# Music through Curve Insights

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Our Results

#### 3.1. Curve Interpretation

**Assertion**

**1.**

**Remark**

**1.**

**Remark**

**2.**

#### 3.2. Curvature as a Tool to Estimate Musical Complexity

**Remark**

**3.**

## 4. Future Work

## 5. Summary and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Additional Explorations

## References

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**Figure 1.**Three Beatles songs which we transformed into $3D$ physical objects, from left to right: Hello Goodbye; All You Need Is Love; Like Dreamers Do. $3D$ printing Pla/Sla, by Lior Bar.

**Figure 2.**Different polygonal curves in the plane. (

**a**) Total curvature $2\pi $. (

**b**) Total curvature zero.

**Figure 3.**Our first experiment to define a sequence of triads into three-dimensional physical objects. The chords as coordinates/vertices embedded in three-dimensional space. The wire moves along the coordinates and represents chord progression, which leads to a polygonal curve.

**Figure 4.**Take both curves counterclockwise. The index of the curves is five and the total curvature is $10\pi $. (

**a**) A curve with four laps. (

**b**) Approximation as a polygonal curve.

**Figure 5.**The Beatles: Across the Universe. The red dot represents the first vertex. The arrows represent the curve direction. From left to right, $3D$ representation, projection $(x,y)$ respective to (root, third), projection $(y,z)$ respective to (third, fifth), projection $(z,x)$ respective to (third, root).

**Figure 6.**The Beatles: Ask Me Why. From left to right, a $3D$ representation and the respective projections. The numbers on the edges represent the direction of movement.

**Figure 7.**The Beatles: Get Back. From left to right: the chorus, the three-dimensional representation, and the three projections. The dot in every curve is the first triad in the chorus. The total curvature of the respective projection in the plane is $(2\pi ,-2\pi ,-2\pi )$.

**Figure 8.**The Beatles: Like Dreamers Do. The respective projection’s total curvature is $(2\pi ,0,-2\pi )$.

**Figure 9.**The Beatles: Hello, Goodbye. The respective projection’s total curvature is $(0,-4\pi ,4\pi )$.

**Figure 10.**The Beatles: All You Need Is Love. The curve reveals the complexity of this song. The respective projection’s total curvature is $(-4\pi ,2\pi ,6\pi )$.

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**MDPI and ACS Style**

Gul, S.
Music through Curve Insights. *Mathematics* **2023**, *11*, 4398.
https://doi.org/10.3390/math11204398

**AMA Style**

Gul S.
Music through Curve Insights. *Mathematics*. 2023; 11(20):4398.
https://doi.org/10.3390/math11204398

**Chicago/Turabian Style**

Gul, Shai.
2023. "Music through Curve Insights" *Mathematics* 11, no. 20: 4398.
https://doi.org/10.3390/math11204398