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Loïc Guégan 2024-10-28 15:49:13 +01:00
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source/music/code/pythagoricienne.R Executable file
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#!/usr/bin/Rscript
do=261.63 # do frequency convention
N=13 # Number de nouvelle notes
quintes=rep(do,N) # Les nouvelles quintes
## Calcule l'ensemble des quintes de la gamme
facteurs=(3/2)^(0:(N-1)) # Facteurs relatif au do 261.63
## On divise par 2 jusqu'à retombé dans notre octave
toOctave=function(x){
if(x<(2*do+10)) # +10 car la dernière quinte n'est pas parfait et dépasse 2*261.63
x
else
toOctave(x/2)
}
notes=sort(sapply(quintes*facteurs,FUN=toOctave)) # Generation des notes de la gamme

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source/music/figures/harmoniques.svg Normal file
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10
source/music/gammes.rst
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Gammes
================================
Tonalité
----------
en musique une tonalité est une note accompagné de son qualificatif (example majeur, mineur) vrai ou faut?
ChatGPT said:
ChatGPT
Vrai ! En musique, une tonalité est effectivement définie par une note fondamentale (comme Do, Ré, Mi, etc.) accompagnée d'un qualificatif qui indique le mode, par exemple majeur ou mineur.
Par exemple, Do majeur et La mineur sont des tonalités.
Le qualificatif (majeur ou mineur) spécifie la couleur et la structure de la tonalité, influençant ainsi le caractère de la musique.

88
source/music/harmoniques.rst
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Harmoniques
================================
------------
Timbre
========
Lorsque l'on joue une note à l'aide d'un intruments, celui-ci sonne à
une frequence :math:`f_0` (appelée *fondamentale*) ainsi qu'à plusieurs
multiples de cette fréquence :math:`f_i=(i+1)\times f_0` (appelées *harmoniques*).
Ainsi, un lorsque l'on joue un do, on entends la superposition de la fondamentale et de ces harmoniques.
Chacunes de ces harmoniques corresponds à l'une des 12 notes de musique jouée à des hauteurs (octaves) différentes.
.. figure:: figures/harmoniques.svg
:width: 500px
:align: center
Représentation des harmoniques d'une fréquence :math:`f_0` (tout en haut)
Le *timbre* corresponds aux caracteristiques propres au son d'un
instruments, ce qui permet de le reconnaitre. Un do joué sur un piano,
une guitare, une flute reste un do. Ce qui permet de savoir si il
s'agit d'un piano, guitare ou flute est le *timbre* de
l'instrument. En terme de fréquences, le timbre ce traduit par une
variation des amplitudes (propre à chaque instruments) des harmoniques.
.. note::
La video suivante illustre bien ce concept: `Les harmoniques d'une note <https://www.youtube.com/watch?v=sPOLR60ADs8>`__
Gamme Pythagoricienne
======================
En musique, un intervalle est la rapport de deux fréquences.
Un octave corresponds à deux fois la fréquence initiale :math:`f_{octave}=2f_0`.
Il s'agit d'un intervalle important car il sonne extrêmement juste lorsque l'on joue :math:`f_0` et :math:`f_{octave}` simulatanement.
On note également que le passage à l'octave :math:`f_{octave_i}=2^i*f_0` est une opération qui ne change pas la note de départ!
Ainsi, diviser/multiplier une fréquence par 2, une ou plusieurs fois, laisse la note inchangée.
Un concept très important sur lequelle repose la gamme Pythagoricienne.
En multipliant par 3 une fréquence :math:`f_0`, on obtient une note
qui correspond quasiment à la quinte de celle-ci. Pourquoi? Car si on
ramène cette quinte sur l'octave the :math:`f_0` (en divisant par 2),
on obtient une fréquence qui correspond quasiment la quinte de cette
octave. Pour rappel, diviser par deux ne change pas la note!
En répétant ce principe, Pythagore ajoute 12 nouvelle notes l'octave.
En partant d'une fréquence :math:`f_0`, il créer une quinte :math:`f_a=3f_0`. Puis il prends à nouveau la quinte de :math:`f_b=3f_a`.
On oubli pas de divisé chacunes de ce nouvelles fréquences par 2 jusqu'a retomber dans l'octave de :math:`f_0`.
C'est à dire une fréquence :math:`f` tel que :math:`f_0<f<2f_0`.
Pourquoi 12 notes? Car si on en créer une :math:`13^{ème}`, on retombe **quasiment** sur :math:`f_0`.
Il est donc inutile de poursuivre.
En effet, :math:`\frac{3}{2}^{12}f_0 = 129.7f_0 \approx 128f_0 = 2^7f_0` ce qui corresponds à :math:`f_0` joué à 7 octaves plus haut.
Ramener cela à l'octave de :math:`f_0` celà corresponds **quasiment** à :math:`f_0` joué 1 octave plus haut.
Voici comment générer la gamme Pythagoricienne:
.. literalinclude:: code/pythagoricienne.R
:language: R
Le fait que l'on ne retombe pas éxactement à l'octave suppèrieure créer ce que l'on appèle une *quinte du loup*.
Ainsi, la dernière quinte de la gamme Pythagoricienne ne sonne pas *juste*.
.. figure:: figures/quinte_loup.jpg
:width: 300px
:align: center
Quinte du loup
De la même manière, l'intervalles de Tierce majeurs (de do à ré) ne tombe pas juste.
Pour résoudre ce problème, on pouvait utiliser le *tempérament mésotonique* (au lieu de multiplier par :math:`\frac{3}{2}`, on multipli par :math:`1.495^4` pour calculer les quintes).
Grace à cela, les tierces tombents juste! Au prix de quintes qui le sont un peu moin.
Par contre, cela agrave la quinte du loup.
En réalité, il y a plusieurs *tempérament mésotoniques*!
Chaque musiciens utilisait celui qui permettait de s'accommoder au mieux à ça pratique.
Historiquement, cela a eu un impacte sur la façon de composé.
On choisisait les accords qui sonnent le mieux avec le *tempérament mésotoniques* utilisé.
.. note::
Excellentes vidéos sur le sujet: `Antoine Houlou-Garcia <https://www.youtube.com/watch?v=RO5qmvgTH5E>`__, `ScienceÉtonnante <https://www.youtube.com/watch?v=cTYvCpLRwao>`__
Gamme Tempéré
==============
game pythagoritienne
game tempéré