Download History of Music Theory - Totally Ratted Limited

Creating The Pythagorean Scale
We are told that Pytharoras experimented with an
device called the Monochord (which literally means
“one string”) by his student Philolaus. This was a
single stringed instrument with a moveable bridge
and by positioning the bridge in different positions it
was possible to play different notes on the string.
His reported aim in his analysis of the vibrations of
the strings was to define the “music of the spheres”
and was thus an attempt to understand the heavens.
He, along with other Greeks, were of the opinion
that the heavenly bodies moved in a form of musical
precision and that by analysing music one gained an
insight into the movement of the heavens. As early
as Ancient Babylon, mathemeticians believed that
the heavens were governed by ratios of integers and
it is perhaps from here that Pythagoras found the
inspiration to experiment with the instrument.
Whatever the inspiration, there is little evidence that
any theory was used in the tuning of musical scales
prior to the life of Pythagoras. Conversely, the start
of music theory being important to the tuning of
instruments almost certainly begins with the words
of Philolaus. It is likely that prior to the advent of
the Pythagorean ratios that musical scales were very
varied and perhaps even unique to the individual.
By using at least two monochords, Pythagoras was
trying to measure the lengths at which the notes
from the strings would ring together in what could
be considered “perfect” harmony. It is highly likely
that he believed that these notes would be found at
whole number ratios along the length of a string. It
is also quite likely that he was happy to find that
this appeared to be the case.
Through his experimentation, he discovered that
strings at a length ratio of 2:1 provided a consonant
sound, an interval the Greeks called a diapason (dia
means across, between or through in Greek). The
second consonance was found at a length ratio of
3:2, which became named the diapente. The final
consonance was found at a length ratio of 4:3 and
was called a diatessaron. These ratios, when stated
together, formed the ratio 1:2:3:4 which will no
doubt have reinforced Pythagoras’ beliefs.
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Music theory in Ancient Greece was based around
the Tetrachord (in Greek, tetra means four). The
four notes were tuned to notes in a descending
order. The first and fourth notes were separated by
the interval of a diatessaron. The two other strings
were tuned to one of a number of intervals, the size
of which depended on the musical scale.
It was soon noticed that the two intervals 3:2 and
4:3 could be multipled together to become the 2:1
ratio. For example:
3
2
x
x
4
3
=
=
12
6
=
=
2
1
It was also noted that the interval between the notes
could be found by dividing one by the other. This
interval was known as the Whole Tone and is
defined as the ratio 9:8. The calculation would be:
3
2
x
x
3
4
=
=
9
8
This definition of a tone persists for centuries and
does not fully disappear from music theory until as
late as the eighteenth century. When two tetrachords
are placed within an octave and separated by a tone
the resulting scale is known as a Diatonic scale. The
two tetrachords in a diatonic scale are often called
diatonic tetrachords.
It is widely believed that Pythagoras constructed a
tuning system which he based on the interval ratios
that he discoved using the monchord. The exact
method he used for achieving this is a matter for
debate between music historians, but the belief most
commonly held is that the intervals are calculated
using the 3:2 ratio.
As the note found at two thirds of the string length
is consonant with the note found at the full length,
then the note found by a further shortening of the
string is consonant with that found at the two thirds
point. By extending the string by a third, we find the
consonant note suggested by the 4:3 ratio. If this
longer string is then extended by a third of its new
length the resulting note is consonant.
Chapter 2: A Condensed History
We can continue the process of both shortening and
lengthening the string by the 2:3 ratio to find a
series of steps which denote musically related notes.
Mathematics at the time saw no reason to go further
than a cubed number as only three dimensions were
visible to the naked eye. Due to this opinion,
Pythagoras probably achieved his seven note scale
by multiplying the ratios by themselves to create a
squared and a cubed ratio - thus making 3 powers of
the ratio 3:2 in each direction.
For the 2:3 , the squared ratio is:
2
3
x
x
2
3
=
=
4
9
2
3
=
=
8
27
And the cubed ratio is:
4
9
x
x
For the 3:2, the squared ratio is:
3
2
x
x
3
2
=
=
9
4
3
2
=
=
27
8
And the cubed ratio is:
9
4
x
x
These fractions describe a set of notes which span
from under half the length of the original string to
almost four times its length. Using the 2:1 ratio to
shorten or lengthen the string, we can place all of
the calculated notes in the same range. The range we
chose is between the original length of the string
and a point described by the 2:1 ratio itself. i.e. half
the original string length.
