The missing fundamentals of harmonic theory: Chord roots and their ambiguity in arrangements of jazz standards

Since Lippius and Rameau, chords have roots that are often voiced in the bass, doubled, and used as labels. Psychological experiments and analyses of databases of Western classical music have not produced clear evidence for the psychological reality of chord roots. We analyzed a symbolic database of 100 arrangements of jazz standards (musical instrument digital interface [MIDI] files from midkar.com and thejazzpage.de ). Selection criteria were representativeness and quality.The original songs had been composed in the 1930s and 1950s, and each file had a beat track. Files were converted to chord progressions by identifying tone onsets near beat locations (±10% of beat duration). Chords were classified as triads (major, minor, diminished, suspended) or seventh chords (major–minor, minor, major, half-diminished, diminished, and suspended) plus extra tones. Roots that were theoretically less ambiguous were more often in the bass or (to a lesser extent) doubled. The root of the minor triad was ambiguous, as predicted (conventional root or third). Of the sevenths, the major–minor had the clearest root. The diminished triad was often part of a major–minor seventh chord; the half-diminished seventh, of a dominant ninth. Added notes (“tensions”) tended to minimize dissonance (roughness or inharmonicity). In arrangements of songs from the 1950s, diminished triads and sevenths were less common, and suspended triads more common, relative to the 1930s. Results confirm the psychological reality of chord roots and their specific ambiguities. Results are consistent with Terhardt’s virtual pitch theory and the idea that musical chords emerge gradually from cultural and historic processes. The approach can enrich music theory (including pitch-class set analysis) and jazz pedagogy.

Uncertainty in tuning evaluation with low-register complex tones of orchestra instruments

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.

Tuning in to Very Fast Events

This chapter demonstrates that entrainment applies to very fast events, namely sounds with high frequencies. To illustrate this, prominent approaches to pitch perception are sketched along with basic pitch perception phenomena (e.g., virtual pitch perception). In this chapter, multiple frequency components comprise a single complex sound, and people must judge the pitch of this collection of frequencies. Both a successful psychoacoustic theory of pitch perception and a dynamic attending approach offer valid explanations of various phenomena surrounding the pitch of such sounds. This suggests the potential of entrainment in describing pitch perception (i.e., entrainments at fast time scales). The perception of consonance and dissonance is also considered, where dissonance is linked to complex synchronicities termed attractors. Finally, this chapter introduces oscillator clusters, a group of endogenously entrained oscillations.