Yahoo Tuning Groups Ultimate Backup tuning Mathematics and Tuning

Hi everyone,

Just dipping in, sorry still not able to take a very active part.

But thought as a mathematician I might be able to say something helpful about the use of maths, and the rather hard to understand maths from the likes of Gene Ward Smith and other posters with understanding and inclinations to abstract maths.

As a mathematician I wish I had the time to study their work. And I can understand how you end up doing things like that.

The thing is with maths when you make it more abstract and general, you can easily see many more connections between things that seemed originally maybe quite separate. You get short and elegant proofs of things. For those with the background and inclination it can actually be much easier to use the more abstract ideas than to use more elementary concepts to prove the same thing or to understand things better.

But that then creates a translation problem for others interested in the same field who haven't got that same background, even other mathematicians from another field of mathematics.

It's obvious that it's possible to talk about tuning maths without use of those advanced algebraic ideas - since we managed fine for a number of years using nothing more than geometry, numbers and fractions. In fact geometry is a rather accessible area of advanced maths, since you can do geometrical drawings which anyone can understand. Just the subject - even in pure advanced maths, then geometrical researches are probably easier to understand by more mathematicians than algebraic type researches. And the subject is really the same just looked at from a different angle - and some proofs and ideas may perhaps be more suited for a geometrical approach even. But neither is better than the other and even geometry can become hard to understand depending how it is approached and become more abstract in various ways.

So - anyway - it's not really the fault of the mathematicians that they get hard to understand, it's more like an irresistible pull, like force of gravity, drags you towards ways of looking at the subject which make things easier for you as a mathematician.

So that may help others to understand the maths researchers better - it's not intentional, it's not trying to make things obscure or complicated or for reasons of prestige or whatever, it is just because that is what happens when mathematicians work on a subject and prove things. Inventing new names for things is also a standard thing in maths, you do that all the time, and you may even invent a new name for something for just the one paper, mathematicians expect that sort of thing to happen, because it is quicker, lets you say something in one word rather than repeat a complicated definition every time you need to use the new concept.

The other way it is important for mathematicians to remember that less complicated geometrical and other types of reasoning can often lead to new results that you miss using the more complex approaches. Just because you have all this powerful algebraic machinery doesn't mean you necessarily are better at seeing how things work.

Also in a practical field like tuning, mathematical elegance doesn't necessarily mean correctness. We are pretty complicated creatures and it is all based on how we hear things, and on how we interpret the world, which is a messy place as well in many ways. So the simplest and most elegant answer isn't necessarily always the right one.

Indeed, any kind of criterion you develop for "goodness" of a tuning or scale could become something that interests a composer who wants to see what happens if you push the parameters in the opposite direction - e.g. what is the "worst possible" tuning in that system - you could then explore those tunings, and who knows, may find something you like the sound of.

E.g. just intonation ideas lead you in this way to the idea of intervals such as the golden ratio which are as far from just as possible in a certain well defined mathematical sense - and which lead to interesting tunings - interesting not just mathematically but also for compositions as well.

So - hard to know what to do about the communications issue really. The tuning maths list has been unreadable for me for many years now just because I haven't had the time to keep up with the mathematics involved, even though I am trained as a mathematician. But clearly powerful, when I have had the need to ask questions from the researchers then they have told me things I could never have discovered myself without their research.

Basically I think the thing is as someone not actively involved in the research, just don't expect to understand it. Mathematicians generally are unable to understand each other's research except in the broadest terms, if not in the same field of research.

But that's no reason to dismiss it. It may seem like nonsense and gobbledegook - but it isn't. Well it might be occasionally, like anyone mathematicians may sometimes talk nonsense and not know it - just because they are using mathematical language doesn't guarantee that they are saying something sensible. But there is a good chance they are.

And if you have questions with clear mathematical type answers, then try asking the mathematicians and they may well be able to answer your questions and provide results you would never have thought of.

But don't give up on the geometrical and other types of easier to understand research methods as they may well still give many new results and ideas.

And if possible try to get more dialog going between the two types of researchers. Many of the tuning research ideas can probably be expressed in less esoteric type methods using terms more familiar to musicians. When there is just mutual incomprehension, well it is worth a bit of effort and try to bear in mind how hard it is to express mathematical concepts and results in non mathematical terms or even just to express them in terms understandable by a mathematician in another field of mathematics. But it is often a very rewarding exercise too.

Hope this helps. Just a few thoughts from me as a mathematician - but one who doesn't understand the tuning maths group theories - so can see a bit of both sides perhaps.

🔗Graham Breed <gbreed@...>

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "robert_inventor5" <robertwalker@...> wrote:
> Just the subject - even in pure advanced maths, then geometrical researches are probably easier to understand by more mathematicians than algebraic type researches.

Some people think it's easier, and some harder. And it's hard to separate the two; very often the simplest models for something geometric is algebraic. A well-known example is the use of real vector spaces to model the affine Euclidean space of synthetic geometry.

> But neither is better than the other and even geometry can become hard to understand depending how it is approached and become more abstract in various ways.

