Introduction
The program "Instant Bach" was made as a project for an unnamed
computer science class,
which was when I was a sophomore computer science major, and a senior music
performance and music composition double-major. Needless to say at the time of
this program's creation I knew far more about musical theory than I did computer
science. When I approached the professor of this class with my project idea his
response was that "other people have tried stuff similar for this class and failed, and the
same will happen to you." Despite this opinion I remained confident that a
four-part harmony writing program could be made using the skills I currently had
available, so I went ahead and made the program for my project. I have recently
decided to resurrect this program to expand its capabilities as
an open source project (http://sourceforge.net/projects/instantbach).
The original program was based of of static rules which limits it to a
particular number of possible progressions:
My most immediate goal for this project is to move all of the rules of voice
leading, doubling, chord structure, and progression to an external rules file,
so that the program can be adjusted to work beyond just being able to voice
major key progressions.
Purpose
The first two years of undergraduate music theory are spent learning and
doing mostly one thing: four-part harmony writing. The purpose of this program
is to be able to do four part harmony writing, with or without a user specified
progression, according to all the rules of four part harmony writing, and have
the ability to transpose the result into any key.
Musical Intervals
The distance between any two musical notes is referred to as an interval, and
is represented by a two-character symbol. The first character denotes the
quality, being P for perfect, A for augmented, D for
diminished, m for minor, and M for major. The second character
denotes the melodic distance with a number of how many lines and spaces exist
between the two notes of the interval. Major and minor refer to
whether an interval occurs within a natural minor scale or within a major scale.
Augmented and diminished are used for intervals that do not exist
in the major or natural minor scales, augmented being greater than, and
diminished being less than. The following is a chart of the most common musical
intervals:
Tertian Harmony
Tertian Harmony refers to simultaneously sounding pitches in which each of
the pitches are separated by the distance of major or minor third, in at least
one permutation of the order of notes. Tertian harmonies consisting of three
distinct notes are referred to as triads, and harmonies consisting of
four distinct notes are referred to as seventh chords. A triad can have
four different classifications depending on the interval relationship between
the notes in the triad. These different types of triads are called major, minor,
augmented, and diminished. Seventh chords can be major-major, dominant, minor,
half-diminished, or fully diminished.
Progression
For any given scale, for example a C Major scale (C, D, E, F, G, A,
B), each of the pitches in that scale represent the root note of a tertian chord
of some type. Each of these harmonies are referred to as chords and are
represented by a roman numeral. The roman numeral indicates the quality of a
chord (major, minor, augmented, diminished), the position of the chord inside
the key, and which note is on the bottom, (the root), of that chord.
The first note of a chord is known as the root, the second is the third, the
third is the fifth, and if there is a fourth note it is called the seventh.
Whenever a chord cannot be spelled in thirds, it is in an inversion, meaning
that a note other then the root is on the bottom of the chord. An inversion is
denoted by one number on top of another and is written to the right of the chord
symbol, but I will write them as fractions for simplicity. In a triad, when the
third is on the bottom of the chord it is in first inversion and the symbol that
is used is 6. When the fifth is on the bottom of a triad it is in second
inversion and is written as 6/4. Seventh chords can have three inversions, with
the third, fifth, and seventh on the bottom, denoted 6/5, 4/3, and 2.
Rules of Progression
The order in which chords can occur have multiple rules, the easiest rule
being that I can move to anything, but rather than explain the progression
behavior with words, I will use a picture of the progression graph, where each
chord symbol is a vertex, and all edges are un-weighted.
Other Chord Symbols
As shown in the graph, there are other chord symbols that have not yet been
described. They are the Neopolitan Sixth Chord (N6), the French Augmented Sixth
Chord (Fr+6), and the secondary dominant (V/x). These chords incorporate notes
that are outside of the given key, and have strict rules for which chord they
can lead to. The N6 and Fr+6 must go to V, and secondary dominants must lead to
chord symbol on the bottom of the fraction. Secondary dominants are considered
to be a borrowed V from another key, and therefore must progress to the I of the
key from which they were borrowed.
Four part Harmony Writing
Four-part harmony writing is when a progression is developed and the notes in
the chords of that progression are assigned to four voices, the bass, tenor,
alto, and soprano. Each voice has to stay within a particular range and is
limited to the distance which the voice can move between chords. Each note has
rules for motion and each chord has rules for its resolution. Because most of
the progression consists of triads, and there are four voices, a member of the
chord has to be doubled, which there are also rules for.
Rules of Doubling
- I, ii, IV, iii, and vi : The root and third can be doubled.
- V and V/x : The root can be doubled.
- V7 and V7/x : no doubling is needed or the root can be doubled and the
fifth omitted.
- viio : The third can be doubled.
- vii07 : No doubling is needed.
- N6 : The third can be doubled.
- Fr+6 : No doubling is needed.
Rules of Resolution
- I, ii, IV, and vi : None.
- iii : If the fifth is in the soprano or bass it must resolve up by a m2.
If the fifth is in the tenor or alto it must either resolve up a m2 or move
down a M3.
- V and V/x : If the third is in the soprano or bass it must resolve up by
a m2. If the third is in the tenor or alto it must either resolve up a m2 or
move down a M3.
- V7 and V7/x : The third must resolve as the same as in the V and V/x and
the seventh must move down by a m2.
