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Message: 7401 Date: Mon, 08 Sep 2003 07:35:59 Subject: Re: Classificiation of musical scales From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "pitchcolor" <pitchcolor@a...> wrote:

> Dave wrote: >

> > > I think it's fairly standard that 'just' means constructed by

> ratios of

> > > (usually small) whole numbers.

> > > > Yes unfortunately, it has become fairly standard. "A recent > > abberation" as I said.

> > > But you've supplied nothing which substantiates this claim.

Hi Aaron, I should have explained. That number at the end of the OED entry, "1811". It's the year of publication of the earliest document that the editors of the OED can find which uses the word "just" with this meaning, in this context, in the English language. When the meaning of a word changes with time, the OED editors show them all, and indicate when each new meaning first entered the language. See for example the entry for "Jazz". Apparently "just" was a synonym for "pure" in this context back in 1811 and nothing has yet convinced them that this has changed. By far the best survey of definitions of just intonation that I know of is Joe Monzo's "dictionary" entry (more like an encyclopedia). Definitions of tuning terms: just-intonation, ... * [with cont.] (Wayb.) You will find "pure" used as a synonym for "just" in several of the quotes there.

> The word 'just' (which translates as 'just' into English) is used as > a theoretical term in South America and in Poland, for example, > where 'fifths' are assumed to be 3/2 and 'fourths' are assumed to > be 4/3. These are theoretical constructs. These terms apply to > piano music, for example. 'Just' is clearly used there as a > _theoretical term. Again, if you need to make the distinction, then > just say either it '_sounds just' or it '_is just'.

The word "sounds" makes it clear that we have perception in mind, but how does the word "is" make it clear that we are talking math or music theory? In fact I have written elsewhere that "Just is as just sounds." So I'm afraid this doesn't work for me. But also, the justness of the fifths and fourths on a piano tuned to 12-equal is not so theoretical. They sound close enough to just for many people, particularly given the tuning "stretch" on a piano.

> > Can you provide any evidence that they have ever been in

> separate spheres? > > No, I can only hold that they _belong in seperate spheres. > People make errors with language and say unclear things all the > time. We have two words there. It's worthwhile to exploit their > connotations. If you choose not to do that, OK, but I think it's > inefficient and unnecessary to make them synonymous.

Sorry Aaron, I didn't mean to come down hard on you. I understand you were trying to give us all a way out of a possible impasse. I agree it's inefficient. But it's also a bad idea to have a discontinuity of meaning for a term in the literature, and there's no doubt in my mind that "pure" and "just" have been used as synonyms in this context for a very long time, both having perceptual "connotations". And besides, we've already got a word meaning "relating to ratios", namely "rational", so if we were to use "just" for this as well, that would be inefficient too.

> What Kyle wrote there is not redundant.

I don't think so either.

> 'Just-intonation' has a > theoretical connotation and 'pure tuning' has a perceptual > connotation. Anyway, it sounds like you may have an axe to grind > here, and I'm not interested in sparring, so this is my reply and it > will stand as my opinion and nothing more.

Sorry Aaron. I think you've got me wrong. Although I must admit that my fondness for logic and consistency sometimes causes me to ride roughshod over others' feelings, I don't think I have any "axe to grind" in this case. Just consciousness to raise. My description of those responses to my earlier raising of this issue on the tuning list, was intended in the nature of a tersely summarised anthropological observation. No hard feelings whatsoever. Regards, -- Dave Keenan

Message: 7403 Date: Thu, 11 Sep 2003 02:03:07 Subject: Re: Classificiation of musical scales From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "pitchcolor" <pitchcolor@a...> wrote:

> I have immediate access to the OED from my office, so here is a > copy of the current entry. Just intonation falls under number 5 as > letter b.

Yes. It's only definition 5b that is relevant here.

> 5. Conformable to the standard, or to what is fitting or > requisite; right in amount, proportion, ęsthetic quality, etc.; > proper; correct.

...

> b. Mus. in just interval, intonation, etc.: Harmonically pure; > sounding perfectly in tune. > > 1850 GEN. P. THOMPSON (title) Theory and Practice of Just > Intonation. 1878 W. H. STONE Sci. Basis Music v. §90 The > differences of the old [mean-tone] and equal systems [of > temperament], and their respective departures from just > intonation. 1881 BROADHOUSE Mus. Acoustics 353 Just > Intonation, where all the Fifths and Thirds are perfect, used only > by singers and theorists. > *** > > At any rate it seems to me that there is an emphasis is on > correctness of proportion, with aesthetic quality listed > secondarily.

I don't think you can conclude anything about relative importance from order of listing.

> In this context consider that 'harmonically pure' may > refer to numbers in correct relation rather than to an aural > perception.

I guess it's _possible_, but I thought you were previously of the opinion that "pure" had a perceptual connotation. If we now have "rational", "just" and "pure" all referring to the same mathematical or theoretical property then we no longer have any word left for the perceptual one! Here's another possibile way out: Yahoo groups: /tuning/message/29654 * [with cont.] where I suggest the adjective "JI-system" as opposed to simply "JI", for intervals like 64:81 or 32:45, and I propose a more sophisticated mathematical model of justness derived from Paul Erlich's Harmonic Entropy. You can see a proposed curve of justness versus interval size in cents at Yahoo groups: /harmonic_entropy/files/Erlich/k... * [with cont.] You may find that you have to join the harmonic_entropy group, at least temporarily, to get to see the above.

