The Spherical Coordinate based Improvisation in Unfolding Sonic Space

<The Spherical Coordinate based Improvisation in Unfolding Sonic Space> is a concept that extends the triple integral of a spherical coordinate system to music/sound, interpreting improvisation as a process within a 3D probability space. A triple integral in a spherical coordinate system is a method used to calculate the volume or total quantity of a distribution in 3D space by integrating its density.

which means,

1. The meaning of this is that the portion of the total sonic energy generated by the performer within a specific time window that is integrated into the Altered region represents the ratio of the Altered color for that segment.

2. In other words, "the relative proportion of the total 'sonic energy' created by the performer during a specific time, which is poured into the 'Altered scale region,' is precisely the weight (or the dominant color of the performance) of the Altered scale."

Musical Meaning of the Coordinates

Each coordinate in this space has a specific musical meaning:

• θ (theta) represents the scale family.

• φ (phi) represents the degree of separation of a tension from the center of the chord (the degree of tension).

• ρ (rho) is the radius, signifying the dynamics (intensity or energy) of the sound.

  
  

In this defined (θ, ϕ, ρ) coordinate system, a single sonic event (like a note or a moment of sound) corresponds to a single point represented by a specific scale context, tension level, and dynamic intensity. Each scale family naturally occupies a unique tension region within this space.

Theoretical Framework and Performer's Mindset

Many jazz-trained musicians approach performance based on one-to-one correspondences. Through training, they reach a 'state' where a specific response is naturally derived for a certain measure, chord, or beat. The spherical coordinate-based approach in an unfolding sonic space is realized from the idea that all possible sounds related to the resonance of a dominant chord are placed within a single, unified space of possibility. By varying θ, a "framework of thought" is created that allows for a continuous spectrum of expression, from dreamy colors to exotic flavors and sharp tensions.

Even when freely moving back and forth between the colors of different tensions within a single performance, it maintains logical consistency as a single path in the coordinate space. Even if the languages of different scales seem to intersect, a single flow of sonic coordinates exists to consistently support the performance. The important perspective here is not to see each note as an isolated decision but to view them as a probability distribution. The improviser is seen as drawing a path through time while exploring that space. The distribution of sonic energy can be represented as a probability density function. In other words, the probability and energy of a certain sound or tension region appearing in the performance are distributed in this space.

By applying the concept of a triple integral, the entire performance can be understood as the sum of microscopic sonic energies generated while passing through this probability space. Just as total mass is found by integrating a density function in physics, the performer scatters notes throughout the sonic coordinate space and combines their energies integratively to create a single, overall musical form. This mathematical metaphor is intended to emphasize that this approach is continuous and integrative. The movement is understood as a sonic flow occurring on a probability space, and thus, the improvisation is completed as one giant sculpture of sonic distribution.

Impact on the Performer's Thought Process

This sonic coordinate approach is intended to fundamentally shift the performer's way of thinking. The performer focuses on sensing their current position while navigating the sonic space.

1. From Scale Selection to Coordinate Flow: The method of picking a scale for each chord progression can be replaced with the mental image of drawing a path within the sonic coordinate system. This image of continuous movement, minimizing discontinuity, allows for an organic understanding of the relationships between notes as a trajectory in space.

2. From Individual Pitches to a Topography: A single note is not understood by its isolated meaning but as part of the energy distribution in the current sonic topography. Therefore, musical judgment can focus on the balance of energy and the contour of tension within the overall flow, rather than the appropriateness of individual pitches.

3. From Limited Choices to Free Expression: Thinking in terms of a coordinate space provides psychological freedom to the performer and enables three-dimensional mental imagery. The performer can concentrate more on rhythmic ideas or changes in timbre, broadening their spectrum of expression. By perceiving improvisation as a single organic flow rather than a series of choices, it becomes possible to exercise creativity moment by moment while also solidifying a sense of overall form. This leads to a way of thinking that places moments of sound within a larger picture, like drawing a course on a map or a picture on a large blank canvas.

   
   

Ultimately, this allows for the execution of a more intuitive and integrated musical imagination. This approach aims for creative acts that occur within free play, much like the concepts of Xiao Yao You (逍遙遊, Free and Easy Wandering) from Zhuangzi and Lila. It becomes a practice on a 'path' (道), aiming for an embodied knowledge that transcends technical skill. By exploring the unfolding sonic space, the performer forges their own musical world and gains insight. The performer will repeatedly gain new insights into music and themselves by exploring this coordinate space. This is believed to gradually change the perception of music and human expression, with the hope of encountering a state where sound, thought, and art become one.