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.


Monday, November 25, 2013

Practical Mnemonics For The Ukulele

This is a longer than typical entry: to summarize, it provides a technique for remembering the association between finger positions and chord names on the ukulele, as well as for remembering note names in the order given by the circle of perfect fifths.  In you are interested in how to accomplish this feat, then read on!

It is that time of year again when I lecture about practical memory methods.  Last year I remembered π to 100 decimal places.  This year, I am more interested in some remembering some practical musical information. I plan to start my Thursday class by walking through the room, ukulele in hand, telling the following story:

I open the front door of my house and step into the vestibule. There on the wall I see a huge saw, obviously for cutting logs.  However, this saw is extraordinarily curved, its entire length bent around so that it takes the form of the letter ‘C’.  I cannot imagine using such a tool.

Stepping beyond the vestibule, I look into the walk-in closet. .  I am surprised; I expect to see my dog in his crate.  Instead, there is a full-grown jersey cow, cheerfully munching on tall grass-like plants that grow from the floor.  The brush-like heads of these plants take the shape of the letter ‘G’

Walking through the house, I look next into the main floor bathroom. There I see a tall, robed, bearded man – he looks like Gandalf!  I realize that he is actually Noah.  He shaves at the sink, using a large D-shaped tool, much like an oversized potato peeler.

Walking towards the kitchen, I hear a loud buzzing sound.  I stop and glance up the staircase that leads to the second floor.  On the landing, I see a large honey bee leaving a hive that is peculiarly shaped like the letter ‘A’. The buzzing bee generates the sound that attracted my attention.

Turning towards the kitchen, I continue my walk.  I glance down the basement stairs.  At the bottom landing, I see an enormous bottle of rye whiskey.  The bottle is exceedingly strange; it has three long horizontal tubes coming from its side, giving it the shape of the letter ‘E’.

I finally reach the kitchen. I see a small child, a tot, working by the gas stove.  He stands on a chair in order to reach the burners.  He skewers the letter ‘B’ onto a long stick, and toasts it over the open flame.  How will it taste?

In the middle of the kitchen is the large, yellow kitchen sink.  I glance into it.  There I see an enormous, braided, rawhide dog chew.  Someone has painstakingly shaped it into an ‘F♯’.  Ah, I think, a new musical dog chew!

At the end of the kitchen is a room that contains the refrigerator, and has a small counter upon which the cats have their containers of water and kibble.  My cat Phoebe is there, watching me.  She is wearing an enormous, long, wide black tie.  The tie has a musical theme, covered with gaudy yellow ‘C♯’s of different sizes. I think that the ‘C’ stands for cat, and the ♯ indicates ‘sharp-dressed’.

I leave the kitchen, and enter the dining room. On the table rests a large bowl filled with red juice, and decorated with the same pattern as a bottle of V8.  I look at the juice in the bowl.  On its surface, perhaps created using sour cream, I see the shape ‘G♯’.  I assume that G means that it is good for me, and that the ♯ warns me that it is very spicy.

Beside the dining room table is my Baldwin piano.  On its bench sits my mother.  She is repeatedly pounding a single, enormous, black key.  The key is ‘D♯’, her favorite note.

I pass from the dining room into the living room. There, on the couch, reclines a woman.  I only see her bare feet.  Her toes, covered in elaborate nail polish, draw my attention; each bright pink toenail has a green ‘A♯’ inscribed on top.

At the end of the living room, I notice my favorite brown recliner.  A tired policeman rests there, his feet raised.  He is in full uniform, with many decorations.  I notice a distinct ‘F’ on the sole of each of his shoes.  I realize that he is Toronto Police Chief Bill Blair, and that he has been stomping on the Fords.  This has marked his shoes.

The images in the story above are novel, bizarre, and dynamic; this makes them highly memorable. In fact, after designing the images, I learned each of them as well as their position in my ‘memory palace’ – the main floor of my house -- after only a couple of walk-throughs.

So what is the purpose of this story?  Why am I carrying my ukulele when I tell it?

First, each part of the story pairs two key images together.  One image is of a ‘peg’; this is an image that is first converted into a word, and then is converted into a number according to the famous Major Method, which maps consonant sounds into digits as follows:

Consonant Sounds
sh, ch, g, j
p, b

In the first image, the ‘peg’ is the saw, whose consonant sound is ‘s’, which is converted into a 0.  The peg of the second image is the cow, whose consonant sound is ‘k’, which is converted into a 7.  The table below lists the full set of peg images in the story.

The second image concept in each part of the story is a shape that maps into a musical note.  For instance, the shape of the saw in the first story image brings to mind the note ‘C’, while the shapes of the heads of grass in the second story image brings to mind the note ‘G’.  The table below also provides the full set of note images.

The purpose of the story is to help me remember key information about the ukulele.  Currently, I am learning ‘closed form’ chords.  These chords involve pressing down each of the four strings of the instrument.  They are practical because one can move the same chord shape up and down the fret board, playing the same kind of chord, but in a different musical key.  For instance, if I use my index finger to press down on each string along the same fret, the result will be a 6 chord.  The specific chord depends upon which fret I use: if I press on the first fret, I will play a C♯6 chord; if I press on the seventh fret, the result is a G6 chord.

Each of the twelve images in the story connects a particular number to a particular musical note, linking the root of the chord (for the subset of chords whose root comes from the C-string on the ukulele) to a fret number.  So, if I want to remember what fret to use to play a chord whose root is A (such as A6), then I remember the A-shaped beehive on the stairs, with the buzzing bee, and realize that I must use fret 9 (because bee = 9).  Each of the twelve images in the story provides musical meaning to my hand positions on the instrument!

‘Peg’ Image
Major Method Logic
Note Image
S = 0
C (saw shape
Front Closet
C = 7
G (grass heads)
N = 2
D (razor)
B = 9
A (bee hive)
Basement stairs
R = 4
E (bottle shape)
T T = 11
B (letter cooked)
Kitchen sink
Ch = 6
F♯ (chew shape)
Cat dish beside fridge
T = 1
C♯ (tie pattern)
Dining Room Table
V = 8
G# (floating in bowl)
M = 3
D# (giant piano key)
T S = 10
A♯ (on toenails)
Leather Chair
L = 5
F (on soles)

Importantly, there is even more to the story.  I used a classic technique, the method of loci, to associate each two-concept image with a particular location in my house.  As I move through the house in my memory, I encounter these images in a particular order.  The order is deliberate: I retrieve the different notes in the same order as given by a key musical concept, the circle of perfect fifths.  That is, the G in the walk-in closet is a perfect fifth higher than the C in the vestibule; the D in the washroom is a perfect fifth higher than the G in the walk-in closet, and so on.   The image below shows the complete circle of fifths; note how it matches the order of the root notes in the rows of the tables above.  Playing chords in the order given by this circle is a standard technique in jazz, and generates particularly pleasing changes from one chord to the next.

I could practice my closed form chords – for instance, all of the 6 chords – simply by moving up one fret at a time (start with C6 (all strings open, fret 0), then C♯ (fret 1), D (fret 2), and so on).  This exercise is excellent for strengthening my index finger, but hard on the ear – it is not musically interesting.  I get the same workout, but one that is much more musical, by playing the same chords in a different order: the order given by the circle of fifths.  I start with C6 (fret 0), move on to G6 (fret 7), then to D6 (fret 2), and so on according to the table above.  By keeping my story in mind, and using its images, I play the entire chord sequence in a musical order, and learn to associate finger positions with chord names.  All without having to look at a single sheet of music!