# Re: George Miller [Magical Number 7 +/-2] Part 2

Date: Mon Mar 16 1998 - 14:02:28 GMT

On Mon, 2 Mar 1998, bejoya rakshit wrote:

> A researcher named Pollack was interested in the ability to
> judge different tones of pitch. Pollack took differences
> tones of frequency ranging from 100-8,000 cps and a different
> number was given to each tone.These tones were arranged at
> equal intervals. A tone was then played and the participant
> had to say which number it was. The participant was told
> straight away if they were right, and if not, what the
> correct number was, before the next tone was played.
>
> The amount of tones in the range was varied to see if there
> was any difference in the amount the participants could
> remember.
>
> If there were just 2 or 3 tones in the range the listeners
> named all the numbers correctly. When there were 4 tones in
> the range there were mistakes but hardly ever. The
> participants frequently made mistakes when the range
> consisted of 5 or more tones. When the range had 14 tones
> the participants made lots of mistakes.

This is just another example of what I had said about how many
subdivisions you could make of a dimension before it was no longer
giving you any information, because there are so many errors.

> As the amount of information increases, the amount that can
> be remembered also increases until it reaches a point at
> which it is not possible for the brain to be able to sort and
> recognise it. The maximum the brain can deal with is known
> as the capacity. In this case the information is measured in
> bits. 2.5 bits is the point whee if it is increased,
> confusion will occur and mistakes will be made. (it is the
> capacity). 2.5 bits is equal to 6 different categories. In
> other words the participants can only identify 6 different
> pitches without making mistakes. (there are exceptions to
> this e.g musicians with perfect pitch).

Musicians with perfect pitch are special; they are able to remember and
identify every semitone (C, C#, D, Eb etc.) from the lowest sounds we
can hear all the way up to the highest. That may seem to violate
the 7 +/1 2 rule, but it's not quite as big a violation as you might
think.

First, there is "octave equivalence," which all of us have. That means
that a pitch sounds almost the same as the pitch an octave above or
below it. This is partly because of the way we hear, but also partly
because of the physics of pitch. (The following explanation should make
sense even if you are not a musician, and hence do not know what the
notes "A" "B" "C#" etc. mean; if you don't know what "octave" means,
I'll sing it for you in class!)

Pitches are sound vibrations; higher pitches are faster vibrations and
lower ones are slower. But some pitches "contain" other pitches in
them, because their speed of vibration is a multiple of the speed of
other speeds of vibration. For example, the note that the orchestra uses
to tune the instrument, the one first played by the oboe because it is
the shrillest instrument, is "A" 440, where 440 is the number of
vibrations per second. The A one octave above is A880, which is exactly
twice as fast. When an oboe plays an A440, the sound is a mixture,
mostly A440, but also A880 and other higher multiples (called
"harmonics") that we recognise.

To prove that the vibrations contain certain other vibrations in them,
test it out on the piano. Hold down the key for, say, A880 without
making a sound (that just releases the string, which is otherwise
held down and muted by the felt hammer with which the piano springs
are struck when you play a note). That string is now free to vibrate.
Next, hit the key for A440 hard, but then let go; don't hold down the
A440 key. What will happen is that you'll hear A440 loud, of course, and
then it will stop abruptly, because you let go of that key, but the A880
string will vibrate "in sympathy," and you'll fiantly hear A880. (This
is why "sympathetic vibrations" have been used as an image for the way
lovers' hearts "resonate" with one another.) The hitting of A440 made
the A880 string vibrate too, because the sound waves carried over to
it, and 440 contains 880.

If you instead hold down the note above A880, namely the B flat just
above A880, and then hit and let go of A440, the B flat string will NOT
vibrate in sympathy, because its vibration frequency (I've forgotten
what it is, but something in the 900's) is NOT a simple multiple of 440.

So, for this reason, what looks like the huge memory of someone for
perfect pitch is really just memory for one octave; the rest of
the pitches you get for free, because of octave equivalence.

