This conversation reminds me of the book "Godel, Escher, Bach" by Douglas Hofstadter. It is relevant enough to read if you are interested in this sort of thing.<br><br><a href="http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach">http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach</a><br>
<br><br><div class="gmail_quote">On Mon, Oct 5, 2009 at 1:18 PM, Andrew Salch <span dir="ltr"><<a href="mailto:asalch@math.jhu.edu">asalch@math.jhu.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
I think it's most useful, for musical purposes, to regard mathematics as the study of (particular kinds of) structure, not necessarily those related to numbers; in higher mathematics one certainly considers many kind of structure that don't necessarily relate in any clear way to any kind of numbers (for example, the semigroups which are used to model the way meaning is formed, in mathematical linguistics). A musical example: the mathematics we can use to classify all the transpositions of a row, in a piece of serial music, is much more about structure than it is specifically about number or numbers.<br>
<br>
I think the questions about number and numbers are interesting but they might be sort of a red herring, if what we're really interested in is the application of mathematics to music.<div><div></div><div class="h5"><br>
<br>
<br>
On Mon, 5 Oct 2009, lea nicholson wrote:<br>
<br>
<blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
I think really we would have to address the question "What is a number?" first. Obviously, Russell and Whiteheads "Principia Mathematica" comes to mind here.<br>
<br>
<br>
On 5 Oct 2009, at 17:01, Manannan Mac Lir wrote:<br>
<br>
<blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<br>
<br>
----- Original Message -----<br>
<br>
I think the question of what the quality of a number is is the interesting one. On this topic I am ignorant.<br>
<br>
<br>
From: "hans w. koch"<br>
To: <a href="mailto:microsound@microsound.org" target="_blank">microsound@microsound.org</a><br>
Subject: [microsound] Subject: Re: Bach and mathematics<br>
Date: Mon, 5 Oct 2009 15:46:36 +0200<br>
<br>
<br>
actually, if one looks close, bach is much more about symbols and<br>
numbers, than about mathematics.<br>
he would e.g. put as many notes into a chorale prelude as was the<br>
sum of his names letters taken as numbers. etc.<br>
what makes people think of mathematics is the structural clearness<br>
of his canons and fugues etc.<br>
but, on the other hand he had quite a reputation in leipzig for<br>
playing very entertaining coffee house music with some friends.<br>
<br>
whereas beethoven, who comes across so emotional, was known to<br>
carefully calculate his pieces on whatever was at hand, up to the<br>
point,<br>
that once he used the window-shutters of his summer vacation<br>
residency to scribble calculations all over, which the owner of<br>
that residency sold for a good price<br>
as a souvenir to some fans.<br>
<br>
in renaissance, when they composed the most complicated canons,<br>
which sound so expressive and lush (e.g."missa prolationum" by<br>
ockhegem), the prevailing idea was<br>
to compose for the greater glory of god. so some aspects of the<br>
composition were supposed to be only intelligble by god, while the<br>
other aspects remained accessible for human listening as well.<br>
<br>
hans<br>
<a href="http://www.hans-w-koch.net" target="_blank">www.hans-w-koch.net</a><br>
<br>
Message: 6<br>
Date: Mon, 5 Oct 2009 14:24:45 +0300<br>
From: Batuhan Bozkurt<br>
To: <a href="mailto:microsound@microsound.org" target="_blank">microsound@microsound.org</a><br>
Subject: Re: [microsound] Bach and mathematics<br>
Message-ID: <<a href="mailto:213FEAC1-79D0-4009-BA02-70C6031BA323@batuhanbozkurt.com" target="_blank">213FEAC1-79D0-4009-BA02-70C6031BA323@batuhanbozkurt.com</a>><br>
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<br>
Hi Ismael, I think this is an interesting subject.<br>
<br>
Could you please provide the source of the article? There we can see<br>
how the article approaches the inner workings of Bach's work and maybe<br>
than can provide a framework for the discussion.<br>
<br>
In my opinion, the notion that integrating mathematics into music,<br>
makes the art form seem more difficult and incomprehensible for others<br>
is flawed. In this particular case, I think composing baroque music<br>
already "needs" know-how, and is difficult regardless of the inclusion<br>
of mathematics into it. It needs previous exposure, ear training,<br>
analysis, studies, experience, many stuff. One simply isn't born with<br>
