Additive Asymmetry and Bichronous Interpretation: A Mathematically Extensible Model.

My musical practice, in addition to visual work, serves as a device to reveal phenomena. The homogeneous 4/4 time signature represents a standard temporal phenomenon shared by common. Its linear flow, guaranteeing predictability and stability, is like a perfectly horizontal horizon in a photograph. However, odd time—asymmetrical rhythms such as 5/4, 7/8, or 11/16—shakes this horizon, creating an unstable and unfamiliar phenomenon. This has its own role within my work's theme of <Exploring Manifolds>.

From a neuroscientific perspective, playing in odd time continuously breaks the familiar patterns established by the brain's internal clock. While the basal ganglia attempt to process regular rhythms automatically, odd time constantly generates prediction errors, demanding the involvement of the prefrontal cortex and working memory. The cerebellum corrects these errors, imprinting new rhythmic patterns into long-term memory through repetition. This process is how a new rhythmic phenomenon becomes embodied in the brain and body, extending music beyond mere calculation of beats into neurological and psychological phenomenology.

Crucially, it is the discomfort provided by odd time that matters. When familiar rhythms collapse, performers and listeners are compelled to experience tension and intense focus. Yet, precisely this discomfort leads to flow, which in turn generates a sense of liberation. This phenomenal tension, where discomfort and liberation intersect, is central to the aesthetics I pursue. It is a genuine communication with the world, grasped not by an intuitive, easily accessible feeling, but by observing phenomena carefully and reflecting upon them. This also symbolizes a worldview that reveals phenomena as they are, despite being distant from the primary intuition one feels when looking at the horizon, especially concerning phenomena on the surface that are difficult to grasp at first glance.

Therefore, in my work, odd time does not merely signify musical technique or difficulty. It is a phenomenon where time reveals itself with different faces, and a sensory field where discomfort and immersion intersect. Within this field, I experience a wavering and entangled flow, breaking free from the framework of linear time, and I consider this process itself to be the essence of art. Ultimately, odd time is the art of phenomenon that makes us re-experience time.

Musical Practice & Tools for my work

Examples include Aksak 7/8 (2+2+3), Chilchae (3+2+2), and even the phase shifts in Reich's Drumming.

• Additive Meter Structure: Defines local units consisting of sequences of short and long beats (2 or 3), and represents the entire cycle as their sum.

• Bichronous Interpretation: Explains the concept of simultaneously encoding global (position on the cycle) and local (phase within a segment) phases in the Torus ($\mathbb{T}^2$) space.

• Unequal Grid Implementation: Presents a method for generating an unequal grid from a uniform grid through time warping.

• Onset Generation and Accent Weighting: Describes an onset generation model including beat pulses and accent weights for each section, and implements part-specific rhythms by adding instrument-specific role weights.

• Perception Model: Proposes a model that represents how listeners distribute attention between the global cycle and local units as a weighted sum.

• Improvisation/Transition Map: Explains phenomena like 'mol-a-gagi' (driving forward) in Samulnori by modeling pattern transitions to the next cycle as Markov transitions or continuous-time vector fields.

• Reich Drumming: Mathematically formalizes the phase shifts and moiré rhythms resulting from subtle tempo differences between two patterns, showing how bichronous interpretation naturally emerges.

• Comparison of Aksak vs. Chilchae: Emphasizes that despite having the same total cycle ($L=7$), their decomposition vectors differ, leading to variations in accent positions and length ratios of local phases.

1) Definition of Additive Meter Structure

• Let the sequence of local units (short and long beats) be $\mathbf{a}=(a_1,\dots,a_K),\quad a_i\in{2,3}$. The total cycle (global length) is $L=\sum_{i=1}^{K} a_i$.

• Example: Aksak 7/8 → $\mathbf{a}=(2,2,3),\quad L=7$.

• Example: Chilchae (7 beats) → $\mathbf{a}=(3,2,2),\quad L=7$.

2) Bichronous Interpretation: Global-Local Phase (Torus $\mathbb{T}^2$)

Separates global cycle phase and local (detailed) phase in continuous time $t$.

• Global phase (position on the cycle): $\Theta(t)=\frac{t \bmod T}{T}\in \mathbb{S}^1$, where T is the physical length of one period (in seconds).

• Local phase is the internal phase of the segment (e.g., 2 or 3) currently located within. Specifically, if $\Theta(t)$ belongs to the interval $\mathcal{I}j=\Big[\frac{b_{j-1}}{L},\ \frac{b_{j}}{L}\Big)$, then $\phi_{\text{local}}(t)=\frac{\Theta(t)-\frac{b_{j-1}}{L}}{\frac{a_j}{L}}\in[0,1)$.Thus, $(\Theta(t),\phi_{\text{local}}(t))\in\mathbb{T}^2=\mathbb{S}^1\times\mathbb{S}^1$ becomes the bichronous coordinate.Intuition: Global is "where am I now in the 7-beat cycle?", Local is "where am I now within this 2/3 beat?".

