Monday, December 02, 2013

Underdetermination, Ukuleles, And The New Look

Cognitive science frequently faces problems of underdetermination.  A problem of underdetermination occurs when some given information is consistent with many different conclusions, only one of which is correct, as illustrated in the following figure. In short, the given information does not uniquely determine the correct solution.
In cognitive science, one example of a problem of underdetermination is the ‘poverty of the stimulus’ faced by children learning their first language.  In this case, the given information is the sample of language to which a child is exposed.  The problem is that this information is in principle consistent with an infinity of different natural grammars, only one of which is correct.

Problems of underdetermination are often found in the study of visual perception.  The information provided directly to our eyes (the proximal stimulus) is actually consistent with an infinite number of different scene interpretations (models of the world).  Only one of these interpretations is correct.

We do not experience problems of underdetermination, suggesting that the human mind has mechanisms that eliminate incorrect conclusions, delivering only the correct result.  It seems that the mind provides additional information that serves as a ‘filter’ that only lets correct conclusions through, as shown in the figure below, which assumes that ‘Conclusion 2’ is correct:

One goal of different theories in cognitive science is to propose mechanisms for solving problems of underdetermination.  For instance, the poverty of the stimulus problem faced by children learning natural grammars is thought to be solved by an innate universal grammar.  This grammar provides the needed additional information.  Instead of learning a whole grammar, children are thought to face a much more tractable problem in which the given information is used to adjust a small set of settings in their universal grammar.

Similarly, perception researchers argue that visual problems of underdetermination are solved by adding required knowledge of the world.  For some, this knowledge is innate: the visual system is wired in such a way that certain general properties (neighbouring points in a scene will have similar color, depth, motion, and so on) are true.  For others – the New Look theorists – this knowledge is general knowledge of the world, which provides context that can be used to solve the problem of underdetermination.  For the New Look, seeing is literally a kind of thinking.

Cognitive scientists are not the only scholars who face problems of underdetermination.  Music theorists are often concerned with analyzing musical scores by assigning chord labels to configurations of notes.  However, different theorists may assign radically different chords to the same score, a classic example of underdetermination.  David Damschroder, in his 2008 book Thinking About Harmony, observes that “analysts guided by contrasting basic principles may offer wildly divergent views concerning a chord’s root; or, the same chord may be interpreted in different ways depending upon its context. … A chord may in certain contexts be understood as an incomplete or modified representative of some other chord” (p. 17).

Ukulele players constantly face this kind of underdetermination.  When learning their instrument they soon realize that a single finger configuration on the fret board can have more than one chord name!  Two of the many examples of this are illustrated in the figure below.  For the first pair (F6 and Dm7), note that the interpretation of the chord’s name depends upon which ukulele string is assumed to provide the chord’s root note (the string associated with the number 1 at the bottom of each chord diagram).

Underdetermination is also encountered with the ukulele because it has only four strings, and therefore can play at most four different notes at the same time.  This causes a problem if one is interested in playing chord extensions, which are defined by more than four notes.  For instance, a ninth chord is defined using 5 different notes, and a thirteenth chord is defined using 7 different notes.  It is obviously impossible to play every note of such chords on a ukulele.

The solution to this problem is to play a subset of a chord’s notes, four notes that are sufficient to provide the musical sense of the chord even when the other notes are absent.  One example of this is the second pairing of chords in the figure directly above.  If one assumes that one of the strings provides the chord’s root, then the chord could be named as Bm75.  However, if one assumes that the root is not one of the notes that is actually played, one can interpret the same set of four notes as a chord extension, G9.

A ukulele player cannot avoid chord underdetermination.  How can they cope with interpreting music, or deciding upon chord names as they compose their own music?

My suggestion is to endorse the position of the New Look theory of visual perception, and rely upon context supplied by musical knowledge.  Consider the three chord progression provided in the figure below.  From the information given above, it would be completely correct to label the first chord as being F6, and the second chord as being Bm75.  However, it is difficult to come up with a basic musical context and key in which these chord labels make sense.

If one instead names the first two chords as Dm7 and G9 (as is done in the figure), then one is really asserting that the three chords are related by a particular musical context: the II-V-I chord progression found in jazz.  In this jazz context, all three chords are related together in the key of C major.

This raises an additional interesting question: when one hears a chord progression, is their experience of the chords affected by the context that they adopt?  Do F6 and Dm7 actually sound like different chords in different contexts, even though they are played in exactly the same way on the ukulele?  The New Look theorists hypothesized that experience is indeed altered by the contexts, beliefs, and expectations that we bring into perception.


Damschroder, D. (2008).  Thinking About Harmony.  Cambridge University Press. Cambridge, UK.


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