Tell me about it. Algebraic geometry in the language of schemes is much hard to understand than anything we get up to on tuning math.

>Inventing new names for things is also a standard thing in maths, you do that all the time, and you may even invent a new name for something for just the one paper, mathematicians expect that sort of thing to happen, because it is quicker, lets you say something in one word rather than repeat a complicated definition every time you need to use the new concept.

Exactly, and there's even more: reformulating old definitions in new terms which extend their reach and clarify their underlying nature is a standard move; in this way, mathematics tends to work itself towards a cleaner and more general presentation of a subject. Objecting that something isn't how a term was used two hundred years ago misses the point so far as mathematicians are concerned, because they are more interested in how it ought to be used.

> Also in a practical field like tuning, mathematical elegance doesn't necessarily mean correctness.

In fact, since simplifying assumptions are likely to be made, it probably doesn't. But this is true of chemistry or physics, or most things which apply mathematics.

🔗robert_inventor5 <robertwalker@...>

Hi Gene,

Yes, I agree.

A couple of practical examples may help.

For an example of something you'd discover sooner from the algebraic point of view I'd suggest your method for transforming tunes which I used in my transforming hexany piece. I dare say it could probably be expressed geometrically, but not likely to think of it.

http://robertinventor.com/musicandvirtualflowers/tunes/tunes.htm#hexany_phrase_transformations

For an example of something that you would discover much sooner from the geometrical point of view I'd suggest Erv Wilson's geometrical tuhnings - and for a particularly clear example, his mapping of pitches to the vertices of a Penrose tiling.

http://www.anaphoria.com/dal.PDF

It does have another interpretation. You can get a Penrose tiling by considering a carefully selected slice in a 5 dimensional lattice. So - you could find the Penrose tiling tuning algebraically, just not likely to think of it with the abstract algebra type approach.

I explored it a bit in Tune Smithy here:

http://robertinventor.com/software/tunesmithy/help/Penrose_tilings.htm

It is one of those things I want to explore much more when I have some time. E.g. to have an actual visual display of the Penrose tiling and a way for the user to play the notes directly on the tiling or for the fractal tune to wander around on the tiling not just follow a "straight" path along one of its rows.

So anyway some people will think naturally one way and some the other way, and that adds to the richness of the subject.

And then others work much more directly and intuitively with the tunings themselves, and not with any particularly complex maths, mainly by ear.

Or, sometimes using the maths, but in "unorthodox" ways, more for inspiration as with some of Jacky ligon's tunings. If you find the maths inspiring, that can lead you to new and interesting tunings, where you use the maths intuitively, like the colours and shapes on a canvas, and something wonderful comes out of it which you can't really explain, but your ears tell you that it is great.

That relates to another discussion I see going on at the moment about whether you can take a piece in a xenharmonic tuning and find it still works in 12-et. I think a composer working in a tuning - gets inspired by features of the tuning which then affects the way the composition develops. Then - if you then retune it to 12-et - then something of the inspiration of the original tuning may in some mysterious way carry through into the 12-et version. It is a 12-et piece, but perhaps you would never have written it in quite that way without first writing it in the xenharmonic tuning. It may still sound best in the xenharmonic tuning, if that was the composer's original intention, but something of it may survive in 12-et.

Yes indeed, the simplifying assumptions, and also - because of the recursive nature of theory about tunings. As soon as you theorise something, then composers can take your theory and use it to explore things they mightn't otherwise, including things which run counter to the theory deliberately, just to see what happens. So - unlike most physics or chemistry etc., you also have the addition of the human response to the theory - more like sociology and politics in that respect. Or like maths itself for that matter, in e.g. the Godel sentences.

Here btw is a video I did of the golden ratio tuning - coupled with the golden ratio rhythm - which is an example you would probably only think of doing as a result of studying tuning theory - and then you look for the most inharmonic tuning instead of the most harmonic possible:

http://www.youtube.com/watch?v=oDMqkGFJsBo

As it is a pleasant sounding interval then I'm sure more sophisticated ideas such as harmonic entropy will find it to be a reasonably harmonious interval - so it is a very simple example, but just shows how the human response to theorising can lead to new ideas that the original theory didn't cover.

So - not attempting to prove it here, but I suspect it is a subject which by its nature can never be completed because of this recursive aspect to it. While with physics for instance, though it seems very unlikely to some, you have that idea in the back of the mind that perhaps, if one just knew enough and had a clever enough theory, you could find all the laws of physics. I don't know if that is possible in physics. whether or not I think it can never happen in tuning theory, though hard to prove conclusively, but because of the human and recursive nature of the subject.

In maths though you go to cleaner and more general presentations - it's possible sometimes to miss things as a result. Things that were familiar to previous generations and which are the reasons the theory got developed in particular ways get forgotten. E.g. all the "pathalogical" examples of nineteenth century analysis get forgotten now that we have theories that get you very quickly to a position where you never need to encounter them.

So anyway I think the main thing is that people should do whatever they find inspiring. And if someone finds the abstract algebra inspiring, others who find it hard to understand need to respect and understand that to some this is a very inspiring way into the subject.

In the same way some find geometrical approaches inspiring and easy.

