Now that we have covered the five basic bluegrass
instruments, it is time to develop just a bit of music theory. I promise to
keep it lightweight and hopefully interesting.
However, before we can get into theory, it is both important and
interesting to understand some of the basic physics behind musical notes and
scales, and that is the subject of this week’s blog post.
The Overtone Series
As you may have learned in high school, sound is nothing
more than vibrations carried by the air.
And a musical note is nothing more than the air vibrating with a
distinct set of frequencies. If we take
a closer look at these vibrations, they can be decomposed into sine waves. OK, I might have just crossed the line here
into math, but stick with me… you should find this interesting. Let me back up a bit. A sine wave, pictured below, can be thought
of as a smooth ripple, sort of like what you might find if you drop a pebble
into a pond.
Let’s say you pick an A note on your guitar. The guitar
string will oscillate at a (primary) rate of 440 times every second, or 440
Hertz, which can be shortened to “440Hz”.
The air molecules right next to your guitar will vibrate at that same
rate, causing a periodic increase and decrease of pressure in the air 440 times
per second. This so-called compression
wave will subsequently propagate at the speed of sound away from
your guitar and be picked up by your eardrum, which is very sensitive to these
types of pressure variations. If
there were a magic camera that could take a snap shot and show air-pressure
readings between your guitar and your ear, it would look similar to the sine
wave pictured above. The peaks of the
wave would be high-pressure areas located about 30” apart and the troughs would
be low-pressure regions also separated from each other by about 30”. This wave would be moving away from your
guitar at 340 meters per second, which is the speed of sound in air.
Things get considerably more interesting when you look very carefully
at this sound wave. It turns out that
the waveform is not just a simple sine wave as illustrated in the figure above,
but many different sine waves superimposed on each other. Since we don’t have the magic camera, I will
use a slightly different picture to show what is happening. The figure below shows a sound wave from a
guitar and plots the pressure as it varies with time (note that this is
something you can easily display on your computer with any modern sound program
such as Pro-tools). Time goes from left
to right on the horizontal axis and the vertical axis is pressure variation.
In this figure, you can clearly see the effect of the pick
striking the strings in the left part of the diagram, which results in strong
pressure variations both up and down. As
the string settles and starts to ring, it is not a perfect sine wave but a
jagged one that develops. Have a look at
the blown up portions on the top right and the bottom right, you can see the jagged
sound wave.
It turns out that the reason these waves are jagged and not
perfectly smooth is that your guitar string was not only oscillating at 440Hz
but was also vibrating with many other different (but related) frequencies at
the same time. These additional
vibrations end up being superimposed on the main 440Hz one and are referred to
as overtones (some people call them “partials”). In the case of the vibrating A-string, there will
be overtones at 880Hz (=2x440) 1320Hz (=3x440), 1760Hz (=4x440) and all the
other integer multiples of 440Hz. These
are illustrated conceptually in the figure below:
The relative strength, or amplitude of each one of the above
individual overtone components will vary from instrument to instrument and
these relative strengths will determine the timbre of the note.
Below is a picture of these relative amplitudes in the
overtones of the guitar note that we have described above. The way to read this chart is as
follows: The horizontal axis is frequency (in Hz
or cycles per second) and the vertical axis is amplitude or loudness of the sound. You can see the overtones clearly as spikes
and that the first, second, fourth and seventh overtones are particularly
prominent in this guitar note.
To illustrate how the relative strengths of these overtones
contribute to the timbre, it is instructive to compare the guitar with another
instrument. For example, the clarinet’s
spectrum (shown in the figure below) is, indeed, most unlike the guitar. Every other partial is almost totally missing
in the clarinet whereas they are very prominent in the guitar’s spectrum.
Each instrument, including the human voice, has a unique sound
or timbre and one of the major reasons is the distinctive set of amplitudes in
the overtones.
Formation of Musical Scales
The overtone series underlies another important concept in
music – that of creating musical scales.
Again, using the example of the A (440Hz) note, the first overtone is 2x440
which is 880Hz – this is an octave of the original note. The second overtone, at 3x440, or 1320Hz is
an E note, which is an octave plus a fifth.
