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\begin{document}
\newtheorem{satz}{Theorem}
\newtheorem{lemma}[satz]{Lemma}
\newtheorem{definition}[satz]{Definition}
\newtheorem{bem}[satz]{Remark}
\newtheorem{beisp}[satz]{Example}
\def\CI{\mathop{\rm CI}\nolimits }
\def\inter{\mathop{\rm Int}\nolimits }

\title{Enumeration in Musical Theory}
\author{Harald Fripertinger \thanks{The author thanks Jens Schwaiger for
helpful comments.}
\\ Voitsberg, Graz }
\date{\empty}
\maketitle
\markboth{\hfill{\sc H. Fripertinger}\hfill}{\hfill{\sc Enumeration in Musical
Theory}\hfill}
\setcounter{page}{29}
\begin{abstract}
Being a mathematician and a musician (I play the flute) I found it very
interesting to deal with P\'olya's	counting theory in my Master's thesis.
When reading about	P\'olya's theory I came across an article, called	"Enumeration in 
Music Theory" by D. L. Reiner \cite{21}. I took up his ideas and tried to
enumerate some other "musical objects".
\par At first I would like to generalize certain aspects of 12-tone music to
$n$-tone music, where $n$ is a positive integer. Then I will explain how to
interpret intervals, chords, tone-rows, all-interval-rows, rhythms, motifs and
tropes in $n$-tone music. Transposing, inversion and retrogradation are
defined to be permutations on the sets of "musical objects". These
permutations generate permutation groups, and these groups induce
equivalence relations on the sets of "musical objects". The aim of this
article is to determine the number of equivalence classes (I will call them
patterns) of "musical objects". P\'olya's enumeration theory is the right
tool to solve this problem.
\par In the first chapter I will present a short survey of parts of P\'olya's
counting theory. In the second chapter I will investigate several 
"musical objects".
\end{abstract}

\section{Preliminaries}
There is a lot of literature about P\'olya's counting theory. For
instance see \cite{18}, \cite{3}, \cite{11}, \cite{19} or \cite{20}. 
Let $M$ be a set with 	$\vert M\vert = m$.
You should know the definition of the type $(\lambda_1,\lambda_2,\dots ,\lambda_m)$ 
of a permutation $\pi \in	S_M$ and the definition	of the cycle index 
$\CI(\Gamma ;x_1,\dots	,x_m)$ of a permutation group	$\Gamma\leq S_M$.
In particular we will use the cycle index of the cyclic group	and of 
the dihedral group.

\section {Applications of P\'olya's Theory in  Musical Theory}
\label {H}
Some parts of this chapter were already discussed by D.L.Reiner in \cite{21}.
Now we are going to calculate the number of patterns of chords, intervals,
tone-rows, all-interval-rows, rhythms, motifs and tropes. Proving any detail
would carry me too far. For further information see \cite{Frip}.

\pagebreak[4]
\subsection{Patterns of Intervals and Chords}

\subsubsection {Number of Patterns of Chords}

\begin{definition}[$n$-Scale]\label {H.1}\em
\begin{enumerate}
\item If we divide one octave into $n$ parts, we will speak of an $n$-scale.
The objects of an $n$-scale are	designated as $0,1,\dots ,n-1$.
\item In twelve tone music we usually identify two tones which are 12 
semi-tones apart. For that reason we define an $n$-scale as the cyclic group
$(Z_n,+)$ of order $n$.
\end{enumerate}
\end{definition}

\begin{definition}[Transposing, Inversion]\label{H.1a}\em
\begin{enumerate}
\item Let us define $T$ the operation of transposing as	a permutation
$T\colon Z_n\to Z_n$, $a\mapsto T(a)\co 1+a$.
The group	$\langle T\rangle$ is the cyclic group $\zeta _n^{(E)}$.
\item Let us define $I$ the operation of inversion as
$I\colon Z_n\to Z_n$, $a\mapsto I(a)\co -a$.
The group	$\langle T,I\rangle$ is the dihedral group $\vartheta _n^{(E)}$.
\end{enumerate}
\end{definition}

\begin{definition}[$k$-Chord]\label{H.2}\em
\begin{enumerate}
\item 
Let $k\leq n$. A $k$-chord in an $n$-scale is a subset of $k$ elements of
$Z_n$.	An interval is a 2-chord.
\item Let $G=\zeta _n^{(E)}$ or $G=\vartheta _n^{(E)}$. Two $k$-chords
$A_1,A_2$ are called equivalent iff there is some $\gamma\in G$ such that
$A_2=\gamma (A_1)$.
\end{enumerate}
\end{definition}

