when optimizing the tuning. Have you been including errors in ratios of 9 in optimizations? I notice you never talk of the 9-limit. Only the 7- and 11- limits. At the 7-limit, the 3-error is counted once. At the 11-limit, it should be counted thrice. -Carl
Message: 6855 - Contents - Hide Contents Date: Thu, 05 Jun 2003 03:02:23 Subject: Re: ...continued... From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> And so forth. I suggest you try sticking each of these pairs into >> my Maple routine "a5val" and seeing what comes forth. >> If I enter "a5val [19, 30, 44] [31, 49, 72];", it just spits back > the input.a5val is a Maple function, so you need to enter a5val([19, 30, 44], [31, 49, 72]);>> If we have linearity, then if 9 is as consonant as 5, 3 will have to >> be twice as consonant. > > Why?Because if a is the approximation to 3, a^2 will be the approximation to 9. Then the error log(a^2/9) will be twice that of log(a/3).
Message: 6856 - Contents - Hide Contents Date: Thu, 05 Jun 2003 06:11:10 Subject: Re: ...continued... From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Have you been including errors in ratios of 9 in optimizations?Of course; otherwise it would not be a 9-limit opitmization.> I notice you never talk of the 9-limit. Only the 7- and 11- limits.I do sometimes.
Message: 6857 - Contents - Hide Contents Date: Fri, 06 Jun 2003 00:35:48 Subject: Re: Interval Database Experiences From: Dave Keenan Hi Alex, Welcome to tuning-math. Here's what Gene was talking about, implemented in an Excel spreadsheet. The page cannot be found * [with cont.] (Wayb.) You enter an interval in cents and it gives a (two-dimensional) series of ever-better ratios with their errors in cents. I remember when I first learned how to calculate convergents and semi-convergents (which I don't think Gene mentioned), to get ratios from cents, it seemed like magic. On the question of how much error is acceptable in just intonation: In my humble opinion, many folks find +-3 cent errors entirely acceptable for most music intended to be played in just intonation, and indeed this is a typical intonation error for many fixed-pitch acoustic intruments, but for others you'd have to be within +-0.5 cents (and not tell them about it) before they would be happy. But there is little point in accepting more than about 3 cents error, or in striving for less than about 0.5 cents. But of course there are extreme and unusual contexts in which these do make sense. There are no sharp cutoffs. Others might be amused to learn that George Secor and I, in our deliberations on a universal microtonal notation system, have coined the terms "mortal" and "olympian" (as both adjective and noun) for these two extremes of accuracy. Obviously a notation for JI can have fewer symbols if greater errors can be tolerated. The general approach is to define symbols only for the most common ratios and to reuse these symbols for less common ratios which are sufficiently close. But we'll never make everyone happy with one such notation. The olympians would complain about the errors or ambiguities in a notation designed for mere mortals, while the mortals would complain about the number and complexity of the symbols in a system designed for olympians. A system about midway between mortal and olympian we call "herculean". A herculean notation will of course have too many symbols for mortals and too much ambiguity for olympians. So we've decided to specify three systems. The trick is in making the transition from one to the other as seamless as possible; to have as much in common between the three as possible. I'm afraid we're progressing rather slowly (but surely) on this at present because we have both had (and are still having) long periods where (heaven forbid) we had to concentrate on our jobs and families. We are also working with Manuel Op de Coul on a Scala implementation of the notation system. Regards, -- Dave Keenan
Message: 6858 - Contents - Hide Contents Date: Fri, 06 Jun 2003 21:14:43 Subject: Re: Need help finding unusual linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <D.KEENAN@U...> wrote:> You have misunderstood me. I'm sorry I wasn't clearer. I hope the > above helps. Octaves are not important.this is going to be a bizarre notation indeed! without octave equivalence???> The single most popular comma in the entire universe (for notational > purposes such as ours) is of course the syntonic comma 81/80. Then the > septimal comma 64/63. Then the double syntonic comma 6561/6400.how did that leapfrog over the diaschisma 2048/2025??
