# Fluids in Music: The Mathematics of Pan’s Flutes

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## Abstract

**:**

## 1. Introduction

## 2. The Relationship between the Frequency and the Length of a Pipe

## 3. The Relationship between the Musical Notes and the Frequencies

## 4. Putting It All Together

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Fletcher, N.H.; Rossing, T.D. The Physics of Musical Instruments; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Feynman, R. Chapter 47: Sound. The wave equation. In Lectures in Physics; Caltech: Pasadena, CA, USA, 1963; Volume 1. [Google Scholar]
- Boyce, W.E.; Di Prima, R.C.; Meade, D.B. Elementary Differential Equations and Boundary Value Problems, 11th ed.; Wiley: Hoboken, NJ, USA, 2017. [Google Scholar]
- Benson, D. Music: A Mathematical Offering; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]

**Figure 1.**An 18-pipe Pan’s flute tuned in G-major and a bar graph representing the lengths of individual pipes in cm. The red bars show the root notes (in this case G’s), the gray bars show the location of whole tone intervals, and the blue bars show the location of semitone intervals.

**Figure 2.**The relation between the notes played by a Pan’s flute, their corresponding frequency, and the length of pipes as a composition of functions.

**Figure 3.**The curve in Equation (9) superimposed on the model of the Pan’s flute. The semitones, represented in blue, were adjusted to indicate the correct musical intervals.

**Table 1.**The notes and their corresponding frequencies (rounded to the nearest integer) produced by the Pan’s flute.

Notes | D4 | E4 | F#4 | G4 | A4 | B4 | C5 | D5 | E5 | F#5 | G5 | A5 | B5 | C6 | D6 | E6 | F#6 | G6 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Frequencies | 294 | 330 | 370 | 392 | 440 | 494 | 523 | 587 | 659 | 740 | 784 | 880 | 988 | 1047 | 1175 | 1319 | 1480 | 1568 |

Note | G4 | A4 | B4 | C5 | D5 | E5 | F#5 | G5 |
---|---|---|---|---|---|---|---|---|

Ratio | 1:1 | 9:8 | 81:64 | 4:3 | 3:2 | 27:16 | 243:128 | 2:1 |

Note | G4 | A4 | B4 | C5 | D5 | E5 | F#5 | G5 |
---|---|---|---|---|---|---|---|---|

Ratios | 1:1 | ${2}^{\frac{1}{6}}:1$ | ${2}^{\frac{1}{3}}:1$ | ${2}^{\frac{5}{12}}:1$ | ${2}^{\frac{7}{12}}:1$ | ${2}^{\frac{3}{4}}:1$ | ${2}^{\frac{11}{12}}:1$ | 2:1 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nita, B.G.; Ramanathan, S. Fluids in Music: The Mathematics of Pan’s Flutes. *Fluids* **2019**, *4*, 181.
https://doi.org/10.3390/fluids4040181

**AMA Style**

Nita BG, Ramanathan S. Fluids in Music: The Mathematics of Pan’s Flutes. *Fluids*. 2019; 4(4):181.
https://doi.org/10.3390/fluids4040181

**Chicago/Turabian Style**

Nita, Bogdan G., and Sajan Ramanathan. 2019. "Fluids in Music: The Mathematics of Pan’s Flutes" *Fluids* 4, no. 4: 181.
https://doi.org/10.3390/fluids4040181