3) Unequal Grid = Implemented with Time Warping

An unequal grid is obtained by applying time warping $g:\to$ to a uniform 7-division (virtual grid) according to local lengths. $g(x)=\frac{1}{L}\sum_{i=1}^{J-1} a_i;+;\frac{a_J}{L}\cdot \frac{x-x_{J-1}}{x_J-x_{J-1}}$, where ${x_j}$ are uniform division points (e.g., $x_j=\frac{j}{K}$), $x\in[x_{J-1},x_J]$. This maps a uniform index $x$ to an unequal global phase $g(x)$, reflecting the $2/3$ length difference.

4) Onset Generation (Beat Pulse) and Accent Weighting

Onsets for each segment as a sequence of impulses: $s(t)=\sum_{n\in\mathbb{Z}} \sum_{j=1}^{K} w_j\ \delta!(t - nT - \frac{T}{L},b_{j-1})$, where $w_j$ is the accent weight of that segment.

For instrument $m$, multiplying by role weight $c_{m,j}$ gives part-specific rhythms:

$s_m(t)=\sum_{n}\sum_{j} c_{m,j}w_j,\delta!(t - nT - \frac{T}{L},b_{j-1})$.

(Example: In Chilchae, the jing has a large $w_j$ at the start of the segment, while the janggu adds sub-onsets within the fine divisions to implement detailed rhythms.)

5) Perceptual Model of Additive Asymmetry (Local-Global Concurrent Encoding)

Listeners distribute attention to both the global cycle and local units. Weighted sum:

$\text{Salience}(t)=\alpha\cdot A_{\text{global}}\big(\Theta(t)\big) + (1-\alpha)\cdot A_{\text{local}}\big(\phi_{\text{local}}(t)\big)$,

$A_{\text{global}}$ is the prediction/accent function at the cycle level, and $A_{\text{local}}$ is the fine beat contrast within 2/3. $\alpha\in$ is the distribution of attention (varies with performance/dance/tempo).

6) Probabilistic-Dynamic Model of Improvisation/Transition Map

Pattern transition to the next cycle (e.g., 2+2+3 → 3+2+2) as a Markov transition:

$\mathbb{P}(\mathbf{a}^{(n+1)}=\mathbf{p}\mid \mathbf{a}^{(n)}=\mathbf{q})=M_{\mathbf{q}\to\mathbf{p}}$.

Or interaction of local accents/energies $v_i$ as a continuous-time vector field:

$\frac{d v_i}{dt}=F_i\big(v,\ \Theta(t),\ \phi_{\text{local}}(t)\big)$, where Sangsoe's gestures/breath signals enter as external input $u(t)$ to induce transitions:

$\frac{d v}{dt}=F(v)+B,u(t)$.

(Mol-a-gagi in Samulnori can be modeled as a simultaneous increase in $|v|$ and tempo $\dot{\Theta}$.)

7) Reich Drumming: Micro-Mismatch of Phase Shift

Let the basic pattern onsets be ${t_k}$, and if the tempo of two groups is $\omega$ and $\omega+\varepsilon$, the relative phase $\varphi(t)$ is $\dot{\varphi}(t)=\varepsilon,\quad \varphi(t)=\varphi(0)+\varepsilon t$.

The resulting (moiré) signal from the superposition of two patterns: $x(t)=\sum_k \delta(t-t_k);+;\sum_k \delta\big(t-t_k-\Delta(\varphi(t))\big)$, where $\Delta(\varphi)$ is the micro-delay due to the phase difference. From this, local-global bichronous interpretation naturally arises (local: misalignment, global: 12/8 period).

8) Aksak vs. Chilchae: Same $L$, Different Decompositions

Aksak (2+2+3) and Chilchae (3+2+2) have the same $L=7$ but different decomposition vectors.

→ Even with the same global phase $\Theta$, the accent positions and length ratios of the local phase $\phi_{\text{local}}$ differ, leading to a different perceived feeling (dance's "limping" vs. "long-short-short" compression).

9) Practical Calculation Routine (Summary)

1. Select pattern $\mathbf{a}$ → Calculate $L, {b_j}$

2. Determine $\Theta(t)$ with $T$ and tempo modulation ($rit./accel.$)

3. Assign segment weights $w_j$, part weights $c_{m,j}$

4. Improvisation/Transition: Update next cycle with $M$ or $F,B$