10.1.8.2. MusicTheory¶
10.1.8.2.1. 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 VaschyBuckingham theorem, we can build a dimensionless constant:
and thus
where \(\beta\) is a dimensionless constant.
In reality, the vibration of a string corresponds to the superposition of standing waves of frequencies:
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.
10.1.8.2.2. 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:
 Closure
 For all a, b in G, the result of the operation, a • b, is also in G.
 Associativity
 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.
Subgroup
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”.

cayley_table
¶

modulo
¶

class
Musica.Math.MusicTheory.
ET12GroupSingleton
[source]¶ Bases:
Musica.Math.MusicTheory.AdditionModuloGroup

subgroups
¶


class
Musica.Math.MusicTheory.
EqualTemperamentPitch
(step_number, number_of_steps)[source]¶ Bases:
object

cent
¶

number_of_steps
¶

step_number
¶


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

fifth
¶

fifth_series
¶ 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)

fifth_series_for_major_scale
¶ 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.

first_pitch
¶

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

group
¶

major_scale
¶ 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)

number_of_steps
¶

perfect_steps
¶

static

class
Musica.Math.MusicTheory.
Frequency
(frequency)[source]¶ Bases:
object

__hash__
= None¶

frequency
¶

period
¶

pulsation
¶


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

denominator
¶

numerator
¶


class
Musica.Math.MusicTheory.
PythagoreanFourth
(numerator_power, denominator_power)[source]¶ Bases:
Musica.Math.MusicTheory.PythagoreanPitch

denominator
¶

numerator
¶
