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\chapter{A Musical Interlude}
\label{ch:interlude}
At this point enough detail about Haskell and Euterpea has been
covered that it is worth developing a small but full application or
two. In this chapter an existing composition will be transcribed into
Euterpea, thus exemplifying how to express conventional musical ideas
in Euterpea. Then a simple form of algorithmic composition will be
presented, where it will become apparent that more exotic things can
be easily expressed as well.
But before tackling either of these, Haskell's \emph{modules} will be
described in more detail.
\section{Modules}
\label{sec:modules}
Haskell programs are partitioned into \emph{modules} that capture
common types, functions, etc.\ that naturally comprise an application.
The first part of a module is called the module \emph{header}, which
declares what the name of the module is, and what other modules
it might import. For this chapter the module's name is |Interlude|,
into which the module |Euterpea| is imported:
\indexkw{module}
\begin{spec}
module Interlude where
import Euterpea
\end{spec}
\syn{Module names must always be capitalized (just like type names).}
Maintaining the name space of modules in a large software system can
be a daunting task. So Haskell provides a way to structure module
names \emph{hierachically}. Indeed, because the |Interlude| module is
part of the overall Euterpea library, the actual module declaration
that is used is:
\begin{spec}
module Euterpea.Examples.Interlude where
import Euterpea
\end{spec}
This says that the |Interlude| module is part of the |Examples| folder
in the overall |Euterpea| library. In general, these hierarchical
names correspond to the folder (directory) structure of a particular
implementation. Similarly, the name of the file containing the module
is generally the same as the module name, plus the file extension (in
this case, the name of the file is |Interlude.lhs|).
If we wish to use this module in another module |M|, say, it may be
imported into |M|, just as was done above in importing |Euterpea| into
|Interlude|:
\begin{spec}
module M where
import Euterpea.Examples.Interlude
\end{spec}
This will make available in |M| all of the names of functions, types,
and so on that are defined at the top-level of |Interlude|.
\index{module!interface}
\index{module!\hkw{import}} \indexkw{import}
But this is not always what the programmer would like. Another
purpose of a module is to manage the overall name space of an
application. Modules allow us to structure an application in such a
way that only the functionality intended for the end user is
visible---everything else needed to implement the system is
effectively hidden. In the case of |Interlude|, there are only two
names whose visibillity is desirable: |childSong6|, and |prefix|.
This can be achieved by writing the module header as follows:
\begin{spec}
module Euterpea.Examples.Interlude(childSong6, prefix) where
import Euterpea
\end{spec}
This set of visible names is sometimes called the \emph{export list}
of the module. If the list is omitted, as was done initially, then
\emph{all} names defined at the top level of the module are exported.
Although explicit type signatures in export lists are
not allowed, it is sometime useful to add them as comments, at least,
as in:
\begin{code}
module Euterpea.Examples.Interlude
( childSong6, -- :: Music Pitch,
prefix -- :: [Music a] -> Music a)
) where
import Euterpea
\end{code}
In this case the list of names is sometimes called the {\em interface}
to the module.
There are several other rules concerning the import and export of
names to and from modules. Rather than introduce them all at once,
they will be introduced as needed in future chapters.
\begin{figure*}
\IfFileExists{pics/ChildSong6.eps}{
\centerline{
\epsfysize=7in
\epsfbox{pics/ChildSong6.eps}}
}{
% We use a parbox here to make sure that the figure takes up a full page,
% just like ChildSong6, so that page numbers will remain consistent.
\parbox[c][8in][c]{\textwidth}{\center{\framebox[1.1\width]{Image omitted due to respository space issues.}}}
}
\caption{Excerpt from Chick Corea's \emph{Children's Songs No.\ 6}}
\label{fig:childsong6}
\end{figure*}
\section{Transcribing an Existing Score}
Figure \ref{fig:childsong6} shows the first 28 bars of Chick Corea's
\emph{Children's Songs No.\ 6}, written for electric piano
\cite{Corea94}. Analyzing the structure of this tune explores several
basic issues that arise in the transcription of an existing score into
Euterpea, including repeating phrases, grace notes, triplets, tempo,
and specifying an instrument. To begin, however, we will define a
couple of auxiliary functions to make our job easier.