> I'm not sure where 1811 came from in your citation.

I was quoting from a 1959 edition of the Shorter Oxford, which is the best that I have ready access to, although I did look up the complete one a few years ago. So thanks for the above. Of course a change from 1811 to 1850 makes no difference to my arguments. I expect everyone else in this list is sick of hearing my arguments for the perceptual definition of "just", and would rather I just gave up. My apologies. I just couldn't resist another tilt at the windmill. :-) Regards, -- Dave Keenan

Message: 7404 Date: Thu, 11 Sep 2003 02:40:36 Subject: Precisely anything can never be achieved by any means whatever From: Dave Keenan While I was looking up that tuning list thread starting with Yahoo groups: /tuning/message/29654 * [with cont.] I noticed something I don't think I ever responded to. Me: I think JIS is a better term than RI [for intervals like 64:81] because it allows that these intervals may have a tolerance too, and don't have to be precisely rational (which can only be achieved by extraordinary digital means in any case). Gene: Precisely anything can never be achieved by any means whatever. Translation by Bob Wendell: Nothing can be achieved with total precision no matter what means are chosen in the attempt. I suppose I agree with this. But what about those electronic musical instruments, such as George Secor's Scalatron, where the frequencies are all obtained by digital frequency division (counting of pulse edges) from a single master oscillator. Do we not then have strictly rational relationships between all the notes, so long as the instrument is functioning correctly? I suppose we have to consider the possibilty that thermal or other noise will eventually cause a miscount somewhere, but I expect one could be built so that, had it been put into operation soon after the big bang, we would expect no such error to have occurred by now. So, even if they are not precisely rational, they are so many orders of magnitude closer to it, that they need to be distinguished from other instruments that do not guarantee phase-locking even when functioning correctly.

Message: 7406 Date: Tue, 16 Sep 2003 18:39:38 Subject: Re: Precisely anything can never be achieved by any means whatever From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "hstraub64" <straub@d...> wrote:

> Hmm, I did not follow that thread, but: do we need to refer to the big > bang? The limitation of human hearing should be enough to make the > qestion whether "precisely rational" or not meaningless, isn't it? > > Hans Straub

I have no doubt that it's psychoacoustically or perceptually meaningless. And indeed one can go further and show that it's immeasurable unless over infinite time. This was more a question of philosophy, in particular epistememology, and as such I suppose it is off topic. But the question is "Can we know that two frequencies are in a precise ratio N:M by knowing that the mechanism that produces them does it by counting every Nth pulse in one case and every Mth in the other, of a single master frequency source?"

Message: 7407 Date: Tue, 16 Sep 2003 18:42:43 Subject: Re: Precisely anything can never be achieved by any means whatever From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "hstraub64" <straub@d...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:

> > > > I suppose we have to consider the possibilty that thermal or other > > noise will eventually cause a miscount somewhere, but I expect one > > could be built so that, had it been put into operation soon after

> the

> > big bang, we would expect no such error to have occurred by now.

> Hmm, I did not follow that thread, but: do we need to refer to the

big

> bang? The limitation of human hearing should be enough to make the > qestion whether "precisely rational" or not meaningless, isn't it?

Certainly if human hearing is the point it will. However even if you are counting events, there is no way to make the events you count precisely of the same duration--for quantum uncertainty reasons at least, if no other.

Message: 7410 Date: Wed, 17 Sep 2003 19:56:48 Subject: Re: Precisely anything can never be achieved by any means whatever From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "pitchcolor" <pitchcolor@a...> wrote:

> We don't need infinite time to show that we can't measure the > true values. The true value of anything measured can never be > known, which is why metrology deals heavily with statistical error > calculation.

OK. Skip the infinite-time thing. But what about the ratio thing? Surely _counting_ can be considered precise, assuming the counting mechanism is not faulty, since it only needs whole numbers.

Message: 7411 Date: Thu, 18 Sep 2003 19:11:11 Subject: Re: Please remind me From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> When calculating generators for Linear Temperaments, is root-mean- > square used, poptimal, or some other method used?

Poptimal isn't a method, but a defintion. As it happens when dealing with it, my usual system is to calculate root-mean-square, root-mean- cube, root-mean-fourth-power, and minimax (root-mean-infinity.)

Message: 7412 Date: Thu, 18 Sep 2003 19:13:05 Subject: Re: Precisely anything can never be achieved by any means whatever From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> OK. Skip the infinite-time thing. But what about the ratio thing? > Surely _counting_ can be considered precise, assuming the counting > mechanism is not faulty, since it only needs whole numbers.

But what you are counting is not absolutely precise, so it doesn't matter.