- viio : If the root is in the soprano or bass it must resolve up by a m2.
If the root is in the tenor or alto it must either resolve up a m2 or move
down a M3. If the chord which the viio is going to is a V, then the leading
tone (the seventh degree of the scale) is already present and there are no
rules of resolution.
- vii07 : The root must resolve the same as in the viio, and even if the
next chord is V, the seventh must move down by a M2.
- N6 : The root must move down by a D3.
- Fr+6 : The A6 (or its inversion d3) interval in the chord must resolve
to P8 or P1.
Rules of Individual Voice Motion
- Bass: The bass generally skips around more than all of the other voices
and does not have any specific restrictions.
- Tenor and Alto: Both these voices generally move very little and rarely
move more then the interval of a third.
- Soprano: This voice moves more then the tenor and alto but is usually
limited to skips of less then a P4.
Range of Voices in Relation to Each Other
The distance between the tenor and the bass can never exceed a P8+ M3, while
the distance between all other voices must remain within a P8.
Cross-Voicing
All voices must always be in order from lowest to highest: bass, tenor, alto,
and soprano. If a voice is ever out of that order it is considered
cross-voicing, which is not allowed. Cross-voicing can also occur in the motion
between two chords.
Illegal Parallel Motion
If two voices move the same interval in the same direction they are in
parallel motion. The parallel motions of a P8, P5, or P1 are illegal parallel
motions. These are called parallel octaves, fifths, and unisons.
Programming the Progression
The progression for the program can be created two ways, either the user can
select the individual chords, or the computer can generate the progression
itself. Harmonic progression can be represented as an unweighted graph, in order
to determine whether a specified progression is valid. The main problem with
this approach is that if the computer were to generate its own progression, it
would have to end at a specific point with a cadence. (A cadence is a series of
chords which end a musical idea or statement). The computer is able to do two of
the most common cadences, an imperfect authentic cadence (IAC), and a perfect
authentic cadence (PAC). Each of the cadences consists of a V then I, but in the
PAC the soprano shares the root with the bass. The problem is that if you have n
chords and start with a random chord, and randomly move through the graph until
n-2, then cadence, there is a good chance that the chord at n-3 will not be able
to go to the chord at n-2. This would mean that the progression would have to be
re-done. To prevent this problem, I built the graph backwards, and the
progression is formed backwards as well. This way the progression never runs
itself into a dead-end because it always starts with I, V, and then the random
chords, so when the progression reaches n, the chord at n will always exist
because there are no verticies in the graph that do not have at least one
adjacent vertex.
Turning the Progression into Musical Notes
After the progression has been created, the computer needs a way to assign
notes of all chords in the progression to the four voices. To assign notes to
the four voices the computer must first generate all possible notes each voice
can have. This is done by using a recursive method that when given the starting
pitch of a chord, and the symbol of that chord, builds a chord consisting of
every possible note in that chord. Because rules of voice leading depend on
individual chords and the position of individual notes within that chord, each
note in a chord maintains whether a given note is a root, third, fifth, or
seventh.
The Chord as a Data Structure
The main object of this program is the chord, which consists of four voices
and a chord symbol, so the obvious main data structure for the program is a
Chord object, consisting of four voices and a chord symbol. A Chord object also
contains methods which allow the individual voices to be viewed as either the
note or the position of that note within the chord, as well as a method to
obtain the chord symbol. This allows other methods to deal with notes
arithmetically and as the position of that note within the chord for voice
leading purposes.
Creating All The Chords
Because of all the rules associated with four-part harmony writing, sometimes
the voicing of a progression leads to a dead-end where the progression is
correct, but because of the way a chord is voiced, it may not be able to
correctly move to the next chord. This is why for the computer to be able to
voice a progression, it must have available all the possible combinations of
four note chords. The combinations have restrictions though, for each voice the
note must be within that voice’s range, and may not overlap that of another
voice.
Checking the Motion Between Chords
To check the motion between chords, the motion itself becomes a Motion object
and contains methods to compare the motion with other Motion objects, thus being
able to check for all the rules of motion. A Motion object is created with the
same voice in two adjacent chords, for all four voices. The object basically
takes the two notes, calculates the distance between the two notes, and
determined the direction that they are moving. Besides the Motion object, which
checks for invalid parallel motion, the program also checks for cross-voicing
within chords, cross-voicing between chords, and invalid interval skips. Within
the same method which checks for all the previous mentioned motions, there are a
series of nested Boolean statements which check for proper resolutions of chords
based on chord symbols. Each chord is unique in its resolution, with the
exception of when a member of a chord has to resolve in the same direction and
distance as another. In this case a method was developed to be used for checking
the resolution on chords that resolve using only step-wise motion. Otherwise if
there is resolution, it directly involves the leading tone or has a completely
unique resolution, and must be dealt with individually (Fr+6 and N6 chords).
Choosing the Most Correct Chord
After all the chords have been created and checked for doubling and motion,
the chords that remain are all correct, but there is always a most optimal chord
to choose. This is done by first looking at the inversion of the chord because
chords are generally preferred to be in root position, with the exception of
chords which cannot exist in root position. If there is more than one chord in
root position, the chord which has the least motion to its already created
adjacent chord is chosen. In the case where a chord does not exist in root
position, the chord with the least motion to its already created adjacent chord
is chosen.
