WHAT IS A SCALE ??

Hi

I originally asked this question to David via an email a few days
ago. I guess I should have posted it on the tuning group but I was
concerned that maybe the question was alittle too basic. David was
very helpful in helping me with an answer as well posting this
question to everyone in the group.

I found a link on thge net that addresses this question and I thought
I might share it with everyone. Here is the link
http://dspace.dial.pipex.com/andymilne/Scales.shtml#subdivision

Please look on the left hand side of the above mentioned link. There
you will find a table of contents. Look for "Page Contents" and
beneath it you will see a topic called "What is a Scale". I pasted
the topic into this post in the event the link does not work for some
tuning members. The link starts out with a topic called, "subdivision
of the Octave" and then continues later with , "What is a Scale".
Please let me know what you think after reading this.

Thank You
Walter Here it is.......... Please scroll down to the

Subdivision of the octave

The mind perceives pitch to be continuously variable - there are
no "quanta" of pitch - but in music, out of the infinite possible
pitches that could be chosen from the pitch continuum, only a limited
number are used.

Usually each octave is subdivided into a small number of steps and
each of these is repeated in every octave. In almost all musical
cultures notes separated by an octave are regarded as somehow
equivalent, so that, for the sake of consistency and simplicity,
divisions in any one octave are repeated in all others. Scales in
which different notes are used in different octaves, or where octaves
are not found at all, are rare.

By choosing a limited number of notes the ear is given a structure
that is simple enough to be understandable and whose notes are spaced
apart enough to be easily heard as different. Ideally within any
octave, each note is perceived to be fundamentally different from
every other note - each note has a unique identity. When that
identity is unique enough it allows for each note's pitch to be
varied with vibrato and other decorative techniques without losing
its identity and becoming confused with other notes.

The pitches used in purely melodic musics - such as classical Arabic
and Indian are generally more flexible and complex than those used in
the tonal harmonic music of common practice classical and popular
music. Within any one scale we will often find more than seven notes
and the distance between consecutive steps can be very small.

But we also find in melodic music the frequent use of both the
pentatonic scale - Celtic and Asian songs, and the seven note
diatonic scale - Native American and African songs. These common
scales often sound very different to how they do in tonal harmonic
music, through generic forms of decoration and pitch variation.

On this site I will be examining in detail only those scales which
are suitable for use in tonal harmonic music.

What is a scale ?

What is it that differentiates a "scale" from simply just
a "collection of notes" ?

A scale should constitute a unified collection of notes - a selection
which is in some sense complete and to which any addition is heard to
be extraneous.

The scale must also fulfil the functions demanded of it. There are
three principal functions that a scale may be asked to fulfil:

1) to serve as a melodic resource
2) to serve as a harmonic resource
3) to be tonally effective

It may fulfil any or all of these three functions - depending on its
intended use.

The scale as a melodic resource

For a scale to be successful as a melodic resource it should be
reasonably smooth and even; without sudden gaps which sound as if a
note has been omitted, or sudden concentrations of notes which sound
as if an extraneous note has been added.

One of the most important measures of the completeness of a scale is
whether or not it can be classed as as a proper mode or not.

A mode (scale) is considered to be proper when all intervals of an
interval class are not smaller than those of lower interval classes.
This means, for example, that if we start on any note in the scale
and move up four notes the interval traversed should be larger (or
the same size as) any other interval made up from traversing three
notes.

The propriety of a scale is a significant factor of scales which are
recognised to be melodically smooth.

Another important measure is consistency of the size of intervals for
each pitch class in the scale. The diatonic scale, for instance, has
just two types of second - a major and a minor second, while the
harmonic minor scale has three types of second - major, minor and
augmented. This makes the diatonic scale melodically smoother than
the harmonic minor.

The scale as a harmonic resource

In any harmonic music which uses major and minor triads, a suitable
scale must be a resource not just for melody (notes in isolation) but
for major and minor triads.

If we take major and minor triads to be the fundamental building
blocks of our harmonic system then this means that if any note is not
part of any major or minor triad then it is serving no harmonic
purpose. It is therefore extraneous to the harmonic function of the
scale, and so cannot be considered to be a unified member of that
scale.

An example of such a "scale" is: c, d, e, f, g, g, a. Here the g is
part of no triad, and so cannot be considered be a unified member of
the scale.

The other requirement for a harmonic scale is that it should not
contain any notes that allow for both a major and a minor triad to be
built on the same root. This is because in any such scale one of
these two possible thirds will always be heard as superfluous
addition.

For example, in the "scale" c, d, e, f, g, a, a, b either the a or
the a is entirely superfluous to the harmonic requirements of the
scale.

This requirement forbids the use of any chromatic semitones in a
fully unified scale.

Remarkably enough, out of all possible scales there are only five
prime scales, in which every single note is a member of at least one
major or minor triad and which contain no chromatic semitones. All of
these scales contain seven notes.

The five prime scales
The prime scales are those scales in which every note is a member of
at least one major or minor triad, but which contain no chromatic
semitones.

Each of the prime scales is best considered as a set of seven
different scales or modes. Each mode of the prime scales contains the
same notes but has a different "home" note or tonic.

