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’ 19th 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 18th 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).
As noted above, changing the length of a musical string changes its pitch. For instance, a string of length AC in the figure above is half the length of string AB. If the former is plucked, it will generate a pitch a full octave higher than is generated by plucking the latter.
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 12th 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!
Given that the perfect fifth is generally confirmed as the most consonant of the musical intervals, it is perhaps not surprising that it is the harmonic mean of the two notes an octave apart that it stands between.