Waveforms and Harmony
There are theoretical compositions that only involve theoretical sound,
such as planetary motion viewed as waves.
Perception by the senses takes us back to the famous philosophical problem regarding
whether a tree falling in a forest makes any sound. Traditionally, the
answer is, "No," because 'sound' requires a hearer. This anthropocentric
solution assumes that animals cannot hear, or that their hearing does
not count. Of course, if there was just a forest fire and all the
animals were gone...well, that would explain the tree falling too, I
In the early days of music synthesizers, only a few other simple waves
were available, largely because they lent themselves to easy electronic
production. They were the square wave , the triangle wave , and the
sawtooth wave . Each of these wave forms has its own recognizable
sound, and users of primitive synthesizers used to combine them to
Music is, first and foremost, sound.1
Sound is defined as what can be perceived by the senses,2
although not necessarily human ones, or those associated with our ears
(deaf people often feel sounds, sometimes, for obvious reasons,
preferring rhythmic music with lots of bass and drums). What ears
perceive are vibrations in the air (so, in spite of what Hollywood may
want you to think, explosions in space are silent). Those vibrations are
usually called 'waves,' and can be represented graphically in images
like the following:
In this graphic (which happens to be a sine wave), the push and pull of
the air are represented by rising and falling areas on the wave (the
center line is not part of the wave; it represents 0, the point at which
the air pressure crosses over its initial state). The only way to
produce a true sine wave is with a synthesizer, but a modern flute comes
The amount of time it takes to complete one cycle is called
frequency. In music, we know that as pitch (note
value). The distance between the centerline and the highest or lowest
point in the wave is called amplitude. We usually call it
volume. Nevertheless, we buy amplifiers, because a
volumizer is something you use on your hair.
Of course, your ear hears lots of types of vibrations, not just
those represented by the sine wave, and we can hear the difference.3
Real-world sound combines all sorts of different things, and anyone who
has used recording software knows the waves look more like this:
Or, in something more akin to real time, like this:
Both of these waves are from a rhythm electric guitar. The irregular
bumps are overtones, other notes, or other instruments in the general
mix of sound. When different waves are combined, we are lost making
visual sense out of them (except for the occasional 0s), but our ears
can hear and separate, say, the flute from the trombone in a duet.
However, in order to continue this discussion with
respect to harmony, it is better to go back to the sine wave, simply
because it is much easier to see what is going on.
Polyphony is the word for two different notes occurring at the
same time. Harmony describes the constantly moving target of what
types of polyphony 'sound good.' Since sounding good is entirely
subjective, what one person regards as harmonious may well come off as
dissonance (polyphony that sounds bad) to someone else, and this has
certainly been true over time. Listeners in the 21st century don't even
raise an eyebrow at some combinations that left 14th century listeners
The most basic harmony, with which even the ancients as far back as we
can know were comfortable (so basic that most of us do not even think of
it as polyphonic), is the octave. The octave is simply a doubling (or
halving) of the frequency of a note. An octave higher vibrates twice as
fast; an octave below does the opposite, vibrating half as fast.
The second image illustrates what happens when you sound the octave
harmonic on a stringed instrument, where the first image represents that
string played open; the vibrating length is divided in half, and so the
frequency of vibration doubles. If you quarter the length of the string
(fifth fret [=4th] harmonic-not illustrated), it sounds two octaves
above the fundamental. You will keep getting higher octaves as you keep
chopping the vibrating length in half. If you are actually doing
harmonics, you create a node at the point where you touch the string,
and the string itself creates a series of other nodes, so that the
string itself is vibrating in a pattern that resembles the waves in the
article lists the harmonic series up to 20. Some have argued that
the fourth appears in the harmonic series as the distance between the
third and fourth harmonics. This same approach would allow us to find a
minor third between 10th and 12th. Personally, I think the harmonic
series only includes intervals from the root. Both, however, may sound
fine to us simply because they are inversions of the fifth and major
sixth respectively. In the 15th c. the fourth was heard as a dissonance
(Johannes Tinctoris in Terminorum musicae diffinitorium, 1473), although
it had been perceived as consonant previously during the middle ages. Modern
Western listeners go both ways, hearing it as consonant when internal to
a chord, but dissonant when exposed: Exposed on top, it seeks to resolve
to the third; on the bottom, it destabilizes a chord, and usually
resolves with the fourth becoming the root of the next chord.
