Mathematical Music

_ by
WaLTEeR F, BopMER

THE concept of a chord as a combination of notes is very familiar
but its general structure is perhaps not so well known. Thus we
associate with the dominant C-major chord CE GC its inversions
EGCE and GCEG, and with each of the twelve notes we
associate such a dominant major chord corresponding to each of
the major scales. In other words, this “chord” can be considered
as a findamental musical element irrespective of the scale or
particular inversion.

It will be seen that the characteristic of the chord is merely the
sequence of intervals 4 3 5, adding to 12, the inversions correspond-
ing to the cyclic permutations 354 and 543. Sucha splitting
of the number 12 is an ordinary partition of 12 into three parts,
ignoring the order of the numbers, or a composition of 12, taking
order into account.

We therefore define the general chord as a composition of the
number 12, together with the cyclic permutations of the composition,
the characteristic type of the chord being the partition given by the
numbers contained in the composition. It is clear that, correspond-
ing to any chord of m notes, there is a chord of 12 — m notes
obtained from all the notes not used in the original chord. This
will be called the “conjugate” chord. For example, the conjugate
chord of the dominant minor chord (345) of three notes, is
(I2I112z1112) of nine notes, partitional type (23 18), (It will
be noticed that the minor and major dominant chords are the
only two of the partitional type (5 43).) Because of the above,
to enumerate all the possible chords of 12 notes, we need only
consider those of 6 or less.

For chords with total interval gteater than twelve, we replace
successively the highest notes of the chord by the equivalent notes
within the first octave after the first note of the chord. Thus the
chords CGD’ A’, CGD” A”, etc., are all equivalent to CDGA.

The following is an enumeration of all the chords of six or less
notes, according to their partitional type.

 
Inote 1 chord of type (12).

2 notes 6 chords, 1 each of type (11 1) (10 2) (9 3) (8 4) G7 5) (© 6).
3 notes 14 chords, 2 each of type (92 1) (831) (743) (73 2)
(65 1) (642) (5 43).
4 chords, 1 each of type (10 1*) (8 22) (6 3?) (5? 2).
r chord of type (4%).

Total 19
4 notes 12 chords, 6 each of type (6 3 2 I) (5421).
24 chords, 3 each of type (8 2 1) (7 3 19) (7 271) (641°)
(5 3° 1) (5 3 2%) (473 1) (4 32).
4 chords, 2 each of type (52 1) (4? 22),
2 chords, 1 each of type (9 13) (6 23),
1 chord of type (3%).

Total 43

5 notes 24 chords, 12 each of type (5 3 2 1°) (4 3 2 1).
18 chords, 6 each of type (6 22 12) (422 12) (4 3? 1%).
20 chords, 4 each of type (7 2 1) (63 15) (5 4.13) (5 2* x)

(33 2 1).
2 chords of type (3? 23),
2 chords, 1 each of type (8 14) (4 24).
Total 66
6 notes 20 chords of type (4 3 2 13).
16 chords of type (3? 2? 12),

20 chords, 10 each of type (5 2? 13) (4 23 1’),
15 chords, § each of type (6 2 14) (5 3 14} (3 242).

4 chords of type (3? 15),

3 chords of type (4? 14).

1 chord of type (7 15).

i chord of type (2§).
Total 80

 
In the above table, the partitions corresponding to chords of
m notes are all the partitions of 12 containing m parts. It will be
noticed that the number of chords of given partitional type depends
only on the sub-partition formed by the indices. Thus, (7 2 15) has
sub-partition (3 12) and all the partitions with this sub-partition
yield 4 chords of ¢ notes.

This is an example of the endless scope which exists for mathe-
maticians to formalise and generalise the rules of harmony. A
major scale, for instance, can be identified with the chord
(2212221), the conjugate being (2 3 2 2 3), and it is not difficult
to formulate rules for operating on these chords in many ways.
For example, a chord can be contracted to one of fewer notes, simply
by adding together any number of consecutive intervals, arpeggios
may be indicated by the consecutive playing of the cyclic forms
of a given chord, or the transition from major to minor chord can
be indicated by the permutation of the intervals (3 4) in (435).
After all, the whole of music consists merely of all the possible
combinations of the twelve notes (or more if we use micro-tones)
amongst the different instruments and octaves! It thus lends
itself to a systematic classification of the kind considered here,
However, I feel confident that music will remain the creative art
of the musician rather than the mathematician.

I am indebted to Roberto Gerhard for suggesting to me the
notion of the general chord, and the problem of enumerating
such chords.

International Congress of Mathematicians

Edinburgh, 1958

Tuts will take place from rq4th to 2tst August. A number of
mathematicians will deliver one-hour and half-hour addresses;
there will also be daily sessions for fifteen-minute communications.
The Congress is divided into eight sections according to subjects.
A programme of entertainments and excursions is planned. Those
who wish to receive further information should communicate direct
with the Secretary of the Congress:

Frank Smithies,
Mathematical Institute,
16, Chambers Street,
Edinburgh 1, Scotland.

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