ï~~attempting to explain aspects of human musical
intelligence, but we will focus here on using the theory
for developing new educational tools.
I
Perfect
5ths
Db F A C# E# Gx
Gb Bb D F# A# Cx
Cb Eb B D# Fx
Fb Ab C E G# B#
key
Bbb Db F A C# E# window
E Bfor key of
Ebb Gb Bb D F# AR CMajor
Abb Cb Eb G B D#
However, for the purposes of educating novices in the
elementary facts of tonal harmony we map LonguetHiggins space onto the twelve note vocabulary of a fixedtuning instrument resulting in what we might call '12 -note two-dimensional Longuet-Higgins harmony space' or
2D harmony space for short. Consequently we lose the
double sharps & double flats of fig 1, and the space now
repeats exactly in all directions (fig 2). Notes with the
same name really are the same note in this space. In fact a
little thought will show that the space is in fact a torus,
which we have unfolded and repeated like a wallpaper
pattern.2 One result of this is that instead of a single
key window we have a repeating key window (fig 2).
Major thirds ---
Fig 1
3.1 Representing the 'Statics' of harmony
3.1.1 Representing key areas and modulation
In diagrams such as Fig 1 all of the notes of the diatonic
scale are "clumped" into a compact region. For example,
all of the notes of C major, and no other notes are
contained in the box or window in Fig 1. If we imagine
the box or window as being free to slide around over the
fixed grid of notes and delimit the set the notes it lies over
at any one time we will see that moving the window
vertically upwards or downwards, for example,
corresponds to modulation to the dominant and
subdominant keys respectively. Other keys can be found
by sliding the window in other directions. Despite the
repetition of note names, it is important to note that
notes with the same name in different positions are not
the same note, but notes with the same name in different
keys (Steedman (72) calls these "homonyms"). This is an
extension, motivated by Longuet-Higgins' theory, of the
standard notational distinction that C double sharp, for
example, is not the same note as D. (Steedman calls such
pairs "homophones".)
perfect
5ths
using
12-note
pitch set
key
windows
for
C major
major thirds using
12-note pitch set
Fig 2
3.1.2 Representing chords and tonal centres
Let us now turn to look at the representation of triads and
tonal centres. In 2D harmony space, major triads
correspond to L-shapes (fig 3). A triad consists of three
maximally close distinct notes in the space. The dominant
and subdominant triads are maximally close to the tonic
triad. We can instantly see from the diagram that the three
primary triads contain all the notes in the diatonic scale.
2We have used arbitrary spellings in these diagrams
(e.g.. F# instead of Gb etc.), but an environment could
equally easily use neutral semitone numbers or any
preferred convention. The convention could even be
dynamically affected by changes in the position of the key
window.
1987 ICMC Proceedings
183
