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We hear sound because our ears can detect
vibrations in the air, which come from sources
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like everyday objects, speakers and other
people talking.
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These vibrations travel through the air as
areas with higher pressure, known as compressions,
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and areas with lower pressure, known as rarefactions,
bounce back and forth, carrying this longitudinal
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wave forward.
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If we graph the air pressure across the space
that the sound is travelling through, we can
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see the wave shape of the sound.
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This is Inspecto, and this video will be all
about sound frequencies, wave shapes and the
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math behind it all.
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You probably know that a sound’s pitch is
dependent on the frequency of its sound wave.
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Frequency and pitch hold an exponential relationship,
meaning that a difference of a certain number
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of hertz, for example a 200 Hz difference
from 200 Hz to 400 Hz, and from 400 Hz to
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600 Hz, will not result in the same perceived
difference in pitch.
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Rather, the same ratio between different frequencies,
for example a 1:2 ratio from 200 Hz to 400
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Hz, then from 400 Hz to 800 Hz, is what will
result in the same perceived difference in
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pitch.
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How does this tie into music, and its familiar
12 note scale?
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Two notes that are one octave apart are named
the same, and kind of sound like they are
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the same note… but one is higher pitch,
and one is lower.
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This important gap of the octave is precisely
a double in frequency.
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All other note intervals are derived from
this – if each semitone is 1/12 of an octave,
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the ratio of a semitone would be the 12th
root of 2, or 2 to the power of 1/12.
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This way, to increase a frequency by one semitone,
it can simply be multiplied by 2 to the one
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twelfth.
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Increasing that frequency by 12 semitones,
or an octave, it would be multiplied 12 times,
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which is the 2 to the one twelfth, to the
power of 12 – in other words, 2 to the power
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of 12/12, which is just multiplying it by
2.
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Combinations of two notes at certain intervals
can sound “consonant”, which means they
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sound like they “agree” with each other,
when the two frequencies have a simple whole
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number ratio.
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For example, the octave has a ratio of 1:2
– really simple ratio.
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Other than the octave, the most consonant
interval is arguably the perfect fifth, which
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is 7 semitones.
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Let’s see what 2 to the power of 7/12 is
– it is approximately 1.498, which is quite
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close to 1.5.
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1:1.5 simplifies to 2:3 – another really
simple ratio.
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The reason a simple whole number ratio sounds
consonant probably has to do with the period
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of the resulting wave when the two waves are
added together.
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With a 1:2 ratio, the resulting wave’s period
is 1x the lower frequency wave and 2x the
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higher frequency wave.
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With a 2:3 ratio, the resulting period is
2x the lower frequency wave and 3x the higher
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frequency wave.
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These simple ratios make it easy for our brains
to notice a regular pattern when the two notes
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are added together.
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Now let’s hear what a dissonant interval
sounds like – one where the notes sound
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like they “disagree” with each other.
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The most dissonant interval is arguably the
tritone, which is 6 semitones.
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2 to the power of 6/12, or one half, is just
square root of 2, equal to about 1.4142.
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This results in an irrational ratio which
isn’t close to any simple ratio, resulting
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in a very irregular pattern when the waves
of notes one tritone apart are added together.
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A chord consists of three or more notes.
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The most consonant chord is arguably the basic
major triad, which is just a root, a perfect
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fifth, and a major third added in the middle,
4 semitones above the root.
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2 to the power of 4/12 is about 1.26, which
is close to one and a quarter.
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This results in, once again, quite a simple
ratio of 4:5.
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However, to see that dissonant intervals also
have their roles, we can look at jazz chords
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such as the dominant 7th chord
and diminished 7th chord, which both have
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dissonant intervals in them, but sound nice.
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Sinusoidal waves are the simplest waves that
us humans can hear – we perceive them to
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only be one frequency.
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Now the Fourier theorem states that periodic
functions that are reasonably continuous can
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often be expressed as the sum of different
sinusoidal functions.