And perfect pitch is even less than that, because we ALL have "perfect
pitch" for a short time after we are given just one reference tone, say,
A440. As long as that one identified pitch is fresh in our minds, we too
can identify any pitch from the lowest to the highest (if we know their
names, of course).

The reason is again, in part, harmonics or "sympathetic vibrations."
There is not only octave equivalence (the octave above a note is called
its "first harmonic") but there is also 5th equivalence: the second
harmonic is the note a fifth above the octave. So if the first,
reference note is A440, the first harmonic is the octave above it,
A880, and then the fifth above that, which happens to be E (but I don't
remember its frequency; it's about half way between 880 and 1760, which
is the next octave [2 x 880]). (Actually, it's not exactly half way, it
is half-way along the logarithm of the frequencies, but never mind;
it's not the maths that matter, but the idea that some pitches are
simple multiples or ratios of one another, and others aren't).

So whenever you hear one "reference" note, for example, A440, as long as
you remember it, you not only get the octaves "for free" (A880, A1760,
even A220, etc.), but you also get the fifths (E) the major thirds (C#)
the minor sevenths (G) and, more weakly, even the 2nds (B) and
eventually even the semitones (D#) for free from the harmonics that are
already contained in A440, because of the physics and maths of
vibrations and sympathetic vibrations. (Your eardrum, which vibrates,
already contains the harmonics, and they are coded in your cochlea
too.)

And this is all a long way of suggesting that "perfect pitch" is not
quite as remarkable as it seems. It would already be there if you could
somehow keep A440 in mind forever.

And people with perfect pitch are not much better than the rest of us
when it comes to remembering pitches that are smaller than semitones
(i.e., something between C and C#). They can hear and remember them as
slightly high or slightly low C#s, but that's about it.

But now we come to the interesting part, about BITS. When I told you the
definition of "getting information" as "getting something that reduces your
uncertainty between alternatives that matter to you," I used examples
like a sandwich machine with six buttons, which is like throwing dice
that have six sides. Tossing a coin involves just two alternatives, and
if there is something special about the number "7," there is something
special about the number "2" too!

First, consider that with the sandwich machine, you could always reduce
the 6 possibilities to 2: For any particular button, there are only two
possibilities: either it is the right one for the sandwich or it is not.
So it's either YES or NO. The same is true for the other buttons.

Let's think of the full set of buttons in this YES/NO way, but to make
the number of bits come out as whole numbers I'll change this to an
8-button machine instead of a 6-button machine:

First, with the 8-button machine, your uncertainty would be 7/8. But do
you need to ask 7 YES/NO questions to reduce that uncertainty to 0?

Well, you could start with the first button and ask "Is it this one, YES
or NO?" If your luck was the worst it could be, then you will need to
ask 7 questions (because if the next to last answer is NO then you know
the last will be YES because the number of possibilities has been fixed
at 8).

(This is related to what I said earlier about the "degrees of freedom"
of the mean, versus the degrees of freedom of the variance, which are
one less than the degrees of freedom of the mean, because the variance
is calculated on the basis of deviations from the mean, hence the mean
has already been fixed and must come out right when you are measuring
the last of the deviations).

But would you really have to ask as many YES/NO questions as that?

Wouldn't it be more sensible to divide the buttons into two groups
(equal sized, if possible, and if you know no better: say, 1-4 vs.
5-8) and then to ask:

QUESTION I. Is it in the first half (1-4): YES or NO?

If the answer is YES, then how many MORE questions do you need to ask?

[It should be obvious that if the answer is NO, the situation is the
same, except you deal with the second group instead of the first.]

You could now divide 1-4 into two groups again, 1-2 vs. 3-4, and again

QUESTION II. Is it in the first half (1-2): YES or NO?

Again, if the answer is YES, you could divide the first group, 1-2 into
two "groups," 1 and 2, and ask about 1 (only):

QUESTION III. Is it 1: YES or NO?

This means that, even with the worst luck, you would always reduce the
uncertainty to zero within 3 YES/NO questions.