Message: 7414 Date: Thu, 18 Sep 2003 20:45:54 Subject: Re: Please remind me From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"

<gwsmith@s...>

> wrote:

> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > > <paul.hjelmstad@u...> wrote: > >

> > > When calculating generators for Linear Temperaments, is root-

mean-

> > > square used, poptimal, or some other method used?

> > > > Poptimal isn't a method, but a defintion. As it happens when

> dealing

> > with it, my usual system is to calculate root-mean-square, root-

> mean-

> > cube, root-mean-fourth-power, and minimax (root-mean-infinity.)

> > Thanks. I look at your lists a lot. However, with respect to Linear > Temperaments, as calculated in Python, for example, I still am not > clear. Per Graham Breed's web site, I've learned that the generator > for a Linear Temperament is (in simple situations) g such that > n*g+1=0 mod d (where m,n are temperaments such that m+n=d) I can

see

> that the generator is always real close the this fraction (for > example 13/31 for 12&19, with a generator of 503.4 is close to

(13/31)

> *(1200) which is 503.2258... How is 503.4 calculated, exactly?

503.4 looks like the 1/4-comma meantone fourth, so graham must be using minimax here. oftentimes minimax is not defined, but you can take the limit as the exponent in the "norm" goes to infinity.

Message: 7416 Date: Thu, 18 Sep 2003 23:18:49 Subject: Re: Please remind me From: Graham Breed Paul Erlich wrote:

>503.4 looks like the 1/4-comma meantone fourth, so graham must be >using minimax here. oftentimes minimax is not defined, but you can >take the limit as the exponent in the "norm" goes to infinity. > >

Yes, the module can do either minimax or RMS, and defaults to minimax. For simple cases, the minimax (which should probably be called maximin for consistency with game theory) can be found by choosing a generator such that each consonance in turn is perfectly tuned. For each choice, score by the most out of tune consonance and take the generator that makes this a minimum. For more details, see the optimizeMinimax method. There are so many complications, it turns out the RMS is simpler, so see optimizeRMS for that. If anybody's following the OCaml, the methods there are optimize_minimax and optimize_rms. Graham

Message: 7417 Date: Thu, 18 Sep 2003 22:57:41 Subject: Re: Please remind me From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> Another meantone that > Python calculates is a 251.7 cent generator, or 1.0, > 0.20975898813907973 basis.

that's meantone? i don't think so! :)

Message: 7418 Date: Thu, 18 Sep 2003 23:01:08 Subject: Re: Please remind me From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Paul Erlich wrote: >

> >503.4 looks like the 1/4-comma meantone fourth, so graham must be > >using minimax here. oftentimes minimax is not defined, but you can > >take the limit as the exponent in the "norm" goes to infinity. > > > >

> Yes, the module can do either minimax or RMS, and defaults to >minimax. > For simple cases, the minimax (which should probably be called >maximin > for consistency with game theory)

why? in game theory you're trying to maximize profits, in optimization or statistics (where the term minimax is found) you're trying to minimize error. maximin is where the smallest possible profit is maximized. minimax is where the largest possible error is minimized.

> can be found by choosing a generator > such that each consonance in turn is perfectly tuned. For each

choice,

> score by the most out of tune consonance and take the generator

that

> makes this a minimum.

so it's minimax, not maximin.

Message: 7420 Date: Fri, 19 Sep 2003 17:54:38 Subject: Re: Please remind me From: Graham Breed Paul G Hjelmstad wrote:

>You didn't read the whole post. This is exactly 1/2 of 1/4-comma >meantone (503.4 cents). It's the calculation that Python does >for 5&19 temperaments, close to 5/24 of an octave with g=5 >being (n*g+1=0 mod d) according to Graham. What I don't see is >how it 'knows' to cut this in half to "match" 5/24 > >

That's contorsion. It's covered in How to find linear temperaments * [with cont.] (Wayb.) If you get 5&19 for the 7-limit, you'll see it is different to meantone. Graham

Message: 7422 Date: Fri, 19 Sep 2003 18:57:02 Subject: Re: Please remind me From: Graham Breed Paul G Hjelmstad wrote:

>Right. Sorry to be a pain, but could you tell me how 497.1 cents >is calculated as the generator for 5&12 (7/17)? Is is based on 81/80 >or something else? > >

It's 200*log2(5.6) by the looks of it. It does temper out 81:80, but also a 7-limit comma such that 7:4 is approximated by two perfect fourths. The minimax optimum is where 7:5 is just. Graham

Message: 7424 Date: Fri, 19 Sep 2003 21:26:41 Subject: Re: Please remind me From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> You didn't read the whole post. This is exactly 1/2 of 1/4-comma > meantone (503.4 cents). It's the calculation that Python does > for 5&19 temperaments, close to 5/24 of an octave with g=5 > being (n*g+1=0 mod d) according to Graham. What I don't see is > how it 'knows' to cut this in half to "match" 5/24

I don't know what Graham does, but these sort of (con)torsion problems are always eliminated if you use wedgies to define temperaments. In the 5-limit case, we have h5^h19 = (81/80)^2 = [-8, 8, -2]. The corresponding wedgie is 81/80, or [-4, 4, -1] in monzo terms, and this is simply (5-limit) meantone.

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