A 3MB PDF slideshow can be
downloaded here:
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slideshow.
If you wish to request a high resolution version or
original Keynote presentation, please contact the author
directly through the contact page.
Below are some additional
animated models to illustrate some of the concepts
described in the book and slideshow. You will need Flash 7
or higher installed on your computer to play the
animations.
Touch here to install the current
version of Flash.
The
INTERFERENCE
Function
This animation provides a step-by-step explanation of the
core function proposed by Harmonic
Interference Theory and how
it can be used to organically describe human perception of
music harmony. The book goes into much greater detail,
citing neurophysiological studies and mathematical systems
to support the concept. For a little background, an excerpt
from the book on this subject can be found
here.
Touch here to view the
INTERFERENCE function animated.
The Harmonic Hierarchy
One of the key principles of Harmonic
Interference Theory is the
idea that coherent wave interference of any kind is
recursive in space and time, nesting the same pattern
inside itself synchronously in order to maintain coherence.
The theory proposes a 5-level 12th-power hierarchy to
explain the natural organization of harmonic sound and,
thus, how we recognize it. The five recursive levels begin
as twelve octaves that cover the entire auditory spectrum,
then replicate itself at one-octave, one-twelfth octave (or
semitone), a single tone and then one-twelfth tone. In this
way, the harmonic waves composing the timbre of a tone
align at Level 2 while also aligning with an octave at
Level 4. The entire hierarchy can be expressed as f(n) =
2^((12^n)/12), n = {-2 .. 2} which means that equal
temperament (with its 2^(1/12) semitones) are actually a
natural property of the harmonic series.
The following animation will take you on a tour of each of
the five levels, showing how they fit as a Gaussian pattern
within one another and how the golden ratio acts as a
constant - a golden pillar, if you will - holding up the
entire auditory structure.
Touch here to view the
Harmonic Hierarchy animated.
The Synesthetic Color Model
The colors used in the various harmonic models defined in
the book are based on the simple idea that the visible
color spectrum represents an octave frequency range just
like a musical octave. Just as a musical octave is a
frequency doubling of x to 2x, such as A440 Hz to A880 Hz,
the color octave is also a frequency doubling from 370 THz
to 740 THz. As a result, the Newton 12-step color wheel may
be assigned as isomorphic spectral proportions to each of
the tones in a musical octave using the Harmonic Center (D
in the key of C) as a polar axis. This is discussed a
little more in the excerpt located here.
Touch here to view the
Synesthetic Model animated.
Touch here to view this
on a black background.
The Keyboard Standing Wave Model
The Keyboard Standing Wave model represents the inspiration
that led to Harmonic Interference Theory, introduced in the
excerpt located here. It
shows the Tritone Function oscillating inside the common
practice V-I chord cadence in the key of A from Dominant
E7 to Tonic A major. This illustrates the symmetry
around the Super Tonic, or what I refer to as the
Harmonic Center, and how the tritone contracts across
two opposing golden ratios in each half octave.
This is significant because Landau damping theory
identifies the vertical silver bars (in particular the
darker thin bars intersecting the stars) as calm regions
where energy jumps between waves. Harmonic
Interference Theory proposes
that it is the flow of energy across these two "Phi-damping
locations" that accounts for perceived qualities in music,
such as consonance, dissonance, tension and resolution. In
short, this model is the heartbeat of harmonic
perception.
Touch here to view
the Keyboard Model animated.
The Diatonic Standing Wave Model
This model defines the 7-step diatonic scale to be a 2:1
coherent pathway of the Keyboard Standing Wave model, which
amplifies the alternating odd-even wholetone harmonic
components. This means that diatonic scales are recognized
as a frequency doubling (or octave harmonic) of the
fundamental. This higher frequency (or "higher order")
pathway is then a new standing wave that alternates between
two major scales exactly a tritone apart while following
symmetrically opposing energy currents. This is discussed
further in the excerpt located here.
Illustrated as first the C major scale and then the F#
major scale in alternation, this animation reveals a
natural harmonic "breathing pattern" over an octave.
Touch here to view
the Diatonic Model animated.
As I was creating this model, I
suddenly realized that it matched the three circular
regions of the human torso introduced in the excerpt
here and in greater detail in
the Idea Generator here.
While Harmonic Interference Theory proposes the
idea that diatonic scales (meaning "through the body")
are perceived naturally as a coherent pathway through
the harmonic series, the human body itself can also be
seen to follow the same standing wave model. The
following animation applies the Diatonic Standing Wave
model to the human torso, suggesting an explanation for
the Indian Hindu chakra system and likely organizing
principle behind human physiology.
Touch
here to view the Diatonic Model in the Human Torso
animated.
Of course, Western medical
science doesn't recognize anything like this, but it may be
how cells communicate to one another to maintain coherence
and pump things through the body.