Message: 6859 - Contents - Hide Contents Date: Fri, 06 Jun 2003 14:21:05 Subject: Need help finding unusual linear temperaments From: David C Keenan Hi guys, George Secor and I could use some help with the sagittal notation project. We're approaching the notation of ratios from a new direction. We expect it will give the same results as before at the level typically used by mere mortals, but we're now going totally off the deep end and attempting a JI notation suitable for use by the gods on Mount Olympus. ;-) It will have a maximum error of around half a cent (say 0.4 to 0.9 c). It will not be based on an equal division of the octave. What we're looking at are planar temperaments. Specifically planar temperaments where one generator is fixed at the just fifth (and of course the octave is fixed too). It seems to me that this is equivalent to a _linear_ temperament where the "interval of equivalence" is the apotome 3^7/2^11 (~ 113.685 cents). In any case, what we really want to know is how best to divide up the apotome in a relatively even (Rothenberg proper?) manner, from minus half an apotome to plus half an apotome, so as to closely approximate many commas for many popular ratios. We've found 3 such temperaments by ad hoc means, but we'd really like to know that we haven't missed any really good ones. What we need is a list of (minimax) generators and periods to examine (where the period is either the apotome or an integer fraction of the apotome, and the generator is smaller than half the period. For example, one such generator is, perhaps unsurprisingly, an approximate syntonic comma (of about 21.57 cents) with a whole apotome period. Another is an approximate minor diesis of about 41.1 cents in a whole apotome, and another is an approximate syntonic comma in a 1/3-apotome period. It is reasonably safe to assume a 23 prime-limit, although 31 would probably be better. And you can assume an odd (actually factor-of-2-and-3-free) limit of 625, although 1225 would be better. If 625 is too hard I'd settle for 385. Ideally I'd supply you with a list of commas, giving each a popularity rating but you can probably come up with a reasonable surrogate mathematical function for this yourself. For example a weighted sum of prime exponents such that these ratios are approximately equally popular 175/1 19/1 245/1 13/7 625/1 23/1 49/25 The problem is, while we have stats on the popularity of ratios (from the Scala archive), what we really need is the popularity of their commas. Take any 2,3-free ratio and add factors of 2 and 3 in such a way as to produce a ratio in the range -apotome/2 to +apotome/2 and you have a comma for the original ratio. There are an infinite number of possible commas for any ratio, however the relative popularity of the different commas will be strongly inversely related to their exponent of 3. Most commas with an absolute 3-exponent greater than 12 can be ignored. All commas with a 3-exponent greater than 17 can be ignored. There are rarely more than 3 commas of significance for any ratio. A good temperament for our purpose then is one that gets close to many popular commas without having to iterate the generator too many times. We're not interested in chains of more than 160 generators, (meaning N chains of 160/N generators). And there is unlikely to be anything useful with fewer than 45 generators. Minimax errors should be less than 0.9 cents for the X most popular commas where X is greater than 20, and less than 0.5 cents for the Y most popular commas where Y is greater than 10, and we are interested in what these values of X and Y are for each temperament. Can anyone help? Do you need more info? Regards,
Message: 6861 - Contents - Hide Contents Date: Fri, 06 Jun 2003 05:57:59 Subject: Re: Need help finding unusual linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> What we're looking at are planar temperaments. Specifically planar > temperaments where one generator is fixed at the just fifth (and of course > the octave is fixed too).Those are pretty common; you just need a planar temperament where the fifth works as a generator, and then use a tuning where it is pure.\ It seems to me that this is equivalent to a> _linear_ temperament where the "interval of equivalence" is the apotome > 3^7/2^11 (~ 113.685 cents).This makes zero sense to me. Should I give examples of planar temperaments?> We've found 3 such temperaments by ad hoc means, but we'd really like to > know that we haven't missed any really good ones. What we need is a list of > (minimax) generators and periods to examine (where the period is either the > apotome or an integer fraction of the apotome, and the generator is smaller > than half the period.If you are using exact octaves, the period can only be an approximate apitome. However, it is certainly possible to look for such beasts.> The problem is, while we have stats on the popularity of ratios (from the > Scala archive), what we really need is the popularity of their commas.Why not simply take lists of commas which pass a goodness test instead?
