| Systematics
of scales adapted to the
equal-tempered tone system.
We imagine a line representing all
frequencies. In music we are interested
in what we can hear at most i.e. frequencies
from about 20 Hz up to 15-20 kHz. Then
every point between these frequency
limits make what we call a tone.
The ratio between two tones is known
as an interval. In
the equal-tempered tone system we
use the interval iHS=1,059463,
the frequency ratio between the two
tones in a half tone step. Whichever
frequency f we may
choose we obtain the frequency for the
next ascending half tone by multiplying
with iHS and the next
descending half tone by dividing with
iHS.
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In the scale systematics we are not
interested in the tone frequencies but
concentrate on the intervals
between the tones. We say that a half
tone step has the size of 1 HS. Now
we measure all other musical intervals
in HS (Half tone Steps). An octave comprises
12 HS and a perfect fifth as we know
is 7 HS.
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We visualise this by a ruler, the HS-ruler.
Simultaneously all intervals up to the
octave are given a logical letter
code. The smallest interval,
the half tone step or the minor second,
is called B as we use
the b accidental when we lower (flat)
a note in written music. Next interval,
the most common in all scales, the whole
tone step or major second we name A.
A minor third is called C
and a major third we call D
and so on.
A scale is a repeated one-dimensional
12 HS pattern of intervals on the frequency
line. Drawing intervals as circles shows
us a two-dimensional representation
of this one-dimensional pattern:
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Now let us look at this 12 HS pattern
expressed in HS-codes:
.....CACBABCACBABCACBABCACBABCACBABCACBABCAC..
the dots and incomplete circles indicate
that, theoretically, the pattern has
no beginning and no ending. To describe
it we only need a 12 HS part which we
can do in six different ways:
BABCAC, ABCACB, BCACBA, CACBAB, ACBABC
or CBABCA.
Remember from the HS-ruler that B=1,
A=2 and C=3 HS. To obtain the six different
possibilities we can either do a parallel
transition along the frequency line
or simply rotate any of the portions.
The rotation concept is made a little
more clear by this picture where the
interval row BABCAC is rotated to the
left (clockwise) by moving the element
B from the left to the right to make
the interval row ABCACB.
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Which one of the six possibilities
shall we choose? Since we use letters
and since we all know the alphabet in
a certain order we simply put them in
alphabetic order and
choose the one that comes first:
ABCACB, ACBABC, BABCAC,
BCACBA, CACBAB and CBABCA
Thus ABCACB quite logically becomes
the name of this pattern. But this description
also comprise a "hidden" description
of all the five other combinations,
so we say that the best literal
order is a scale texture
and all the possibilities are scales
in this texture. We apply the rotation
method and use an index for the rotational
order:
ABCACB(1)=ABCACB, ABCACB(2)=BCACBA,
ABCACB(3)=CACBAB, ABCACB(4)=ACBABC,
ABCACB(5)=CBABCA and ABCACB(6)=BABCAC
This index we name the position
of the texture.
Every combination of consecutive intervals
in a texture we call a motive
or an interval row.
The smallest possible motive we need
to describe a texture we call a period.
In the examples above the texture and
the period equal in size, but there
are cases where the period is shorter.
In such cases we write the period code
in parenthesis followed by an index.
An example is the pattern ...ABABABABABABABABAB...
where we only need to write (AB)4
to describe the texture and (AB)4(1)
respectively (AB)4(2) to describe
the two possible scales in this texture.
The period is interesting from two reasons.
First it tells us how many
scales there are in a texture and secondly
it immediatelly tells us how many times
a scale can be transposed,
namely as many times as the period size
in HS. Thus the scales (AB)4 can be
transposed a total of three times.
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Animated rotation
example.
This animation shows interval rotation
to the left of the common diatonic major
texture A3BA2B from
the scale A3BA2B(1) Lydian
to the scale A3BA2B(2)
Mixolydian and vice versa.
The two vertical lines represent the
octave notes. (Your web browser
must be able to show animation pictures)
Summing up on
the road:
A scale texture or
texture is a one-dimensional pattern
of intervals that contains information
on a number of scales.
A scale is a texture
in a certain position.
An interval row (motive)
is any number of consecutive intervals
on the frequency line.
The period is the smallest
required part needed to describe a texture.
Numbers of scales in a texture
= number of intervals in the period.
Number of possible transpositions=number
of HS in the period.
