Mozart: Musikalisches Würfelspiel

>>> import abjad
>>> from abjad.demos import mozart

Mozart’s dice game is a method for aleatorically generating sixteen-measure-long minuets. For each measure, two six-sided dice are rolled, and the sum of the dice used to look up a measure number in one of two tables (one for each half of the minuet). The measure number then locates a single measure from a collection of musical fragments. The fragments are concatenated together, and “music” results.

Implementing the dice game in a composition environment is somewhat akin to (although also somewhat more complicated than) the ubiquitous hello world program which every programming language uses to demonstrate its basic syntax.

../_images/mozart-tables.png

Part of a pen-and-paper implementation from the 20th century.

Note

The musical dice game in question (k516f) has long been attributed to Mozart, albeit inconclusively. Its actual provenance is a musicological problem with which we are unconcerned here.

The materials

At the heart of the dice game is a large collection, or corpus, of musical fragments. Each fragment is a single 3/8 measure, consisting of a treble voice and a bass voice. Traditionally, these fragments are stored in a “score”, or “table of measures”, and located via two tables of measure numbers, which act as lookups, indexing into that collection.

Duplicate measures in the original corpus are common. Notably, the 8th measure - actually a pair of measures represent the first and second alternate ending of the first half of the minuet - are always identical. The last measure of the piece is similarly limited - there are only two possibilities rather than the usual eleven (for the numbers 2 to 12, being all the possible sums of two 6-sided dice).

How might we store this corpus compactly?

Some basic musical information in Abjad can be stored as strings, rather than actual collections of class instances. Abjad can parse simple LilyPond strings via p, which interprets a subset of LilyPond syntax, and understands basic concepts like notes, chords, rests and skips, as well as beams, slurs, ties, and articulations.

>>> staff = abjad.Staff("""
...     c'4 ( d'4 <cs' e'>8 ) -. r8
...     <g' b' d''>4 ^ \marcato ~ <g' b' d''>1
...     """)
>>> f(staff)
\new Staff {
    c'4 (
    d'4
    <cs' e'>8 -\staccato )
    r8
    <g' b' d''>4 ^\marcato ~
    <g' b' d''>1
}
>>> show(staff)

So, instead of storing our musical information as Abjad components, we’ll represent each fragment in the corpus as a pair of strings: one representing the bass voice contents, and the other representing the treble. This pair of strings can be packaged together into a collection. For this implementation, we’ll package them into a dictionary. Python dictionaries are cheap, and often provide more clarity than lists; the composer does not have to rely on remembering a convention for what data should appear in which position in a list - they can simply label that data semantically. In our musical dictionary, the treble voice will use the key ‘t’ and the bass voice will use the key ‘b’.

>>> fragment = {'t': "g''8 ( e''8 c''8 )", 'b': '<c e>4 r8'}

Instead of relying on measure number tables to find our fragments - as in the original implementation, we’ll package our fragment dictionaries into a list of lists of fragment dictionaries. That is to say, each of the sixteen measures in the piece will be represented by a list of fragment dictionaries. Furthermore, the 8th measure, which breaks the pattern, will simply be a list of two fragment dictionaries. Structuring our information in this way lets us avoid using measure number tables entirely; Python’s list-indexing affordances will take care of that for us. The complete corpus looks like this.

We can then use the p() function we saw earlier to “build” the treble and bass components of a measure like this:

Let’s try with a measure-definition of our own:

>>> my_measure_dict = {'b': r'c4 ^\trill r8', 't': "e''8 ( c''8 g'8 )"}
>>> treble, bass = mozart.make_mozart_measure(my_measure_dict)
>>> print(format(treble))
{
    e''8 (
    c''8
    g'8 )
}
>>> print(format(bass))
{
    c4 ^\trill
    r8
}

Now with one from the Mozart measure collection defined earlier. We’ll grab the very last choice for the very last measure:

>>> my_measure_dict = mozart.make_mozart_measure_corpus()[-1][-1]
>>> treble, bass = mozart.make_mozart_measure(my_measure_dict)
>>> print(format(treble))
{
    c''8
    c'8
    r8
}
>>> print(format(bass))
{
    c4
    c,8
}

The structure

After storing all of the musical fragments into a corpus, concatenating those elements into a musical structure is relatively trivial. We’ll use the choice() function from Python’s random module. random.choice() randomly selects one element from an input list.

>>> import random
>>> list_ = [1, 'b', 3]
>>> result = [random.choice(list_) for i in range(20)]
>>> result
['b', 'b', 1, 'b', 3, 'b', 'b', 'b', 'b', 'b', 3, 1, 3, 1, 'b', 1, 1, 3, 'b', 3]

Our corpus is a list comprising sixteen sublists, one for each measure in the minuet. To build our musical structure, we can simply iterate through the corpus and call choice on each sublist, appending the chosen results to another list. The only catch is that the eighth measure of our minuet is actually the first-and-second-ending for the repeat of the first phrase. The sublist of the corpus for measure eight contains only the first and second ending definitions, and both of those measures should appear in the final piece, always in the same order. We’ll have to intercept that sublist while we iterate through the corpus and apply some different logic.

The easist way to intercept measure eight is to use the Python builtin enumerate, which allows you to iterate through a collection while also getting the index of each element in that collection. Note that In mozart.choose_mozart_measures() we test for index 7, rather then 8, because list indices count from 0 instead of 1.

