From 481dc5eb867effe6afda9959f562c2eed4050348 Mon Sep 17 00:00:00 2001 From: Martin Potier Date: Wed, 8 Aug 2012 10:53:17 +0200 Subject: [PATCH] Done with correction, continuing with filling, yum! --- content.tex | 250 +++++++++++++++++++++++++++++++----------- img/chemin-rythm1.png | Bin 0 -> 104486 bytes img/chemin-rythm2.png | Bin 0 -> 11773 bytes 3 files changed, 187 insertions(+), 63 deletions(-) create mode 100644 img/chemin-rythm1.png create mode 100644 img/chemin-rythm2.png diff --git a/content.tex b/content.tex index efa5afb..b53335d 100644 --- a/content.tex +++ b/content.tex @@ -253,9 +253,9 @@ Par exemple, est-il possible d'avoir des mélodies caractéristiques de l'état \subref{fig:desordonnee}, \subref{fig:part-des} et \subref{fig:reguliere} de la figure \ref{fig:mousses-space}, permettant de les distinguer ? \item{Peut-on écouter les épisodes catastrophiques lors de l'évolution -emporelle du système ?} Par exemple, est-il possible d'avoir des mélodies ayant -des variations fortes correspondantes aux épisodes catastrophique, les mettant -en évidence ? +temporelle du système ?} Par exemple, est-il possible d'avoir des mélodies +ayant des variations fortes correspondantes aux épisodes catastrophiques, les +mettant en évidence ? \end{enumerate} Ces deux questions ont orienté notre exploration lors de la @@ -263,13 +263,13 @@ sonification/musification du système. La première est illustrée par les trois états de la figure \ref{fig:mousses-space} ; la seconde est illustrée par les états en fonction du temps de la figure \ref{fig:mousses-time} et le graphe de l'évolution temporelle d'un paramètre de la figure \ref{fig:mousses-graph}. -Dans ce graphe, on peut noter trois moment important~: +Dans ce graphe, on peut noter trois moments importants~: \begin{enumerate} \item une phase initiale : le système semble statique du point de vue du -paramêtre représenté ; -\item une phase catastrophique : on trouve plusieurs marches à chaque -épisode catastrophique et ils sont difficiles à trouver quand on ne connaît -pas le \emph{bon} paramêtre ; +paramètre représenté ; +\item une phase intermédiaire : on trouve plusieurs marches à chaque épisode +catastrophique et ils sont difficiles à trouver quand on ne connaît pas le +\emph{bon} paramètre ; \item une phase dite de « scaling state » : le système continue à évoluer mais de manière similaire dans le temps (on ne peut plus distinguer deux images prises à des moments différents de deux dont la seconde est un agrandissement @@ -364,11 +364,11 @@ en nous fondant sur la set-theory. \label{subsec:music} Nous resterons très général sur cette théorie musicale. La notion importante utilisée tout au long de ce mémoire est celle d'\emph{intervalle} : c'est la -\emph{hauteur} entre deux notes. Le plus petit intervalle considéré est le -demi-ton. Il y a 12 demi-tons dans la gamme occidentale et ils sont répartis -sur 7 notes (figure~\ref{fig:gamme}). On peut altérer la hauteur d'une note, -donc l'intervalle ayant pour une de ses bornes cette note, en la faisant -précéder d'une altération : ♯ (dièse, +1 demi-ton) ou ♭ (bémol, -1 demi-ton). +hauteur entre deux notes. Le plus petit intervalle considéré est le demi-ton. +Il y a 12 demi-tons dans la gamme occidentale et ils sont répartis sur 7 notes +(figure~\ref{fig:gamme}). On peut altérer la hauteur d'une