For this reason, the AMOUNT of information that you get is not "7"
but 3 BITS. "Bits" means "binary alternatives," which is the same as the
number of YES/NO questions. "Binary" just means two possibilities,
which is also the simplest possible case of uncertainty.

Now you can understand what Miller meant when he spoke about identifying
the pitches:

> In this case the information is measured in
> bits. 2.5 bits is the point where if it is increased,
> confusion will occur and mistakes will be made. (it is the
> capacity). 2.5 bits is equal to 6 different categories. In
> other words the participants can only identify 6 different
> pitches without making mistakes.

With eight alternatives, the amount of information needed is exactly 3
bits. With 6 alternatives, as with the pitch identification [and the
sandwich machine], the amount of information is 2.5 bits. (With 4
alternatives it would be exactly 2 bits, and with 2 alternatives,
exactly 1 bit.)

If 2.5 bits of information is all that you can handle or remember, then
your "channel capacity" is 2.5 bits.

But as my explanation of octave equivalence and harmonics suggests, even
people with perfect pitch don't need unusually big channel capacities,
because the number of alternatives is much smaller once you allow for
octave equivalence and other harmonics:

In fact, let's count: In Western music, we have a total of 12 semitones
in an octave: A, B-flat, B, C, C sharp, D, E-flat, F, F-sharp, G,
G-sharp, A-flat [then A again...]. 12 alternatives means:

QUESTION I: Is it the first 6? YES or NO?

That's 1 bit, plus what we already know is the number of bits for 6
alternatives, which is 2.5 bits.

So 12 alternatives would be a total of 3.5 bits.

But it is actually enough to remember only the 7 whole tones -- C, D,
E, G, A, B -- because the sharp and flat notes are heard (and named) as
raised or lowered versions of these basic 7; so if you remember a C, you
automatically remember C-sharp as a just a C a half-tone higher.

And 7 alternatives are already within the Miller-limit. (Actually, the
alternatives are really even fewer than that, but this is close enough

Notice that the trick of remembering a "C-sharp" as just a raised "C"
involves RECODING. What changes is the number of "chunks" we need
to remember.

If we recode C and C-sharp and even C-flat into just one "chunk,"
namely C, plus or minus a half tone, then we don't have to remember all
three of them, only the C, and how to raise or loser it by a half tone.

All of the ways of beating the Miller limit are variants of this: The
number of bits is large, but by recoding more bits into bigger chunks,
the number of chunks is kept within the Miller limit of 7 +/- 2,
with a channel capacity of around 3 bits.

Think of recoding as a way of packing more bits into fewer chunks.

> Garner did similar research on identifying different
> intensities of loudness. In this case the capacity was 2.3
> bits or 5 categories.

But with loudness, there is nothing like octave equivalence or
harmonics. There is nothing special about a sound that is exactly twice
as loud as another sound. That is because loudness is not vibration, it
is intensity. It is the fact that vibrations can be multiples of one
another that makes pitch different.

And, unlike with "perfect pitch," there is no ability called "perfect
loudness" detection. No recoding is possible with loudness because
loudness, unlike pitch, cannot be broken into components, like
harmonics, that can be named or recombined into bigger chunks.

By the way, there is another way to play with harmonics using other
simple instruments that produce pitch: If you pluck a guitar string, it
will produce a sound. If you then put your finger on the fingerboard so
as to divide the string in half and pluck again, you'll here the sound
one octave above the first one.

This is because vibrations, like all waves, have "wave-lengths": Slow
vibrations consist of long waves (like the waves surfers surf on) and
fast vibrations consist of short waves (like the ripples in a brook).
If you halve the length of a guitar string, it is like halving the
wave-length of its vibrations, hence doubling their frequency.

The other subdivisions marked by the frets on the guitar give you not
octaves but fifths, thirds, and eventually tones. Semitones are then
halfway between the tones.

The same is true of a pipe. like a flute. Cut it in half and the sound
goes up one octave, etc.

