Klarinet Archive - Posting 000470.txt from 2004/06

From: Joseph Wakeling <joseph.wakeling@-----.net>
Subj: Re: [kl] "Tunes create context like language"
Date: Mon, 28 Jun 2004 18:14:23 -0400

Ormondtoby Montoya wrote:

>My next question (since WebTV can't access the actual text) is whether
>the author claims that "musical context" is unique to just language and
>music, or is it an example of how "context" influences **all** sorts of
>perception and thought?
>

Well, I use "context" in a rather loose sense in my previous post. That
said, the sort of mechanism which is at work here in this piece is
probably *very* generic (more on this later).

A very appealing model to explain the Zipf Law (and used in the Zanette
paper to fit the data there) was proposed by a man called Herbert Simon
who is (was, sadly) something of an interesting figure in science---very
influential in a wide variety of fields including economics and
artificial intelligence. This particular work is one of his earlier
pieces, dating from 1955, and titled "On a class of skew distribution
functions" (again, more on this later).

The model works like this. Suppose we have a sequence of words forming
a text. We want to add an additional word to the text. We do so
randomly, as follows:

(i) there is a (small) constant probability that the additional word is
a new word that has never before appeared in the text.

(ii) if the word is *not* a new word, then we must select a word already
existing in the text. We pick this at random but the probability is
biased: the probability that we pick a given word is proportional to the
number of times it has already appeared. So a word that has appeared 10
times is 10 times more likely to be picked than a word which has only
appeared 1 time.

These two random processes give rise to a distribution of words (or
pitches) that matches that found in both spoken and written language,
and (according to Zanette) music. You can start with just one word and
work from there; you could (I think) start with any initial group of
words and the long-term behaviour, the more words you add, would be the
same.

(The behaviour described by Simon's model has been called a
"rich-get-richer" situation because, if we add one more word to our
text, the chances are it will go to a word which has already appeared
numerous times.)

So "context" here has a pretty specific meaning that the random process
is biased towards already-existing words, and the number of times they
have appeared. However, that doesn't mean it's not a very generic
process. Simon's title, "On a class of skew distribution functions", is
a clue in this direction: he is trying to provide a general mechanism
for producing a particular type of probability distribution which is
found in many different real-world situations. He cites 5 examples, all
of which probably do have an origin in the mechanism he proposes: (A)
the distribution of words in text; (B) distribution of scientists
according to number of papers published; (C) distributions of cities by
population; (D) distribution of income sizes; (E) distributions of
biological genera by number of species.

I can add at least one more major example to that: the distribution of
websites according to the number of links they receive also follows such
a distribution. Indeed, models closely related to Simon's have been
proposed to model quite a variety of real-world networks including the
World Wide Web, the *physical* internet (i.e. computers and
connections), social networks, and a number of other biological phenomena.

And just as a footnote: none of this tells you exactly what new words
(or pitches) you might add. For example, in tonal music one might start
with one note, then when a new note is created, it might be an octave
different; the next new note, the fifth; the double octave; and so on.
All of this is irrelevant to Simon's mechanism. So there can be outside
influences which require additional information to model effectively.
This is the price you pay for being generic: lots of detail is lost.
But, potentially, quite a lot of insight can also be gained.

-- Joe

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