\begin{bem}\em
\begin{enumerate}
\item We want to work with P\'olya's Theorem, therefore I identify each $k$-chord
$A$	with its characteristic function $\chi_A$.
Two $k$-chords $A_1,A_2$ are equivalent iff the two functions $\chi_{A_1}$
and $\chi_{A_2}$ are equivalent in the sense of P\'olya's Theorem.
\item Let us define two finite sets: $P\colon =Z_n$ and
$F\colon =\{ 0,1 \} $. Each function $f\in F^P$ will be identified with
$A_f\colon =\{ k\in Z_n\big\bracevert f(k)=1\}$.
\item Let $w\colon F\to {\cal R}\co {\bf Q}[x]$ be a mapping with
$w(1)\colon =x$ and $w(0)\colon =1$, where $x$ is an indeterminate.
Define the weight $W(f)$ of a function $f\in F^P$ as 
$$W(f)\colon =\prod	_{k\in Z_n}w\bigl(	f(k)\bigr). $$
We see that the weight of a $k$-chord is $x^k$.
The weight of a pattern $W([f])\co W(f)$ is well defined.
\end{enumerate}
\end{bem}
Applying P\'olya's Theorem	of \cite{3}, we derive:
\begin{satz}[Patterns of $k$-Chords]\label {H.3}
\begin{enumerate}
\item Let $G$ be a permutation group on $Z_n$. The number of patterns of
$k$-chords in the $n$-scale $Z_n$ is the coefficient of $x^k$ in
$$\CI (G;1+x,1+x^2,\dots ,1+x^n).$$
\item If $G=\zeta _n^{(E)}$, the number of patterns of
$k$-chords is
$\displaystyle \frac{1}{n}\sum _{j\mid \eggt (n,k)}\varphi (j){{\ \frac{n}{j}\ }\choose 
{\frac{k}{j}}}$, where $\varphi $ is Euler's $\varphi $-function.
\item If $G=\vartheta _n^{(E)}$, the number of patterns of
$k$-chords is
$$\cases {\frac{1}{2n}\Bigl( {\displaystyle\sum _{j\mid \eggt (n,k)}}\varphi
(j){\frac{n}{j}\choose \frac{k}{j}}+n{\frac{(n-1)}{2}\choose [\frac{k}{2}]}
\Bigr) & if $n\equiv 1\bmod 2$	\cr 
\frac{1}{2n}\Bigl( {\displaystyle\sum _{j\mid \eggt (n,k)}}\varphi (j){\frac{n}{j}\choose
\frac{k}{j}}+n{\frac{n}{2}\choose \frac{k}{2}}\Bigr) & if $n\equiv 0\bmod 2$
and	$k\equiv 0\bmod 2$ \cr 
\frac{1}{2n}\Bigl( {\displaystyle\sum _{j\mid \eggt (n,k)}}\varphi (j){\frac{n}{j}\choose
\frac{k}{j}}+n{\frac{n}{2}-1\choose [\frac{k}{2}]}\Bigr) & if $n\equiv 0\bmod 2$ 
and $k\equiv 1\bmod 2$.\cr }$$
\item In the case $n=12$ and	$G=\zeta _n^{(E)}$, we get the numbers in
table \ref{zykzwtonmusik}.% on page \pageref{zykzwtonmusik}.
\begin{table}[htb]
\centering 
$\begin{array}[c]{c|*{12}{c@{\hspace{3 mm}}}}
k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline 
\mbox {\rm $\#$ of patterns} & 1 & 6 & 19 & 43 & 66 & 80 & 66 & 43 & 19 & 6 & 1 & 1\\
\end{array}$
\caption{Number of patterns of $k$-Chords in 12-tone music with regard
to $\zeta _n^{(E)}$.}
\label{zykzwtonmusik}
\end{table}
\item In the case $n=12$ and	$G=\vartheta _n^{(E)}$, we get	the numbers in
table \ref{diederzwtonmusik}.% on page \pageref{diederzwtonmusik}.
\begin{table}[htb]
\centering 
$\begin{array}[c]{c|*{12}{c@{\hspace{3 mm}}}}
k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline 
\mbox {\rm $\#$ of patterns} & 1 & 6 & 12 & 29 & 38 & 50 & 38 & 29 & 12 & 6 & 1 & 1\\
\end{array}$
\caption{Number of patterns of $k$-Chords in 12-tone music with regard
to $\vartheta _n^{(E)}$.}
\label{diederzwtonmusik}
\end{table}
\end{enumerate}
\end{satz}

\subsubsection {The Complement of a $k$-Chord}

\begin{definition}[Complement of a $k$-Chord]\label{H.16}\em
Let $A\subseteq Z_n$ with $\vert A\vert =k$ be a $k$-chord. The
complement of $A$ is the $(n-k)$-chord $Z_n\setminus A$.
\end{definition}

\begin{bem}				\em
\begin{enumerate}
\item Let $G=\zeta _n^{(E)}$ or $G=\vartheta _n^{(E)}$ be
a permutation group on $Z_n$ and let $1\leq k<n$. 
There exists a bijection between the sets of	patterns of $k$-chords and
$(n-k)$-chords.

\item If $n\equiv 0\bmod 2$, the complement of an 
$\frac{n}{2}$-chord is an $\frac{n}{2}$-chord. 
Now I want to figure out the number of patterns of $\frac{n}{2}$-chords
$[A]$ with the property	$A\sim Z_n\setminus A$. Applying the Theorem of 
\cite{18}	we	get:
\end{enumerate}
\end{bem}
\begin{satz}\label{H.17}
\begin{enumerate}
\item Let $n\equiv 0\bmod 2$. The number of patterns of $\frac{n}{2}$-chords
which	are equivalent to their complement, is	$\CI(G;0,2,0,2,\dots )$.
\item If $n=12$ and  $G= \zeta _n^{(E)}$,	there are 20 patterns of 6-chords
which are equivalent to their complement.
\item If $n=12$ and  $G= \vartheta _n^{(E)}$,	there are 8 patterns of 6-chords
which are equivalent to their complement.
\end{enumerate}
\end{satz}

\subsubsection {The Interval Structure of a $k$-Chord}	
\label{H.19}
In this section we use $\vartheta _n^{(E)}$	as the permutation group acting on $Z_n$.
The set of all possible intervals between two differnet tones in $n$-tone
music will be called $\inter(n)$, thus 
$$\inter(n)\co \{x-y\big\bracevert x,y\in Z_n,\ x\ne y\}=\{1,2,\dots,n-1\}.$$
\begin{definition}[Interval Structure]\label{H.20}\em
On $Z_n$ we define a linear order $0<1<2<\dots <n-1$.	Let $A\co \{ i_1,i_2,\dots 
,i_k\} $		be a $k$-chord. Without loss of generality let $i_1<i_2<\dots
<i_k$. The interval structure of $A$ is defined as the pattern	$[f_A]$,
wherein	the function $f_A$ is defined as
$$f_A\colon \{ 1,2,\dots ,k\} \to \inter(n)$$
$$f_A(1)\co i_2-i_1,\ f_A(2)\co i_3-i_2,\dots ,\ f_A(k-1)\co i_k-i_{k-1},\ 
f_A(k)\co i_1-i_k,$$
and two functions $f_1,f_2\colon \{ 1,2,\dots ,k\} \to \inter(n) $ are called
equivalent, iff	there exists some $\varphi \in \vartheta
_k^{(E)}$ such that $f_2=f_1\circ \varphi $. The group $\vartheta	_k^{(E)}$
is generated by $\tilde T$ and $\tilde I$ with $\tilde 
T(i)\co i+1\bmod k$ and $\tilde I(i)\co k+1-i$ for $i=1,\dots ,k$.
The differences $i_{j+1}-i_j$ must be interpreted as differences in $Z_n$.
They are the intervals between	the tones $i_j$ and $i_{j+1}$.
\end{definition}


\begin{satz}\label{H.22}
Let $A_1\co \{ i_1,i_2,\dots ,i_k\}$ and $A_2\co \{j_1,j_2,\dots ,j_k\}$ be
two $k$-chords with $i_1<i_2<\dots <i_k$ and $j_1<j_2<\dots <j_k$.
Furthermore let $f\co f_{A_1}$ and $g\co f_{A_2} \colon 
\{ 1,2,\dots ,k\} \to \inter(n) $ be constructed as in Definition \ref{H.20}.
Then
$$[f]=[g]\Longleftrightarrow [\{ i_1,i_2,\dots ,i_k\} ]=[\{ j_1,j_2,\dots
,j_k\} ].$$
\end{satz}
I omit the proof of this theorem.
\begin{bem}\em\label{H.23}
If the permutation group acting on $Z_n$ is the cyclic group $\zeta _n^{(E)}
$, then the interval structure of $A\co \{ i_1,i_2,\dots ,i_k\} $ must be
defined as the pattern $[f_A]$ in regard to $\zeta _k^{(E)}\co \langle \tilde 
T\rangle	$ with $\tilde T(i)\co i+1\bmod k$. The function $f_A$ is defined
as in Definition \ref{H.20}.
\end{bem}

