Dissonance
in Music Explained : Learn How Dissonance in Music Theory and Math is Quantized
To
define and explain the meaning of dissonance in music theory, we emphasize the
dramatic distinction between complex harmonic notes and simple tones. Complex
harmonic notes are the familiar notes we hear when we strike a piano key or
pluck a guitar string. Complex harmonic notes consist of a fundamental tone
(root tone) along with an associated attendant set of upper harmonics (higher
frequency partials) that are simple tones that sound together along with the
fundamental. Within a complex harmonic note, its fundamental characteristics
are identified by the fact that the upper harmonic simple tone partials have
frequencies that are integer multiples of the fundamental tone frequency. In
contrast, simple tones are pure single frequency tones that have no associated
upper harmonics.
Hermann
von Helmholtz proposed a simple model for the sensory dissonance of two simple
tones when sounded together.
When two
complex harmonic notes are sounded concurrently -- such as within intervals,
triads and chords played on guitars or pianos or wind instruments -- the
interaction between the upper harmonics of the different notes will have a major
influence upon how we identify the quality of the harmony of the combined sound
that we hear.
The
graph below shows the same six upper harmonics along with the attached
associated Helmholtz dissonance envelope which is symmetrical about each of the
upper partial simple tones. The symmetry of the dissonance envelope about each
partial shows a peak in dissonance at the minor second interval on either side
of the partial -- right or left. This can easily be demonstrated on a piano
keyboard when a minor second is struck on either side of the note.
We
discover what is commonly known. Intervals of complex harmonic notes with
fundamental tone frequency ratios that are small integers (say 3/2 -- the
perfect fifth ratio), provide a greater number coincidents of the upper
partials of the two notes and we identify consonance. On the other hand,
intervals of complex harmonic notes with fundamental tone frequency ratios that
are two large numbers (say 16/15 -- the minor second ratio), provide a lesser
number coincidents of the upper partials of the two notes and we identify
dissonance. Basically, this notion is incorporated in the familiar Pythagorean
principle -- the quality of the harmony – consonant ( consonance ) or dissonant
-- is dependent upon the number of coincidents of the upper partials of the two
notes. The smaller the two numbers in the ratio, the greater the number of
coincidents and the more consonant the sound. Likewise, the larger the two
numbers in the ratio, the lesser the number of coincidents and the more
dissonant the sound.
Covered
in greater detail with audio examples are the following:
Complex
Harmonic Notes and Simple Tones, Intervals of Simple Tones, Perfect Fifth
Interval Simple Tones and Minor Second Interval Simple Tones.
Complex
Harmonic Notes, Intervals of Complex Harmonic Notes, The Complex Harmonic
Frequency Spectrum, The Perfect Octave Interval Ratio, The Perfect Fifth
Interval Ratio and The Minor Second Interval Ratio.
The
Helmholtz Dissonance Envelope Curve and the Helmholtz Dissonance Envelope
Overlap
The
Quantization of Dissonance of Intervals of Complex Harmonic Notes
The
above examples demonstrate that the amount of overlap between the secondary
note partials with the dissonance envelopes of the partials of the root note
gives a measure or quantization of the dissonance of the intervals of the
complex harmonic notes. The above graphs show that the greater the magnitude of
the overlap with the dissonance envelops, the greater the dissonance. From the
above examples, the overlap depends upon the magnitude of the interacting
partials, the interval ratio between the two notes, the mathematical
characteristic shape of the Helmholtz dissonance envelope curve, and the number
of interacting partials that are taken into consideration. The equation shown
is an evaluation of the magnitude of the overlap.
Overlap
Dissonance Intensity Equation for Intervals of Two Complex Harmonic Notes
This
equation shows a double summation over all sets of independent (h) pairs of
upper harmonic partials associated with each of the two complex notes 1 and 2
that make up the interval. Here, r is the dimensionless frequency ratio between
the fundamentals of each primary note in the interval, and n and m are the
orders of the associated upper harmonics with amplitudes A1(n) and A2(m)
respectively. The mathematical character of the Helmholtz dissonance envelope
curve -- as shown earlier above -- is embedded into this equation. The graph of
this equation is shown below for a total of 10 interacting upper partials whose
magnitudes decay exponentially as given in the examples above.
Quantization
of Normalized Sensory Dissonance for Complex Harmonic Note Intervals
Table of
Evaluated Normalized Helmholtz Dissonance for Intervals in One Octave is Based
on this Mathematical Equation.
All
materials are based book by Mathematical Concepts in Music, Scales, Harmony and
Ratios by George A. Articolo.
The book
is now available here:
https://www.amazon.com/Mathematical-Concepts-Music-Scales-Harmony/dp/0615834094
https://www.RensselaerPress.com/
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