\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{music theory}

[[!redirects music]] [[!redirects music]]

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{philosophy}{}\paragraph*{{Philosophy}}\label{philosophy}

[[!include philosophy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{links}{Links}\dotfill \pageref*{links} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The application of mathematics in the theory of music in the West is generally thought to start with Pythagoras and progress though the works of Euler, Mersenne, Helmholtz, and others. Recently, \textbf{mathematical music theory} has gained impetus through, among others, the Neo-Riemannian theory of harmony, David Lewin's research on transformational theory, and Guerino Mazzola's application of topos theory.

As well, the application of science and math to the theory of human perception of sound is known as \emph{pyschoaccoustics} and has had a major impact in today's musically digital world. All of these viewpoints have greatly enhanced the variety of mathematical concepts employed in the study of music that today include tools from [[set]], [[group]], and [[category theory]].

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[dihedral group]]


\item [[harmonic analysis]]


\item [[Music of the Spheres]]


\item [[canonical formula of myth]]



\end{itemize}
\hypertarget{links}{}\subsection*{{Links}}\label{links}

\begin{itemize}%
\item John Baez: \emph{This week's find week 234} , June 12, 2006. (\href{http://math.ucr.edu/home/baez/week234.html}{link})


\item n-caf\'e{}: \emph{The Mathematics of Music at Chicago} , May 31, 2009. (\href{https://golem.ph.utexas.edu/category/2009/05/the_mathematics_of_music_at_ch.html}{link})


\item For applications of mathematics to music, see \href{http://www-personal.umd.umich.edu/~tmfiore/1/music.html}{Thomas M. Fiore's page}.


\item For discussions on the interaction of category theory and music, see the \href{http://recherche.ircam.fr/equipes/repmus/mamux/}{MaMux seminar} at IRCAM, Paris.



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A comprehensive anthology of Riemannian and Neo-Riemannian music theory is:

\begin{itemize}%
\item Edward Gollin, Alexander Rehding (eds.), \emph{The Oxford Handbook of Neo-Riemannian Music Theories} , Oxford UP 2014.

\end{itemize}
An excellent textbook on the mathematics of music that ranges from Fourier analysis over digital music to compositional symmetries is

\begin{itemize}%
\item David J. Benson, \emph{Music - A Mathematical Offering} , Cambridge UP 2007. (\href{https://homepages.abdn.ac.uk/d.j.benson/pages/html/music.pdf}{pdf})

\end{itemize}
Comparable in range though presumably more digestible for the non mathematical reader is

\begin{itemize}%
\item Gareth Loy, \emph{Musimathics - the mathematical foundations of music. Vols. 1,2} , MIT Press 2007.

\end{itemize}
Some canonical music theory texts that rely on mathematics:

\begin{itemize}%
\item Allen Forte, \emph{The Structure of Atonal Music}, Yale University Press 1977.

\end{itemize}
\begin{quote}%
``The Structure of Atonal Music ranks as one of the most important contributions to music theory in the twentieth century. His use of the computer as well as traditional means of analysis has led to fuller knowledge of musical structure and thereby enhanced both the understanding and the enjoyment of music.''


\end{quote}
\begin{itemize}%
\item David Lewin, \emph{Generalized Musical Intervals and Transformations}, Oxford University Press 2010.

\end{itemize}
\begin{quote}%
``David Lewin's Generalized Musical Intervals and Transformations is recognized as the seminal work paving the way for current studies in mathematical and systematic approaches to music analysis.''


\end{quote}
\begin{itemize}%
\item Joseph N. Straus, \emph{Introduction to Post-Tonal Theory (third edition)}, Pearson 2004.

\end{itemize}
\begin{quote}%
``The third edition stays abreast of recent theoretical developments by including discussions of transformational networks and graphs, contour theory, atonal voice leading, triadic post-tonality (including neotonality), inversional symmetry, and interval cycles.''


\end{quote}
\begin{itemize}%
\item Arnold Sch\"o{}nberg, \emph{Theory of Harmony: 100TH ANNIVERSARY EDITION}, University of California Press 2010.

\end{itemize}
A nice first introduction to music theory and what groups can do for it is

\begin{itemize}%
\item [[Alissa Crans|Alissa S. Crans]], [[Thomas M. Fiore]], Ramon Satyendra, \emph{Musical Actions of Dihedral Groups} , arXiv:0711.1873 (2008). (\href{http://arxiv.org/abs/0711.1873}{abstract})

\end{itemize}
More mathematically inclined are the following

\begin{itemize}%
\item Emilio Lluis-Puebla, Ocatavio A. August\'i{}n-Aquino (eds.), \emph{Memoirs of the Fourth International Seminar on Mathematical Music Theory} , Sociedad Matem\'a{}tica Mexicana 2010. (\href{http://sociedadmatematicamexicana.org.mx/SEPA/ECMS/resumen/PME1_1.pdf}{pdf})


\item Guerino Mazzola, \emph{Geometrie der T\"o{}ne} , Birkh\"a{}user Basel 1990.


\item Guerino Mazzola, \emph{The Topos of Music} , Birkh\"a{}user Basel 2002.


\item Guerino Mazzola, \emph{Mathematical Music Theory: Status Quo 2010} , pp.11-42 in \hyperlink{LPAA10}{Lluis-Puebla, August\'i{}n-Aquino (2010)}.


\item Dimitri Tymoczko, \emph{A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice} , Oxford University Press 2011.