\begin{bem}\em\label{H.27}
Let $f$ be a function $f\colon \{ 1,2,\dots ,k\} \to \inter(n)$. The pattern
$[f]$ is the interval structure of a $k$-chord, iff
$\sum _{i=1}^kf(i)=n$.  
One must interpret this sum as a sum of intervals, thus as a sum of
positive integers. 
\end{bem}


\begin{bem}\em
Let $x,y_1,y_2,\dots ,y_n$ be indeterminates over {\bf Q} and let ${\cal R}$ be the
ring ${\cal R}\co {\bf Q}[x,y_1,y_2,\dots ,y_n]$.
Now I want to define a weight function
$w\colon \inter(n) \to {\cal R}$, $i\mapsto w(i)\colon =x^iy_i$.
The weight of a function $f\colon \{ 1,2,\dots ,k\} \to \inter(n) $
is the product weight
$$W(f)\co \prod _{i=1}^kw\bigl(f(i)\bigr)=
\prod _{i=1}^kx^{f(i)}y_{f(i)}=x^{\sum
_{i=1}^kf(i)}\prod _{i=1}^ky_{f(i)}.$$
Now we can define $W([f])\co W(f)$. According to Remark \ref{H.27}	the
pattern $[f]$ is the interval structure of a $k$-chord, iff
$\sum _{i=1}^kf(i)=n$.This is true, iff 
$W(f)=x^n \prod _{i=1}^k y_{f(i)}$.
The indices of the $y$'s in $W(f)$ show, which intervals occur in the
$k$-chord.
\end{bem}
An Application of P\'olya's Theorem	of \cite{3} is 
\begin{satz}\label{H.28}
The inventory of interval structures of $k$-chords in $n$-tone music is the
coefficient of $x^n$ in
$\displaystyle \CI\Bigl( \vartheta _k^{(E)};\sum _{i=1}^{n-1}x^iy_i,\sum
_{i=1}^{n-1}x^{2i}{y_i}^2,\sum	_{i=1}^{n-1}x^{3i}{y_i}^3,\dots ,\Bigr)$.
\end{satz}

\begin{beisp}\em\label{H.29}
The inventory of the interval structures of 3-chords	in 12-tone music is the
coefficient of $x^{12}$ in
$$\CI\Bigl( \vartheta _3^{(E)};\sum _{i=1}^{11}x^iy_i,\sum _{i=1}^{11}x^{2i}
{y_i}^2,\sum _{i=1}^{11}x^{3i}{y_i}^3\Bigr) .$$
This is
$${y_1}^2 y_ {10} + y_1 (y_2 y_9 + y_3 y_8 + y_4 y_7 + y_5 y_6) + {y_2}^2 y_8
+ y_2 (y_3 y_7 + y_4 y_6 + {y_5}^2) + {y_3}^2 y_6 + y_3 y_4 y_5 + {y_4}^3	$$
If you are interested in the number of patterns of 3-chords with intervals
$\geq k$, then put	$y_1\co y_2\co \dots \co y_{k-1}\co 0$ and
$y_k\co y_{k+1}\co \dots \co y_n\co 1$. In the case $k=2$ there are 7
patterns of 3-chords with intervals greater or equal 2.
\end{beisp}

\subsection{Patterns of Tone-Rows}

\begin{definition}[Tone-Row, $k$-Row]\em\label{H.5}
\begin{enumerate}
\item 
Arnold Sch\"onberg introduced		the so called tone-rows. Here I am going to 
give a mathematical form of his definition.
Let $n\geq 3$. A tone-row in an $n$-scale	is a bijectiv mapping
$f\colon \{ 0,1,\dots ,n-1\} \to Z_n$, $i\mapsto f(i)$.
$f(i)$ is the tone which occurs in $i^{\mbox{\rm\small th}}$		position in
the  tone-row.
\item Let $n\geq 3$ and $2\leq k\leq n$.	A $k$-row in $n$-tone music is an
injective mapping	$f\colon \{ 0,1,\dots ,k-1\} \to Z_n$.
\end{enumerate}
\end{definition}

\begin{bem}\em\label{H.12}
\begin{enumerate}
\item A $k$-row with $k=n$ is a tone-row.
\item Two $k$-rows $f_1,f_2$ are equivalent if $f_1$ can be written as
transposing, inversion, retrogradation or an arbitrary sequence of these
operations of $f_2$. 
\par Transposing of a $k$-row $f$ is $T\circ f$, Inversion of $f$ is $I\circ
f$. According to Definition \ref {H.1a}, we know that $T$ and $I$
are permutations on $Z_n$, and that $\langle T,I\rangle =\vartheta _n^{(E)}$.
Actually inversion of a $k$-row $f$ should be defined as
$T^{f(0)}\circ I\circ T^{-f(0)}\circ f$.
Retrogradation $R$, is a permutation 
$R \in S_{\{ 0,1,\dots	,k-1\} }$ defined as: 
$$R \co \cases {(0,k-1)\circ (1,k-2)\circ \dots \circ
(\frac{k}{2}-1,\frac{k}{2}) & if $k\equiv 0\bmod 2$\cr (0,k-1)\circ
(1,k-2)\circ \dots \circ (\frac{k-3}{2},\frac{k+1}{2})\circ (\frac{k-1}{2}) 
& if $k\equiv 1\bmod	2$. \cr }$$
Let $\Pi \co \langle R \rangle \leq S_{\{ 0,1,\dots ,k-1\} }$, then 
$\vert \Pi\vert =2 $.
Retrogradation of a $k$-row $f$ is defined as $f\circ R$.
\item Since $\Pi\colon  =\langle R \rangle $, the cycle index of $\Pi $ is
$$\CI(\Pi ;y_1,y_2,\dots ,y_k) =\cases
{\frac{1}{2}({y_1}^k+{y_2}^{\frac{k}{2}})& if $k\equiv 0\bmod 2$ \cr 
\frac{1}{2}({y_1}^k+y_1{y_2}^{\frac{k-1}{2}})& if $k\equiv 1\bmod 2$.\cr}$$
\end{enumerate}
Thus two $k$-rows $f_1,f_2$ are equivalent iff 
$\exists \varphi \in \vartheta _n^{(E)}\exists \sigma \in
\Pi$ such that $f_1=\varphi \circ f_2\circ \sigma $.
\end{bem}