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How many textures
and scales are there?
The scales.
We remind of the condition precedent
that every octave compass 12 equidimensional
half-tone steps. The number of possible
scales equals the number of possibilities
to pass from one tone to the next octave
tone up or down. We have eleven possible
tone positions to choose or skip. Thus
the total number of scales is 211=2048.
Many authors propose a limit here between
what should be considered as a scale
or a chord let alone the simple octave
jump where all tone positions are skipped.
The textures.
The number of possible textures was
generated in a Pascal computing program
using recursion. The total number of
textures was 351 comprising 2048 scales.
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Back to music...
Finally we show the first three scales
from the example texture ABCACB as written
music (without rhythmic element), interval
codes and interval circles. You are
encouraged to work trough all other
scales in this texture.
ABCACB(1) This is
the first position of the texture. There
is no connection between music and the
property of being in the first position,
but the systematics allow us to always
recognise this scale wherever it may
appear. The starting motive AB is a
typical minor beginning and the end
is a common leading note up to the tonic.
ABCACB(2).
ABCACB(3).
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The reality
Music and scales need not to be created
in the equal-tempered tone system where
all intervals at least theoretically
should be adapted so you can play them
on a piano. On the contrary, in practice
most intervals are quite different from
the ideal values in the HS-ruler. We
say that the ideal intervals B=1HS,
A=2HS, C=3HS etc. make an outer
interval size while the real
intervals have an inner interval
size that can vary quite a
lot. We experience for instance a whole
tone step whether it is 1,7 HS or 2,4
HS. Many scales have an inner interval
size that must be modified to fit into
the tempered system. This is applicable
to all western major and minor scales
and especially all scales that we call
exotic, blues, jazz etc.
Now we will describe two instances,
a Javanese "exotic" pelog
scale where the inner intervals have
been carefully measured and a constructed
scale where also the outer interval
size has been modified.
Laras Ati,
a Javanese pelog scale
This is a scale described by Kunst
having five intervals of the sizes 2,55
2,12 2,60 2,26 and 2,47
HS. These intervals differ quite a lot
from the tempered values A=2HS (whole-tone
step) and C=3HS (minor third) but most
people listening to this scale use to
experience the outer intervals C, A,
C, A and A i.e. the scale A2CAC(3) =
CACAA.
This pentatonic scale (in the top of
the picture) is described as a Japanese
Banshiki-Cho. The next
scale (drawn with dots) is the Laras
Ati pelog scale. As you can
see there are big differences from the
tempered tone positions. Most likely
also the Banshiki-Cho
should have other inner intervals. The
differences are still more evident in
this picture where the scales have been
put on top of each other:
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The
synthetic 7-A scale,
an aural experiment.
If we divide an octave in seven equal
parts we get seven intervals of the
size 12/7 = 1,714.. HS. The interval
ratio i becomes
i = 10(1/7) log 2 =
1,1040895...
Everybody that listens to this interval
experiences a whole tone step. But what
happens if we use this interval as an
imaginary outer interval A=1,7.. and
play the entire scale from one tone
to the same tone in the next octave?
The picture shows
this construction of seven equal-sized
intervals compared to the common major
scale A3BA2B(5) = AABAAAB. This scale
can not be played on any normal musical
instrument so it was synthesised and
played through a computer. The listeners
hear a septatonic scale starting with
a few whole tone-steps. Most people
also experience that the scale ends
in at least one major second. Many people
also seem to experience the rather correct
tone position of the fifth. The most
common interpretations then are AABABAA
and AAABBAA. The former
is known as
<<Back
the Mixolydian
b6 or Hindu
scale:
AAABBAA
is mentioned on the WWW as Lydian minor
or Melarisabhapriya by anonymous sources:
Trivial
names
Trivial names are non-systematic names
of various composition. They can be
quite useful sometimes because they
are easy to use (pure minor, major pentatonic)
but also because they may say something
about the construction of the scale
(see the example above Mixolydian
b6). Systematic names however
immediately inform us of which intervals
(and thereby tones) referred to
and give an unmistakeable description
derived from the scale itself.
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Calculating
HS
We are looking for a
number i that
multiplied by itself twelve times
becomes 2 and get
i12 = 2
This gives us i
= the twelfth root of two which
after logaritmation is
log i = (1/12) log 2
i = 10(1/12) log 2
= 1,059463..
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