The wave equation for stiff strings and piano tuning

We study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing on the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.Peer ReviewedPostprint (published version

X_System

The X_System makes the playing, writing, and learning of music – even when using unconventional tunings – more intuitive, more logical, more expressive, and better sounding. The X_System allows for: • different temperaments to be chosen at the flick of a switch; • tunings to be dynamically altered at the push of a lever; • the use of a special hexagonal button-field that allows for any given interval or chord always to have the same shape on that button-field; • consonant chords to have their consonance maximised, whatever the tuning actually chosen; • radical changes to be made to the timbral character of tones using a minimal number of controls; • a choice of keyboard mappings, which enable for the balance between number of intervals and octaves to be altered

Quantifying Harmony and Dissonance in Piano Intervals and Chords

The level of dissonance in piano intervals and chords was quantified using both experimental and computational methods. Intervals and chords were played and recorded on both a Yamaha YPT-400 portable keyboard and a Steinway & Sons grand piano. The recordings were run through spectral analyses, and dissonance values were calculated using a dissonance equation. The result was a ranking of comparative dissonance levels between each chord and interval. Though the goal was to find a universal ranking of chords, it was instead determined that such a ranking cannot exist. The non-universal rankings revealed that the transition from least dissonant to most dissonant was gradual

The perception of melodic consonance: an acoustical and neurophysiological explanation based on the overtone series

The melodic consonance of a sequence of tones is explained using the overtone series: the overtones form "flow lines" that link the tones melodically; the strength of these flow lines determines the melodic consonance. This hypothesis admits of psychoacoustical and neurophysiological interpretations that fit well with the place theory of pitch perception. The hypothesis is used to create a model for how the auditory system judges melodic consonance, which is used to algorithmically construct melodic sequences of tones

Harmonic duality : from interval ratios and pitch distance to spectra and sensory dissonance

Dissonance curves are the starting point for an investigation into a psychoacoustically informed harmony. Its main hypothesis is that harmony consists of two independent but intertwined aspects operating simultaneously, namely proportionality and linear pitch distance. The former aspect is related to intervallic characters, the latter to ‘high’, ‘low’, ‘bright’ and ‘dark’, therefore to timbre. This research derives from the development of tools for algorithmic composition which extract pitch materials from sound signals, analyzing them according to their timbral and harmonic properties, putting them into motion through diverse rhythmic and textural procedures. The tools and the reflections derived from their use offer fertile ideas for the generation of instrumental scores, electroacoustic soundscapes and interactive live-electronic systems.LEI Universiteit LeidenResearch in and through artistic practic