Justin Sgalio Music

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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