note, donc +l'intervalle ayant pour une de ses bornes cette note, en la faisant précéder +d'une altération : ♯ (dièse, +1 demi-ton) ou ♭ (bémol, -1 demi-ton). \begin{figure}[p] \centering @@ -442,8 +442,9 @@ qui constituent une première formalisation algébrique. \caption{Intervalles et opérations algébriques sur un cercle} \end{figure} +\bigskip Une autre représentation qui nous intéresse est l'organisation spatiale des -intervalles au sein d'un \emph{tonnetz} (figure~\ref{fig:tonnetz}), décrit en +intervalles au sein d'un \emph{tonnetz} (figure~\ref{fig:tonnetz}), décrite en premier par Leonhard Euler. Ce dernier a choisi une disposition spatiale compacte valorisant les intervalles\footnote{% (((----------------------------- L. Euler utilise une notation allemande où le $H$ correspond au $B$ anglo-saxon @@ -453,8 +454,8 @@ pour de plus amples détails.}% )))--------------------------------------------- ~de tierce majeure (4 demi-tons, en progressant sur l'axe horizontal vers la droite) et de quinte juste (7 demi-tons, en progressant sur l'axe vertical vers le bas). Cette représentation est équivalente à la donnée d'un groupe cyclique -d'ordre 12 (comme précédemment) mais exprimé sous forme d'un graphe planaire. -Ces deux intervalles sont les plus consonnants (après l'octave) ; il est donc +d'ordre 12, comme précédemment, mais exprimée sous forme d'un graphe planaire. +Ces deux intervalles sont les plus consonnants après l'octave ; il est donc agréable et pratique de pouvoir passer d'une note à une autre en les privilégiants. @@ -464,26 +465,27 @@ privilégiants. \subfloat[Tonnetz de L. Euler (1739)]{ \includegraphics[width=.45\textwidth]{img/eulers-tonnetz} \label{fig:tonnetz}} -\subfloat[Tonnetz de L. Euler vu comme une partie du graphe de Cayley du groupe -$\mathbb{Z}_{12}$ avec pour partie génératrice $\{4,7\}$]{ +\subfloat[Tonnetz de L. Euler vu comme une partie du graphe de Cayley de la +présentation finie $<~4,~7;~3\cdot4~=~0,~12\cdot7~=~0~>$ du groupe +$\mathbb{Z}_{12}$]{ \begin{tikzpicture} -[note/.style={draw,black,circle,inner sep=2mm}, +[note/.style={draw,black,circle,inner sep=.5mm,minimum size=8mm}, label distance=-1mm,label position=below left, double distance=.5mm] -\node[note,double] (C) [label=Do ] {}; -\node[note,left=of C] (F) [label=Fa ] {}; -\node[note,right=of C] (G) [label=Sol ] {}; -\node[note,right=of G] (D) [label=Ré ] {}; - -\node[note,above=of F] (A) [label=La ] {}; -\node[note,right=of A] (E) [label=Mi ] {}; -\node[note,right=of E] (B) [label=Si ] {}; -\node[note,right=of B] (Fd) [label= Fa♯] {}; - -\node[note,above=of A] (Cd) [label= Do♯] {}; -\node[note,right=of Cd] (Gd) [label=Sol♯] {}; -\node[note,right=of Gd] (Dd) [label= Ré♯] {}; -\node[note,right=of Dd] (Ad) [label= La♯] {}; +\node[note,double] (C) {Do }; +\node[note,left=of C] (F) {Fa }; +\node[note,right=of C] (G) {Sol }; +\node[note,right=of G] (D) {Ré }; + +\node[note,above=of F] (A) {La }; +\node[note,right=of A] (E) {Mi }; +\node[note,right=of E] (B) {Si }; +\node[note,right=of B] (Fd) { Fa♯}; + +\node[note,above=of A] (Cd) { Do♯}; +\node[note,right=of Cd] (Gd) {Sol♯}; +\node[note,right=of Gd] (Dd) { Ré♯}; +\node[note,right=of Dd] (Ad) { La♯}; \draw (F) -- (C) -- node[above,midway] {+7} (G) -- (D); \draw (A) -- (E) -- (B) -- (Fd); @@ -528,24 +530,26 @@ respectivement, comme on peut le voir sur la