MusicTheory String Vibration

There are four physical quantities involved in the string vibration phenomenon :

  • the frequency \(f\) of dimension \(s^{-1}\),
  • the linear density \(\mu\) of the string of dimension \(kg \cdot m^{-1}\),
  • the tension \(T\) applied to the string of dimension \(kg \cdot m \cdot s^{-2}\),
  • the length \(L\) of the string of dimension \(m\).

According to the Vaschy-Buckingham theorem, we can build a dimensionless constant:

\[\alpha = \frac{T}{\mu L^2 f^2}\]

and thus

\[f = \frac{\beta}{L} \sqrt{\frac{T}{\mu}}\]

where \(\beta\) is a dimensionless constant.

In reality, the vibration of a string corresponds to the superposition of standing waves of frequencies:

\[f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}}\]

where \(n\) is an integer greater than 0.

This series of frequencies is called an harmonic series. The first one is called fundamental frequency, \(n=1\), and the others overtones.

We can notice from this formulae:

  • the shorter the string, the higher the frequency of the fundamental,
  • the higher the tension, the higher the frequency of the fundamental,
  • the lighter the string, the higher the frequency of the fundamental. Octave and Fifth

Let be a string of length L with a movable bridge. We denote \(f_0\) the fundamental frequency of the string.

  • If we place the mobile bridge at the middle of the string, \(L/2\), the string fundamental will now ring at \(f_8 = 2 \times f_0\), thus at the higher octave of \(f_0\).
  • If we place the mobile bridge at \(2/3 L\), which is the next simpler subdivision of the string, the larger string part will ring at \(f_5 = \frac{3}{2} \times f_0\), thus at the higher perfect fifth of \(f_0\).

We define the fourth as the ratio of \(f_4 = f_8 / f_5 = \frac{4}{3}\), which is the complement to the fifth to match the octave.

class Musica.Math.MusicTheory.AdditionModuloGroup(modulo)[source]

Bases: object

Group of Addition Modulo a Positive Integer

Group Definition

A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:

For all a, b in G, the result of the operation, a • b, is also in G.
For all a, b and c in G, (a • b) • c = a • (b • c).
Identity element
There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique.
Inverse element
For each a in G, there exists an element b in G, commonly denoted a−1 (or −a, if the operation is denoted “+”), such that a • b = b • a = e, where e is the identity element.


H is a subgroup of G if the restriction of • to H × H is a group operation on H. This is usually denoted H ≤ G, read as “H is a subgroup of G”.

operation(a, b)[source]
class Musica.Math.MusicTheory.Cent(cent)[source]

Bases: object

classmethod from_frequency(frequency1, frequency2)[source]
classmethod from_frequency_ratio(frequency_ratio)[source]
__hash__ = None
class Musica.Math.MusicTheory.ET12GroupSingleton[source]

Bases: Musica.Math.MusicTheory.AdditionModuloGroup

class Musica.Math.MusicTheory.EqualTemperamentPitch(step_number, number_of_steps)[source]

Bases: object

class Musica.Math.MusicTheory.EqualTemperamentTuning(number_of_steps, fifth_step_number)[source]

Bases: object

static fifth_approximations(number_of_steps_max=20)[source]

Compute the best perfect fifth approximations having the form \(2^{i/j}\).

12-TET is based on \(2^{7/12} \approx 1.498\) versus 1.5 for the perfect fifth.


Return the complete fifth series up to the 7th octave.

(7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, 0) (G, D, A, E, B F#, C#, G#, D#, Bb, F, C) (Sol, Ré, La, Mi, Si, Fa#, Do#, Sol#, Ré#, Sib, Fa, Do)


Return the fifth series that constitute the major scale.

(7, 2, 9, 4, 11) (G, D, A, E, B ) (Sol, Ré, La, Mi, Si)

and sorted

(2, 4, 7, 9, 11) (D, E, G, A, B) (Ré, Mi, Sol, La, Si)

Note the 6th fifth 6 / F#, is substituted by the fourth 5 / F since the major scale is made of 7 notes, the first note, the fifth, the fourth, and 4 fifths to complete the set.


Return the fourth (F / Fa) which is the inversion of the fifth to the octave (5 = 12 - 7).


Return the major scale.

The major scale is made of 7 notes, the first note, the fifth, the fourth, and 4 fifths to complete the set.

(0, 2, 4, 5, 7, 9, 11) (C, D, E, F, G, A, B) (Do, Ré, Mi, Fa, Sol, La, Si)

class Musica.Math.MusicTheory.Frequency(frequency)[source]

Bases: object

__hash__ = None
class Musica.Math.MusicTheory.FrequencyRatio[source]

Bases: object

et12 = 1.0594630943592953
fifth = 1.5
fourth = 1.3333333333333333
octave = 2
unisson = 1
class Musica.Math.MusicTheory.PythagoreanFifth(numerator_power, denominator_power)[source]

Bases: Musica.Math.MusicTheory.PythagoreanPitch

class Musica.Math.MusicTheory.PythagoreanFourth(numerator_power, denominator_power)[source]

Bases: Musica.Math.MusicTheory.PythagoreanPitch

class Musica.Math.MusicTheory.PythagoreanPitch(numerator_power, denominator_power)[source]

Bases: object

class Musica.Math.MusicTheory.PythagoreanTuningSingleton[source]

Bases: object

static add_fourths(fifth_series)[source]
static generate_fifth_series()[source]