# Rhythm-Based Attention Analysis: A Comprehensive Model for Music Hierarchy
## 3 Proposed Methods
### Quantization Scheme for Rhythmic Attention
#### 3.1.1 Attention Allocation for Basic Beats  
[PERSON] (1963) explains that the primary meter in music tends to organize and be perceived at the lowest structural level where it shows regularity. He emphasizes that meter, like other musical elements, has an architectural nature. This means that the beats defined by the meter in the time signature can be broken down into equal units or combined into larger metric units, with certain beats being more accented than others. As a result, most musical compositions present a hierarchical structure of metric organizations [38]. Inspired by pink noise (\(1/f\) noise) [39], whose power spectral density (PSD) is inversely proportional to frequency, we derive principles for attention allocation over basic beat types. The relationship is provided by Equation (1):  
\[S(f)\propto\frac{1}{f^{\alpha}}. \tag{1}\]The value of alpha tends to vary with the genre of the music, as well as the composer's preference [34]. According to [PERSON]'s theory of fractal time series and long-range correlation analysis, when \(\alpha=0.5\), the evolution of a system exhibits weak long-range memory effects [40], which aligns with the long-range correlations found in music. Furthermore, [PERSON]'s research has shown that musical rhythms follow a \(1/f\) power law distribution when \(\alpha\) is around 0.5 [34]. To align our model with empirical observations in music theory and psychology, we set \(\alpha\) to 0.5. This specific value of \(\alpha\) reflects the balance between complexity and predictability in musical rhythms. A value of \(\alpha=0.5\) in the \(1/f^{0.5}\) noise model captures the natural variability observed in rhythmic patterns while maintaining a structure that is perceptually appealing and cognitively manageable. Considering a musical beat cycle, there is both a master cycle and its sub-cycles. For instance, in a measure of triple meter, the master cycle would be the entire measure, while the sub-cycles would consist of individual beats within that measure. Similarly, a beat might be subdivided into two half-beats. The first note of the whole cycle will be dominated by the master cycle, while the other notes will be dominated by their sub-cycles. The frequency of sub-cycles is typically an integer multiple of the master cycle's frequency. Given the self-similarity across different scales for the \(1/f^{\alpha}\) function, we can establish a multiplicative relationship between the power intensity of the master cycle and its sub-cycles. Specifically, if the frequency of the master cycle is \(f_{m}\) and the frequency of the sub-cycle is \(kf_{m},k\in\mathbb{N}^{+}\), then the relative power intensities can be normalized to distribute the attentional intensity across the cycle. Figure 2 will illustrate the master cycle and its sub-cycles both in the view of a measure and a beat. For normalization, we consider the individual contributions of the master cycle and the sub-cycles maintaining their relative proportions to the total attention distribution, so we use the proportionate normalization method and set the sum to 1. The normalized value of attention for the master cycle can be expressed as Equation (2):  
\[A(f_{m})=\frac{S(f_{m})}{S(f_{m})+(k-1)S(kf_{m})}. \tag{2}\]  
Similarly, the proportional attention for each sub-cycle can be expressed as Equation (3):  
\[A(kf_{m})=\frac{S(kf_{m})}{S(f_{m})+(k-1)S(kf_{m})}, \tag{3}\]  
where \(A(f_{m})\) and \(A(kf_{m})\) are the normalized attentional intensities for the master cycle and sub-cycles, respectively. Since the fractal property (self-similarity across scales) of \(S\), only the value of integer \(k\) is needed to calculate \(A(f_{m})\) and \(A(kf_{m})\), then we define \(m_{k}=A(f_{m})\) as the master cycle  
Figure 2: Illustration of the master cycle and sub-cycles. power intensities and \(s_{k}=A(kf_{m})\) as sub-cycle power intensities. Table 2 provides the attentional intensities for common scenarios of \(k\) values.