The sound feature diagram and musical kaleidoscope are good at
capturing the individual wave cycles and their variation.
At longer time spans, it is better to see the sound in its
frequency aspect just like what our ears do. In engineering,
we can use spectral analysis.
The human ear can discern very small difference
in tonal frequency of about 1%. In contrast, the scale of
spectral analysis is very imprecise. The mathematical cochlea
is a spiral shaped scale that provide high precision in the
frequency domain. What¡¦s more, this graphing method makes
the analysis of harmonics very easy.
The frequency scale of the mathematical cochlea
is one octave per circle. So notes which are multiples of
an octave apart will appear along the same direction. These
notes also give about the same audio perception. So this graphing
method helps us group similar sounds together. What¡¦s more,
the harmonics of a tone now appear at fixed angular positions
relative to the fundamental frequency. This is very helpful
when designing chords and composing music.
In the past, the working of the ear in sound
perception is invisible to us. With the mathematical cochlea,
the perception of sound as they excite the different parts
of our cochlea becomes visible constellation patterns which
can be easily recognized by our eyes.
The diagram on top is the mathematical cochlea
map which shows the location of harmonics. Taking a fundamental
frequency of 100, you can easily find the harmonics. The centre
circles show the locations of notes for scales of different
temperaments. It is easy to make a slide rule of musical notes
based on the mathematical cochlea by cutting out the middle
circles and allow them to rotate relative to the outer scale
to find the frequency of notes for different temperaments
and tones.
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Musical Instrument Sound Profile
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Each musical instrument has its own characteristics of harmonics
structure. Some of them has very little outside of the fundamental
whereas others are rich in harmonics. String instruments consist
mainly of odd harmonics. These are shown as different patterns
on the mathematical cochlea graphing of sound.
Some musical instruments generate sound that is not based on a
fundamental frequency. Most notable is the drum in which the
vibration of the diaphragm has multiple frequency components
that are not in integer ratios. But they are still having
fixed frequency ratios. So its signatory pattern is still
fixed and easily recognizable. With the help of this graphing
method, we can even match the pitch of drums to other musical
instruments.
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Attack, ustain & Decay Profile
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Musical instruments also have differing sound profiles over the duration
of a single note. Percussion instruments has short attack
and decay without noticeable sustain period and wind instruments
create sound consisting mainly of sustain period. What's
more interesting is that the different harmonics of a note
have different attack ¡V decay characteristics. Harmonics
will appear and die out in its own pattern. So the resulting
mix of harmonics vary over time. This is shown as transitions
in the constellation patterns of frequencies in the mathematical
cochlea.
Among all sound generating devices, the bell perhaps gives the most
intriguing sound over time. Like the drum, it creates frequency
components not in integer ratios. What¡¦s more interesting
is that the vibration modes of the bell will vary over time
such that at different times, different frequency components
become dominant. So the mathematical cochlea display will
appear as bright points that hop from point to point.
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Musical Composition
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When musicians design melodies with the help of the mathematical cochlea, they can see the interaction of the harmonics precisely.
It is also very helpful in what-if analysis. Mixing different
musical instruments will give different sound effects which
can be traced back to the composition of the frequency components
and their variation over time.
The diagram left hand side is an example output of the Mathematical Cochlea with different tones varying
their loudness over time.
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Technology