Applying	Theorem 5.2 of \cite{3}, we get
\begin{satz}[Number of Patterns of $k$-Rows]\label{H.13}
The number of patterns of $k$-rows in $Z_n$ is
$\displaystyle\CI\Bigl( \Pi ;\frac{\partial }{\partial x_1},\frac{\partial }{\partial
x_2},\dots ,\frac{\partial }{\partial x_k}\Bigr) \CI(\vartheta
_n^{(E)};1+x_1,1+2x_2,\dots ,1+nx_n)\big\arrowvert _{x_1=x_2=\dots= 
x_n=0}$.
\linebreak[4]This is
\begin{enumerate}
\item 
$$\frac{1}{2}\Biggl( \frac{1}{4}\biggl(
(2)_k+2^{\frac{k}{2}}(\frac{k}{2})!\Bigl( {{\frac{n}{2}}\choose {\frac
{k}{2}}}+{{\frac{n-2}{2}}\choose{\frac{k}{2}}}\Bigr) \biggr)
+\frac{1}{2n}\biggl(
{{n}\choose{k}}k!+2^{\frac{k}{2}}(\frac{k}{2})!{{\frac{n}{2}}\choose{\frac{k}{2}}}
\biggr) \Biggr) ,$$ if $n\equiv 0\bmod 2$ and $k\equiv 0\bmod 2 $. 
For integers $k,v$, $v\geq 0$ the expression $(k)_v$ is definied as:
$$(k)_v\co k\cdot (k-1)\cdot\dots\cdot	\bigl(k-(v-1)\bigr).$$

\item 
$$\frac{1}{2}\biggl( \frac{1}{4}\cdot 2\cdot
2^{\frac{k-1}{2}}{{\frac{n-2}{2}}\choose{\frac{k-1}{2}}}(\frac{k-1}{2})!+\frac
{1}{2n}{{n}\choose{k}}k!\biggr) ,$$ 
if $n\equiv 0\bmod 2$ and $k\equiv 1\bmod 2 $. 

\item 
$$\frac{1}{2}\biggl(
\frac{1}{2n}{{n}\choose{k}}k!+\frac{1}{2}2^{\frac{k}{2}}{{\frac{n-1}{2}}\choose{
\frac{k}{2}}}(\frac{k}{2})!\biggr) ,$$ 
if $n\equiv 1\bmod 2$ and $k\equiv 0\bmod 2 $. 

\item 
$$\frac{1}{2}\biggl(
\frac{1}{2n}{{n}\choose{k}}k!+\frac{1}{2}2^{\frac{k-1}{2}}{{\frac{n-1}{2}}\choose{
\frac{k-1}{2}}}(\frac{k-1}{2})!\biggr) ,$$
if $n\equiv 1\bmod 2$ and $k\equiv 1\bmod 2 $. 
\end{enumerate}
In the case $n=12$ the number of patterns of $k$-rows is in table
\ref{kReihen}.%	on page \pageref{kReihen}.
\begin{table}[htb]
\centering 
$\begin{array}[c]{c|*{6}{c@{\hspace{3 mm}}}}
k & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline 
\mbox {\rm $\#$ of patterns} & 6 & 30 & 275 & 2\,000 & 14\,060 & 83\,280 \\
\end{array}$
\centering 
$\begin{array}[c]{c|*{5}{c@{\hspace{3 mm}}}}
k & 8 & 9 & 10 & 11 & 12\\
\hline 
\mbox {\rm $\#$ of patterns} & 416\,880 &
1\,663\,680 &	4\,993\,440 & 9\,980\,160 & 9\,985\,920\\
\end{array}$
\caption{Number of patterns of $k$-rows in 12-tone music.}
\label{kReihen}
\end{table}
\end{satz}
The special case of Theorem \ref {H.13} for $k=n$ is
\begin{satz}[Number of patterns of Tone-Rows]\label{H.14}
Let $n\geq 3$. The number of patterns of tone-rows in $n$-tone music is
$$\cases {\frac{1}{4}\Bigl( (n-1)!+(n-1)!!\Bigr) & if $n\equiv 1\bmod 2$ \cr
\frac{1}{4}\Bigl( (n-1)!+(n-2)!!(\frac{n}{2}+1)\Bigr) & if $n\equiv 0\bmod 
2$.\cr	}$$
If $n$ is in ${\bf N}$ then 
$$n!!=\cases{n\cdot (n-2)\cdot \dots \cdot 2& if	$n\equiv 0 \bmod 2$\cr 
n\cdot (n-2)\cdot \dots \cdot 1& if $n\equiv 1 \bmod 2$.\cr }$$
Especially there are 9\,985\,920 patterns of tone-rows in 12-tone music.
\end{satz}

\def\inter {\mathop{\rm Int}\nolimits }
\def\zili{{\rm Inj}\bigl( \{ 0,1,\dots ,\linebreak[2]n-1\} ,Z_n\bigr) }
\def\mimi{{\rm Inj}\bigl( \{ 1,2,\dots ,\linebreak[2]n-1\} ,\inter (n)\bigr) }
\def\all{\mathop{\rm Allint}\nolimits}

\subsection{Patterns of All-Interval-Rows}\label{Kapallint}

Let $A$ and $B$ be two finite sets. The set of all injective functions
$f\colon A\to B$ will be denoted by ${\rm Inj}(A,B)$. For that reason the set
of all tone-rows is $\zili$. In this chapter let $n\geq 3$.

\begin{definition}[All-Interval-Rows]\em\label{H.49}
Let us define a mapping
$$\alpha \colon \zili \to \{ g\big\bracevert g\colon \{ 1,2,\dots ,n-1\} \to
\inter (n)\} $$
$$f\mapsto \alpha (f)$$
and $\alpha (f)(i)\co f(i)-f(i-1)$ for $i=1,2,\dots ,n-1$. This is 
subtraction in $Z_n$. The function $\alpha (f)$ is called all-interval-row, iff 
$\alpha (f)$ is injective, that means $\alpha (f)\in \mimi$. In other words a
tone-row induces an all-interval-row, iff all possible intervals occur
as differences between two successive tones of	the tone-row. 
The set of all all-interval-rows	will be denoted as $\all(n)$.
\end{definition}