> The results of these two experiments were very close ad it is
> not possible to establish if they are significantly different
> from one another as they used different methods of obtaining
> and analysing data and they were done in different lab
> conditions. The conclusion is that we are slightly more
> accurate in our judgement of pitch than that of loudness.

Probably because pitch is not really just a one dimensional sound: it is
made up of components, the harmonics, which can be recombined to
increase our channel capacity.

> Beebe-Center, Rogers and O'Connell experimented with
> judgements of taste intensities using different
> concentrations of salt solutions. The results for this
> showed that auditory senses are more accurate then taste.
> The capacity for taste was 1.9 bits or 4 categories.

We don't know much about taste and smell, but I suspect that they have
components too, which we could learn to use, if we had to rely more on
taste and smell. And probably there are people who are more like
"musicians" with tastes and smells: I'm thinking of the great wine
tasters, perfume makers, and cooks of the world, who can analyse tastes
and smells into the ingredients out of which they are composed. Wine
tasters and parfumiers, by the way, have a "vocabulary" for tastes and
smells, and this is a sign that they are doing some recoding into
bigger chunks and/or are using more of the available dimensions
than we are.

> Hake and Garne tested for the capacity of visual judgements.
> Participants were told to say where a pointer was between two
> markers. In one experiment they could use any number between
> 0 and 100 to describe the position of the pointer. in the
> second experiment the participants were restricted with the
> numbers hey could use to describe the position. There was no
> real real difference in the results of the two experiments.
> The capacity for visual judgement was 3.25 bits.

It's important to understand this: It means you can only subdivide a
one-dimensional continuum into about 7 categories, regardless of how
long the continuum is.

Think of line lengths. If I show you lines that vary from the shortest
at 1 inch and the longest at 1 foot, you will be able to subdivide them
reliably into about 7 categories: shortest, very short, slightly short,
"luke-long" [neither long nor short], slightly long, very long, longest.
[This means that you could not subdivide them to within an inch, because
12 subdivisions are too much to remember or identify.]

But if you now had to subdivide lengths from an inch to a yard, you
could still only subdivide them into 7 categories; you could not "carry
over" the 7 1/7-foot categories from when the range was from an inch to
a foot, to give you 21 1/27-yard categories when the range is from an
inch to a yard.

This means that your one-dimensional channel capacity of about 3 bits is
the same no matter what scale of measurement you use (inches, yards,
miles). The limit is "scale-invariant." It keeps re-appearing at all
scales.

> Coonan and Klemmer later repeated the experiment and found
> 3.2 bits to be the capacity when the pointer position was
> only exposed for a short time. When the pointer was exposed
> for a longer time the capacity was 3.9 bits.

Probably with more time you can use tricks, like eye-balling the
distance and mentally subdividing, etc.; it doesn't help you much.

> These results show visual judgements to be the most accurate
> of the senses mentioned so far.

Probably because they are not one-dimensional. We see things, even
lines, in two (or even three) dimensions, so we can sometimes use the
extra dimensions to increase our channel capacity.

> Other research has looked at the ability to judge the sizes
> of squares (2.2 bts, 5 categories), brightness (2.3bits) and
> hue (3.1 bits).
>
> The mean and standard deviation of all the results were
> calculated and showed the total range of categories we can
> judge is from 3 to 15 depending on the sense in question.
> The cause of the limitation we have for judging categories is
> unknown but we all have it. Maybe it is learnt or it could
> be biological.

The cause is that it would require too big a brain to remember
everything exactly, and to notice every detail exactly. Also, it would
not be helpful, because when we identify categories we are ABSTRACTING
-- selectively noticing and remembering some features and ignoring or
forgetting the rest. It's much better that we should be flexible in this
way, letting our lives and experiences pick out which categories are
important for us -- which alternatives "matter" -- rather than having it
all rigidly built into our brains in advance. [For one thing, if that
were true, we would all be exactly the same in our tastes and our
experiences, and we could never handle any unexpected new categories.]

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