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Elastic overtones: an equal temperament 12 tone music system with "perfect" fifths

X. Hernandez, Luis Nasser, Pablo Garcia-Valenzuela

Abstract

The impossibility of a transposable 12 semitone tuning of the octave arises from the mathematical fact that $2 \times 2^{7/12} \neq 3$ i.e., the second harmonic of the fifth can not exactly match the third harmonic of the fundamental. This in turn, stems from the whole number harmonic structure of western music, and the subsequent fundamental character of the octave interval as multiples of 2 in frequency, a property inherited by our music system from the physics of instruments with vibrating elements being to a good approximation one dimensional. In the current era of electronic music, one can relax the above assumptions to construct an analogous music system where all the structural properties of the standard music system are preserved, but where harmonics are not whole number multiples of the fundamental frequency, and the octave is no longer a factor of 2 in frequency. This now allows to construct a transposable 12 semitone music system where the second harmonic of the fifth exactly matches the third harmonic of the fundamental. The enhanced harmonic qualities of this system recover to a good approximation the musical qualities of Just Intonation, whilst retaining by construction all the versatility and modulating ability of 12TET.

Elastic overtones: an equal temperament 12 tone music system with "perfect" fifths

Abstract

The impossibility of a transposable 12 semitone tuning of the octave arises from the mathematical fact that i.e., the second harmonic of the fifth can not exactly match the third harmonic of the fundamental. This in turn, stems from the whole number harmonic structure of western music, and the subsequent fundamental character of the octave interval as multiples of 2 in frequency, a property inherited by our music system from the physics of instruments with vibrating elements being to a good approximation one dimensional. In the current era of electronic music, one can relax the above assumptions to construct an analogous music system where all the structural properties of the standard music system are preserved, but where harmonics are not whole number multiples of the fundamental frequency, and the octave is no longer a factor of 2 in frequency. This now allows to construct a transposable 12 semitone music system where the second harmonic of the fifth exactly matches the third harmonic of the fundamental. The enhanced harmonic qualities of this system recover to a good approximation the musical qualities of Just Intonation, whilst retaining by construction all the versatility and modulating ability of 12TET.
Paper Structure (13 sections, 13 equations, 4 figures, 3 tables)

This paper contains 13 sections, 13 equations, 4 figures, 3 tables.

Table of Contents

  1. Abstract
  2. Introduction
  3. Proposal of a perfect fifth$^{\star}$ music system that can modulate.
  4. Implementation
  5. Conclusion
  6. Supporting information
  7. Acknowledgments

Figures (4)

  • Figure 1: Score for sound File 1. This Figure shows the score for sound file 1, a chromatic scale, which is played in the standard 12TET system first and in the proposed elastic overtone system after.
  • Figure 2: Score for sound File 2. This Figure shows the score for sound file 2, a series of parallel fifths, which are played in the standard 12TET system first and in the proposed elastic overtone system after.
  • Figure 3: Score for sound File 3. This Figure shows the score for sound file 3, a series of chords, which are played in the standard 12TET system and the proposed modified system, respectively.
  • Figure 4: Score for sound File 4. This Figure shows the score for sound file 4, a series of tritones, which are played in the standard 12TET system and the proposed modified system, respectively.