figure~\ref{fig:trig}. \begin{figure}[p] \centering +\subfloat[Triangulation d'accords sur un graphe de Cayley, en exemple Do Majeur +(gris clair) et La mineur (gris foncé)]{ \begin{tikzpicture} -[note/.style={draw,black,circle,inner sep=2mm}, -label distance=-1mm,label position=below left, -double distance=.5mm] -\node[note,double] (C) [label=Do ] {}; -\node[note,left=of C] (F) [label=Fa ] {}; -\node[note,right=of C] (G) [label=Sol ] {}; -\node[note,right=of G] (D) [label=Ré ] {}; - -\node[note,above=of F] (A) [label=La ] {}; -\node[note,right=of A] (E) [label=Mi ] {}; -\node[note,right=of E] (B) [label=Si ] {}; -\node[note,right=of B] (Fd) [label= Fa♯] {}; - -\node[note,above=of A] (Cd) [label= Do♯] {}; -\node[note,right=of Cd] (Gd) [label=Sol♯] {}; -\node[note,right=of Gd] (Dd) [label= Ré♯] {}; -\node[note,right=of Dd] (Ad) [label= La♯] {}; +[note/.style={draw,black,circle,inner sep=.5mm,minimum size=8mm}, +label distance=-1mm,label position=above right, +double distance=.5mm,scale=.30\textwidth/7.2cm] +\node[note] (F) at (0cm,0cm) {Fa }; +\node[note,double] (C) at (2cm,0cm) {Do }; +\node[note] (G) at (4cm,0cm) {Sol }; +\node[note] (D) at (6cm,0cm) {Ré }; + +\node[note] (A) at (0cm,2cm) {La }; +\node[note] (E) at (2cm,2cm) {Mi }; +\node[note] (B) at (4cm,2cm) {Si }; +\node[note] (Fd) at (6cm,2cm) { Fa♯}; + +\node[note] (Cd) at (0cm,4cm) { Do♯}; +\node[note] (Gd) at (2cm,4cm) {Sol♯}; +\node[note] (Dd) at (4cm,4cm) { Ré♯}; +\node[note] (Ad) at (6cm,4cm) { La♯}; \draw (F) -- (C) -- (G) -- (D); \draw (A) -- (E) -- (B) -- (Fd); @@ -581,10 +585,96 @@ double distance=.5mm] \draw[dashed] (Fd.east) -- +(6mm ,0cm ); \draw[dashed] (D.east) -- +(6mm ,0cm ); \end{tikzpicture} +\label{fig:trig}} +\quad +\subfloat[Dual du graphe de Cayley mettant en exergue une structure +hexagonale]{ +\begin{tikzpicture} +[note/.style={draw,black,circle,inner sep=2mm}, +hex/.style={}, +label distance=-1mm,label position=below left, +double distance=.5mm,scale=.50\textwidth/9.2cm] +\begin{scope}[opacity=.5] +\node[note] (F) at (-1cm,0cm) {}; +\node[note,double] (C) at ( 1cm,0cm) {}; +\node[note] (G) at ( 3cm,0cm) {}; +\node[note] (D) at ( 5cm,0cm) {}; -\caption{Triangulation d'accords sur un graphe de Cayley, en exemple Do Majeur -(gris clair) et La mineur (gris foncé)} -\label{fig:trig} +\node[note] (A) at ( 0cm,2cm) {}; +\node[note] (E) at ( 2cm,2cm) {}; +\node[note] (B) at ( 4cm,2cm) {}; +\node[note] (Fd) at ( 6cm,2cm) {}; + +\node[note] (Cd) at ( 1cm,4cm) {}; +\node[note] (Gd) at ( 3cm,4cm) {}; +\node[note] (Dd) at ( 5cm,4cm) {}; +\node[note] (Ad) at ( 7cm,4cm) {}; + +\draw (F) -- (C) -- (G) -- (D); +\draw (A) -- (E) -- (B) -- (Fd); +\draw (Cd) -- (Gd) -- (Dd) -- (Ad); + +\draw (F) -- (A) -- (Cd); +\draw (C) -- (E) -- (Gd); +\draw (G) -- (B) -- (Dd); +\draw (D) -- (Fd) -- (Ad); +\draw (Cd) -- (E) -- (G); +\draw (Gd) -- (B) -- (D); +\draw (Dd) -- (Fd); +\draw (A) -- (C); + +\node (1u) at (barycentric cs:A=1,Cd=1,E=1) {}; +\node (2u) at (barycentric cs:Gd=1,B=1,E=1) {}; +\node (3u) at (barycentric cs:B=1,Dd=1,Fd=1) {}; +\node (4u) at (barycentric cs:F=1,A=1,C=1) {}; +\node (5u) at (barycentric cs:E=1,G=1,C=1) {}; +\node (6u) at (barycentric cs:B=1,G=1,D=1) {}; +\node (1d) at (barycentric cs:Cd=1,Gd=1,E=1) {}; +\node (2d) at (barycentric cs:Dd=1,Gd=1,B=1) {}; +\node (3d) at (barycentric cs:Dd=1,Ad=1,Fd=1) {}; +\node (4d) at (barycentric cs:A=1,E=1,C=1) {}; +\node (5d) at (barycentric cs:G=1,E=1,B=1) {}; +\node (6d) at (barycentric cs:D=1,Fd=1,B=1) {}; + +\draw[dashed] (Cd.north) -- +(0cm ,6mm ); +\draw[dashed] (Gd.north) -- +(0cm ,6mm ); +\draw[dashed] (Dd.north) -- +(0cm ,6mm ); +\draw[dashed] (Ad.north) -- +(0cm ,6mm ); +\draw[dashed] (F.south) -- +(0cm ,-6mm); +\draw[dashed] (C.south) -- +(0cm ,-6mm); +\draw[dashed] (G.south) -- +(0cm ,-6mm); +\draw[dashed] (D.south) -- +(0cm ,-6mm); +\draw[dashed] (F.west) -- +(-6mm,0cm ); +\draw[dashed] (A.west) -- +(-6mm,0cm ); +\draw[dashed] (Cd.west) -- +(-6mm,0cm ); +\draw[dashed] (Ad.east) -- +(6mm ,0cm ); +\draw[dashed] (Fd.east) -- +(6mm ,0cm ); +\draw[dashed] (D.east) -- +(6mm ,0cm ); +\end{scope} + +\draw[hex] (1u.center) -- (1d.center) -- (2u.center) + -- (2d.center) -- (3u.center) -- (3d.center); +\draw[hex] (4u.center) -- (4d.center) -- (5u.center) + -- (5d.center) -- (6u.center) -- (6d.center); +\draw[hex] (1u.center) -- (4d.center); +\draw[hex] (2u.center) -- (5d.center); +\draw[hex] (3u.center) -- (6d.center); +\draw[hex,dashed] (1d.center) -- +(0, 1.5cm); +\draw[hex,dashed] (2d.center) -- +(0, 1.5cm); +\draw[hex,dashed] (3d.center) -- +(0, 1.5cm); +\draw[hex,dashed] (4u.center) -- +(0,-1.5cm); +\draw[hex,dashed] (5u.center) -- +(0,-1.5cm); +\draw[hex,dashed] (6u.center) -- +(0,-1.5cm); + +\draw[hex,dashed] (1u.center) -- +(150:1.0cm); +\draw[hex,dashed] (4u.center) -- +(150:1.0cm); +\draw[hex,dashed] (3d.center) -- +(-30:1.0cm); +\draw[hex,dashed] (6d.center) -- +(-30:1.0cm); +\end{tikzpicture} +\label{fig:dual}} + +\caption{Triangulation d'accords et dual se rapportant à un graphe de Cayley} +\label{fig:cayley-use} \end{figure} \section{Formalisation} @@ -600,14 +690,21 @@ G~: \item un arc étiqueté $e$ setrouve entre les sommets $U$ et $V$ si $U + e = V$. \end{itemize} -Un tonnetz peut être vu comme le graphe de Cayley du groupe cyclique -$\mathbb{Z}_{12}$ des 12 demi-tons de la gamme occidentale muni de l'addition -comme loi commutative et d'une partie génératrice $S$ ; pour garder l'analogie -dans la figure \ref{fig:cayley}, nous utilisons deux générateurs libres : la -tierce Majeur et la quinte juste. Le sommet à l'origine du graphe de Cayley est -l'élément \emph{neutre} du groupe. +Un tonnetz peut être vu comme le graphe de Cayley de la présentation finie +d'un groupe, en l'occurence du groupe cyclique $\mathbb{Z}_{12}$ des 12 +demi-tons de la gamme occidentale, muni de l'addition comme loi commutative et +d'une partie génératrice $S$. Pour garder l'analogie dans la figure +\ref{fig:cayley}, nous utilisons deux générateurs : la tierce Majeure +(\texttt{4}) et la quinte juste (\texttt{7}). Le sommet à l'origine du graphe +de Cayley est l'élément \emph{neutre} du groupe. Dans nos exemples, nous +utilisons la présentation finie suivante : +\begin{quote} +\texttt{g = < 4, 7; 3.4=0, 12.7=0 >} +\end{quote} +Dans le graphe de Cayley associé à cette présentation, l'espace se replie sur +lui même après 4 « sauts » de tierce Majeure ou 12 « sauts » de quinte juste. -Chemins hamiltoniens dans le tonnetz \cite{albini_hamiltonian_2009}. +%Chemins hamiltoniens dans le tonnetz \cite{albini_hamiltonian_2009}. \subsection{Quelques mappings} Pour apporter des éléments de réponse aux questions des physiciens (§~% @@ -630,16 +727,42 @@ Nous utilisons cette