Re: Miller: Magical Number 7 +/- 2

From: Harnad, Stevan (
Date: Fri Nov 10 1995 - 21:33:44 GMT


George Miller:

Good summary by Denise!

> From: "Baden, Denise" <>
> Date: Fri, 10 Nov 1995 16:12:38 GMT
> Unidimensional stimuli: Listeners were asked to identify tones
> varying from 100-8000 cps, by assigning each frequency with a
> number. At up to 3 tones, responses were all correct. at 4 different
> tones, occasional confusions occurred. 5+ tones led to frequent
> confusion. It was found that the amount of transmitted info
> increased linearly up to about 2 bits (4 tones) and plateaued at
> around 2.5 bits (6 tones). This means that, exceptional musical
> ability aside, we cannot pick more than 6 different pitches that the
> listener will never confuse.

Actually, given octave equivalence, this pretty much covers the whole
tone scale: C D E F# G# Bb = 6... We do here semitones, but maybe we
hear them as half-way between whole-tones. People with perfect pitch can
then tell you whether a semitone is high or low, but I doubt they can
cut the absolute pie any finer than that.

> N.B. I was interested to hear how little this varied according to the
> gaps between the stimuli i.e. whether a narrow or a wide range of
> frequencies were used.

Yes and no. If you keep presenting the same range of stimuli over and
over again, then you get the magic number of 7 categories whether the
range is big or small. For example, the two lines:




could each be divided into 7 absolute categories, but you can't have your
cake and eat it too: You couldn't use a big range AND a small range
together, because that would just be the same as the big range alone.

And there are bottom limits too: If the range is only a few jnd's
(just-noticeable differences) long, you can't get 7 absolute categories
out of it either.

> Miller speculates that this narrow range may be built into us either
> by learning or the design of our nervous systems: "we possess a
> finite and rather small capacity for making such unidimensional
> judgements and this capacity does not vary a great deal from one
> simple sensory attribute to another". However, we can identify
> thousands of faces, objects, words etc. but this may be because
> these stimuli vary in many dimensions.

The difference between the minimal relative difference (the jnd) and the
minimal absolute difference (7 categories) in one dimension is the
difference between being an much more extreme version of S, who
remembers everything but can abstract no categories, can not select or
generalise, and ourselves with our vocabulary of objects and
increasingly abstract categories. The price we pay for the abstracting,
categorising capacity is information REDUCTION: No INDUCTION
(abstraction of patterns or regularities) without REDUCTION.

A lot of this reduction is probably already done for us by our brains
from birth (so the world it not really a blooming, buzzing confusion
even in the beginning), but perhaps the most interesting information
reduction is the kind induced by learning.

> 2-dimensional judgements: the cc for determining the position of a
> dot in a square was 4.6 bits (24 positions). Saltiness + sweetness
> judgements gave 2.3 bits. The cc for saltiness only was 1.9 bits.
> loudness + pitch yielded 3.1 bits (2.5 and 2.3 bits were obtained
> for pitch and loudness respectively). So the extra dimension
> increases the cc, but not by a factor of 2.
> Miller concludes that the addition of varying attributes to the
> stimulus increases the channel capacity, but at a decreasing rate.
> So as we add more variables we increase the total capacity, but
> decrease accuracy for any particular variable. This is probably
> adaptive, as in a changing world it is better to have a little info
> about a lot of things than have a lot of info about a small segment
> of our environment.

In fact, if you do a reductio ad absurdum: if there is no information
reduction, you have infinite information about nothing (about infinitely
unique instants: even Funes the Memorious does not go that far!).

> When it comes to immediate memory, an experiment by Pollack showed
> that the amount of info transmitted increases almost linearly as the
> amount of info per item is increased.

What you left out was the counterpart in the memory for a series of
items in time that corresponds to increasing the number of items in
space by adding more dimensions: In time you can "recode" into bigger
"chinks": Miller's example was that if you heard lists of 0's and 1's
you could remember a string of about 7 of them, but if you recoded them
as the decimal names of binary strings:


then if you used the three digit code for 4-7, you could remember three
times as many digits as if you used 0-1. This is another form of
abstracting and combinatory power that is provided by language. Whereas
invariants (regularities) in the stimulus variance need to be
abstracted, these higher-order codes need to be imposed. In practice,
in language, concrete words name objects (categories, the bottom-level
chunks), for which we need to abstract an invariance (ignoring and
forgetting, reducing information) so we know what is and isn't in the
category named by the word. Then when we combine words into
sentences that define new words: we create new, higher-order chunks.
Both involve selection and abstraction: remembering some things at the
expense of others.

> To return to my point in the seminar about 'The Mind of a
> Mnemonist', if one considers that the amount of info S received in
> each stimulus was high (eg words/ numbers also conjured up shapes,
> colours, texture, sounds, smells, tastes etc) then it follows from
> Millers paper that he should have an exceptional memory. S, for
> example sees '8' as naive, milky blue and 'red' as a man in red
> shirt coming towards him. If one broke down his experience of a
> stimulus into the number of dimensions it had for him, the number
> would be phenomenally high. Pollacks experiment showed that 1 bit of
> info per item led to about 8 items being retained; 5 bits of info
> led to 35 items being retained. Therefore if the stimulus contained
> 50 bits of information, the linear scale suggests he should be able
> to record 350 items. I don't suggest that S's synaesthesia can explain
> all of his abilities, such as his long-term recall and difficulty in
> forgetting, but I believe his synaesthesia is related to a great extent
> to his phenomenal memory.

I agree that this may explain part of it, but certainly not all of it,
since S's memory seems to have been unlimited -- though one would like to
see whether he could absolutely identify right down to categories the
size of a jnd. If so, it would seem there wasn't enough dimensionality
available to explain all of that through synesthesia.

This archive was generated by hypermail 2b30 : Tue Feb 13 2001 - 16:23:56 GMT