A couple of years ago, succumbing to a certain degree of
sentimentality, I had a growing urge to take up the ukulele once again. My wife was kind enough to give me a concert
ukulele for my birthday (a Samich UK-50), and my noodling resumed. This time around, grounded in a great deal
more musical theory and practice than in my youth, and with a growing interest
in the cognitive science of music, my ukulele skills have advanced much further
than was the case in the past. I have
discovered barre chords, and have been able to use them to explore a variety of
progressions that we have stumbled on via our
explorations of jazz using neural networks.
It probably hasn’t hurt that my hands are bigger now too!

Over the past few days, I have found another use for my
ukulele: as a cognitive scaffold for exploring the mathematics of musical
sound. In August, I read Hermann von
Helmholtz’ 19

^{th}century classic*On The Sensations Of Tone*, which covered a great deal of material on musical intervals and chords. Often, when Helmholtz discusses such topics, he refers the reader to the 18^{th}century work on harmony by Jean-Philippe Rameau. A couple of days ago I acquired a translation of Rameau’s 1722 important*Treatise On Harmony*– and found myself working my way through early parts of this book with my ukulele in hand.
Rameau begins with some core ideas that date back to
Pythagorean studies of music. First, a
taught string of a given length, when plucked, generates a tone of a specific
pitch. Second, if one shortens the
length of this string, holding other string properties constant, the generated
pitch is higher.

Fretted instruments like the ukulele are founded upon this
basic principle. A ukulele is an instrument
whose four strings are typically tuned to the musical notes G, C, E, and A,
where the C is middle C and the A is the first A above middle C – that is the A
associated with a frequency of 440 hz, called concert pitch.

Consider the A string, which is the bottom-most string in
the figure below. Once the ukulele is
properly tuned, if one plucks this string, one hears concert pitch. One can change the pitch of this string by
pressing down on one of the ukulele frets, which results in the string becoming
shorter. For instance, if you press down
on the ukulele’s second fret and pluck this string, the result is hearing a
note a full tone higher than A, B.
Pressing down on the fourth fret produces a note two full tones higher
than A, C#. Pressing down on the twelfth
fret produces a tone an octave higher than A, represented as A’ (or A
880). The first figure below indicates the
sequence frets to press to play the succession of notes in the A major scale
using this one ukulele string, beginning with the open string (i.e. plucking
the string without holding it down anywhere).

Rameau’s theory begins by considering different ways in which the length of a set string (like the A string of the ukulele) could be divided into segments of equal length. Consider the figure below. The full string, labeled ‘1’ in the figure, has length AB. The simplest way to divide it into two segments is to find its midpoint (C in the string labeled 2). The line segment AC is exactly half the length of line segment AB, as is the line segment CB. The figure shows how the original string can be divided into three equal segments (line 3) as well as into four equal segments (line 4).

This claim of Rameau’s is essentially a citation of a
Pythagorean discovery, and not surprisingly it can be easily confirmed with the
ukulele. I measured the length of the A
string on my ukulele, which (rounding to decimal places) was 37.47 cm from
saddle to nut (see the figure above). To
generate A’ an octave higher, I require a string that is half this length, or
18.73 cm. I measured this distance up
from the saddle of my ukulele, and found that it took me exactly to its 12

^{th}fret. As shown in the first figure above, one produces A’ by pressing the A string down on this fret.
Rameau goes on to consider strings of lengths in between AB
and AC. For instance, what if one
plucked a string that was average in length between these two?

Interestingly, the result of doing this depends on how one
defines ‘average’. One approach is to
take the

__arithmetic mean__of the lengths AB and AC. This is equal to (AB + AC)/2 = (37.47 + 18.73)/2 = 28.10 cm. If one measures this distance along the A string from the ukulele’s saddle, the endpoint is at the ukulele’s fifth fret. Pressing this fret down and plucking the string produces the note D (see the first figure). This is the fourth note of the A major scale, and is five semitones above A (a musical interval of a perfect fourth).
Mathematically, all of this makes sense. The Pythagoreans observed that the length
ratio between a string that produced one pitch and a string that produced a pitch
a perfect fourth higher was 4:3. Note
that 28.10 = ¾ * 37.47. That is, if a
string of length AB produces the note A, then a string of length ¾ AB produces the note D a perfect fourth
higher. In the line diagrams given
earlier, the segment AH in the line labeled ‘4’ (i.e. the line divided into four
equal segments) has a length of ¾ AB. Thus the line segment AH is the arithmetic
mean of AB, and is associated with a tone that is a perfect fourth higher than
the tone associated with a string of length AB.

A second approach to defining ‘average’ is to compute the

__harmonic mean__, which is a measure of central tendency that is less familiar than the arithmetic mean, and is used to compute averages for sets of numbers that have a few outliers. The harmonic mean of two numbers x and y is (2xy)/(x+y).
What happens if we compute the harmonic mean of the two
lengths AB and AC? Using the equation
for the harmonic mean given above, we compute (2 * 37.47 * 18.73)/(37.47 +
18.73) = 24.98. That is, a string with a
length of 24.98 cm is the harmonic mean of the lengths of AB and AC. Note that this value is substantially smaller
than the arithmetic mean calculated earlier.
Indeed, if we measure this distance from the ukulele’s saddle, we reach
the seventh fret, not the fifth fret!
Earlier, the first figure demonstrated that by pressing the seventh fret
of the A string we produce the note E, which is a perfect fifth above A (or seven
semitones).

Again, all of this makes
mathematical sense. The Pythagoreans
observed that the length ratio between a string that produced one note and
another that produced a note a perfect fifth higher was 3:2. Note that the harmonic mean that we computed,
24.98, = 2/3 * 37.47. In other words, a
string that has a length of 2/3 AB produces a tone a perfect fifth higher. In the line diagram given earlier, the line
segment AE in the line labeled ‘3’ has a length that is 2/3 AB, which is equal
to the harmonic mean of AB and AC, and which produces a tone a perfect fifth
higher than the tone associated with AB – a fact that was confirmed with the
ukulele!

__harmonic__mean of the two notes an octave apart that it stands between.

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