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Musical Tension Curves and Its Applications

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Musical Tension Curves and Its Applications Musical Tension Curves and its Applications Min-Joon Yoo and In-Kwon Lee Department of Computer Science, Yonsei University [email protected], [email protected] Abstract Madson and Fredrickson 1993; Krumhansl 1996). In these attempts, the tension at each part of music has been mea- We suggest a graphical representation of the musical tension sured by manual inputs of the participants in the experiments. flow in tonal music using a piecewise parametric curve, which In our work, we developed several methods to automatically is a function of time illustrating the changing degree of ten- compute the tension of a chord by combining known experi- sion in a corresponding chord progression. The tension curve mental results. Then the series of tension values correspond- can be edited by using conventional curve editing techniques ing to the given chord progression in the input music are in- to reharmonize the original music with reflecting the user’s terpolated to construct a smooth piecewise parametric curve. demand to control the tension of music. We introduce three We exploit the B-spline curve representation that is one of the different methods to measure the tension of a chord in terms most suitable method to model a complex shape such as ten- of a specific key, which can be used to represent the tension sion flow. Once the tension curve is constructed, we can edit of the chord numerically. Then, by interpolating the series of the tension flow of the original music by geometrically edit- numerical tension values, a tension curve is constructed. In ing the tension curve, where the tension curve editing also this paper, we show the tension curve editing method can be generates the new reharmonization of the original chord pro- effectively used in several interesting applications: enhanc- gression. ing or weakening the overall feeling of tension in a whole In this paper, we show our tension curve editing method song, the local control of tension in a specific region of music, can be effectively used in several interesting applications: en- the progressive transition of tension flow from source to tar- hancing or weakening the overall feeling of tension in a whole get chord progressions, and natural connection of two songs song, the local control of tension in a specific region of music, with maintaining the smoothness of the tension flow. the progressive transition of tension flow from source to tar- get chord progressions, and natural connection of two songs with maintaining the smoothness of the tension flow. All of 1 Introduction these application results are achieved by appropriate geomet- rical curve editing methods such as curve translation, local Musical tension or dissonance is an important term in music curve shape control with space-time constraints, curve mor- analysis. In most music analysis books (Schoenberg 1954; phing, and shape blending. Piston 1987), musical tension is analyzed by an examination Consequently, our work shows the possibility of control- of the intervals, the harmonic context, and the tonal motion. ling the perceptual factor (tension) in music by using numer- We divide musical tension in three categories: melodic ten- ical methods. Note that most of the computations used in sion (melody versus chord), harmonic tension (chord versus this paper are not expensive so they can be executed in real- key), and chord tension (simultaneous tones) and we focus on time. Thus, the real-time modifications of the tenseness in harmonic tension and chord tension in this paper. background music could be possible according to the user’s we suggest a new method to represent the tension flow emotion or current scenario in the interactive environments using a smooth piecewise parametric curve. The resulting such as games. tension curve, a numerical model, can be used to control the degree of tension in music. By applying various conventional curve editing techniques to the tension curve, we can con- struct a flexible model for locally or globally controlling the tension of music. The attempts to illustrate the tension in music has been executed mainly in the area of psychology (Nielson 1983; 482 2 Computing Tension of Chords Shepard 1979). She suggested a numeral rating assigned by human subjects to certain pitches, that is, the rating of the 2.1 Using Lerdahl’s Tonal Pitch Space tonal stability of certain pitches with respect to a given scale. We thought that this