Let's define some mappings:
\begin{enumerate}
\item 
$$\beta \colon \mimi \to \{ g\big\bracevert g\colon \{ 0,1,\dots ,n-1\} \to
Z_n\} $$
$$f\mapsto \beta (f)$$
$\beta (f)(0)\co 0$ and $\beta (f)(i)\co \beta (f)(i-1)+f(i)\bmod n$ for
$i=1,2,\dots ,n-1$. You can easily derive that for $i=0,1,\dots ,n-1$
$$\beta	(f)(i)\equiv \sum _{j=1}^if(j)\bmod n.$$ 
\item	Let $l\in Z_n$.
$$\tilde \beta \colon \mimi \to \{ g\big\bracevert g\colon \{ 0,1,\dots ,n-1\} \to
Z_n\} $$
$$f\mapsto \tilde \beta (f),\qquad \tilde \beta (f)(i)\equiv \sum _{j=1}^if(j)+l\bmod n.$$
\end{enumerate}
\begin{satz}\label{H.49a}
Let $f$ be a mapping	$f\colon \{ 1,2,\dots ,n-1\} \to	\inter(n)$. The
following statements are equivalent:
\begin{enumerate}
\item $f$ is an all-interval-row.
\item $f\in\mimi$ and $\beta (f)\in \zili$.
\item $f\in\mimi$ and $\tilde\beta (f)\in \zili$.
\end{enumerate}
\end{satz}
The proof is omitted.	\newline
You can easily prove the following results:
\begin{enumerate}
\item If $n\equiv 1\bmod 2$, there are no all-interval-rows.
\item If $n\equiv 0\bmod 2$ the function $f$ defined as
$$f(i)\co \cases {i & if $i\equiv 1\bmod 2$ \cr -i &  if $i\equiv 0\bmod
2$ \cr }	$$
is an all-interval-row.
\par For the rest of this chapter let $n\geq 4$ and $n\equiv 0\bmod 2$.
\item $f\in\all(n)$ implies	$\beta (f)(n-1)=\frac {n}{2}$.
\item $f\in\all(n)$ implies	$f(1)\ne \frac {n}{2}$ and $f(n-1)\ne \frac
{n}{2}$.
\end{enumerate}

\begin{bem}\em	\label{H.50}
\begin{enumerate}
\item 
On $\inter(n)$ we have the following permutations:
$$I\colon \inter (n)\to \inter (n),\qquad j\mapsto I(j)\co n-j.$$
$I$ stands for inversion. $I$ is of the type $(1,\frac {n}{2}-1,0,\dots )$.
\par In the case $n=12$ there is a further permutation called 
$$Q\colon \inter (n)\to \inter (n),\qquad	j\mapsto Q(j)\colon \equiv 5\cdot j\bmod 12.$$
$Q$ stands for quartcircle symmetry. Since $\eggt(5,12)=1$, $Q$ is a
permutation on $Z_n$, and since $5\cdot 0=0$,  $Q$ is a
permutation on  $\inter (n)$. 
$Q$ is of the type $(3,4,0,\dots ,0)$. 
You can easily prove that $(I\circ Q)(j)=(Q\circ I)(j)=7\cdot	j\bmod 12$ and
that it is of the type $(5,3,0,\dots ,0)$.
$I\circ Q$ is called quintcircle symmetry.
\item On the set $\{ 1,2,\dots ,n-1\} $ retrogradation $R$ is a permutation,
defined as 
$$R\colon =(1,n-1)\circ (2,n-2)\circ \dots \circ (\frac {n}{2}-1,\frac
{n}{2}+1)\circ (\frac {n}{2}).$$
\item If $f\in \all(n)$, then $I\circ f$,	$f\circ R$	are in $\all(n)$.
Furthermore if $n=12$ then $Q\circ f\in \all(12)$.
\item For that reason we can define the following permutations on $\all(n)$.
$$\varphi _I,\varphi _R,\varphi _Q\colon \all(n)	\to \all(n)$$
$$f\mapsto \varphi _I(f)\co I\circ f,\ f\mapsto \varphi _R(f)\co f\circ R,\ 
f\mapsto \varphi _Q(f)\co Q\circ f.$$
For $\varphi _Q$ we need the assumption that $n=12$.
\item It is easy to prove that these permutations commute in pairs
and that ${\varphi _I}^2={\varphi _R}^2={\varphi	_Q}^2=\id$.
\item 	In \cite {Eimert}	 there is a further	permutation $E$ called exchange at
$\frac{n}{2}$. It is defined as
$$E\colon \all(n) \to \all(n),\qquad f\mapsto E(f)$$	and
$$E(f)(i)\co\cases {f\bigl( f^{-1}(\frac {n}{2})+i\bigr) & if $i<n-f^{-1}(\frac
{n}{2})$ \cr \frac {n}{2} & if $i=n-f^{-1}(\frac {n}{2})$ \cr
f\bigl(i-n+f^{-1}(\frac {n}{2})\bigr) & if $i>n-f^{-1}(\frac {n}{2})$. \cr
}$$
I have already mentioned, that $f(1)\ne \frac	{n}{2}$ and $f(n-1)\ne \frac 
{n}{2}$. Since $f\in \all(n)$ is bijective, there exists exactly one 
$j$, such that $1<j<n-1$ and $f(j)=\frac {n}{2}$. The values of the  
function $E(f)(i)$ for $i=1,2,\dots	,n-1$ are 
$f(j+1),f(j+2),\dots ,f(n-1),f(j)=\frac {n}{2},f(1),f(2), \dots	,f(j-1)$.
The permutation $E$ is defined for $n\geq 4$, but in the case $n=4$ we have
$E=\varphi_R$. 
\item The following formulas hold:
$E\circ \varphi_I=\varphi_I\circ E$, $E\circ \varphi_Q=\varphi_Q\circ E$,
$E\circ \varphi_R=\varphi_R\circ E$ and $E^2=\id$.
\item Let us define three permutation groups on $\all(n)$.
\newline $G_1\co \langle \varphi _I,\varphi _R\rangle ,G_2\co \langle
\varphi _I,\varphi _R, E\rangle $ und $G_3\co \langle \varphi _I,\varphi _R,
E,\varphi _Q\rangle $.	For $G_2$ we must assume $n\geq 6$, and 
for $G_3$ we must assume  $n=12$.	We calculate that 
$\vert	G_1\vert =4,\ \vert G_2\vert =8, \ \vert G_3\vert =16$.
\end{enumerate}
\end{bem}

