Luis de la Barrera
Abstraction of Consonance
ABSTRAC
This work stems from and is based on a previous one entitled “Harmonic Musical Systems,” completed at the end of 2004 and subsequently published online. In that earlier work, starting from the physical-harmonic principle, a deduction and description was made of the main harmonic musical systems that have shaped the history of Western music from its origins to the present day. This initial approach allowed for a new perspective on how the various harmonic musical systems originate, from a purely mathematical point of view, independent of the historical moment and musical or aesthetic reasons that led to their emergence and transformations. The realization that each interval, whether a component of a musical system or not, originates from the interaction of two harmonics of a single basic series, which serves as its foundation and differentiates it from all others by its unique qualities, led to the need to obtain a procedure, as objective as possible, to determine the degree of consonance or dissonance of these intervals.
The topic of consonance and dissonance has been approached from numerous perspectives, depending on the scientific discipline within which it has been studied: music, physics, acoustics, psychoacoustics, and so on. Although all of these fields share similar principles and foundations, the resulting findings are ultimately imbued with subjectivity and influenced by both the experimenters and the subjects being studied, as it involves a sensory experience. However, it is true that these results are the closest to and most consistent with reality.
In this work, the aim has been to completely eliminate the subjective element, basing all conclusions on almost exclusively mathematical reasoning derived from the application of laws established from the physical-harmonic principle. This has allowed for the establishment of a procedure for quantifying intervals, whether simple or compound, in which all possible subjective elements disappear, achieving a complete abstraction from the concept of "consonance and dissonance" as it has been understood throughout history.
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Introduction
1. Harmonic Meaning of the Interval Ratio.
2. Harmonic Fundamentals.
3. Consonance–Dissonance.
4. The Physical–Harmonic Series as an Integrative Element.
5. Ordering of Intervals According to Frequency.
6. Ordering of Intervals based on the Harmonic Interaction of their Constituent Harmonics.
7. Consonance–Dissonance Grading within the Octave.
8. Determination of the Equations of the Primary Curves.
9. Consonance–Dissonance beyond the Octave Range.
10. Interval Grading in Terms of Consonance beyond the Octave.
11. Interval Filtering According to the Frecuency Resolution of the Human Ear.
12. Consonance and Dissonance in Just Intonation and Equal Temperament Systems.
13. Recapitulation.