it, and occasional listening just won't cut it for anyone except the<br>
extremely talented.<br>
<br>
And the case is similar with mathematics. Here I must say that some of<br>
my favorite artists are mathematicians, architects, physicists,<br>
philosophers etc. (they don't necessarily have to do anything else) so<br>
I don't discriminate between the sides of an artificially constructed<br>
border which separates sciences and fine arts. I see nothing wrong<br>
with pursuing a mathematical integrity in a particular work or between<br>
a body of works, on various time scales; this is just another approach<br>
to artistic composition and the approaches are governed by personal<br>
preferences (i.e. what an individual thinks is worthy of taking<br>
inspiration from).<br>
<br>
Mathematics is accessible to anyone, just like music. Taking<br>
inspiration from it, and using it as a basis of artistic work does not<br>
necessarily make things more difficult for anyone. It just might make<br>
it "look" difficult for those who are not interested enough in<br>
mathematics to study it in more depth. But the same situation is there<br>
even if there is no mathematics involved. Composing, (for example)<br>
baroque music might also look difficult to anyone who is not<br>
interested enough in studying the stylistic details of the era, this,<br>
in the same sense make things "look" difficult for others. Art doesn't<br>
come out of thin air, and everyone has their inspiration sources<br>
whether they are conscious about it or not. And approaching the<br>
analysis of ones work from different perspectives (mathematical,<br>
sociological etc.) would not hurt anyone I guess, I don't see a<br>
problem with that.<br>
<br>
Because of this, judging the quality of artworks by means of the<br>
difficulty of production doesn't feel right for me, because difficulty<br>
of something is subjective, depends on the choices (and by effect<br>
training) of the individual. I see this also makes you feel<br>
uncomfortable but it seems that this uneasiness is there only for<br>
mathematics. Because I see that there is a little contradiction in<br>
what you've just said; you say that you prefer some other artists over<br>
Escher and some of your reasons for this preference includes "usage of<br>
color by those people is far more difficult therefore they produce<br>
real art". This is highly subjective territory. The works of Escher<br>
has its own difficulties and others have their own. I see no sensible<br>
way of comparing them objectively, there can only be preferences. And<br>
I personally feel closer to Escher's works not because I think he<br>
makes more "difficult and real art", but because the way he approaches<br>
to material, source, form and other things appeal more to me as an<br>
individual, I also care about similar stuff. That is my preference as<br>
an individual, but I can't say that Escher makes "real art" just<br>
because we care about similar things...<br>
<br>
<blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
- Also many people talk about mathematics when they simply see<br>
repetitive patterns and simetry. For many people "mathematics" is<br>
simply "arithmetics", and for me mathematics is a far deeper science.<br>
Why people only talk of mathematics referring to baroc music like<br>
Bach's and not referring to Liszt Transcendental Studies, which sure<br>
also contain a lot of mathematics and a lot more sophisticated ones?<br>
</blockquote>
<br>
I'm pretty sure, mathematical integrity is not considered only for<br>
Bach's music. In my opinion, any time you analyze a work by using some<br>
sort of abstract thinking, logical reasoning and try to reduce the<br>
vast amount of musical information by grouping similarities etc. you<br>
are essentially doing some sorts of maths on it. I can only speculate<br>
about your question here, but in the case of Bach's music (and in the<br>
body of some other baroque music too), the mathematical integrity on<br>
some of the works tend to stand out more, because the creator of the<br>
particular work seems to be mainly inspired by abstract thinking.<br>
Sometimes you can really see that the artist tried to limit him/<br>
herself to pursue a mathematical integrity in a particular work. One<br>
can approach analyzing, say, Escher's repetitive, self-similar tile<br>
based works by abstract thinking and it immediately becomes obvious<br>
what he tried to achieve, how he tried to be creative between the<br>
borders of self imposed limits for creating something. Similarly, one<br>
can also do the same while trying to analyze how Picasso dissects and<br>