\begin{bem}[Counting of All-Interval-Rows]\label{H.51}\em
Let $$x_1,x_2,\dots ,x_{n-1},y_1,y_2,\dots ,y_{n-1},z_1,z_2,\dots
,z_{n-1}$$ be indeterminates over ${\bf Q}$.	Furthermore let $f$ be a mapping 
$f\colon 	\{ 1,2,\dots ,n-1\} \to \inter(n)$.
We define ${\cal R}\co {\bf Q}[x_1,x_2,\dots ,x_{n-1},z_1,z_2,\dots ,z_{n-1}]$
and
$$W(f)\co \prod _{i\co 1}^{n-1}w_i\bigl( f(i)\bigr).$$ 
The functions $w_i$ are defined as
$w_i\colon \inter(n)\to {\cal R}$, $j\mapsto w_i(j)\co z_j\prod _{\nu \co i}^{n-1}{x_{\nu
}}^j$.
After calculating $W(f)$ you have to replace terms of the form ${x_{\nu }}^j$ 
by $y_{j\bmod n}$. Then you get $\tilde W(f)\in {\bf Q}[y_1,y_2,\dots ,y_{n-1},z_1,z_2,\dots
,z_{n-1}]$. According to Theorem \ref{H.49a}	$f$ is an all-interval-row, if and
only if, $\tilde	W(f)=\prod _{i=1}^{n-1}y_iz_i$.
Consequently the number of all-interval-rows in $n$-tone music is the
coefficient of $\prod _{i=1}^{n-1}y_iz_i$	in
$$\prod _{i=1}^{n-1}\Bigl( \sum _{j=1}^{n-1}z_j\prod
_{k=i}^{n-1}{x_k}^j\Bigr) \Big\arrowvert _{{x_{\nu }}^j=y_{j\bmod n} }.$$
\end{bem}
\begin{bem}\em\label{H.52}
For $\varphi \in G_1$ or $G_2$ or $G_3$ we want to calculate 
$$\chi(\varphi)\co \vert \{f\in\all(n)\big\bracevert \varphi(f)=f\}\vert.$$
After some calculations we can derive that there are only 4 permutions $\varphi$ 
such that $\chi(\varphi)\ne 0$. In Remark \ref{H.51} we calculated
$\chi(\id)$. The value of $\chi (\varphi _I\circ \varphi _R)$ is the 
coefficient of $\prod _{i=1}^{n-1}y_iz_i$	in
$$\prod _{i= 1}^{\frac {n}{2}-1}\Bigl( \sum _{{j= 1}\atop {j\ne \frac
{n}{2}}}^{n-1}z_jz_{n-j}\prod _{k= i}^{n-1}{x_k}^j\prod
_{k= n-i}^{n-1}{x_k}^{n-j}\Bigr) z_{\frac {n}{2}}\prod _{k= \frac
{n}{2}}^{n-1}{x_k}^{\frac {n}{2}}\Big\arrowvert _{{x_{\nu }}^j= y_{j\bmod n}
}.$$
\par Now let $n\geq 6$.
The value of $\chi (\varphi _I\circ V)$ is the coefficient of 
$\prod _{i= 1}^{n-1}y_iz_i$	in
$$\prod _{i= 1}^{\frac {n}{2}-1}\Bigl( \sum _{{j= 1}\atop {j\ne \frac
{n}{2}}}^{n-1}z_jz_{n-j}\prod _{k= i}^{n-1}{x_k}^j\prod _{k= (\frac
{n}{2}+i)}^{n-1}{x_k}^{n-j}\Bigr) z_{\frac {n}{2}}\prod _{k= \frac
{n}{2}}^{n-1}{x_k}^{\frac {n}{2}}\Big\arrowvert _{{x_{\nu }}^j= y_{j\bmod n}
}.$$
\par Now let 	$n= 12$.
In order to calculate 	$\chi (\varphi  _{Q}\circ V\circ \varphi _R)$
you must compute
$$\sum _{i= 1}^5\Biggl( z_6\prod _{j= 2i}^{11}{x_j}^6z_3z_9\Bigl( \prod
_{j= i}^{11}{x_j}^3\prod _{j= i+6}^{11}{x_j}^9+\prod
_{j= i}^{11}{x_j}^9\prod _{j= i+6}^{11}{x_j}^3\Bigr) \cdot $$
$$\cdot\prod _{{j= 1}}^{i-1}\Bigl( \sum _{{k= 1}\atop {k\not\in \{
3,6,9\} }}^{n-1}z_kz_{5k\bmod 12}\prod _{l= j}^{11}{x_l}^k\prod
_{l= 2i-j}^{11}{x_l}^{5k\bmod 12}\Bigr) \cdot $$
$$\cdot \prod _{{j= 2i+1}}^{i+5}\Bigl( \sum _{{k= 1}\atop {k\not\in \{
3,6,9\} }}^{n-1}z_kz_{5k\bmod 12}\prod _{l= j}^{11}{x_l}^k\prod
_{l= 12+2i-j}^{11}{x_l}^{5k\bmod 12}\Bigr) \Biggr) .$$
Then substitute $y_{j\bmod 12}$ for ${x_{\nu }}^j$ and find the 
coefficient of $\prod _{i= 1}^{11}y_iz_i$. 
\end{bem}
\begin{satz}[Number of Patterns of All-Interval-Rows]\label{H.53}
For $i=1,2,3$ the number of patterns of all-interval-rows in regard to $G_i$ is
\begin{enumerate}
\item $\frac {1}{4}\bigl( \chi (\id )+\chi (\varphi _I\circ \varphi
_R)\bigr) $ for $i=1$.

\item $\frac {1}{8}\bigl( \chi (\id )+\chi (\varphi _I\circ \varphi _R)+\chi
(\varphi _I\circ V)\bigr) $ for $i=2$.

\item For $i=3$ we calculate
$$\frac {1}{16}\bigl( \chi (\id )+\chi (\varphi _I\circ \varphi _R)+\chi
(\varphi _I\circ V)+\chi (\varphi _Q\circ \varphi _R\circ V)\bigr) =$$
$$=\frac	{1}{16}(3\,856+176+120+120)=267.$$
\end{enumerate}
\end{satz}
This is an application of the Lemma of Bunside of \cite{3}.

\subsection {Patterns of Rhythms}
\begin{definition}[$n$-Bar, Entry-time, $k$-Rhythm]\em\label{H.30}
A bar is an important contribution in a composition. Usually a lot of bars of
the same form follow one another. If you know the smallest rhythmical
subdivision of a bar, you can figure out how many entry-times (think of
rhythmical accents played on a drum) a bar holds. If there are $n$
entry-times in a bar, I call it an $n$-bar. In mathematical terms an $n$-bar
is expressed as the cyclic group $Z_n$. We can define cyclic temporal
shifting $S$ as a permutation
$S\colon Z_n\to Z_n$, $t\mapsto S(t)\colon =t+1$.
Retrogradation $R$	(temporal inversion) is defined as
$R\colon Z_n\to Z_n$, $t\mapsto R(t)\colon =-t$.
The group $\langle S\rangle $ is	$\zeta _n^{(E)}$ and $\langle S,R\rangle =
\vartheta _n^{(E)}$. A $k$-rhythm in an $n$-bar is a subset of $k$ elements
of $Z_n$. The permutation groups $\zeta _n^{(E)}$ or $\vartheta _n^{(E)}$
induce an equivalence relation on the set of all $k$-rhythms. Now we want to
calculate the number of patterns of $k$-rhythms. We get the same numbers as
in Theorem \ref{H.3}.
\end{definition}

\subsection {Patterns of Motifs}

\begin{definition}[$k$-Motif]\label{H.44}\em
\begin{enumerate}
\item Now I want to combine both rhythmical and tonal aspects of music.