The result will be a seventeen-item-long list of measure definitions:

>>> choices = mozart.choose_mozart_measures()
>>> for i, measure in enumerate(choices):
...     print(i, measure)
... 
0 {'b': '<c e>4 r8', 't': "g''8 c''8 e''8"}
1 {'b': '<c e>4 r8', 't': "g''8 e''8 c''8"}
2 {'b': '<b, d>4 r8', 't': "g''16 fs''16 g''16 d''16 b'16 g'16"}
3 {'b': 'c4 r8', 't': "e''16 c''16 b'16 c''16 g'8"}
4 {'b': 'c8 c8 c8', 't': "<fs' d''>8 <d'' fs''>8 <fs'' a''>8"}
5 {'b': '<b, d>4 r8', 't': "g''16 b''16 g''16 d''16 b'8"}
6 {'b': 'c8 d8 d,8', 't': "e''16 a''16 g''16 b''16 fs''16 a''16"}
7 {'b': 'g,8 g16 f16 e16 d16', 't': "<g' b' d'' g''>4 r8"}
8 {'b': 'g,8 b16 g16 fs16 e16', 't': "<g' b' d'' g''>4 r8"}
9 {'b': '<d fs>4 <c fs>8', 't': "d'''8 a''16 fs''16 d''16 a'16"}
10 {'b': '<b, d>4 <b, g>8', 't': "g''16 b''16 g''8 d''8"}
11 {'b': 'e4 e16 c16', 't': "c''16 g'16 c''16 e''16 g''16 <c'' e''>16"}
12 {'b': 'g4 g,8', 't': "b'16 d''16 g''16 d''16 b'8"}
13 {'b': '<c e>4 r8', 't': "g''8 f''16 e''16 d''16 c''16"}
14 {'b': '<c g>4 <c e>8', 't': "e''16 d''16 e''16 g''16 c'''16 g''16"}
15 {'b': 'f4 g8', 't': "f''16 a''16 a'8 b'16 d''16"}
16 {'b': 'c8 g,8 c,8', 't': "c''4 r8"}

The score

Now that we have our raw materials, and a way to organize them, we can start building our score. The tricky part here is figuring out how to implement LilyPond’s repeat structure in Abjad. LilyPond structures its repeats something like this:

\repeat volta n {
    music to be repeated
}

\alternative {
    { ending 1 }
    { ending 2 }
    { ending n }
}

...music after the repeat...

What you see above is really just two containers, each with a little text (“repeat volta n” and “alternative”) prepended to their opening curly brace. To create that structure in Abjad, we’ll need to use the LilyPondCommand class, which allows you to place LilyPond commands like “break” relative to any score component:

>>> container = abjad.Container("c'4 d'4 e'4 f'4")
>>> command = abjad.LilyPondCommand('before-the-container', 'before')
>>> abjad.attach(command, container)
>>> command = abjad.LilyPondCommand('after-the-container', 'after')
>>> abjad.attach(command, container)
>>> command = abjad.LilyPondCommand('opening-of-the-container', 'opening')
>>> abjad.attach(command, container)
>>> command = abjad.LilyPondCommand('closing-of-the-container', 'closing')
>>> abjad.attach(command, container)
>>> command = abjad.LilyPondCommand('to-the-right-of-a-note', 'right')
>>> abjad.attach(command, container[2])
>>> f(container)
\beforeTheContainer
{
    \openingOfTheContainer
    c'4
    d'4
    e'4 \toTheRightOfANote
    f'4
    \closingOfTheContainer
}
\afterTheContainer

Notice the second argument to each LilyPondCommand above, like before and closing. These are format slot indications, which control where the command is placed in the LilyPond code relative to the score element it is attached to. To mimic LilyPond’s repeat syntax, we’ll have to create two LilyPondCommand instances, both using the “before” format slot, insuring that their command is placed before their container’s opening curly brace.

Now let’s take a look at the code that puts our score together:

>>> score = mozart.make_mozart_score()
>>> show(score)

Our instrument name got cut off! Looks like we need to do a little formatting.

The document

As you can see above, we’ve now got our randomized minuet. However, we can still go a bit further. LilyPond provides a wide variety of settings for controlling the overall look of a musical document, often through its header, layout and paper blocks. Abjad, in turn, gives us object-oriented access to these settings through the its lilypondfiletools module.

We’ll use abjad.tools.lilypondfiletools.LilyPondFile.new() to wrap our Score inside a LilyPondFile instance. From there we can access the other “blocks” of our document to add a title, a composer’s name, change the global staff size, paper size, staff spacing and so forth.

>>> lilypond_file = mozart.make_mozart_lilypond_file()
>>> print(lilypond_file)
LilyPondFile(comments=[], date_time_token=DateTimeToken(date_string='2017-08-02 22:03'), global_staff_size=12, includes=[], items=[<Block(name='header')>, <Block(name='layout')>, <Block(name='paper')>, <Block(name='score')>], lilypond_language_token=LilyPondLanguageToken(), lilypond_version_token=LilyPondVersionToken(version_string='2.19.44'))
>>> print(format(lilypond_file.header_block))
\header {
    title = \markup {
        \bold
            \sans
                "Ein Musikalisches Wuerfelspiel"
        }
    composer = #"W. A. Mozart (maybe?)"
}
>>> print(format(lilypond_file.header_block))
\header {
    title = \markup {
        \bold
            \sans
                "Ein Musikalisches Wuerfelspiel"
        }
    composer = #"W. A. Mozart (maybe?)"
}
>>> print(format(lilypond_file.layout_block))
\layout {
    ragged-right = ##t
}
>>> print(format(lilypond_file.layout_block))
\layout {
    ragged-right = ##t
}
>>> print(format(lilypond_file.paper_block))
\paper {
    markup-system-spacing = #'((basic_distance . 8))
    paper-width = #180
}
>>> print(format(lilypond_file.paper_block))
\paper {
    markup-system-spacing = #'((basic_distance . 8))
    paper-width = #180
}

And now the final result:

>>> show(lilypond_file)

Explore the abjad/demos/mozart/ directory for the complete code to this example, or import it into your Python session directly with from abjad.demos import mozart.