\item Dmitri Tymoczko, \emph{Generalizing Musical Intervals} , Journal of Music Theory \textbf{53} no.2 (2009) pp.227-254. (\href{http://dmitri.mycpanel.princeton.edu/files/publications/lewin.pdf}{draft})



\end{itemize}
An introduction to group theory with applications to Neo-Riemannian theory is

\begin{itemize}%
\item Ocatavio A. August\'i{}n-Aquino, Janine du Plessis, Emilio Lluis-Puebla, Mariana Montiel, \emph{Una introducci\'o{}n a la Teor\'i{}a de Grupos con aplicaciones en la Teor\'i{}a Matem\'a{}tica de la M\'u{}sica} , Sociedad Matem\'a{}tica Mexicana 2009. (\href{http://pesmm.org.mx/Serie%20Textos_archivos/T10.pdf}{pdf})

\end{itemize}
Besides the monumental \href{(#Mazzola02}{Mazzola (2002)}, the following employ category theory:

\begin{itemize}%
\item Thomas Noll, \emph{The Topos of Triads} , pp.1-26 in Fripertinger, Reich (eds.), \emph{Colloquium on Mathematical Music Theory} , Grazer Math. Ber. \textbf{347} (2005). (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.134.5669}{citeseer})


\item Alexandre Popoff, \emph{Towards A Categorical Approach of Transformational Music Theory} , arXiv:1204.3216 (2014). (\href{http://arxiv.org/abs/1204.3216}{abstract})



\end{itemize}
Methods of physics are employed to study tonality in

\begin{itemize}%
\item Peter beim Graben, Reinhard Blutner, \emph{Toward a Gauge Theory of Musical Forces} , pp.99-111 in LNCS \textbf{10106} (2017). (\href{http://www.blutner.de/GrabenBlutnerQI16.pdf}{draft})

\end{itemize}
Hegel's philosophy of music can be found in his lectures on aesthetics though these reflect to a large extent the view of his pupil Heinrich Gustav Hotho who compiled the posthumeous publication:

\begin{itemize}%
\item [[Georg Hegel|G. W. F. Hegel]], \emph{Vorlesungen über die Ästhetik II} , Verlag Das Europäische Buch Westberlin $^4$1985. (pp.258-326)

\end{itemize}
A development of the basic concepts of music theory along his ideas on dialectics can be found in

\begin{itemize}%
\item Moritz Hauptmann, \emph{Die Natur der Harmonik und Metrik} , Breitkopf\&Härtel Leipzig 1853. (\href{https://de.wikisource.org/wiki/Moritz_Hauptmann}{wikisource})

\end{itemize}
The following introductory texts on psychoaccoustics help provide a foundation to better understand how sound becomes emotion becomes music:

\begin{itemize}%
\item Arthur H. Benade, \emph{Fundamentals of Musical Acoustics} , Dover New York 1976.


\item Hugo Fastl, Eberhard Zwicker, \emph{Psychoaccoustics: facts and models} , Springer Heidelberg 1990.


\item Juan G. Roederer, \emph{The Physics and Psychophysics of Music} , Springer Heidelberg 2008$^4$.



\end{itemize}
A theory of emotions which applies beyond music in narrative arts is in

\begin{itemize}%
\item David Huron, \emph{Sweet Anticipation - Music and the Psychology of Expectation} , MIT Press 2006.

\end{itemize}
An introduction to [[Haskell]] programming systematically employing examples from music is

\begin{itemize}%
\item Paul Hudak, \emph{The Haskell School of Music - From Signals to Symphonies} , ms. Yale University 2012. (\href{http://www.cs.yale.edu/homes/hudak/Papers/HSoM.pdf}{pdf})

\end{itemize}
Other ``mathymusical'' topics include:

\begin{itemize}%
\item Milton Babbit, \emph{The function of set structure in the twelve-tone system} , Ph.D. dissertation Princeton University 1946.


\item E. Chew, \emph{Towards a Mathematical Model of Tonality} , Ph.D. dissertation MIT Cambridge 2000.


\item David Clampitt, Thomas Noll, \emph{Modes, the Height-Width Duality, and Handschin's Tone Character} , Music Theory Online \textbf{17} no.1 (2011).


\item [[Thomas M. Fiore]], Ramon Satyendra, \emph{Generalized Contextual Groups} , Music Theory Online \textbf{11} no.3 (2005).


\item J. Hook, \emph{Uniform triadic transformations} , Journal of Music Theory \textbf{46} no.1-2 (2002) pp.57--126.


\item T. A. Johnson, \emph{Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals} , Scarecrow Press 2008.


\item Fred Lerdahl, \emph{Tonal Pitch Space} , Oxford University Press 2004.


\item Miguel F. M. Lima, J. A. Tenreiro Machado, Ant\'o{}nio C. Costa, \emph{A Multidimensional Scaling Analysis of Musical Sounds Based on Pseudo Phase Plane} , Abstract and Applied Analysis Vol. 2012. (Article ID 436108)


\item Thomas Noll, \emph{Sturmian sequences and morphisms: a music theoretical application} , Journ\'e{}e annuelle (2008) pp.79--102.


\item R. N. Shepard, \emph{Geometrical approximations to the structure of musical pitch. Psychological Review, \textbf{89} no.4 (1982) 305.}


\item J. Tennenbaum, \emph{The Foundations of Scientific Musical Tuning} , in Staff, Sigerson (eds.), \emph{A Manual on the rudiments of tuning and registration} , Schiller Institute Washington 1992.


\item J. S. Walker, D. W. Gary, \emph{Mathematics and Music: Composition, Perception, and Performance} , Taylor\&Francis Boca Raton 2013.



\end{itemize}
[[!redirects music]] [[!redirects mathematical music theory]]



\end{document}