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Ratios
The Octave
Of
all the notes, in all the octaves.....
Intervals
Early Musical Scales
The
Western Equitempered Musical Scale
Major Scales and Keys
How many Keys are there?
Finale
Encore
Further Reading
(NOTE: If you are not familiar with basic
sound-wave parameters
such as frequency, read about them on the "Sound Waves" page
before continuing).
Introduction
On this page, with the non-musician
in mind, an outline is presented of the origins of the present-day Western Equitempered
Musical Scale, the basis of virtually all
modern western music.
The story starts with ratios.
Ratios
With
the exception of a few (possibly unfortunate) people who have perfect pitch
("pitch" is a musical term used instead of "frequency"), we
humans do not detect the absolute frequency of a tone very
precisely. We are aware of "high notes" and "low notes" but
little more. We can, however, compare two tones with
considerable precision. In doing this our ears and brains
are sensing frequency ratios, not frequency
differences. We find tones
whose frequencies bear a simple ratio to each other to be particularly harmonious
while for more complicated ratios we generally find the sound to be less pleasant.
So ratios are important. Let us start with the simplest and
most important ratio of all.
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The Octave
Tones
whose frequencies are in the ratio 2:1 sound so harmonious
that most people, musicians and non-musicians alike, would agree that they in
some sense are the "same" note, even though one is clearly higher than the other.
This special ratio, where one tone (or note) is twice the frequency of the other, is
called an octave (you can learn where this name came from later on).
Julie Andrews illustrated the "sameness" of two
notes an octave apart when in "The Sound of
Music" she sang "doh, a deer, a female
deer.......". The song ends with "........which will bring us back to doh".
The two 'doh's' are an octave apart, but because of their "sameness" it seems entirely correct to use the same name for
both.
To hear this octave "sameness", click the image on
the left to hear two piano notes
one octave apart - first one note, then a note one octave higher,
then the two notes played at the same time.
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Of
all the notes, in all the octaves.....
Since an octave involves a
2:1 frequency ratio, it follows that we could start at the lowest note
humans can
hear, then double the frequency repeatedly to move up first one octave,
then another and so on. Because of the way doubling works we
don't have to go through too many octaves to cover the whole range of
human hearing. This range
is 20 Hz - 20,000 Hz, so the entire range of audible
tones - all the notes there are - covers a range of just under 10 octaves. You can work this
out for yourself quite easily by starting at 20 Hz and then doubling over and
over to see how many doublings are needed to reach 20,000 Hz.
To illustrate the idea, you can hear seven notes
spanning six octaves if you click the image below.
Because of the repetitive nature of octaves, we feel
that all octaves are the "same", apart from being higher or lower.
So the design of a musical scale (a series of notes we can choose
from when we
create music) comes down to deciding on a suitable series of notes to
fit into each octave.
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Intervals
One final detail before we go on: to a
musician that word 'ratio'
brings to mind complicated things like multiplications and divisions.
Musicians can count but they don't like anything more complicated, so they use the term
interval instead of ratio. So we had better use interval instead of
ratio from now on.
If you are mathematically inclined you will recognise
that what we are dealing with here is a logarithmic scale, which is what
the musical scale actually is. If musicians knew this they
would be
seriously alarmed, but happily no one ever mentions this during
their training.
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 For all cultures for which records exist, it is clear
that the octave, important ratio that it is, was central to all their musical systems. What
varied was how many notes were fitted into each octave, and how they
were arranged. The ancient Persians used 23 notes per octave,
the Chinese 5. Indian music used, and still uses, 31 notes to
the octave.
In the context of music for brass bands we are concerned with the
western tradition. Early western music used a system of tuning ascribed to Pythagoras (of
right-angled-triangle fame) whereby repeated application of the
ratio 3:2 was used to define up to 7 notes within the octave.
This means there were up to 8 intervals in each octave, which
is where the name "octave" came from. These notes
were not, in ratio terms, spaced equally within the octave. The
number count was extended later to
include 11 notes (12 intervals) per octave, which is where it
currently stands. The tuning system based on these early ideas
is sometimes referred to as Just Intonation.
During the medieval era the tunings were adjusted
partly as a result of musical fashion until in 1691 Andreas
Werckmeister suggested a tuning system which led to the present-day
Equitempered musical scale used in all modern western music.
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The
Western Equitempered Musical Scale
All
western music has for more than two centuries been based on the Equitempered
scale, in which each octave is divided into twelve equal intervals. For
historical reasons these intervals are called semitones and
two such intervals taken together constitute a whole tone.
For the mathematically inclined this means that a semitone interval
must represent a frequency ratio equal to the 12th root of 2 (12√2,
or 21/12)
Each note is given a label based on letters of the
alphabet. From this one might expect the notes to be labelled
A, B, C, D, E, F, G, H, I, J, K before starting again at the next
octave with A again. Instead, again for historical reasons,
something rather more complicated is done:
Firstly, only the letters A - G are used. That
is only seven letters, of course, so to the extra five labels we
need are obtained by using five of the letters twice, with a
secondary character attached the second time each letter is used.