So, the calculation for the first power of the 3:2
ratio would be:
3
2
x
x
1
2
=
=
3
4
This gives us the third of the “perfect” intervals that found by the 3:4 ratio - and this is within the
range we require. When the result is still outside the
range the 2:1 ratio is applied a second time
Metal In Theory
Ratio
(3 : 2) ^
(3 : 2) ^
(3 : 2) ^
1:1
(2 : 3) ^
(2 : 3) ^
(2 : 3) ^
3
2
1
1
2
3
Calculation
27
9
3
1
2
4
8
/
/
/
/
/
/
/
8
4
2
1
3
9
27
Adjustment
1:4
1:4
1:2
1:1
1:1
2:1
2:1
Adjusted Ratio
27 : 32
9 : 16
3:4
1:1
2:3
8:9
16 : 27
Fig 2.01 : Pythagorean Series using the 3:2 ratio
Fig 2.01 shows the calculations, the adjustment
ratios and the adjusted lengths of a string. Seven
was an especially important number to the Greeks
as it denoted the number of heavenly bodies, not
including stars, that they were aware of. These were
the Sun, the Moon, the Earth, Mercury, Venus, Mars
and Jupiter. To Pythgoras, the idea that his newly
defined scale had seven notes was very appealing
and this fact was perhaps the reason that the scale
became an accepted part of music theory.
Because there are seven notes, the scale is referred
to as a heptatonic scale - hepta means seven and
tonos means tone. Additionally, the eighth note
when ascending through the scale is defined by the
2:1 ratio and is called the octave (octa means eight).
Now, despite our extensive use of a twelve note
scale, the interval between two notes at a 2:1 ratio is
known by a name which has its origin in a seven
note scale which is more than 2,000 years old.
When the seven calculated string lengths are
reordered from longest to shortest, they define what
is referred to as the Pythagorean Heptatonic scale
(see Fig 2.02). It stands as the first scale built on a
basis in mathematics and as such begins a process of
mathematical analysis in music that lasts to this day.
Ratio
(2 / 3)
(2 / 3)
(3 / 2)
(3 / 2)
(2 / 3)
(2 / 3)
(3 / 2)
(1 / 2)
^
^
^
^
^
^
^
^
0
2
3
1
1
3
2
1
Calculated
Ratio
1
8
27
3
2
16
9
1
/
/
/
/
/
/
/
/
1
9
32
4
3
27
16
2
Decimal
1.0000
0.8889
0.8438
0.7500
0.6667
0.5926
0.5625
0.5000
Ratio
to Next
8/9
243 / 256
8/9
8/9
8/9
243 / 256
8/9
8/9
Roman
Letter
D
C
B
A
G
F
E
D
Fig 2.02 : Pythagorean Heptatonic Scale values in order
3
It can be seen in Fig 2.02 that many of the intervals
between notes are described as a ratio of 8:9, which
we have already seen was defined as a Whole Tone.
In this table is another ratio of a special significance
- the ratio 243:256 - known as the Semitone. We can
see how this figure is arived at by divining the ratio
between the second and third degrees of the scale.
27
32
/
/
8
9
=
=
27
32
x
x
9
8
=
=
243
256
The letters assigned to the notes in Fig 2.02 are at
first counter-intuitive to the modern musician - the
letters seem to be the mirror image of what would
be expected in a modern scale. This is because the
Ancient Greeks listed their scales in descending
order as opposed to the ascending order which is
commonly used today. The letters themselves are
actually derived from Roman music theory, although
the Romans used fifteen letters to describe their
scale to the modern system’s seven.
Whilst the mathematical theory is of importance to
the modern musician, a full scale analysis of the
Greek theory system lies beyond the scope of this
book. That said, a short detour into the outlines of
the system is worthwhile as the concepts which
drive it also drive the theory behind the modern
musical system defined fifteen hundred years later.
Tetrachord
Hyperbolaiôn
Diezeugmenôn
Mesôn
Hypatôn
Roman
Note
P
O
N
M
L
K
I
H
G
F
E
D
C
B
A
Greek Note
Nêtê
Paranêtê
Tritê
Nêtê
Paranêtê
Tritê
Paramesê
Mesê
Likhanos
Parhypatê
Hypatê
Likhanos
Parhypatê
Hypatê
Proslambanomenos
Planet
Saturn
Jupiter
Mars
Sun
Venus
Mercury
Moon
-
Saturn
Jupiter
Mars
Sun
Venus
Mercury
Moon
Fig 2.03: The Greater Perfect System of Ancient Greece
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The Greater Perfect System (Systêma Teleion
Meizon) was a Greek scale that was built on a set of
four stacked tetrachords called the Hypatôn, Mesôn,
Diezeugmenôn and Hyperbolaiôn tetrachords. Each
of these tetrachords contains the two fixed notes that
bound it. Fig 2.03 shows these notes in the darker
shade of grey, whilst the tetrachords are highlighted
in a lighter shade.
The cousin of the Greater Perfect, the Lesser Perfect
System, was built on three stacked tetrachords - the
Hypatôn, Mesôn and Synêmenôn. The first two of
these are the same as the first two tetrachords of the
Greater Perfect, whilst the third tetrachord is placed
above the Mesôn. When viewed together, with the
Synêmenôn tetrachord placed between the Mesôn
and Diezeugmenôn tatreachords, they make up the
Immutable System (Systêma Ametabolon) which is
also referred to as the Unmodulating System.