reduces a form of something to its essentials, and might conclude that<br>
while there is some deterministic direction in how he tries to achieve<br>
the final form of something, his intentions are not directly guided by<br>
mathematical constructs. That would mean that he mainly relies on<br>
other inspirational sources (and/or self imposed limits for artistic<br>
expression) which might be obvious for someone who knows what he is<br>
really concerned about. It might be very easy to see it for someone,<br>
but really difficult for others who are not familiar with it.<br>
Essentially the same with how the integration of mathematics in<br>
analysis makes a work seem like for others.<br>
<br>
That said, as a last note, I don't really believe that Bach was a<br>
hardcore mathematician in any sense, and relied primarily to that<br>
while creating his pieces. His ability to take really simple,<br>
seemingly natural mathematical constructs and use them in really<br>
efficient and striking ways astonishes me, and one can see that in<br>
some pieces he really tried to achieve a strict mathematical<br>
integrity. But most of those mathematical constructs are more or less<br>
common for the baroque era, I personally care about how he used them<br>
to create such beautiful music.<br>
<br>
I haven't seen the paper you've mentioned, so I must say that, while<br>
looking for hidden patterns, little mathematical wits are fun and<br>
educating, but searching for very advanced stuff and attributing them<br>
to Bach's conscious compositional thinking model would be highly<br>
speculative in my opinion (thought I can't cite anything about this,<br>
I'm speculating). Those constructs might really be there, but after<br>
all, there must be a formal way of explaining why one likes a<br>
particular piece of music anyway (which probably will never be<br>
expressed with an elegant mathematical formula). As an example, one<br>
might be able to find "golden ratio" in effect in just about any<br>
artistic creation; but not all artists know what golden ratio is<br>
formally, it might be here and there, just because of exposure and<br>
familiarity. Similarly one might also analyze a Bach piece to death,<br>
to find advanced mathematical constructs that makes it sound<br>
beautiful, but finding them doesn't necessarily mean that the artist<br>
put them into the piece by making rigorous mathematical calculations<br>
consciously. Nonetheless, I think there is no problem in approaching<br>
analysis in that way unless the results of findings are attributed to<br>
the artist in that way.<br>
<br>
Best,<br>
Batuhan Bozkurt<br>
/* <a href="http://www.earslap.com" target="_blank">http://www.earslap.com</a> */<br>
<br>
<br>
<br>
<br>
On Oct 5, 2009, at 11:34 AM, Ismael Valladolid Torres wrote:<br>
<br>
<blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
Recently we have discussed about an article in a very popular spanish<br>
blog talking about the relationship between J.S. Bach's music and<br>
mathematics. It's very common to relation both, but that relationship<br>
has always made feel uncomfortable, mainly because of two reasons.<br>
<br>
- Many people suffer because they feel they don't understand art (as<br>
if art were understandable at all!) and they often search for ways to<br>
"measure" art. This makes them feel comfortable, as thus they can call<br>
"artist" to someone that simply makes use of his know-how to make<br>
something apparently difficult for the rest. Internet people often<br>
treat Escher as the best painter ever. Escher's drawings were tricky<br>
and enjoyable, but i.e. usage of color by people like Picasso, Miro,<br>
Malevich, Kandinsky, etc. is for me far more difficult and real art.<br>
<br>
- Also many people talk about mathematics when they simply see<br>
repetitive patterns and simetry. For many people "mathematics" is<br>
simply "arithmetics", and for me mathematics is a far deeper science.<br>
Why people only talk of mathematics referring to baroc music like<br>
Bach's and not referring to Liszt Transcendental Studies, which sure<br>
also contain a lot of mathematics and a lot more sophisticated ones?<br>
<br>
Nonetheless I'd like to know the truth about the relationship between<br>
Bach and mathematics, even if he really worked as a mathematician as<br>
some say. Also of course I'd like to know your opinion about the<br>
relationship between Bach's music (and others' music!) and<br>
mathematics.<br>
<br>
Any comments, ideas, welcome, so thanks in advance.<br>
<br>
Cordially, Ismael<br>
-- Ismael Valladolid Torres Hey there! ivalladt is using Twitter.<br>
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