Tuning-Math Archive Section 1: 200

Message: 200 - Contents - Hide Contents Date: Mon, 11 Jun 2001 12:48:05 Subject: Re: Bobs one true tuning for the rest of his life... From: Paul Erlich I wrote:

> I think superparticulars are the smallest unison vectors for a given taxicab distance in the triangular > lattice, if the lattice is constructed Kees' way.

This seems to be true until you run out of superparticulars for the given prime limit. This happens at 81:80 for the 5-prime-limit. The first smaller unison vector obtained by searching slightly larger regions of the lattice is 2025:2048. 2048 - 2025 = 23, so it's not too surprising that the numbers in this ratio are on the order of 23 times the numbers in 80:81. In the 7-prime-limit, this happens at 4374:4375. The first smaller unison vector obtained by searching slightly larger regions of the lattice is 250000:250047. 250047 - 250000 = 47, so it's not too surprising that the numbers in this ratio are on the order of 47 times the numbers in 4374:4375. Make sense?

Message: 201 - Contents - Hide Contents Date: Mon, 11 Jun 2001 12:52:11 Subject: Re: linear approximation From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> What I'm trying to do is approximating a set of intervals and then if > the line doesn't pass through the origin and 1/1 shifts, then _all_ the > intervals are affected. So I chose to keep it fixed in order to avoid that.

But if the line passes through the origin, and 1/1 is at the origin, 1/1 shifting by +x would be tantamount to 1/1 remaining fixed and all each of the other pitches shifting by -x. If you think of it that way, shouldn't _that_ affect all the intervals?

Message: 203 - Contents - Hide Contents Date: Mon, 11 Jun 2001 13:00:43 Subject: Re: linear approximation From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> > Yes, this is exactly what I meant. > > Manuel

So we agree . . . and yet we disagree. I'm confused.

Message: 204 - Contents - Hide Contents Date: Mon, 11 Jun 2001 14:08 +0 Subject: Re: Bobs one true tuning for the rest of his life... From: graham@m... In-Reply-To: <9g2dms+eaf5@e...> Paul wrote:

>> I haven't checked back, but I think it's >> one of the unison vectors I originally gave. >

> As I recall, those didn't seem to give Fokker periodicity blocks that > quite agreed with the MIRACLE MOSs. I wonder why that is?

You mean the hyperparallelopiped doesn't agree? I don't see why it should.

> I think superparticulars are the smallest unison vectors for a given > taxicab distance in the triangular lattice, if the lattice is > constructed Kees' way. I was conjecturing to Dave Keenan that tempering > out superparticular unison vectors, with number-size proportional to N, > generally cause the constituent consonant intervals to be tempered by > an amount proportional to 1/N^2. If a unison vector is not > superparticular, but is instead K-particular, where K is the difference > between numerator and denominator, then the amount of tempering of the > consonances will be more like K/(N^2) -- i.e., more tempering.

Don't know about this, it's getting complex. But a superparticular ratio will tend to be the simplest way of expressing a relationship. The thing is, are all temperaments expressible in terms of superparticulars? I think septimal schismic is 5120:5103 and 225:224, so am I missing a superparticular? Miracle temperament has so many of them: 225:224, 243:242, 385:384, 441:440, 540:539, 2401:2400, and I'm sure there are more. Does the 11-limit make it easier to get them? Graham

Message: 206 - Contents - Hide Contents Date: Mon, 11 Jun 2001 13:39:28 Subject: Re: Bobs one true tuning for the rest of his life... From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:

> In-Reply-To: <9g2dms+eaf5@e...> > Paul wrote: >

>>> I haven't checked back, but I think it's >>> one of the unison vectors I originally gave. >>

>> As I recall, those didn't seem to give Fokker periodicity blocks that >> quite agreed with the MIRACLE MOSs. I wonder why that is? >

> You mean the hyperparallelopiped doesn't agree? I don't see why it > should.

It did when I used 224:225 and 2400:2401! Did you miss that?

> > The thing is, are all temperaments expressible in terms of > superparticulars?

No -- Pythagorean pentatonic has a unison vector of 256:243; tempered out, that leads to 5-equal. Also, very large temperaments will lie beyond the reach of the superparticulars in a given prime limit (they go up to 81:80 in 5-prime-limit and 4374:4375 in 7-prime limit, so very large temperaments may involve 2025:2048, 250000:250047, etc.).

> > Miracle temperament has so many of them: 225:224, 243:242, 385:384, > 441:440, 540:539, 2401:2400, and I'm sure there are more. Does the > 11-limit make it easier to get them?