Message: 6862 - Contents - Hide Contents Date: Fri, 06 Jun 2003 23:58:45 Subject: Re: Need help finding unusual linear temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <D.KEENAN@U...> > wrote: >>> You have misunderstood me. I'm sorry I wasn't clearer. I hope the >> above helps. Octaves are not important. >> this is going to be a bizarre notation indeed! without octave > equivalence???Oh dear. I almost despair of communicating my intentions here if _you_ are not getting it Paul. But this certainly is not the kind of "linear temperament" any of us have ever considered before. So I shouldn't be surprised. Let me reassure you that the notation has octave equivalence. It is the same sagittal notation whose essential details were worked out on this list over a year ago and have not changed since. The letter-names and sharps and flats are based on octaves and fifths as you would expect. What I am investigating here is what happens _after_ you have chosen a letter and sharps or flats for a ratio, and deals only with the comma inflections. For this purpose, I am looking only at the region within half an apotome either side of any 3-limit pitch and looking at dividing up this space by considering the proper moments-of-symmetry of a comma-sized generator iterated modulo this apotome.>> The single most popular comma in the entire universe (for notational >> purposes such as ours) is of course the syntonic comma 81/80. Then > the>> septimal comma 64/63. Then the double syntonic comma 6561/6400. >> how did that leapfrog over the diaschisma 2048/2025??It seems to me that folks prefer to notate ratios of 25 as stacked ratios of 5, e.g. 1:5:25 as C E\ G#\\ rather than C E\ Ab'\ where \ is syntonic comma down and '\ is diaschisma down. (By the way, the apostrophe ' can also be read, when on its own, as schisma up). But lets not get bogged down in details like this. I don't mind if you want to rank the diaschisma as more popular (for comma-inflected notation) than the double-syntonic comma. It is still likely to benefit from the same kind of apotome-MOS or linear-temperament-of-the-apotome. Regards, -- Dave Keenan
Message: 6863 - Contents - Hide Contents Date: Fri, 06 Jun 2003 06:27:01 Subject: An example? From: Gene Ward Smith Is this at all what you are looking for? 4375/4374 7-limit planar temperament generator 1: 2 generator 2: 3 generator 3: 4.999748162, or 2786.226514 cents period matrix: [[1, 0, 0, 1], [0, 1, 0, 7], [0, 0, 1, -4]]
Message: 6864 - Contents - Hide Contents Date: Fri, 06 Jun 2003 07:01:32 Subject: Re: Need help finding unusual linear temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote: > It seems to me that this is equivalent to a>> _linear_ temperament where the "interval of equivalence" is the apotome >> 3^7/2^11 (~ 113.685 cents). >> This makes zero sense to me. Should I give examples of planar > temperaments?OK. Forget the planar temperament stuff. The things I'm interested in are mathematically like a _linear_ temperament, but instead of repeating at the octave they repeat at the apotome, and instead of approximating simple-ratio JI harmonies they approximate popular notational commas. This is for the purpose of notating JI by means of comma-inflection symbols applied to a chain of Pythagorean fifths. We want maximum accuracy for the maximum number of popular commas with the minimum number of symbols. We are just concentrating on the region within +-1/2-apotome of a single note in the chain of fifths.>> We've found 3 such temperaments by ad hoc means, but we'd really > like to>> know that we haven't missed any really good ones. What we need is a > list of>> (minimax) generators and periods to examine (where the period is > either the>> apotome or an integer fraction of the apotome, and the generator is > smaller>> than half the period. >> If you are using exact octaves, the period can only be an approximate > apitome. However, it is certainly possible to look for such beasts.You have misunderstood me. I'm sorry I wasn't clearer. I hope the above helps. Octaves are not important. Exact apotomes are. We could divide the apotome into equal pieces and call the resulting object an EDA (equal division of the apotome), but equal divisions are not necessary, all we need is an approximately even (proper) division, and we can probably get greater accuracy that way.>> The problem is, while we have stats on the popularity of ratios > (from the>> Scala archive), what we really need is the popularity of their commas. >> Why not simply take lists of commas which pass a goodness test instead?I'm certainly open to suggestions. Goodness for what? Actually I don't need to know what. Just post a list and I'll tell you if I think they are roughly in popularity order. The single most popular comma in the entire universe (for notational purposes such as ours) is of course the syntonic comma 81/80. Then the septimal comma 64/63. Then the double syntonic comma 6561/6400. After that it starts to get a little hazy with regard to the commas, but decreasing popularity of the ratios which the commas are meant to notate start off like this (with factors of 2 and 3 removed). 1/1 5/1 7/1 25/1 7/5 11/1 35/1 125/1 49/1 13/1 11/5 11/7 17/1 25/7 49/5 13/5 175/1 19/1 245/1 13/7 625/1 23/1 49/25 55/1 77/1 17/5 19/5 35/11 13/11 31/1 343/1 29/1 125/7 55/7 17/11 77/5 19/7 385/1 55/49 17/7 1225/1 37/1 121/1 23/5 19/13 17/13 23/7 25/11 Another reason for looking at these linear temperaments of the apotome is that they may provide us with a series of proper divisions of the apotome that increase in resolution but belong to a common "family". This is so that we can provide a number of notations that trade off accuracy for number-of-symbols, in different ways, while making it easy to move between them. Each notation would correspond to a proper MOS of the same LTA (linear temperament of the apotome). I note that the first 7 ratios above account for 71% of all ocurrences of ratios in the Scala archive (when factors of 2 and 3 are removed). So ideally their commas would all be represented in a reasonably short chain of generators (say 20 or less).