capacité de reconnaissance pour reconnaître et différencier différentes organisations spatiales des bulles dans une mousse en deux dimensions et plus tard reconnaître leur évolution. -\subsubsection{Organisation rythmique} +La synthèse modale est un cas de sonification classique. Nous associons un +résonnateur à chaque bulle de la mousse, ainsi les paramètres de la bulle +sert à déterminer les paramètres du résonnateur. Ce dernier est modélisé très +simplement par une fonction oscillante atténuée : +$$ cos(a\cdot t)\cdot e^{-k\cdot t} $$ +Ce premier mapping se veut très simple afin de déterminer quelles informations +sont très facilement accessibles à l'ouïe. L'Implémentation +(§~\ref{sec:implementation}) a été menée en utilisant \modalys. + +\subsubsection{Chemins rythmiques} À l'image d'un exemple de sonification du §~\ref{subsec:sonification}, le compteur Geiger, nous pouvons utiliser le rythme comme lien à l'organisation spatiale des bulles d'une mousse liquide en deux dimensions. +\begin{figure}[ht] +\centering +\subfloat[$(\Delta)$, en rouge, traverse la mousse. Les centres des bulles +proches sont sélectionnés]{ + \includegraphics[width=.45\textwidth]{img/chemin-rythm1} + \label{fig:rythm1}} +\qquad +\subfloat[Projection orthogonale des centres de bulles sur $(\Delta)$ pour +obtenir une phrase rythmique]{ + \includegraphics[width=.45\textwidth]{img/chemin-rythm2} + \label{fig:rythm2}} + \caption{Extraction d'une phrase rythmique dans une mousse en deux + dimensions} +\end{figure} + On parcours par balayage le long d'une segment de droite $(\Delta)$ l'image d'une mousse en sélectionnant tous les centre de bulle étant à une distance $d$ de la droite. Ces échantillons récoltés sont ensuites projetés orthogonalement sur $(\Delta)$. On sonifie ensuite la distance entre chaque point projeté pour -obtenir un motif rythmique. +obtenir un motif rythmique : nous avons ainsi une information en une dimension +en traversant un échantillon et nous pouvons détecter une symétrie axiale d'axe +orthogonal à $(\Delta)$. Cette technique est mise en pratique par S. Adhitya dans \textsc{Sum} \cite{adhitya_audio-assisted_2011}, un outil permettant de sonifier @@ -663,7 +786,7 @@ voisins, en prenant les directions \texttt{a}, \texttt{b}, \texttt{a+b}, \texttt{-a}, \texttt{-b} et \texttt{-(a+b)}. Plongé dans un tonnetz, ceci correspond à une suite de notes. -\subsubsection{Chemins sonores augmentés} +\subsubsection{Chemins augmentés} Nous avons rajouté des extensions au dessus des chemins sonores tels que décrits dans la section précédente : accords, mélodies plus complexes et rythme. @@ -675,6 +798,7 @@ différences de distance entre le parcours dans un espace \emph{régulier} et déformé. \section{Implementation} +\label{sec:implementation} Tous les mappings ont été implémentés grâce aux outils présents à l'\ircam, notamment \modalys, pour la synthèse modale, et \openmusic\ comme environnement de programmation principal. diff --git a/img/chemin-rythm1.png b/img/chemin-rythm1.png new file mode 100644 index 0000000000000000000000000000000000000000..3f748d39de7b708f26be6f9e70d5e9862ec31b5e GIT binary patch literal 104486 zcmXtf1z1*Vu=PiGN_U5Vgp|^ubci4zoq}|COSd4MQc@Bk0#brVm!b&LDIL;z=iLAP z`*}Q%a=zGm-kCLPtr?;<)fMotsjv|U1fG(jycPn1Gys1jF_GaXq1U~y5C}3pTRAyR 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