Some find working directly with the tunings themselves inspiring, and some use maths, but in an intuitive way that often irritates the mathematicians who can't understand why it works - but can't really deny that it does because the tunings they produce as a result speak for themselves.

Then there is the very rich approach based on history and tradition and cultural ways that tunings have developed over many generations in various cultures.

So I think we need to respect that there are many experts on these lists with different backgrounds, and to work together and respect other's special approaches to the subject.

BTW nice to see you back on the lists again Gene, and sorry I can't take part more. In fact at present I'm nearly 100% thinking about rhythms rather than tunings with my Bounce Metronome Pro program, well - also on the music therapy aspect of Tune Smithy. And not doing any microtonal composing either, not done any for a few years now just a phase I'm going through.

I'll be working on Tune Smithy microtonally again maybe some time next year, and also hopefully get back to my microtonal composition again which I so enjoy doing, if so will probably get more active on these lists again.

Robert

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "robert_inventor5" <robertwalker@> wrote:
> > Just the subject - even in pure advanced maths, then geometrical researches are probably easier to understand by more mathematicians than algebraic type researches.
>
> Some people think it's easier, and some harder. And it's hard to separate the two; very often the simplest models for something geometric is algebraic. A well-known example is the use of real vector spaces to model the affine Euclidean space of synthetic geometry.
>
> > But neither is better than the other and even geometry can become hard to understand depending how it is approached and become more abstract in various ways.
>
> Tell me about it. Algebraic geometry in the language of schemes is much hard to understand than anything we get up to on tuning math.
>
> >Inventing new names for things is also a standard thing in maths, you do that all the time, and you may even invent a new name for something for just the one paper, mathematicians expect that sort of thing to happen, because it is quicker, lets you say something in one word rather than repeat a complicated definition every time you need to use the new concept.
>
> Exactly, and there's even more: reformulating old definitions in new terms which extend their reach and clarify their underlying nature is a standard move; in this way, mathematics tends to work itself towards a cleaner and more general presentation of a subject. Objecting that something isn't how a term was used two hundred years ago misses the point so far as mathematicians are concerned, because they are more interested in how it ought to be used.
>
> > Also in a practical field like tuning, mathematical elegance doesn't necessarily mean correctness.
>
> In fact, since simplifying assumptions are likely to be made, it probably doesn't. But this is true of chemistry or physics, or most things which apply mathematics.
>

🔗robert_inventor5 <robertwalker@...>

Hi Graham!

Yes - and I dare say it can happen the other way around too, to spend a lot of time solving a problem geometrically and then find a simple algebraic solution.

Yes that's natural, a lot of the maths is rather simple once you understand it. Once you "get it" you wonder what all the fuss was but then when you try to explain it to anyone else you realise it is still just as hard to understand for a newbie as it ever was. Which is why the texts teaching it tend to be so hard to understand.

Some maths is truly intricate, and many proofs are, but many of the basic concepts are really simple, not more complicated than e.g. the concept of a line, or point, in geometry, just unfamiliar until you "get it".

I've done a fair bit of algebraic stuff too, and enjoyed a lot of it. Some of the foundational stuff I studied e.g. model theory, was highly abstract, one of the most abstract areas of maths in a way.

In fact only really got involved in geometry myself after I started doing research into non periodic tilings and discovered how rich the modern subject of geometry now is - that was as a graduate so quite late in life. Then after that of course after I started finding out about tuning theory and finding lots of connections there e.g. with my previous interest in four dimensional and higher dimensional geometry. Not particularly interested in geometry at school or as undergraduate.

Anyway glad to see you are still active in these researches and keep it up!

Robert

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "robert_inventor5" <robertwalker@...> wrote:
>
> Hi Robert! That's an interesting post. What I'll say now
> is to this:
>
> > It's obvious that it's possible to talk about tuning
> > maths without use of those advanced algebraic ideas -
> > since we managed fine for a number of years using nothing
> > more than geometry, numbers and fractions. In fact
> > geometry is a rather accessible area of advanced maths,
> > since you can do geometrical drawings which anyone can
> > understand. Just the subject - even in pure advanced
> > maths, then geometrical researches are probably easier to
> > understand by more mathematicians than algebraic type
> > researches. And the subject is really the same just
> > looked at from a different angle - and some proofs and
> > ideas may perhaps be more suited for a geometrical
> > approach even. But neither is better than the other and
> > even geometry can become hard to understand depending how
> > it is approached and become more abstract in various ways.
>
> A lot of it is geometry, and we have talked about the
> geometry on tuning-math. Personally, I don't find geometry
> easy to understand. I'm not even good at visualizing the
> usual three dimensions. So I put a lot of work into
> solving an algebraic problem, and then discovered that it
> led to a simple geometry. If you want to learn more about
> it, of course, you should ask on tuning-math.
>
> Another thing I'll state is that I'm not a mathematician in
> the sense that you and Gene are. I've spend a lot of time
> going through almost impenetrable graduate-level texts to
> try and get what I need. The results tend not to be that
> complicated but there may be because I have to understand
> them.
>
>
> Graham
>