Message: 6866 - Contents - Hide Contents Date: Fri, 6 Jun 2003 16:39:28 Subject: Re: Interval Database Experiences From: Manuel Op de Coul Alex, see this: Stichting Huygens-Fokker: List of intervals * [with cont.] (Wayb.) There's a link on the bottom to another page of Dave for more info. Manuel
Message: 6867 - Contents - Hide Contents Date: Mon, 09 Jun 2003 17:08:31 Subject: equal-beating temperaments From: Carl Lumma Gene, So what's the deal with the synchronous versions of temperaments? Can you give a rundown of of some of them -- metameantone, wreckmeister (?), wendell nat. sync well, sync. vallotti... are they all based on the beat ratio btwn. maj and min 3rds? Have you been able to hear a difference? I think metameantone is the best-sounding meantone I've heard, but I haven't given a close listen to other sync. temperaments yet. Maybe we should have a midi file comparing triads with various beat ratios? -Carl
Message: 6868 - Contents - Hide Contents Date: Wed, 11 Jun 2003 14:27:34 Subject: Re: Interval Database Experiences From: Manuel Op de Coul Alex wrote:>Now what's your oppinion, should the terminology be redefined and >form a new tradition, like Dave is working on, or should one bother >on the collection of historycal names that are available out there?No opinion about what should be done, but I felt more for making a list of historical names, and Dave for making a consistent terminology.>and that other names >belonging to not that important terminologies are irrelevent to you >people, is that right?Well there's an infinite amount of interval ratios, there's no point in trying to find a name for all of them. Manuel
Message: 6870 - Contents - Hide Contents Date: Wed, 11 Jun 2003 13:52:33 Subject: Re: Transformation From: monz hi Ben,> From: "bdenckla" <bdenckla@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, June 11, 2003 12:58 PM > Subject: [tuning-math] Re: Transformation > > > If you're interested in Regener, my Master's thesis > presents his work, among other things: > > No Title * [with cont.] (Wayb.) > > I've struggled to get a small subset of this work, > in particular my streamlined presentation of Regener's > work, published with no luck.wow, thanks! i'll enjoy reading this! good luck with publication ... keep trying. -monz
Message: 6871 - Contents - Hide Contents Date: Fri, 13 Jun 2003 14:55:11 Subject: Baked Alaska From: Gene Ward Smith Responding to a question from Carl, I present Baked Alaska, an alaska-style temperament with brats of exactly 3/2 and 2. This isn't as nifty as it sounds, because since the octave is no longer pure, what happens with close position no longer applies to wider spacings. Anyway, it supplies one answer to what exact values we might use for these things. If we take a scale with steps a, b, b, repeated, then we have two situations with major thirds and fifths--either we have a third equal to a^2 b^2 and a fifth equal to a^3 b^4, or we have a third equal to a b^3 and a fifth equal to a^2 b^5. If we set the brat for the first equal to 3/2 and for the second equal to 2, and solve the resulting system of two nonlinear equations in two unknowns via the resultant, we get that a satisfies 320000*a^7-288000*a^4-1728*a^2+32400*a-151875 = 0 and b satisfies 1125*b^7+48*b^5+1800*b^4-4000 = 0 This now gives us the following alaska-style scale: ! alabake.scl Baked alaska, with brats of 2 and 3/2 12 ! 102.565223 201.130054 299.694886 402.260110 500.824942 599.389774 701.954998 800.519830 899.084662 1001.649885 1100.214717 1198.779549
Message: 6872 - Contents - Hide Contents Date: Fri, 13 Jun 2003 18:17:54 Subject: Re: Interval Database Experiences From: Manuel Op de Coul>but I still wonder if that >fokker list is complete for this matter, any clues?I'm not aware of missing important intervals, and open to any corrections. Manuel
Message: 6874 - Contents - Hide Contents Date: Fri, 13 Jun 2003 19:24:26 Subject: Re: Interval Database Experiences From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:>> but I still wonder if that >> fokker list is complete for this matter, any clues? >> I'm not aware of missing important intervals, and open > to any corrections. > > Manuelunfortunately, some of the "big names" in tuning history, such as rameau, helmholtz, etc., have given different, mutually inconsistent names to different intervals. in the following table, i attempted to tabulate the most common names for important small 5-limit intervals, and give the details of the temperaments that result when these intervals vanish. some of the names of the latter were jovial creations of this (tuning-math) list. this is meant to replace the similar table on monz's "equal temperament" dictionary page: Yahoo groups: /tuning/database? * [with cont.] method=reportRows&tbl=10&sortBy=3
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