Each higher limit will make it easier to get them, of course -- they'll be denser, and they'll extend farther out, in the lattice.

Message: 208 - Contents - Hide Contents Date: Mon, 11 Jun 2001 16:19:37 Subject: Re: high volume on tuning@yahoogroups.com From: Robert Walker Hi there, I wonder if anyone has thought about extrapolating to next year. Four times as many posts as last year for May, and this month seems as busy, already more posts this month in 11 days than in the entire month last year (and no special fantastic new discovery to boost the numbers). Just suppose for a moment that we are entering a period of exponential growth, then next year one would expect 10000 posts per month! Robert

Message: 209 - Contents - Hide Contents Date: Sun, 11 Jun 2000 13:42:22 Subject: Re: superparticular unison vectors (was: Re: Bobs ...) From: monz ----- Original Message ----- From: Paul Erlich <paul@s...> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Monday, June 11, 2001 5:48 AM Subject: [tuning-math] Re: Bobs one true tuning for the rest of his life...

> I wrote: >

>> I think superparticulars are the smallest unison vectors >> for a given taxicab distance in the triangular >> lattice, if the lattice is constructed Kees' way. >

> This seems to be true until you run out of superparticulars > for the given prime limit. This happens at 81:80 for the > 5-prime-limit. <etc.>

Wow, Paul... this is *interesting* stuff! -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 212 - Contents - Hide Contents Date: Wed, 13 Jun 2001 02:06:20 Subject: Antti Karttunen From: Pierre Lamothe I'm glad to see << tuning-math lives on . . . >>. It was a motivation to invite on a such specialized list a mathematician like Antti Karttunen. I asked to him if he would be eventually interested to post on tuning-math. His answer was << Why not >> Antti has many contributions in the "Neil J.A. Sloane's Encyclopedia of Integer Sequences", some of them about the Stern-Brocot tree. He has actually a Maple module in development (gammoid.txt) about /harmoids/gammoids/gammiers. Even if he has first to complete projects before to post here, I wanted to signal its existence and refer to its web page. I would add I greatly appreciate, in private email, his constant research of the substance behind the words. See <404 Not Found * [with cont.] Search for http://www.megabaud.fi/~karttu/ in Wayback Machine> Pierre

Message: 213 - Contents - Hide Contents Date: Wed, 13 Jun 2001 02:11:54 Subject: Re: Math models (about Hypothesis) From: Pierre Lamothe Robert C Valentine wrote: << The way I tend to think of things (musically directed) is that a scale is the set of distinct intervals and a mode is a specific rotation of that set. Therefor, if a two dimensional matrix were made of all the rotations (modes), viewing a column (class) would have no more than two intervals. >> It's almost what I presumed from the Paul's answers. These structured notions invite to algebraic treatment. I don't begin inside this post and give only here an example showing how to transform the implied notions in mathematical objects like matrices and lattices. d(M) = (s L)(0 1 1 1 1 0 1 1 1 1) == "LssssLssss" (1 0 0 0 0 1 0 0 0 0) o o o o o o o o o o . . . . o . . . . ----- As alternative to the matrix where the classes would appear in columns, I would use however the interval matrix where the classes appear in diagonals, for that is the fundamental object from which the other notions are derived and on which we have to ask questions about the implied musical representation. [ ... Is the space of intervals spanned by a scale simply a space for modulation and transposition or also a space for harmony (I mean that the total chromatic of the scale would be also acceptable as a chord)? How to interpret in that context the historical treatment of the triton? Would there exist subspaces matching more adequately varied choices of harmony in time? ... ] I forget for the moment the meaning and work only on mathematical structures. We could not talk of harmony anyway without to specify the basis (s L) but I want to show eventually we can do sophisticated maths without to know the values of s and L. ----- I would propose to call _formal scale_ that category of scale defined by an elementary chain of _formal steps_ where the term "formal" means that the value of the steps, like (s L) in a bivalent system, are not determined, and where the important thing about the concept is the link with the _formal classes_ obtained by rotation (of the tonic). Since neither pitch height values (Hz) nor interval values (cents) are given, the formal scale concept would be obviously almost empty without this implicit reference at such formal classes. A formal scale has very few determined microtonal properties (we know only unison and octave - supposing we talk about octaviant system) but very much macrotonal properties. I imagine someone somewhere has already formalized something about that but I presume the algebraic properties implied by such concepts perhaps have not been yet well explicitated. It will be a good opportunity to show here the utility of an abstract theory like the one defining the chordoid structure. The interval set, at this level, has not to be numerical values, neither frequency ratios nor octave ratios. The simple set of coordinate vectors representing the intervals, in the step generator acting as a basis, may have a chordoid structure. More, I think to define a new object, the f-gammier or "formal gammier", using only such coordinates. It appears to me easy to define and handle a such formal structure since almost conditions of gammoids are already implicit. I will explain that later. For the moment, I would like only to discuss about the _class_ concept applied to formal scale. In precedent discussion with Robert Walker about propriety and Paul Erlich about constant structure I had problem with the use of the term class, for instance in Monzo's definition. I recognize that this current use matches the sense of "equivalence class" in mathematics, but only if we interpret the term class as a formal class. If values are given to (s L) then the formal classes don't become forcely numerical classes. For instance, the formal space S\S corresponding to the formal scale S == LLsLLLs have true formal classes, since there exist a true formal partition. If we use s = 4 and L = 9, we have true (tempered) classes for the same reason. With s = 256/243 and L = 9/8, we have yet (rational) classes. However with s = 1 and L = 2, there exist an ambiguity : the presumed classes 3 and 4 have in common the value 6. Thus there not exist a partition and the term class, in mathematical sense of "equivalence class", is not applicable. So we have to distinguish that about the concept of constant structure which corresponds to the concept of congruity condition in gammoids. We have seen that a formal property is not conserved for any set of values. Now I would like to show that another formal property, the symmetry, is conserved and has consequence for any set of values. In the example of LssssLssss, there exist no possibility to give rational values (with multiplication) to (s L) for class 5 has to be sqrt(2). The corresponding property, for any set of integer values (with addition) in (s L), is there exist a subspace which is a group. I refer here to the distinction between symmetric s-gammier (which have a such group as part) and assymetric a-gammier in "Pourquoi les gammes naturelles ne sont-elles pas symétriques ?" on my web page. The concrete things will follow. Hoping someone interested to dig in that direction. Pierre