\subsection{Auxiliary Functions}
For starters, note that there are several repeating patterns of notes
in this composition, each enclosed in a rectangle in Figure
\ref{fig:childsong6}. In fact, the bass line consists \emph{entirely}
of three repeating phrases. In anticipation of this, a function can
be defined that repeats a phrase a particular number of times:
\begin{spec}
timesM :: Int -> Music a -> Music a
timesM 0 m = rest 0
timesM n m = m :+: timesM (n-1) m
\end{spec}
\syn{Note that pattern-matching can be used on numbers. As mentioned
earlier, when there is more than one equation that defines a
function, the first equation is tried first. If it fails, the
second equation is tried, and so on. In the case above, if the
first argument to |timesM| is not 0, the first equation will fail.
The second equation is then tried, which always succeeds.
}
%% An expression \hs{if pred then cons else alt} is called a {\em
%% conditional expression}. If \hs{pred} (called the {\em
%% predicate}) evaluates to |True|, then \hs{cons} (called the {\em
%% consequence}) is the result; if \hs{pred} evaluates to |False|,
%% then \hs{alt} (called the {\em alternative}) is the result.
So, for example, |timesM 3 b1| will repeat the baseline |b1| (to be
defined shortly) three times.
To motivate the second auxiliary function, note in Figure
\ref{fig:childsong6} that there are many melodic lines that consist of
a sequence of consecutive notes having the same duration (for example
eighth notes in the melody, and dotted quarter notes in the bass). To
avoid having to write each of these durations explicitly, we will
define a function that specifies them just once. To do this, recall
that |a 4 qn| is a concert A quarter note. Then note that, because of
currying, |a 4| is a function that can be applied to any
duration---i.e.\ its type is |Dur -> Music a|. In other words, it is
a note whose duration has not been specified yet.
With this thought in mind, we can return to the original problem and
define a function that takes a duration and a \emph{list} of notes with
the aforementioned type, returning a |Music| value with the duration
attached to each note appropriately. In Haskell:
\begin{code}
addDur :: Dur -> [Dur -> Music a] -> Music a
addDur d ns = let f n = n d
in line (map f ns)
\end{code}
(Compare this idea with Exercise \ref{ex:fuse} in Chapter
\ref{ch:poly}.)
Finally, a function to add a grace note to a note is defined. Grace
notes can approach the principal note from above or below; sometimes
starting a half-step away, and sometimes a whole step; and having a
rhythmic interpretation that is to a large extent up to the performer.
In the case of the six uses of grace notes in \emph{Children's Songs
No.\ 6}, we will assume that the grace note begins on the downbeat
of the principal note, and thus its duration will subtract from that
of the principal note. We will also assume that the grace note
duration is 1/8 of that of the principal note. Thus the goal is to
define a function:
\begin{code}
graceNote :: Int -> Music Pitch -> Music Pitch
\end{code}
such that |graceNote n (note d p)| is a |Music| value consisting of
two notes, the first being the grace note whose duration is
$\nicefrac{d}{8}$ and whose pitch is |n| semitones higher (or lower if
|n| is negative) than |p|, and the second being the principal note at
pitch |p| but now with duration $\nicefrac{7d}{8}$. In Haskell:
\begin{code}
graceNote n (Prim (Note d p)) =
note (d/8) (trans n p) :+: note (7*d/8) p
graceNote n _ =
error "Can only add a grace note to a note."
\end{code}
Note that pattern-matching is performed against the nested
constructors of |Prim| and |Note|---we cannot match against the
application of a function such as |note|. Also note the error
message---programs are not expected to ever apply |graceNote| to
something other than a single note.
(In Chapter~\ref{ch:more-music} a slightly more general form of
|graceNote| will be defined.)
The only special cases that will not be handled using auxiliary
functions are the single staccato on note four of bar fifteen, and the
single portamento on note three of bar sixteen. These situations will
be addressed differently in a later chapter.
\subsection{Bass Line}
With these auxilary functions now defined, the base line in
Figure~\ref{fig:childsong6} can be defined by first noting the three
repeating phrases (enclosed in rectangular boxes), which can be
captured as follows:
\begin{code}
b1 = addDur dqn [b 3, fs 4, g 4, fs 4]
b2 = addDur dqn [b 3, es 4, fs 4, es 4]
b3 = addDur dqn [as 3, fs 4, g 4, fs 4]
\end{code}
Using |timesM| it is then easy to define the entire 28 bars of the
base line:
\begin{code}
bassLine = timesM 3 b1 :+: timesM 2 b2 :+:
timesM 4 b3 :+: timesM 5 b1
\end{code}
\subsection{Main Voice}
The upper voice of this composition is a bit more tedious to define,
but is still straightforward. At the highest level, it consists of
the phrase |v1| in the first two bars (in the rectangular box)
repeated three times, followed by the remaining melody, which will be
named |v2|:
\begin{code}
mainVoice = timesM 3 v1 :+: v2
\end{code}
The repeating phrase |v1| is defined by:
\begin{code}
v1 = v1a :+: graceNote (-1) (d 5 qn) :+: v1b -- bars 1-2
v1a = addDur en [a 5, e 5, d 5, fs 5, cs 5, b 4, e 5, b 4]
v1b = addDur en [cs 5, b 4]
\end{code}
Note the treatment of the grace note.
