bob_dunham
03-08-2002, 08:47 AM
this might be more interesting for someone who is into making music,
but anyway,i think "creating" music is like a science in many ways.
"harmonic analysis" point of view is useful to understanding why
certain
instruments sound different from others.
first, the whole emphasis on harmonic analysis grew out of early work
on the
mathematics of sound by pythagoras and culminated when fourier
created his
theory that any waveform could be represented by the sum of a series
of
simple sine waves each related to the other by a simple integer ratio.
pythagoras was studying musical scales while fourier was a pure
mathematician. the scale thing is noteworthy because why our ears like
certain chords and certain sounds is all coming from the same ear-
brain
phenomena.
the single frequency sine wave is the simplest waveform and is said
to have
a pure tone - because it only has that one frequency present.
canonical
synthesizer waves like the saw, triangle, and square are all
variations on
the harmonic series. the saw is considered the wave of strongest
harmonic
content of the three (and why it is used for phatt synth sounds) as
it has
all the harmonics present, with an amplitude equal to 1/nth of the
fundamental frequency sine wave. so the 2nd harmonic (frequency = 2x)
is
1/2 the amplitude, the 3rd is 1/3 and so on. discussing this in terms
of
filters, the saw wave forms harmonics are decreasing at 6db per
octave, so
it is as if a simple one pole low pass filter at the fundamental
frequency
was applied.
the square wave is next up in harmonic intensity. it is like a saw
but is
missing all of the even harmonics (2x, 4x, etc.) so it sounds a bit
hollow.
one thing to note is that waveforms that are symmetric around an axis
cutting it in two in time are always missing the even harmonics. one
of
those little handy factoids.
the triangle is like a toned-down square. it's harmonics reduce in
amplitude by the square of the harmonic number. so the 3rd harmonic is
1/9th as loud as the fundamental. so one could achieve a triangle by
simply
filtering a square with a one pole low pass at the fundamental
frequency.
another handy thing to know.
one of the useful other waveforms that is common is the pulse. in
fact the
square is just a special case of the pulse with a duty cycle (up to
down
ratio) of 50% (half up, half down). pulses have amplitudes in the
harmonics
that are governed by an important mathematical function called the
sinc,
(sine(x)/x) which comes from calculus. sinc functions are a series of
decreasing amplitude "lobes" with nulls (zero amplitudes) every n
frequencies. n, in this case, is the duty cycle expressed as a number
less
than 1/2 and in the form 1/n. so for the case of a square wave, which
has a
duty cycle of 50%, n is 2. so as you see the square is missing all of
the
even harmonics because of the nulls in the sinc function. armed with
this
factoid one can predict the harmonic series of any pulse wave. pulses
with
33% duty cycle are missing every 3rd harmonic, 25% missing every 4th
and so
on. this is why the pulse is so versatile as it can produce a whole
family
of different harmonic spectrums.
filtering is used to alter, by subtracting, harmonics from the basic
starting waveform. this is why it is called subtractive synthesis.
starting with only sine waves does not lead to much variety in timbre!
hopefully the reason is pretty clear.
but anyway,i think "creating" music is like a science in many ways.
"harmonic analysis" point of view is useful to understanding why
certain
instruments sound different from others.
first, the whole emphasis on harmonic analysis grew out of early work
on the
mathematics of sound by pythagoras and culminated when fourier
created his
theory that any waveform could be represented by the sum of a series
of
simple sine waves each related to the other by a simple integer ratio.
pythagoras was studying musical scales while fourier was a pure
mathematician. the scale thing is noteworthy because why our ears like
certain chords and certain sounds is all coming from the same ear-
brain
phenomena.
the single frequency sine wave is the simplest waveform and is said
to have
a pure tone - because it only has that one frequency present.
canonical
synthesizer waves like the saw, triangle, and square are all
variations on
the harmonic series. the saw is considered the wave of strongest
harmonic
content of the three (and why it is used for phatt synth sounds) as
it has
all the harmonics present, with an amplitude equal to 1/nth of the
fundamental frequency sine wave. so the 2nd harmonic (frequency = 2x)
is
1/2 the amplitude, the 3rd is 1/3 and so on. discussing this in terms
of
filters, the saw wave forms harmonics are decreasing at 6db per
octave, so
it is as if a simple one pole low pass filter at the fundamental
frequency
was applied.
the square wave is next up in harmonic intensity. it is like a saw
but is
missing all of the even harmonics (2x, 4x, etc.) so it sounds a bit
hollow.
one thing to note is that waveforms that are symmetric around an axis
cutting it in two in time are always missing the even harmonics. one
of
those little handy factoids.
the triangle is like a toned-down square. it's harmonics reduce in
amplitude by the square of the harmonic number. so the 3rd harmonic is
1/9th as loud as the fundamental. so one could achieve a triangle by
simply
filtering a square with a one pole low pass at the fundamental
frequency.
another handy thing to know.
one of the useful other waveforms that is common is the pulse. in
fact the
square is just a special case of the pulse with a duty cycle (up to
down
ratio) of 50% (half up, half down). pulses have amplitudes in the
harmonics
that are governed by an important mathematical function called the
sinc,
(sine(x)/x) which comes from calculus. sinc functions are a series of
decreasing amplitude "lobes" with nulls (zero amplitudes) every n
frequencies. n, in this case, is the duty cycle expressed as a number
less
than 1/2 and in the form 1/n. so for the case of a square wave, which
has a
duty cycle of 50%, n is 2. so as you see the square is missing all of
the
even harmonics because of the nulls in the sinc function. armed with
this
factoid one can predict the harmonic series of any pulse wave. pulses
with
33% duty cycle are missing every 3rd harmonic, 25% missing every 4th
and so
on. this is why the pulse is so versatile as it can produce a whole
family
of different harmonic spectrums.
filtering is used to alter, by subtracting, harmonics from the basic
starting waveform. this is why it is called subtractive synthesis.
starting with only sine waves does not lead to much variety in timbre!
hopefully the reason is pretty clear.