Message: 219 - Contents - Hide Contents Date: Sat, 16 Jun 2001 00:04:34 Subject: Re: First melodic spring results From: Paul Erlich --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> I really don't know! But my guess is that the reason I like 34-tET is > vertical, not melodic. I'm modifying the program so that "tuning file > free" versions can target intervals other than exact JI, and maybe I'll > post a blind A/B (one JI-targetted tuning file free, the other with > slightly widened thirds and fifths (== narrowed fourths and sixths)), > much like 34-tET. Could be interesting! Hi John,

Perhaps you prefer wide major thirds and fifths to just ones. Brian McLaren often cites a preference for stretched intervals in the psychoacoustic literature. But then you should probably stretch the octaves quite a bit too, especially since you'll probably prefer just (or perhaps wide) fourths and minor sixths to narrow ones. Don't you think? -Paul

Message: 221 - Contents - Hide Contents Date: Sat, 16 Jun 2001 20:46:47 Subject: Re: First melodic spring results From: Paul Erlich

> But then how to explain the sensation of 34-tET? It _does_ have narrow > fourths (4 cents) and sixths (2 cents apiece).

Maybe the fourths and minor sixths are "less important" than the fifths and major thirds.

Message: 223 - Contents - Hide Contents Date: Sat, 16 Jun 2001 22:45:34 Subject: Re: First melodic spring results From: Robert Walker Hi John,

> Oh, and, at Robert Walker's request, I've allowed negative melodic > spring constants. As predicted, this can cause the matrix to go > unstable beyond some critical value, but smaller values cause the > melodic steps to spread (though why Robert wants this, I'll let _him_ > explain!). Thanks!

I'm not that directly aware of equalness of step sizes when listening to music, but when quite a fair amount is played, I think do get some impression of equalness to it. As it happens, I find a kind of raggedness of step size rather attractive. Or at least, I think I do, hard to really say if one isn't that directly aware of it as such. So, as I like your original retuning of Chopin with the irregular step sizes, would be intereseting to see what it sounds like with the steps even more irregular. That's my explanation, as far as it goes, and I'll be interested to hear it if you do retune the Chopin like that. While on the subject of adaptive tuning: I've seen some posts suggesting targetting a particular temperament as well as adaptively tuning. I wonder if one could do that by taking some particular 12 tone temperament and making semitone and tone springs that depend on the position in the scale, e.g. maybe stronger for C - C# than C# to D, or vice versa, etc, and adjust the strength of all those springs so that when on their own, with rest set to 0, causes the tuning to settle to the desired temperament. No idea if this is feasible / desirable etc., just suggesting it as an idea that occurred in case it suggests anything useful or interesting. Robert