Returning to the Pythagorean mathematics behind
the system, we can further analyse the intervals
between any two notes in the scale. The results of
which are found in ratio form in Fig 2.04. The titles
of the columns indicate the note from which the
interval should be measured whilst the titles of the
rows indicate the target notes. The ratios are stated
in the form:
Target Length : Initial Length
We can see from the table that all but one of the five
note intervals are defined by the ratio 2:3 (a true
harmonic fifth). The remaining interval between the
notes F and B can be found by analysis to be one
semitone smaller than a harmonic fifth. Fig 2.04
also shows us that all but one four note interval is
defined by the ratio 3:4 (a true harmonic fourth).
The exception to the rule is that found between B
and F which is found to be a semitone larger than
the harmonic fourth.
These two exceptions when considered together
should, like the harmonic fourth and fifth, define the
ratio of an octave (1:2). If we analyse this we find:
729
1024
*
*
512
729
=
=
1
2
Chapter 2: A Condensed History
D
1:1
8:9
27 : 32
3:4
2:3
16 : 27
9 : 16
1:2
D
C
B
A
G
F
E
D
C
9 : 16
1:1
243 : 256
27 : 32
3:4
2:3
81 : 128
9 : 16
B
16 : 27
128 : 243
1:1
8:9
64 : 81
512 : 729
2:3
16 : 27
A
2:3
16 : 27
9 : 16
1:1
8:9
64 : 81
3:4
2:3
G
3:4
2:3
81 : 128
9 : 16
1:1
8:9
27 : 32
3:4
F
27 : 32
3:4
729 : 1024
81 : 128
9 : 16
1:1
243 : 256
27 : 32
E
8:9
64 : 81
3:4
2:3
16 : 27
128 : 243
1:1
8:9
D
1:2
8:9
27 : 32
3:4
2:3
16 : 27
9 : 16
1:1
Fig 2.04: Pythagorean Heptatonic Scale intervals
When we consider the difference between the two
we would expect to find them identical - a fifth
minus a semitone should equal a fourth plus the
same semitone. What we find is somewhat different:
729 /
1024 /
512
729
= 729 x
= 1024 x
729
512
=
=
531441
524288
This ratio describes an interval fractionally over an
octave and demonstrates a fault in Pythagorean
mathematics. Known as the Pythagorean Comma, it
was the focus of much analysis over the next fifteen
hundred years as mathematically inclined theorists
attempted to solve the problem it defined.
Intervals between two consecutive notes all measure
either a tone (8:9) and a semitone (243:256). Three
note intervals also have two possible ratios, of
which the larger is two tones (64:81) and is referred
to as a Ditone in Greek theory. The smaller of the
two three note intervals in a tone and a semitone
(27:32). The four note interval found between the B
and the F (512:729) is three tones and as such is
known as a Tritone, as is the interval between the
notes F and B (729:1024). Because there are two
possible tritones, the interval was avoided.
Ratio
(2 / 3)
(2 / 3)
(3 / 2)
(2 / 3)
(3 / 2)
(1 / 2)
^
^
^
^
^
^
0
2
1
1
2
1
Calculated
Ratio
1
9
4
3
16
2
/
/
/
/
/
/
1
8
3
2
9
1
Decimal
1.0000
1.1250
1.3333
1.5000
1.7778
2.0000
Ratio
to Next
9/8
32 / 27
9/8
32 / 27
9/8
9/8
Roman
Letter
Fig 2.05: Pythagorean Pentatonic Scale values in order
Metal In Theory
D
E
G
A
C
D
The six note intervals in the table can be found in
two forms - four tones (81:128) or four tones plus a
semitone (16:27). Finally, the seven note intervals
also appear in two forms - five tones (9:16) or five
tones plus a semitone (128:243).
If the last step of both the ascending and descending
series is ignored, the scale generated contains five
notes. This is a Pentatonic scale and can be found in
many styles of music, not least far eastern (Chinese,
Japanese et al.) and European folk music. The
Pentatonic version of the scale, shown in Fig 2.05,
removes the two least correct members of the tonal
series. This version of the table also inverts interval
ratios to list the note letters in a modern context.
The use of a Pentatonic scale removes the Tritone
and hence removes the inaccuracy within the system
- all fourths are now “perfect” fourths and all fifths
are similarly “perfect”.
Our understanding of Ancient Greek music itself is
rather limited by the fact that no notational system
existed with which to record the note sequences.
However, a number of Ancient Greek scales have
become part of the lexicon of modern music, mostly
thanks to the work of Kathleen Schlesinger in her
1939 book “The Greek Aulos” (an aulos is a type of
flute). From archaeological evidence she lists a set
of scales which form the basis of Ancient Greek
music. More recent research has since brought the
accuracy of her work into question, but the scales in
her book still persist. Whatever the truth of
Schlesinger’s analysis, her work helps to cement the
link between Ancient music theory and that of the
modern era. As with much of music theory, her
work will probably be augmented by the discoveries
of future generations rather than being lost.
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