The remainder of the main voice, |v2|, is defined in seven pieces:
\begin{code}
v2 = v2a :+: v2b :+: v2c :+: v2d :+: v2e :+: v2f :+: v2g
\end{code}
with each of the pieces defined in Figure \ref{fig:bars7-28}. Note that:
\begin{itemize}
\item The phrases are divided so as to (for the most part) line up
with bar lines, for convenience. But it may be that this is not the
best way to organize the music---for example, we could argue that
the last two notes in bar 20 form a ``pick-up'' to the phrase that
follows, and thus more logically fall with that following phrase.
The organization of the Euterpea code in this way is at the
discretion of the composer.
\item The stacatto is treated by playing the qurater note as an eighth
note; the portamento is ignored. As mentioned earlier, these
ornamentations will be addressed differently in a later chapter.
\item The triplet of eighth notes in bar 25 is addressed by scaling the
tempo by a factor of |3/2|.
\end{itemize}
\begin{figure}
\cbox{\small
\begin{code}
v2a = line [ cs 5 (dhn+dhn), d 5 dhn,
f 5 hn, gs 5 qn, fs 5 (hn+en), g 5 en] -- bars 7-11
v2b = addDur en [ fs 5, e 5, cs 5, as 4] :+: a 4 dqn :+:
addDur en [ as 4, cs 5, fs 5, e 5, fs 5] -- bars 12-13
v2c = line [ g 5 en, as 5 en, cs 6 (hn+en), d 6 en, cs 6 en] :+:
e 5 en :+: enr :+:
line [ as 5 en, a 5 en, g 5 en, d 5 qn, c 5 en, cs 5 en]
-- bars 14-16
v2d = addDur en [ fs 5, cs 5, e 5, cs 5,
a 4, as 4, d 5, e 5, fs 5] -- bars 17-18.5
v2e = line [ graceNote 2 (e 5 qn), d 5 en, graceNote 2 (d 5 qn), cs 5 en,
graceNote 1 (cs 5 qn), b 4 (en+hn), cs 5 en, b 4 en ]
-- bars 18.5-20
v2f = line [ fs 5 en, a 5 en, b 5 (hn+qn), a 5 en, fs 5 en, e 5 qn,
d 5 en, fs 5 en, e 5 hn, d 5 hn, fs 5 qn] -- bars 21-23
v2g = tempo (3/2) (line [cs 5 en, d 5 en, cs 5 en]) :+:
b 4 (3*dhn+hn) -- bars 24-28
\end{code}}
\caption{Bars 7-28}
\label{fig:bars7-28}
\end{figure}
\subsection{Putting It All Together}
In the Preface to \emph{Children's Songs -- 20 Pieces for Keyboard}
\cite{Corea94}, Chick Corea notes that, ``Songs 1 through 15 were
composed for the Fender Rhodes.'' Therefore the MIDI instrument
|RhodesPiano| is a logical choice for the transcription of his
composition. Furthermore, note in the score that a dotted half-note
is specified to have a metronome value of 69. By default, the |play|
function in Euterpea uses a tempo equivalent to a quarter note having
a metronome value of 120. Therefore the tempo should be scaled by a
factor of |(dhn/qn)*(69/120)|.
These two observations lead to the final definition of the
transcription of \emph{Children's Songs No.\ 6} into Euterpea:
\begin{code}
childSong6 :: Music Pitch
childSong6 = let t = (dhn/qn)*(69/120)
in instrument RhodesPiano
(tempo t (bassLine :=: mainVoice))
\end{code}
The intent is that this is the only value that will be of interest to
users of this module, and thus |childSong6| is the only name exported
from this section of the module, as discussed in Section
\ref{sec:modules}.
This example can be played through the command |play childSong6|.
