Low Energy and Equal Spacing The multifactorial evolution of tuning systems

Studies of tuning systems established in different historical and ethnic cultures suggest that their evolution is mainly dependent on two different, mathematically incompatible, principles: equidistance, based on a mostly logarithmic perception of the audible frequency range, and harmonic consonance generated by simple integer pitch ratios (SIPR). Since for tuning systems the number of relevant SIPRs was overestimated in older theories (often mixing a scientific approach with irrational numerology), a new perceptual model based on both, harmonic consonance and pitch strength, the latter is closely related to timbre, is presented here. A method for numeric evaluation of the harmonic consonance of any interval in pitch space allows a mathematical and graphical representation of harmonic energy. The quality and stability of tuning systems thus can be calculated by combining the energy values for each element of their interval set. The combinatorial energy for tempered systems is plotted in order to determine the system with the lowest value, yielding different systems for different pitch strengths. The high pitch-strength plots are dominated by the most common system, 12-tone equal temperament, whereas the low pitch-strength plots overwhelmingly give priority to the equal tempered pentatonic system a finding which suggests a strong relationship between the historical evolution of tuning systems and a culturally determined focus on different musical parameter like melody or rhythm.