Then comes 4x440, which is another octave. The following frequency at 5x440 is pretty
close to a D – a fourth. And so on. If I continue to enumerate the overtones in
this fashion and then list out all the unique notes and rearrange them in
alphabetical order, I’ll end up with the following list:
A-A#-B-C-C#-D-D#-E-F-G-G#
This list is all the notes in a western chromatic scale. And if I choose the first seven (unique)
notes from the list of the most prominent overtones, it will be an A major
scale:
A-B-C#-D-E-F#-G#
That’s right, an A note played on (say) the guitar, or
mandolin, or sung with a human voice, contains all the other notes of the
scale, in varying degrees, imbedded within its overtones. This is also true for all the other notes in
the scale – they all turn out to be related to each other, and so they will
sound harmonious with each other. This
is the fundamental basis of musical scales.
Chords and Blues Scales
The notes (or more properly, intervals) corresponding to frequencies
lower in the overtone series are much more important than the ones higher in
the series. For example, if I look at the
several of the unique note values from early in our overtone series I get this:
A-C#-E
This is an A Major chord and we can now deduce why major chords
exist. Since the three notes of a major
chord come from the intervals that are low in the overtone series, playing them simultaneously will result in a harmonious
sound. This is because the frequencies
of each of these notes are closely related and the overtone series of each of
these individual notes are similar enough to reinforce each other in an
additive fashion. For a more complete explanation please see the footnote1
If you were to look carefully at the C# note, you’d find
that the corresponding frequency is a little bit flat of the C#. In fact it is between a C and a C# so I
should be able to choose either note for my chord. If I were to choose the C instead of the C#,
I’d have this:
A-C-E
This is an A Minor chord.
And nothing feels better than bending or sliding a “minor third” note towards
the major third when you are in a bluesy break.
Your bend goes right thru where the overtone-series-derived interval
lies – between the major and minor third.
There is a similar phenomenon associated with the 7th note of
the scale, giving us both a minor and a major seventh interval.
Let’s continue on and take the first five notes of the overtone
series. This is the pentatonic (i.e.
five noted) scale. Since these five
notes are close to each other in the overtone series and near the fundamental,
they have lots of related harmonics and will sound especially good when played
in sequence. Both sad sounding blues and
happy sounding country music songs are based on these so-called minor and major
pentatonic scales. If you are playing
the blues, the odds are that you are using mostly the first five notes in the
overtone series.
For similar reasons as above, playing notes at the same time
that are far apart instead of close together in the overtone series will result
in a highly dissonant sound. Let me use
the example of playing an A and A# simultaneously. These notes are close together in the scale,
but the A# is way, way up there on the A overtone series and very far removed
from the A fundamental. This means that
the overtone series of these two notes are composed of much different harmonic
components and tend not to reinforce each other. A highly complex and disagreeable sound will
result if they are played together. And
just like making up with your girlfriend after a fight, resolving those
dissonances feels especially good in musical scores.
That is enough physics for now. Next time, we will build on
this foundation and discuss the Nashville numbering system. This will provide us with tools to figure out
chords to songs and also to be able to construct harmony vocal parts to our
favorite bluegrass choruses.
Keep on pickin’
Jeff
Footnote:
Footnote:
1A major chord is made up of three unique notes
that are all early in the overtone series.
Let’s take a more careful look at this.
Using the A chord as an example, these notes would be A, C# and E. To see why a major chord is harmonious, it is
instructive to look at the intervals instead of just the notes. An interval is simply defined as the spacing
between two notes of a scale. To fully
illustrate the intervals that occur in our major chord, we will need to add an
octave “A” note at the top end of our chord (an octave above the root) giving
us A-C#-E-A’ (I’ll denote the octave A as A’)
Let’s list all the intervals:
A to A': Octave
A to E: Dominant
E to A’: Sub-dominant
A to C#: Major third
C# to E: Minor third
The most harmonious interval in a major chord is the octave
– these notes have exactly the same overtone series, so when played
simultaneously, all the overtones add up and reinforce each other. The next most harmonious interval in our
major chord is the dominant (A to E).
This is the next interval (after the octave) that occurs in the overtone
series, which means these notes have lots of overtones in common. It is referred to as the “dominant” interval
for that reason. Then comes the other intervals listed above – they are all
very early in the overtone series and each pairing of notes have a ton of
harmonics in common, which is why a major chord sounds so harmonious.
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