\item Assume we have an $n$-scale and an $m$-bar, then	the set $M$
$$M\co \{ (x,y)\big\bracevert x\in Z_m, y\in Z_n\}=Z_m\times Z_n $$
is the set of all possible combinations of entry-times in the $m$-bar	$Z_m$
and pitches in the $n$-scale $Z_n$. Furthermore let $G$ be a permutation group
on $M$. In Remark \ref{H.46} we are going to study two special groups
$G$. The group $G$ defines an equivalence relation on $M$:
$$(x_1,y_1)\sim (x_2,y_2)\colon \Longleftrightarrow \exists g\in G\ \mbox
{\rm with}	\ (x_2,y_2)=g(x_1,y_1). $$
In addition to 	this we have $\vert M\vert = m\cdot n$.

\item Let $1\leq k\leq m\cdot n$. A {\bf $k$-motif} is a subset of $k$
elements of $M$. 
\end{enumerate}
\end{definition}

\begin{satz}[Number of Patterns of $k$-Motifs]\label{Motivschema}
\label{H.45}
The number of patterns of $k$-mo\-tifs in an $n$-scale and in an $m$-bar is
the coefficient of $x^k$ in
$$\CI(G;1+x,1+x^2,\dots ,1+x^{m\cdot n}).$$
\end{satz}
This completely follows from P\'olya's Theorem of \cite{3}.

\begin{bem}[Special Permutation Groups]\label{Gruppe}	\label{H.46}	\em
Now I want to demonstrate two examples for group $G$.
\begin{enumerate}
\item 
In Definition \ref{H.1a}	we had a permutation group $G_2=\zeta _n^{(E)}$
or $G_2=\vartheta _n^{(E)}$ acting on the $n$-scale $Z_n$. Moreover in
Definition \ref{H.30} there	was a permutation group $G_1=\zeta _m^{(E)}$ or 
$G_1=\vartheta	_m^{(E)}$ defined on the $m$-bar $Z_m$. For that reason, we
define the group $G$ as $G\co G_1\otimes G_2$. Two elements $(x_1,y_1),(x_2,y_2)\in
M$ are called equivalent with respect to $G$, iff there exist
$\varphi \in G_1$ and $\psi \in G_2$, with
$$(x_2,y_2)=(\varphi ,\psi )(x_{1},y_1)=(\varphi (x_1),\psi (y_1)).$$
Because of the fact that we know how to calculate the cycle index of 
$G_1\otimes G_2$, we can compute the number of patterns of $k$-motifs.
\item 
In the case $m=n$, we can define another permutation group $G$, as it is done
in \cite{Maz}. The group $G$ is defined as
$G\co \langle T,S,\varphi _A\big\bracevert A\in \GL(2,Z_n)\rangle $,
with
$$T\colon M\to M,\qquad \pmatrix {x\cr y\cr }\mapsto T\pmatrix {x\cr y\cr }	\co \pmatrix {x\cr y+1
\cr }	$$
$$S\colon M\to M,\qquad \pmatrix {x\cr y\cr }	\mapsto S\pmatrix {x\cr y\cr }	\co \pmatrix {x+1\cr
y\cr }	$$
$$\varphi _A\colon M\to M,\qquad \pmatrix {x\cr y\cr }	\mapsto \varphi _A \pmatrix {x\cr y\cr }	\co A\pmatrix
{x\cr y\cr }	.$$
The multiplication $A\cdot \pmatrix {x\cr y\cr }$ stands for matrix
multiplication. The set $\GL(2,Z_n)$ is the group of all regular 
$2\times 2$-matrices	over $Z_n$.
\par You can easily derive the following results:
\begin{enumerate}
\item $T^n=S^n=\id _M\ \mbox {\rm and}	\ T^j\ne\id _M\ \mbox {\rm and}	\
S^j\ne \id _M\ \mbox {\rm for}	\ 1\leq j<n.$

\item $T\circ S=S\circ T$. In addition to this $T\not\in \langle S\rangle $ 
and $S\not\in	\langle T\rangle $.

\item Let $0\leq i,j<n$, then:
$T^i\circ S^j\not\in \langle \varphi _A\big\bracevert	A\in
\GL(2,Z_n)\rangle$, iff $i\ne 0$ or $j\ne 0$.

\item Let $A\co \pmatrix {a & b\cr c & d\cr }	$, then:
$\varphi _A\circ T^k\circ S^l=T^{(cl+dk)}\circ S^{(al+bk)}\circ \varphi	_A$.

\item 
$G$ is the group of all affine mappings ${Z_n}^2\to {Z_n}^2$.
\end{enumerate}

Although	we know quite a lot about the group $G$, I could not find a
formula for the cycle index of $G$ for arbitrary	$n$.
\end{enumerate}
\end{bem}

\begin{beisp}\label{H.48}\em
Let us consider the case, that $n=m=12$.
\begin{enumerate}
\item If $G$ is defined as $G\co \vartheta _n^{(E)}\otimes \vartheta
_n^{(E)}$, then we derive 
$$\CI (G;x_1,x_2,\dots ,x_{144})=$$
\begin{flushleft}
$=\frac{1}{576}(x_1^{144}+12 x_1^{24} x_2^{60}+36 x_1^4 x_2^{70}+
147 x_2^{72}+	8 x_3^{48}+24 x_3^8 x_6^{20}+60 x_4^{36}+
96 x_6^{24}+	192 x_{12}^{12})$.
\end{flushleft}
By applying Theorem \ref {Motivschema}, the number of patterns of $k$-motifs is
the coefficient of $x^k$	in
$1+ x+ 48 x^2 + 937 x^3 + 31\,261 x^4 + 840\,006 x^5 19\,392\,669 x^6 +
381\,561\,281 x^7 + 6\,532\,510\,709 x^8 + 98\,700\,483\,548 x^9	+
1\,332\,424\,197\,746 x^{10}+ \dots $. 