Both the major scale and the natural minor scale are drawn from the
same prime scale, and that prime scale is the diatonic scale. The
difference between the major scale and the natural minor is
their "home" or tonic note. If we take the notes c, d, e, f, g, a, b
and treat c as the home note then we are using the scale of c major.
If, however we take a as our home note then we are using the scale of
a natural minor.

Indeed we can construct seven different scales from the diatonic
scale by choosing each note as the home note. These seven scales are
known as the seven diatonic modes. If we use the c major scale above,
then the modes of it are as follows:

Tonic note Name of mode
f f Lydian
c c Ionian (or major)
g g Mixolydian
d d Dorian
a a Aeolian (or natural minor)
e e Phrygian
b b Locrian

As I've stated above there are five prime scales. Many of them do not
have conventional names, so I have had to use the following
descriptive terms.

The five prime scales are:

the diatonic
the harmonic minor
the harmonic major
the melodic
the double harmonic

Tonal harmonic scales
There is one final requirement for a scale that is to be used as a
resource for tonal harmonic music. Not only must it be a suitable
resource for melody and triads but it must also be able to support a
tonic triad. That is, it must have a chord which serves as a chord of
rest and completion, as the tonal centre against which all the other
triads are measured and towards which all gravitate.

Within each of the prime scales only one or, at most, two triads are
actually capable of functioning as tonics.

So although any of the modes of the prime scales are suitable in a
melodic music, in a tonal-harmonic music only one or, at most, two
modes of each of the prime scales are suitable.

The diatonic scale, for instance, has only two triads which are
perceived to be totally at rest, resolved and final. In the scale c,
d, e, f, g, a, b these triads are C major and a minor.

This means that there are only two tonally effective scales to be
taken from the diatonic prime - the major scale and the natural minor
(or aeolian) scale.

In total there are eight tonal harmonic scales. I will examine in
detail each of these scales under the prime scales from which they
are derived:

The diatonic scales
The major scale
The aeolian mode
The harmonic minor scale
The harmonic major scale
The melodic scales
The (ascending) melodic minor scale
The (descending) melodic major scale
The double harmonic scales
The double harmonic major scale
The double harmonic minor scale

For an overview of effective cadential progressions in each of these
tonal scales go to The Cadence Page.

END END END

Regards
Walter

Hey Walter!

>
http://dspace.dial.pipex.com/andymilne/Scales.shtml#sub
division

I'd be careful about accepting as the ultimate truth a
few of the claims the author makes there. I am
concerned that a composer could throw out a lot of
valuable resources a priori if he blindly accepted some
of the things said in that article.

> Subdivision of the octave

> Usually each octave is subdivided into a small number of
> steps and each of these is repeated in every octave. In
> almost all musical cultures notes separated by an octave are
> regarded as somehow equivalent, so that, for the sake of
> consistency and simplicity, divisions in any one octave are
> repeated in all others.

Ug not this octave stuff again. If octave is 1200.000
cents, then his claim are certainly false.

> Scales in which different notes are used in different
> octaves, or where octaves are not found at all, are rare.

So happy that he at least acknowledges these exist. Not
sure where he gets his idea about what 'rare' means. I
would say 'common' or 'ubiquitous'.

> By choosing a limited number of notes the ear is given a
> structure that is simple enough to be understandable and
> whose notes are spaced apart enough to be easily heard as
> different.

I agree this is a very important issue to consider in
scale design. Not a concrete rule, depending on your
purposes, but something to keep in mind.

> The pitches used in purely melodic musics - such as classical
> Arabic and Indian are generally more flexible and complex
> than those used in the tonal harmonic music of common
> practice classical and popular music. Within any one scale we
> will often find more than seven notes and the distance
> between consecutive steps can be very small.

Right -- here is an example where his statement "whose
notes are spaced apart enough to be easily heard as
different" might find exception.

> On this site I will be examining in detail only those scales
> which are suitable for use in tonal harmonic music.
--- -------- --------------------

YIKES!!!

(as if)

OK, that is an extremely biased statement he just made
and I want to make sure you don't just gloss over it
and say 'why sure!' Also keep in mind these rules he is
defining in his own abstract system and don't think
he's instead outlining some sort of general musical
truth.

> A scale should constitute a unified collection of notes
> - a selection which is in some sense complete and to which
> any addition is heard to be extraneous.

Hm. And why is this, what does it mean and how is it
defined?

> The scale must also fulfil the functions demanded of it.
----
> There are three principal functions that a scale may be asked
> to fulfil:
>
> 1) to serve as a melodic resource
> 2) to serve as a harmonic resource
> 3) to be tonally effective

1 - Agreed unless scale is being used only for harmony
2 - Quick disproof: nonincidental harmony did not
appear on the scene until after many common scales were
already in use. Yet some of these scales ended up being
fine for harmony anyway.
3 - huh?

> It may fulfil any or all of these three functions - depending
> on its intended use.

Uh, OK. Intended usage is a relevant factor.

> The scale as a melodic resource

> For a scale to be successful as a melodic resource it should
> be reasonably smooth and even; without sudden gaps which
> sound as if a note has been omitted, or sudden concentrations
> of notes which sound as if an extraneous note has been added.

Well, maybe. Maybe not.

Oh well I'll stop now. The rest of the stuff could be
useful depending on what you want to do, but you should
realize that counterexamples can be and have been made
to disprove many of these claims that involve 'should'
and 'must'.

- Jeff