Let's get the technicalities over with first.
A
group [Wikipedia]
is a set G with members denoted by a, b, c, ...,
that is endowed with a binary operation, denoted here by #,
that maps pairs of elements (a, b) of G to elements of G.
Write, for example:
#: (a, b) -> c
or more simply,
a # b = c
The binary operation is associative:
(a # b) # c = a # (b # c)
Furthermore, there is an element E in G called the identity
for the binary operation for which it is true that
E # x = x
for every element x in G.
And, for every x in G there exists an inverse of x, x* with respect to the identity:
x # x* = E
That's all there is to it.
One can start with what looks like a set of weaker axioms like
assuming right and left identities, but wind up proving that
they must be the same element.
The axioms or assumptions for a group are so simple and unrestricted that every part of mathematics is filled with examples of them.
Every time one encounters a group, it is, almost always, as a group of transformations of some set or object. Sometimes an encountered group has the special property of being in addition, commutative:
a # b = b # a
for all a and b in G.
Less descriptively, these are sometimes called Abelian groups,
after a mathematician whose last name is Abel.
Unfortunately, another mathematician M. Jordan has called a
certain class of groups 'Abelian' that are not at all commutative.
Care must be taken in reading older mathematical literature,
but the modern usage of "Abelian" means commutative.
Mathematics has been called the science of quantity, primarily because its fundamental objects are numbers, of one type or another, (numbers which measure quantity, amount, ..., and maybe more) and groups appear automatically in various kinds of numbers. For example the countably infinite set of integers
... -3, -2, -1, 0, +1, +2, +3, ...
satisfy the group axioms, where the binary operation is addition,
the identity for addition is 0, and the inverse of "a" is "-a".
The element "0" is its own additive inverse,
0 + 0 = 0
and one understands that
-(-a) = a
where the unary L handed operation (-) is the same as the
R handed operation to inverse (*), as above.
Clearly the integers form a commutative group.
To vizualize this group as a group of translations is easy; it is a
group of discrete translations of the set (think line) of integers itself.
The set of integers is, of course, NOT a group with respect to the operation of multiplication, because only +1 and -1 have inverses, and 0 does not and will not have a consistant inverse, even if all the other possible inverses are added.
One might correctly guess that the set of real numbers is then also a commutative group of translations of the reals, that similarly the rational numbers and the complex numbers [Wikipedia] form groups with respect to addition.
As sets, these groups are very big, infinite, in fact,
and one wonders whether there are groups with a finite number of elements.
The answer is yes.
The number of elements in a finite group is called its order.
There is a group for every order greater than 0, i.e. for every natural
number. For order 1, there is only one element '0', which is
decidedly uninteresting and therefore dismissed as "trivial".
For order 2, two binary operation tables can be constructed
# E a # E a
|------------ |------------
E | E a E | E E
a | a E a | E a
The table on the left is an abstract way of writing down
the additive properties of of {0, 1} mod 2, or the
multiplicative properties of {1, -1}, while the table
on the right is an abstraction of the multiplicative
properties of {0, 1} whether mod 2 or not, which
fortunately or unfortunately does NOT form a group.
Lesson one here is that simply assuming a binary operation is not sufficient to guarantee group structure.
Lesson two is that an abstract group structure can be represented (or interpreted) in more than one way.
There is only one group structure for order 3:
# E a b
|-----------
E | E a b
a | a b E
b | b E a
and it is commutative.
It happens to be true, as a group theoretical theorem, that if the order of a group is a prime number, the group must be commutative.
Now let's try order 4. There are only two possible group structures (and many more binary operative structures). So, the order of a group does not necessarily determine its structure.
# E a b c # E a b c |--------------- |--------------- E | E a b c E | E a b c a | a E c b a | a E c b b | b c a E b | b c E a c | c b E a c | c b a EThe table on the left is an abstraction for the multiplicative group of {1, -1, i, -i}, where i is the square root of (-1). The table on the right decribes what is often called the Klein 4-group, after Felix Klein. Both groups are commutative, as one can see by the symmetry of the table about the diagonal from upper left to lower right.
It is clear that the idea of building tables to represent group structures becomes impractical as the order increases to very large number. The order of a group is the cardinality or number of elements in the set S of its members. Then, the idea of describing groups by other structural properties becomes important. The order of a group has number theoretic properties that influence the structure of the group; one of these is the prime decomposition of the order.