\vspace{.1in}\hrule
\begin{exercise}{\em
Find a simple piece of music written by your favorite composer, and
transcribe it into Euterpea. In doing so, look for repeating patterns,
transposed phrases, etc.\ and reflect this in your code, thus revealing
deeper structural aspects of the music than that found in common
practice notation.}
\end{exercise}
\vspace{.1in}\hrule
\section{Simple Algorithmic Composition}
\label{sec:alg-comp}
\emph{Algorithmic composition} is the process of designing an
algorithm (or heuristic) for generating music. There are unlimited
possibilites, with some trying to duplicate a particular style of
music, others exploring more exotic styles; some based on traditional
notions of music theory, others not; some completely deterministic,
others probabilistic; and some requiring user interaction, others being
completely automatic. Some even are based simply on ``interpreting''
data---like New York Stock Exchange numbers---in interesting ways! In
this textbook a number of algorithmic composition techniques are
explored, but the possibilities are endless---hopefully what is
presented will motivate the reader to invent new, exciting algorithmic
composition techniques.
To give a very tiny glimpse into algorithmic composition, we end this
chapter with a very simple example. We will call this example
``prefix,'' for reasons that will become clear shortly.
The user of this algorithm provides an initial melody (or ``motif'')
represented as a list of notes. The main idea is to play every proper
(meaning non-empty) prefix of the given melody in succession. So the
first thing we do is define a polymorphic function |prefixes :: [a] ->
[[a]]| that returns all proper prefixes of a list:
\begin{code}
prefixes :: [a] -> [[a]]
prefixes [] = []
prefixes (x:xs) = let f pf = x:pf
in [x] : map f (prefixes xs)
\end{code}
We can use this to play all prefixes of a given melody |mel| in
succession as follows:
\begin{spec}
play (line (concat (prefixes mel)))
\end{spec}
But let's do a bit more. Let's create two voices (each using a
different instrument), one voice being the reverse of the other, and
play them in parallel. And then let's play the whole thing once, then
transposed up a perfect fourth (i.e.\ five semitones), then repeat the
whole thing a final time. And, let's package it all into one
function:
\begin{code}
prefix :: [Music a] -> Music a
prefix mel = let m1 = line (concat (prefixes mel))
m2 = transpose 12 (line (concat (prefixes (reverse mel))))
m = instrument Flute m1 :=: instrument VoiceOohs m2
in m :+: transpose 5 m :+: m
\end{code}
Here are two melodies (differing only in rhythm) that you can try with
this algorithm:
\begin{code}
mel1 = [c 5 en, e 5 sn, g 5 en, b 5 sn, a 5 en, f 5 sn, d 5 en, b 4 sn, c 5 en]
mel2 = [c 5 sn, e 5 sn, g 5 sn, b 5 sn, a 5 sn, f 5 sn, d 5 sn, b 4 sn, c 5 sn]
\end{code}
Although not very sophisticated at all, |prefix| can generate some
interesting music from a very small seed.
Another typical approach to algorithmic composition is to specify some
constraints on the solution space, and then generate lots of solutions
that satisfy those constraints. The user can then choose one of the
solutions based on aesthetic preferences.
As a simple example of this, how do we choose the original melody in
the prefix program above? We could require that all solutions be a
multiple of some preferred meter. For example, in triple meter (say,
$\nicefrac{3}{4}$ time) we might wish for the solutions to be
multiples of 3 quarter-note beats (i.e. one measure), or in
$\nicefrac{4}{4}$ time, multiples of 4 beats. In this way the result
is always an integer number of measures. If the original melody
consists of notes all of the same duration, say one beat, then the
prefixes, when played sequentially, will have a total duration that is
the sum of the numbers 1 through |n|, where |n| is the length of melody
in beats. That sum is $\nicefrac{n*(n+1)}{2}$. The first ten sums in
this series are:
\[1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...\]
The second, third, fifth, sixth, eighth, and ninth of these are
divisible by 3, and the seventh and eighth are divisible by 4. When
rendering the result we could then, for exaple, place an accent on the
first note in each of these implied measures, thus giving the result
more of a musical feel. (Placing an accent on a note will be
explained in Chapters \ref{ch:more-music} and \ref{ch:performane}.)
\vspace{.1in}\hrule
\begin{exercise}{\em
Try using |prefix| on your own melodies. Indeed, note that the list of notes
could in general be a list of any |Music| values.}
\end{exercise}
\begin{exercise}{\em
Try making the following changes to |prefix|:
\begin{enumerate}
\item Use different instruments.
\item Change the definition of |m| in some way.
\item Compose the result in a different way.
\end{enumerate} }
\end{exercise}
\vspace{.1in}\hrule