Octave stretching phenomenon with complex tones of orchestral instruments

Abstract

For decades, the phenomenon of subjectively enlarged octaves has been investigated using sinusoidal and synthesized complex tones. The present study elaborates the topic with samples of real orchestra instruments in successive tone listening experiments. Compared to previous research, this study also included a substantially larger number of subjects (N = 36). Examined instrument tones were categorized into five groups based on their acoustic principles. In addition, each group was assessed at three dynamic levels (pp-mf-ff). Collected data were analyzed with tuning stretch curves by applying generalized additive models in the manner of the Railsback curve used to characterize piano tuning. Although the tuning curve modeled for the orchestra instruments was observed to differ slightly from the Railsback curve and typical Steinway D grand piano tuning (Steinway, New York), the stretching trends were qualitatively similar. Deviation from a mathematical equal-tempered scale was prominent. According to statistical analyses, dynamics or musical background of the participant did not affect results significantly, but some instrument groups exhibited differences in the curve extremities. In conclusion, the stretched scale is natural for a human listener and should be used as a reference scale in tuning machines instead of the mathematical equal-tempered scale.