It is a theorem that every natural number can be expressed uniquely as a product of powers of primes. This expression is called its prime decomposition.
Another such structural property of the group is commutativity, since not all groups are inherently commutative.
It is a theorem, that any finite group can be represented as a group of permutations of some set [Burnside 1911]. Burnside also examines how properties of the group structure can be determined by the prime decomposition of its order.
Another highly important structural aspect of groups that I happen to be ignoring here is the possession of subgroups, and the relations between them. One can, in fact, structurally distinguish the two groups of order 4 abstractly by their subgroup structure.
Now let's look at some groups related to the piano keyboard. A keyboard is divided into a recurring pattern called and octave. A piano has approximately seven such octave patterns. For notation of notes I'll use {A, B, C, D, E, F, G} and the so called accidental signs (b for flat) and (# for sharp); later, we'll also use (x for ##, the double sharp). The repeating octave pattern can be visualized as
C# Eb F# Ab Bb
|-| |-| |-| |-| |-| Black Keys
|---|---|---|---|---|---|---|
C D E F G A B White Keys
As far as a modern keyboard with well tempering goes:
C# = Db, D# = Eb, F# = Gb, G# = Ab, A# = Bb
B = Cb, E = Fb, C = B#, F = E#,
The "chromatic sequence"
... C C# D Eb E F F# G Ab A Bb B (C) ...
is a sequence of pitches with equal consecutive intervals
in the sense that if [X] is the frequency of any note in the
sequence and [X+1] denotes the frequency of the note to the right
of it, then their ratio
[X+1] / [X] = twelfth root of 2
Notice that by speaking in terms of sequences of intervals,
we have eliminated the reference to specific pitches and think
of a more abstract "musical object" which can be realized by
any of its transpositions. One frequently thinks of a 12 tone
row in this way by thinking more in terms of the sequence
of intervals rather than the sequence of pitches.
As it turns out, this is how strictly dodecaphonic music is
also mostly heard, unless you happen to have perfect pitch.
The idea is a democracy of pitches in the first
place, and the system achieves it purpose.
This uniform ratio of consecutive pitches, in fact, defines well tempering by a kind of homogenization or democratization of our modernized musical alphabet.
Dodecaphony is a way of bringing order to the atonal use of this democracy, by allowing an orderly enforcement of a principle of "no tonal center"; this is the actual meaning of the word "atonal". It does not mean "a bunch of unpleasant sounds" and is not synonymous with "cacaphony".
The distance between consecutive notes in the chromatic sequence is called a "semitone" or "half-step". From the keyboard outline and the equivalences it is easy to understand postfix "#" and "b" as operators acting on pitches.
b = "go down (left) one semitone" - Flat
# = "go up (right) one semitone" - Sharp
To develop more musical terminology of intervals using the semitone as the fundamental interval
INTERVAL NAME SEPARATION
minor second = 1 semitone
major second = 2 semitones
minor third = 3 semitones
major third = 4 semitones
fourth = 5 semitones
tritone = 6 semitones
fifth = 7 semitones
minor sixth = 8 semitones
major sixth = 9 semitones
minor seventh = 10 semitones
major seventh = 11 semitones
octave = 12 semitones
minor ninth = 13 semitones
major ninth = 14 semitones
Also
augmented fourth = tritone = diminished fifth
Now consider what to a musician is called transposition, which simply means changing the tonal center in which a piece is written. This is a translation of all the notes of the musical composition by a fixed number of semitones. On an idealized infinite keyboard, the set of transpositions forms a commutative group that is structurally equivalent to the additive group of integers.
Instead of considering an infinite keyboard, allow that transpositions by an octave or any multiple thereof are all equivalent. The keyboard is partitioned into equivalence classes that are labeled by the designating symbols of the chromatic sequence. Assuming Octave Equivalence, there are only 11 distinct transpositions and one identity transposition. The set of transpositions is now a cyclic (commutative) group of transformations of order 12.
Think of the transpositions as being affected by repeated application of the operator '#' or of 'b'. The full group is generated by powers of either operator, which is what is meant by a cyclic group.
We have considered in this aside, the concept of group only as it relates to the conceptually equivalent notions of the modern keyboard and the well tempering that defines it.
For more mathematics, generally, consult Mathematical Atlas: A gateway to Mathematics
The Dog School of Mathematics Presents: An Introduction to Group Theory
For a brief history of groups in mathematics see GROUP THEORY!HISTORY
Email me, Bill Hammel at