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 ¼ 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. V

The consensus in piano tuning philosophy explains the stretched tuning scale by the inharmonicity of piano strings. This study aimed to examine how variable inharmonicity influences the result of the piano tuning process, compare the tuning curves of aurally tuned pianos with the curves derived from subjective octave enlargement experiments, and evaluate whether the pitches of inharmonic or harmonic versions of the same tone are perceived differently. In addition, the influence of strings of other piano keys on the measured inharmonicity of a single piano string was investigated. The inharmonicity of all individual strings was measured on a Steinway D grand piano. Variable inharmonicity was implemented by additive synthesis with frequency-adjusted sinusoidal partials. Fifteen piano tuners and 18 orchestra musicians participated in the experiments. The results indicate that the inharmonic piano tones produced a keyboard tuning curve similar to the Railsback curve and differed significantly from the harmonic counterpart. The inharmonic tuning curve was reminiscent of the subjective octave enlargement curve. Inharmonic tone pitches were perceived to be higher than harmonic tones up to [Formula: see text]. The covibrating strings of the other keys did not exhibit any meaningful effect on the measured inharmonicity of a single string of the played key.

The relationship between perceived pitch and harmonic spectrum in complex tones is ambiguous. In this study, 31 professional orchestra musicians participated in a listening experiment where they adjusted the pitch of complex low-register successively presented tones to unison. Tones ranged from A0 to A2 (27.6–110 Hz) and were derived from acoustic instrument samples at three different dynamic levels. Four orchestra instruments were chosen as sources of the stimuli; double bass, bass tuba, contrabassoon, and contrabass clarinet. In addition, a sawtooth tone with 13 harmonics was included as a synthetic reference stimulus. The deviation of subjects’ tuning adjustments from unison tuning was greatest for the lowest tones, but remained unexpectedly high also for higher tones, even though all participants had long experience in accurate tuning. Preceding studies have proposed spectral centroid and Terhardt’s virtual pitch theory as useful predictors of the influence of the envelope of a harmonic spectrum on the perceived pitch. However, neither of these concepts were supported by our results. According to the principal component analysis of spectral differences between the presented tone pairs, the contrabass clarinet-type spectrum, where every second harmonic is attenuated, lowered the perceived pitch of a tone compared with tones with the same fundamental frequency but a different spectral envelope. In summary, the pitches of the stimuli were perceived as undefined and highly dependent on the listener, spectrum, and dynamic level. Despite their high professional level, the subjects did not perceive a common, unambiguous pitch of any of the stimuli. The contrabass clarinet-type spectrum lowered the perceived pitch.

Musical intonation is a complex phenomenon. It can be broadly divided into three distinct parts: a psychoacoustic stretched tuning, a context-bound distinction of harmonic and melodic intonation, and an affective characteristic intonation.  The stretched tuning, which has been verified in psychoacoustic listening experiments, where the notes above the reference (A4≈442Hz) are more up-tuned than their mathematic equal-tempered (octave ratio 2:1) counterparts and the notes below the reference are correspondingly more down-tuned, forms the basis of the tuning. Harmonic and melodic in-tuning is applied on top of this, rather than in relation to the mathematical equal temperament tuning. Characteristic intonation is used to accentuate the effects of certain intervals. Before the mainstreaming of equal temperament thinking, the separation of enharmonic tones also had a significant impact on intonation. The principle of counterbalance, proposed to stabilize orchestral tuning, is based on a stretched tuning and the fixation of tuning reference always to the middle register (A4) instead of the bass register. In this way, the downward expansion of the tuning of low register instruments allows a moderate increase in the tuning level of high register instruments. Well-tuned intonation is not a question that can be unambiguously defined. It is also influenced by learned and cultural factors, differences in musical aesthetics and neurophysiological differences in the auditory system between perceivers.