A maximum weighted independent set of a permutation graph is a maximum subset of noncrossing chords in a matching diagram (i.e., a set of chords with end-points on two horizontal lines). The problem of nding, among all noncrossing subsets of with density at most k, one with maximum size is considered, where the density of a subset is the maximum number of chords crossing a vertical line and k i...

When musical notes are combined to make a chord, the closeness of fit of the combined spectrum to a single harmonic series (the 'harmonicity' of the chord) predicts the perceived consonance (how pleasant and stable the chord sounds; McDermott, Lehr, & Oxenham, 2010). The distinction between consonance and dissonance is central to Western musical form. Harmonicity is represented in the temporal ...

This paper makes a harmonic analysis of Bach BWV871, hoping that players can understand the function every part and purpose creation from Angle harmony. The harmony also allows player to identify development phrases, making performance more coherent. Harmonic help practice logically interestingly, so as improve efficiency speed up progress memorizing music. In process practice, you learn Bach's...

Some musical chords sound pleasant, or consonant, while others sound unpleasant, or dissonant. Helmholtz's psychoacoustic theory of consonance and dissonance attributes the perception of dissonance to the sensation of "beats" and "roughness" caused by interactions in the auditory periphery between adjacent partials of complex tones comprising a musical chord. Conversely, consonance is character...

Musical ensemble performance requires the synchronization of multiple performers, resulting in sequences of chords containing multiple tones with multiple onsets. Experiments 1 and 2 investigate whether sensorimotor synchronization with chord sequences containing tone-onset asynchronies is affected by (1) the magnitude of these asynchronies (25, 30, or 50 msec) and (2) the pitch of the leading ...

We present a computational framework for chord invention based on a cognitive-theoretic perspective on conceptual blending. The framework builds on algebraic specifications, and solves two musicological problems. It automatically finds transitions between chord progressions of different keys or idioms, and it substitutes chords in a chord progression by other chords of a similar function, as a ...

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram E with n chords is called an n-crossing if all chords of E are mutually crossing. A chord diagram E is called nonintersecting if E contains no 2-crossing. For a chord diagram E having a 2-crossing S = {x1x3, x2x4}, the expansion of E with respect to S is to replace E with E1 = (E\S...

In this study, I propose a theory of voice leading for use with the tonal space of the justintonation Tonnetz. The theory is based largely upon established theories of diatonic voice leading and chromatic harmony, and also draws upon the principles of just intonation within the tonal system. The unambiguous expression of intervals as untempered ratios is dependent upon an established seven-step...

Whilst travelling in South Africa a couple of years ago, I was introduced by family friends to a late middle aged woman, the mother of a musician and a bit of a hippy with a deep interest in jazz, with whom it was assumed I would have much in common, and in the ensuing conversation she breathlessly proclaimed that there were an infinite number of chords. Of course, I immediately disagreed and t...

A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly sy...