The Bertrand paradox question is: “Consider a unit-radius circle for which the length of a side of an inscribed equilateral triangle equals 3 . Determine the probability that the length of a ‘random’ chord of a unit-radius circle has length greater than 3 .” Bertrand derived three different ‘correct’ answers, the correctness depending on interpretation of the word, random. Here we employ geomet...

To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let gn denote the expected genus of a randomly chosen oriented chord diagram of order n. We show that gn satisfies:

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A pair of chords is called a crossing if the two chords intersect. A chord diagram E is called nonintersecting if E contains no crossing. For a chord diagram E having a crossing S = {x1x3, x2x4}, the expansion of E with respect to S is to replace E with E1 = (E \S)∪{x2x3, x4x1} or E2 = (E \S)∪{x1x...

A linear chord diagram of size n is a partition of the set {1, 2, . . . , 2n} into sets of size two, called chords. From a table showing the number of linear chord diagrams of degree n such that every chord has length at least k, we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence. We prove that this holds for all diagonals, and identify when the...

This paper presents a technique of disambiguation for chord recognition based on a-priori knowledge of probabilities of chord voicings in the specific musical medium. The main motivating example is guitar chord recognition, where the physical layout and structure of the instrument, along with human physical and temporal constraints, make certain chord voicings and chord sequences more likely th...

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...

We establish an explicit formula for the number of chord diagrams on 2n points with k short chords and compute the expected number of short chords in a chord diagram.

This paper presents a system named ChordSequenceFactory for automatically generating chord arrangements. A key element of musical composition is the arrangement of chord sequences because good chord arrangements have the potential to enrich the listening experience and create a pleasant feeling of surprise by borrowing elements from different musical styles in unexpected ways. While chord seque...

The increasing accuracy of automatic chord estimation systems, the availability of vast amounts of heterogeneous reference annotations, and insights from annotator subjectivity research make chord label personalization increasingly important. Nevertheless, automatic chord estimation systems are historically exclusively trained and evaluated on a single reference annotation. We introduce a first...

We show that traditional music information retrieval tasks with well-chosen parameters perform similarly using computationally extracted chord annotations and groundtruth annotations. Using a collection of Billboard songs with provided ground-truth chord labels, we use established chord identification algorithms to produce a corresponding extracted chord label dataset. We implement methods to c...