stability rating can also be applied In Lerdahl’s theory of tonal pitch space (Lerdahl 1988), he to calculate the amount of the tension of chord because each suggested a model to calculate the psychological distance of chord consists of several pitches and the tension is made from any pitch, chord, or region (key) from a given reference point. them. Table 1 shows normalized instability values. Let us consider the two chords x = {p1, p2, · · ·} in key kx and y = {q1, q2, · · ·} in key ky, where {pi} and {qi} (i = C C# D D# E F F# G G# A A# B 1, 2, ···) are the sets of pitch classes in the chord x and y, re- Major 0 100 70 97 48 55 94 27 97 67 100 85 spectively. Lerdahl defined the distance between two chords Minor 0 96 73 27 96 73 100 42 62 96 77 85 x and y by: d(x → y) = a + b + c, (1) Table 1: The normalized instability values of each pitch in the C key. where a is the number of steps from the key kx to ky on the circle of fifths, b is the number of steps from the chord x to y on the circle of fifths, and c is a specially weighted Hamming Although we got a good rating of each chord by the probe distance between the two sets of pitch classes x and y. If any tone, this was not enough because the tension is made by not of both chords has a seventh and/or a tension note, an addi- only the pitches but also the intervals among the pitches in tional value is added to the distance by Lerdahl’s suggestion the chord. So we also considered the rating of intervals using for calculating the surface dissonance (Lerdahl 1996). the traditional ranking of harmonic consonance. We assigned In our work, we consider a tension of an arbitrary chord linearly increased tension values to the pitch intervals in the x in terms of a specific key k by assuming that the higher order: Unison, Perfect 5th, , Perfect 4th, Major 3rd, Major the distance between a chord x and a tonic triad chord kt of 6th, minor 3rd, minor 6th, Major 2nd, Minor 7th, Major 7th, the key k, the more tense the chord x. Thus, the tension of minor 2nd, and tritone (see Table 2). the chord x in the key k is defined by d(kt → x) computed by the Equation (1). We calculated the distance from C triad Interval P1 m2 M2 m3 M3 P4 TRI P5 m6 M7 m7 M7 value 0 91 64 46 37 18 100 9 55 36 73 82 to twenty-four selected chords in the C key including triads, seventh, sixth, substitute chords, and etc., according to this Table 2: The normalized instability values of each interval rule. (P:perfect, m:minor, M:major, and TRI:tritone) 2.2 Using Chew’s Spiral Array Model In this model, the degree of tension of a chord is calcu- Chew (Chew 2000) suggested a 3D spiral array model of the lated as follows: circle of fifth that represents a hierarchical musical factors X X such as pitch, chord, and key in a geometric point of view. d(x) = w1 s + w2 r, (3) In this model, the distance between two musical factors can be measured by Euclidean distance between two points cor- where s is the instability value of the pitches in the chord x responding to the factors. Let x = {p1, p2, · · ·} be a chord (Table 1), r is the rank of intervals in the chord x (Table 2), in a key k. According to the Chew’s work, the pitch classes and w1 and w2 are weight constants for the two terms, respec- p1, p2, ··· and the key k can be represented as corresponding tively. In our implementation, we set w1 = w2 in order that 3D points v1, v2, ··· and w in the spiral array model, respec- two factors have influence on tension of the chord x equally. tively. In our work, we define the tension of the chord x in the key k as the sum of all Euclidean distances between the 2.4 Comparison of Three Methods each pitch point and key point: Table 3 shows the ranking of the 24 selected chords in C X d(x) = kvi − wk. (2) key according to the tension values computed from the three i methods. The amount of tension are normalized to the range [0, 1] for easy relative comparison. 2.3 Using Pitch-Interval Method Lerdahl’s method focus on harmonic tension and Chew’s method is related to chord tension (remember the definitions Psychologist Krumhansl introduced the probe tone technique of harmonic and chord tension in Section 1). Pitch-Interval to quantify the hierarchy of tonal stability (Krumhansl and method, however, can consider both tensions. 483 Lerdahl’s Chew’s Pitch-Interval chords tension chords tension chords tension 3 Constructing Tension Curves C 0.00 C 0.00 C 0.00 CM7 0.07 Am 0.06 Am 0.08 The tension value of each chord in an original song is com- C6 0.21 Em 0.50 Em 0.17 puted using one of the methods in the previous section.
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