^{1}

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^{2}

^{2}

In this paper, a simple mathematical model was proposed to answer such a question: what’s the relationship between the finger pressing position and the corresponding pitch for stringed instruments? Furthermore, we compared the theoretical results with measurement results which were obtained from a guitar to prove that the model we propose is reliable. Result shows that relative errors of theoretical results and measurement results are from 0.18% to 1.8%. The mathematics model is more concise, direct and clear than physical model.

Stringed instrument with frets is a big member in the stringed instrument family. In the west, we have guitar and mandolin, in China, Pipa, Liuqin and Ruan, in India, Sitar. It seems that, in the whole world, all the appearances of these stringed instruments are similar: the frets are not uniform distributed, whereas interval of adjacent frets decreases close to the plucking region. How about the frets distributed in mathematic?

Pure mathematical model is rare to be found, people tend to study the stringed instruments physically and acoustically as [

Right hand plays in the plucking region (in

The whole string is full of pitches just as the real number axis is full of real number. So, the principle could extend to stringed instruments without frets, such as violin and cello. The pitches in one octave are listed as

There are 13 pitches (labeled from 0 to 12) in one octave on E string of guitar. Pitch name is a musical name. According to 2.1.1, there are infinite pitch notes between “E” and “F”, “F” and “^{#}F”, and so on.

The pitch of the string played without left hand is called “free string”. When the left hand finger presses in the middle of the string, the pitch is one octave higher than free string. So, we can get two boundary conditions:

For two tight strings of different length (see

A string OZ and frets

Two strings of different length

. Pitch and its order number

Pitch name | E | F | ^{#}F | G | ^{b}A | A | ^{b}B | B | C | ^{b}D | D | ^{b}E | E |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

The finger pressing position close to the plucking region determines the pitch. For example, in

The pitches of three adjoining points P, Q, R are respectively

The deducting process is:

So, we have:

Let

Fortunately, we can get the accurate solution according to (1), (2), (4):

where

And interval of fret could be expressed as:

where r(p) and Dr(p) are respect the fret ratio-position and ratio-interval of the whole string.

Interestingly, based on Equation (5) and Equation (6), if

Fret positions of three arithmetic progression pitches

A real guitar is measured to get measurement results. Results of Equation (5) and Equation (6) are compared with measurement results, which are presented in _{1}(p) is average value of twice measurements; the relative error is computed by

What’s the mathematical relationship between pitch and position for stringed instruments? We answer this

Comparison of measurement result and theoretical result. The curve is theoretical result of Equation (5). If, then. The red points are measurement results of 20 frets of a real guitar

The ratio-interval of adjacent frets. The curve is the theoretical result of Equation (6).,

. Theoretical result and measurement result

p | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

r (p) | 0 | 0.05613 | 0.10910 | 0.15910 | 0.20630 | 0.25085 | 0.29289 |

r_{1} (p) | 0 | 0.05717 | 0.11056 | 0.16079 | 0.20731 | 0.25206 | 0.29411 |

Error (%) | 0 | 1.8 | 1.3 | 1.05 | 0.48 | 0.48 | 0.41 |

p | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

r (p) | 0.33258 | 0.37004 | 0.40540 | 0.43877 | 0.47027 | 0.50000 | 0.52806 |

r_{1} (p) | 0.33431 | 0.37111 | 0.40637 | 0.43994 | 0.47141 | 0.50135 | 0.52920 |

Error (%) | 0.52 | 0.29 | 0.23 | 0.27 | 0.24 | 0.27 | 0.22 |

p | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

r (p) | 0.55455 | 0.57955 | 0.60315 | 0.62542 | 0.64645 | 0.66629 | 0.68502 |

r_{1} (p) | 0.55559 | 0.58059 | 0.60433 | 0.62703 | 0.64848 | 0.66824 | 0.68675 |

Error (%) | 0.19 | 0.18 | 0.21 | 0.26 | 0.31 | 0.29 | 0.25 |

question in this paper. The mathematical model is simple, direct and meaningful, only based on the general music knowledge and performing experiences. Measuring the pitch-position data from a guitar, the calculated values agree with the measurement values so well.

This work was supported by China Post-doctor Foundation No. 2012M520572, Tianjin Municipal Education Commission Grant No. 20120401 and Tianjin Municipal Science and Technology Commission Key Grant (2014).