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For sound, this means that the sounds we hear
can be broken down into a combination of many
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simple sinusoidal waves, each with their respective
frequencies.
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This information is very important to understanding
the sounds produced by musical instruments,
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or when synthesising our own musical sounds.
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The ratios of intervals within the 12-note
scale range from 1 to 2, in 2 to the 1/12
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geometric intervals.
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But if we look at frequencies that are integer
multiples of the fundamental frequency…
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like 2x… 3x… 4x… 5x… etc, those form
a whole nother system known as the harmonic
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series.
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The significance of the harmonic series is
that adding on these harmonics doesn’t change
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the period of the fundamental.
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As a result, the combined sound sounds tonal
– it’s very clear what pitch or note the
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sound is – it’s just the pitch of the
fundamental frequency.
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On the flip side of this, when a sound contains
frequencies that are not in the harmonic series
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of its fundamental frequency, known as inharmonic
frequencies, it sounds more atonal – in
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other words, it’s harder to tell its pitch
or note.
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Comparing the frequencies of a piano to a
drum, we can see that the tonal instrument
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does have more harmonic frequencies.
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In fact, we can theoretically recreate a musical
instrument digitally, if we use a synthesiser
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to match all its frequencies, including its
fundamental, harmonic and inharmonic frequencies
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– there are of course other factors to match
as well, but those are the most important.
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The science of sound waves also applies to
the world of sound design, which is the artificial
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creation of instruments and sound effects,
such as in electronic music.
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Sound design is all about the combination
and manipulation of wave shapes, which come
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in four basic forms: the sine wave, triangle
wave, square wave and saw wave.
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The way that these waves each sound can be
explained by how they can be broken down into
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simpler sinusoidal waves, each with their
own frequency.
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First of all, the sine wave sounds the simplest,
as it only represents one frequency.
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However, here’s an equation that represents
an infinite sum of waves.
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Since it’s a wave, it’s a function of
time, with t being the time, and f being the
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fundamental frequency.
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The sigma means that many copies of this expression
are being added up, with increasing values
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of k.
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This scales up the input of the sine function,
increasing the frequency of successive waves,
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but scales down the output of the sine function,
decreasing the amplitude of successive waves.
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This specific equation is for the square wave.
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As you can see, the expression uses 2k – 1
instead of k, which gives all the odd integers.
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This means that all the odd harmonics are
present in a square wave, in addition to the
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fundamental – which is why it sounds so
full compared to the sine wave.
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Starting as a one sine wave and adding more
and more sine waves following this equation,
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we can see that the sum does approach the
shape of a square wave.
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The triangle wave can be made by also adding
up the odd harmonics, but this time making
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every second one, that is the 1st, 5th, 9th
etc, negative, and the denominator is squared,
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making the successive harmonics decrease in
amplitude much faster.
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That’s why if we look at an analyser, the
harmonics of a triangle wave are not as rich
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as those of a square wave, and you can hear
it too – here’s a triangle wave, and here’s
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a square wave.
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The saw wave can be made by adding up all
the harmonics, not just the odd ones, and
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making every second one negative.
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Having all harmonics is what makes the saw
wave the richest sounding wave out of all
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the basic wave shapes, and that’s why it
is often used for elements in electronic music
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that need to have a big, rich sound.
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This could be a supersaw, which is used to
voice thick chords… a pad, which is used
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to fill up the atmosphere… or even a saw
bass, which can be used for an aggressive
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low end.
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Altogether, they sound like this.
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The other waves of course also have their
place.
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For example, a square wave could be used to
make 8-bit, chip tune sounds.
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A triangle wave could also be used to make
some unique lead sounds.
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And lastly, a distorted sine wave can be used
to make an 808.
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So that was some of the basic math behind
music and sound synthesis, which hopefully
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can now give you a fresh perspective when
encountering music and instruments.
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Hope you enjoyed the video, until next time.