Taking the top wave in the above illustration as the base, the middle
wave is an octave higher, and the bottom one is an octave lower. You
probably noticed that some of the nodes (what I have been calling 0s)
fall in the same places. This congruence of nodes is what creates
But, wave congruence doesn't always have to be in halves. If you were to
divide string producing the first wave into thirds, you would also get
nodes at the beginning and end of the fundamental wave, as in the
You will no longer perceive this as a version of the same note. In fact
it is a fifth higher (if the lower note were a C, this would
be a G).
Pythagoras (6th c. Greece) is credited with taking time away from making
theorems about triangles to discover this property in vibrating strings.
He used what he called a 'monochord,' which was an instrument with two
strings, tuned the same. He successively divided the length of one of
the strings and noted what tones were produced relative to the other. We
will return to him in a minute, but this is what you will get if you
repeat his experiment:
1/2 is one octave up
1/3 is an octave + a fifth up.
1/4 is two octaves up
1/5 is two octaves and a third up.
1/6 is two octaves and a fifth up.
1/7 is two octaves and a dominant seventh up (audibly flat).
1/8 is three octaves up
1/9 is three octaves and a second up.
1/10 is three octaves and a third up
If you have been practicing your scales, you may notice that this does
not really correspond to what westerners consider normal (the major
diatonic scale). There is no dominant seventh in our western major
scale, and we make much use of the fourth and the minor third, which are
notably absent.5 But western practice is partly molded by
cultural expectation, as with all the different ethnic expressions of
music in the world. If the dominant seventh (b♭ in the
chd>C scale) or the second (d in C) sound
mildly dissonant to you, try moving them up a couple of octaves and see
if that helps (leave the fundamental where it was).
Here is a diagram from a Wikipedia article that illustrates some of
Not everyone agrees that these fundamental harmonies are neurologically
perceived, and so built into the human experience. Although evidence
from neurology (Tramo, et al, 2001), musical perception in babies
Trainor, et al., 2002) and cross cultural similarities worldwide
although some world music seems quite different), do suggest that there
are fundamental features of harmony. There is also evidence that we
share a common perception of dissonance (Cousineau, et al., 2012). In
addition, on most instruments, the note sounded actually includes the
other notes listed above, although much more quietly than the
fundamental, and varying in relative strength from instrument to
instrument. The existence of these overtones within sounded
notes may not be audible to our conscious minds, but our ears and our
brains hear them, which may reinforce our tendency to perceive those
notes as harmonious when we hear them separated out.
Preference for extremely dissonant music may also be a psychological reaction to, or
expression of, loss of center in the modern Westerner's self-image.
However, in real life, where our natural responses are reshaped by
culture, we may come to perceive some things as appealing which push the
perception of beauty beyond natural consonance. Modern westerners prefer
many musical combinations that would have sounded dissonant to people in
our own culture in the not so distant past. Western music has
consistently 'pushed the edge'-partly as composers look for new sounds,
and partly as we reach a level of boredom with the same old
thing.6 At the same time, many of us
in the West are still uncomfortable with much Middle-Eastern,
North-Indian, and far-East Asian music, because they push perceived
consonance into areas to which we have not yet become accustomed. Often
what sounds strange about these foreign-to-us expressions of music
represents the same kinds of edge-pushing we see in the modern West.
They are wanderings from the expected natural harmonies which listeners
in those cultures have come to expect and enjoy.
As with harmony, one man's dissonance is another man's
sweet sounds. This doesn't always mean that the second guy (or gal)
doesn't know dissonance when they hear it-they may simply have come to
appreciate it. But we generally hear certain intervals as not quite so
harmonious. Minor seconds, in particular, usually grate on our nerves.
Less than a minor second does so even more. Some folks would say the
same about a major second; others don't have an issue with it. The
bottom line is that the longer time your ear has to wait for the waves
to come together, the less comfortable it generally is with the sound.
But much of music, and Western music in particular, is based on the
creative use of some level of aural discomfort, which, when it resolves
into closer harmony, makes us feel good. Sort of like a happy ending at
the conclusion of a tense movie-it makes the tension seem worth it. In
both movies and music, we call this 'resolution.'
At some point, as dissonant notes approach each other, our brain no
longer hears them as different notes. At first they just sound out of
tune, and then the approaching notes come together. Different people
perceive this unity at different times, but a trained listener can hear
when notes are almost, but not quite there. When that happens we
perceive that as 'beats' in otherwise harmonious notes. This is what you
get when the notes are so close that their coming in and out of phase
with each other is actually audible-a sort of tremolo in the note. As
the notes get closer, the 'beat' gets slower, until if finally
Many people can hear these beats even between different notes,
especially with fifths and thirds. It is listening to, and timing, those
beats that allowed people to develop the equal tempered scale which is
the standard for modern Western music (next section).