\item If $G\co \langle T,S,\varphi _A\big\bracevert A\in \GL(2,Z_n)\rangle
$, I computed the cycle index of $G$ with a Turbo Pascal program as
$$\CI(G;x_1,x_2,\dots ,x_{144})=\frac{1}{663\,552}(x_{1}^{144} + 18 x_{1}^{72}
x_{2}^{36} + 36 x_{1}^{48} x_{2}^{48} + \dots ).$$
By applying Theorem \ref {Motivschema}, the number of patterns of $k$-motifs is
the coefficient of $x^k$	in
$	1 + x + 5 x^2+ 26 x^3	+ 216 x^4 + 2\,024 x^5 + 27\,806 x^6 + 417\,209 x^7 +
6\,345\,735 x^8 + 90\,590\,713 x ^9 +	1\,190\,322\,956	x^{10} + \dots$.

For	$k=1,2,3,4$ these numbers are the same as in \cite{Maz}.	In the case
$k=5$ however, it is stated that	there exist 2\,032	different patterns of
5-motifs, while here we get 2\,024	of these patterns.
\end{enumerate}
\end{beisp}

\subsection{Patterns of Tropes}

\begin{definition}[Trope]\em	\label{H.8}
\begin{enumerate}
\item If you divide the set of 12 tones in 12-tone music into 2 disjointed
sets, each containing 6 elements, and if you label these sets as a first and
a second set, we will speak of a trope. This definition goes back to Josef
Matthias Hauer. Two tropes are called equivalent, iff transposing, inversion,
changing the labels of the two sets or arbitrary sequences of these
operations transform one trope into the other.
\item For a mathematical definition	let $n\geq 4$ and $n\equiv 0\bmod 2$.
A trope in $n$-tone music is a function 
$f\colon Z_n\to F\colon =\{ 1,2\}$ such that $\vert f^{-1}(\{ 1\}
)\vert =\vert f^{-1}(\{ 2\} )\vert =\frac{n}{2}$.
$f(i)=k$ is translated into: The tone $i$ lies in the set with label $k$.
Furthermore $T$ and $I$ are permutations on $Z_n$ as in Definition
\ref{H.1a}.  The group 
$\langle T,I\rangle$ is $\vartheta_{n}^{(E)} $. Two
tropes $f_1,f_2$ are called equivalent, if and only if, 
$\exists \pi \in \vartheta _{n}^{(E)}\	\exists\varphi\in S_2$ such that 
$f_2=\varphi^{-1}\circ f_1\circ\pi$.
\item Let $x$ and $y$ be indeterminates over {\bf Q}. Define a function 
$w\colon	F\to {\bf Q}[x,y]$ by $w(1)\colon =x$ and $w(2)\colon =y$. For 
$f\in F^{Z_n}$ the weight of $f$ is defined as product weight
$$W(f)\colon =\prod _{x\in Z_n}w\bigl( f(x)\bigr).$$
A function $f\colon Z_n\to F\colon =\{ 1,2\}$ is a trope, iff 
$W(f)=x^{\frac{n}{2}}y^{\frac{n}{2}}$.
\end{enumerate}
\end{definition}
\begin{satz}[Patterns of Tropes]\label{H.9}
Let $\varphi$ be Euler's $\varphi $-function. The number of patterns of 
tropes in regard to $\vartheta _{n}^{(E)}$ is
$$\cases {\frac{1}{4}\biggl( \frac{1}{n}\Bigl( {\displaystyle\sum _{t\mid
\frac{n}{2}}}\varphi (t){\frac{n}{t}\choose \frac{n}{2t}}+{\displaystyle\sum
_{{t\mid n}\atop {t\equiv 0\bmod 2}}}\varphi (t)2^{\frac{n}{t}}\Bigr)
+{\frac{n}{2}\choose \frac{n}{4}}+2^{\frac{n}{2}-1}\biggr) & if $n\equiv
0\bmod 4$ \cr \frac{1}{4}\biggl( \frac{1}{n}\Bigl( {\displaystyle \sum
_{t\mid \frac{n}{2}}}\varphi (t){\frac{n}{t}\choose
\frac{n}{2t}}+{\displaystyle\sum _{{t\mid n}\atop {t\equiv 0\bmod 2}}}\varphi
(t)2^{\frac{n}{t}}\Bigr) +{\frac{n-2}{2}\choose
\frac{n-2}{4}}+2^{\frac{n}{2}-1}\biggr) & if $n\equiv 2\bmod 4$. \cr }$$
In 12-tone music there are 35 patterns of tropes. (See \cite{Florey}.) Hauer
himself calculated that there are 44 patterns of tropes, because in his work
the permutation group acting on $Z_n$ was the cyclic group $\langle T\rangle$.
\end{satz}
This is an application of of the {\em Power Group Enumeration Theorem in
polynomial Form\/} of \cite{22}.

\subsection{Special Remarks on 12-tone music}\label{Kap12ton}

In addition to the operations of transposing $T$ and of inversion $I$ we can
study	quartcircle- and quintcircle symmetry in 12-tone music.

\begin{bem}[Quartcircle Symmetry]\em	\label{H.38}
The quartcircle symmetry $Q$ is defined as
$$Q\colon Z_{12}\to Z_{12},\qquad x\mapsto Q(x)\co 5x.$$
$Q$ is a permutation on $Z_{12}$, since $\eggt(5,12)=1$.
Furthermore 
$Q\not \in \langle I,T\rangle $, $Q\circ T=T^5\circ Q$, $Q^2=\id _{Z_{12}}$ 
and $Q\circ I=I\circ Q=7x$, which is called	the quintcircle symmetry.
\newline
Let $G$ be $G\colon =\langle I,T,Q\rangle $. Each element
$\varphi \in G$ can be written as	$\varphi =T^k\circ I^j\circ Q^l$ 
such that $k\in \{ 0,1,\dots ,n-1\}$, $j\in \{ 0,1\} $, and $l\in \{ 0,1\} $.
The cycle index of $G\colon =\langle I,T,Q\rangle $ is
$$\CI(G;x_1,x_2,\dots ,x_{12})=$$
$$=\frac{1}{48}\Bigl(\sum _{t\mid 12}\varphi (t){x_t}^{\frac{12}{t}} +
2x_1^6x_2^3 + 3x_1^4x_2^4 + 6x_1^2x_2^5 + 11x_2^6 + 4x_3^2x_6
+ 6x_4^3 + 4x_6^2\Bigr) .$$
This group $G$ is an other permutation group acting on $Z_{12}$	with a musical
background.
The question arises, how to generalize the quartcircle symmetry of 12-tone
music to $n$-tone music.	Should we take any unit in $Z_n$ or only those
units	$e$ such that $e^2=1$ ?
\end{bem}

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\bibliographystyle{plain}

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\end{thebibliography}

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\begin{minipage}{7cm}
{\sc Address of the author:}\\
\small
H. Fripertinger\\
Universit\"at Graz\\
Institut f\"ur Mathematik\\
Heinrichstr. 36\\
A--8010 Graz, Austria
\end{minipage}
\end{document}