Secondly, and again for historical reasons, there are
actually two secondary characters which may be used to identify the
five re-used letters. One of these is called a "sharp" and is
written "#" and indicates the next note above in the musical scale.
The other is called a "flat", is written " "
(which we will write as "b" in case of possible font restrictions in
your browser) and indicates the next note below in the musical
scale.
We thus have two ways of writing the musical scale;
using sharps we have:-
A A# B C C# D D# E
F F# G G# A A# B
C.........................
<-------------------one octave-------------------->|<-------------next
octave.......
and using flats we have:-
A
Bb
B
C
Db
D
Eb
E
F
Gb
G
Ab A
Bb
B
C........................
<-------------------one
octave-------------------->|<-------------next octave......
In either case the 12-note sequence is repeated for
each octave throughout the entire range of hearing.
The term "chromatic" is used to describe this sequence
of notes one semitone apart, and one would speak of a chromatic
scale starting on.....for example, C.
Here is a piano-style keyboard with just such a scale
marked on the keys.
in manuscript form, an ascending followed by a
descending chromatic scale starting at the lower C might look like
this:
If you click the above manuscript image, you can hear
what this sounds like.
However - because all the intervals in the
sequence are the same (semitones) there is no "focus" - no natural
start and stop. It is just a sequence of notes.
Something more is needed to provide the necessary "focus"
to allow us to write melodies.
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Major Scales and
Keys
During the early development of the musical scale a
variety of ways to fit up to 7 notes into the octave came into and
out of fashion and eventually what we now call the major scale
became established and has remained in use ever since.
This scale will be familiar to many
non-musicians as well as musicians if they have been exposed to
western music, and is the "doh, ray, mi...." scale used by Julie Andrews in that famous song. It consists
of notes selected from the chromatic scale above so that the
intervals between notes are in the following sequence:
tone-tone-semitone-tone-tone-tone-semitone
 If you examine
this sequence you will see that it extends over 12 semitones, thus
fitting exactly into one octave. And if you look at the
keyboard image again, you will see that you get just this sequence
if you start on C and play only the white notes - the sequence C, D,
E, F, G, A, B and finally C again. This is called "the scale
of C major" or "the C major scale" and is the most obvious way for beginners to play their first major scale on a
keyboard. Click on the picture on the left to hear it played.
The pattern of whole tones and semitones encapsulated
in a major scale establishes
that necessary focus, or frame of reference, for melody-writing which is not present in the chromatic scale. Partly
because of the way human hearing works and partly because of western
culture and upbringing, most people with that tradition feel that the first note of the
major scale (the "doh" in the familiar sequence) is a natural
beginning and that the second "doh" an octave higher is a natural
end. The "doh" note is the key note of the sequence, and for
that reason we speak of playing in the "key of C" if we play a melody
based mainly on the notes in the C major scale.
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How many Keys are there?
If you have followed the story this far, you might
think that there must be eleven other major scales, or "keys" in
which we could play, starting on any of the other notes in the
octave. And you would be quite right - your ear will tell you
that there are twelve distinct major keys, or scales.
The trouble is that musicians, who can't leave well alone, will tell
you there are fifteen major keys - three more than your ear tells
you there are! They will tell you that these are:
Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, G#
Despite the
fact that musicians came up with this list, there is actually a sort of logic
to it - which we won't go into here but suffice to say there
are
musical works written in all of these fifteen keys. So how can
there be fifteen? The answer is that in this musicians' list there are three
pairs of duplicates - the three at each end of the
list. Thus, when the scale of Cb major is played you hear
the same sequence of notes as when the scale of B major is
played (Cb is, after all, the same note as B). Likewise Gb and F# are the same note, as are Db and C#. So your ear was right
and there really are only twelve differently-sounding major scales -
but don't try to tell that to a musician!
Finale
There are also other forms of scale, most importantly the minor
scales. They all follow the same general idea, though, of
using a sequence which gives a natural focus - a key.
One could go on, and there is certainly a lot more
involved when it comes to writing good music. But In the
modern western musical tradition it is all based on the Western Equitempered Scale.
Encore
 And
as a reward for getting this far -
you can click the keyboard on
the left to go to a keyboard simulator where you can have a go at
playing some tunes of your own.
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Further Reading
http://www.medieval.org/emfaq/harmony/pyth.html
This website contains an extensive discussion of the
Pythagorean and other scales including Just Intonation and the
Equitempered Scale.
http://cnx.org/content/m11639/latest/#s2
This site contains a lighter treatment of Pythagorean
and other scales, and associated topics.
http://www.musicalintervalstutor.info/.
At this tutorial website you can hear some intervals played and find
out more about
intervals and scales.
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