There are other ways of describing his comma, which I am not
going to burden the reader with, in hopes that she may be able to stay
awake just a little longer.
Just when Pythagoras thought he had it all worked out,
he tried to continue the overtone series around in a circle.
Theoretically, when you take the fifth of a fifth of a fifth of a
fifth... etc., until you have gone all the way around twelve notes, you
should be back where you started. More specifically, 12 perfect fifths
should get you to the seventh octave. He discovered that this was not
the case. When you get done, you are about a quarter of a tone off. This
difference is called the Pythagorean Comma.7 It is one
reason why your guitar never seems to be quite in tune. This may be when
he decided to go back to good old predictable triangles, but, truth be
told, it wasn't that big a deal—the instruments they had
available didn't cover seven octaves anyway. However, by Bach's day,
this was no longer the case, and to complicate things further, composers
wanted to be able to change keys all the time. You could get the clavier
tuned for one key, but then it was all wrong for the next. This was
exacerbated by the fact that the 'perfect third' in the overtone series
was even harder than the perfect fifth on the key change problem (or
even just changing chords within any given key).
Two cents = two hundredth of a semitone. Since the notes are obtained
logarithmically, this does not correspond to a specific change in
This statement depends on how you define "straight." It is a logarithm,
so the actual calculation to get the next note in the series is the
frequency of the current note times the twelfth root of two
(f1 = f0 * 21/12). If you regard this
as straight math, well then, there you have it!
Adopted by ANSI (the American National Standards Institute) in 1936, and
ISO (the International Organization for Standardization) in 1955.
The solution that the musical community arrived at in the
17th century (which still holds for today) was to fudge the
tuning a little. Nowadays, a piano tuner will tune all the fifths just a
hair flat8—close enough that most of us wouldn't notice.
This results in a straight mathematical division of the octave into 12
semitones,9 by means of which she can make the instrument go
around the circle of fifths and end up back where it started, and you
can play the piano in any key you want. This solution is called the
equal tempered scale. Other instrument makers followed suit.
The third is also a little off in this system, but again, not so you
notice, unless you have a very good ear; but if you actually tune a note
to what sounds like a good third, you are likely to be out of tune the
minute you change chords.
The Tuning Standard
Related to the issue of relative tuning is the question
of standard pitch. By tradition the oboe sets the standard pitch for an
orchestra by playing A above middle C, and then
everyone else tunes to that note. But what exact pitch that is has
varied over time, and still does from place to place. In the US and
Great Brittan it has been standardized at 440 Hz10
(Hertz—a measure of frequency that equals cycles [e.g.
complete waves] per second). On the Continent, it is often two or three
Hz higher. Over the last five centuries it has ranged between 400 Hz and
480 Hz. There is a movement afoot to re-standardize it at 432 Hz. The
difference is hard to hear, and the reasons given are usually
semi-religious, having to do with natural rhythms and ratios found in
the universe, but many people simply claim it has a more natural sound.
Those same people will also sometimes push for a return to Pythagorean
tuning (based on overtones, rather than a mathematical division of the
octave). It is true that the natural thirds and fifths sound a little
sweeter, as long as you stay on one key and don't use chords. This is
not a plug, just an observation.
There are, of course, similar music theories underpinning other world
expressions of music, but not much of that is likely to come up here.
That pretty much hits the physics of harmony issues
that are useful for this series. 'Music Theory' usually refers to the
theoretical underpinning of common practice in the West.11 The
issues I have discussed here are rarely of more than academic interest
to actual practitioners of music (either composers or performers), and
certainly not to listeners. This is, to some extent, why I have left it
to last. However, if you are like me, by now you have built up some
A Few Sources to which I have referred:
Cousineau, Marion, Josh McDermott, & Isabelle Peretza (2012). The basis of
musical consonance as revealed by congenital amusia. Proceedings of
the National Academy of Sciences of the United States of America.
Landman, Y. (n.d.). Overtone
series illustration used in the Wikipedia article,
Seeger, Charles. 1975 . "Toward a Unitary Field Theory for Musicology."
In Studies in Musicology, 102-138. Berkeley: University of California
Tramo MJ, PA Cariani, B Delgutte, & LD Braida (2001). Neurobiological
foundations for the theory of harmony in western tonal music. Annals
of the New York Academy of Science, 2001 Jun;930:92-116.
Trainor, Laurel, Christine Tsang, & Vivian Cheung (2002). Preference for sensory consonance in 2- and 4- month-old infants.
Music Perception: An Interdisciplinary Journal, Vol. 20, No. 2 (Winter
2002), pp. 187-194.
Wikipedia. Harmonic series (music). Accessed 3/2013.