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Title: Development of Chords from Scale Tones in Thirds
Level: Beginner
Style: Chord theory
Instructor: Dennis O'Neill
10 February 1993
Introduction.
Many chords can be developed by extracting alternate scale tones,
i.e., using tones that are major thirds or minor thirds apart within a
scale. In this series of exercises, you will begin by building
three-note chords from the major and three minor scales relative to C
major and progress to building seven-note (thirteenth) chords. You
will learn to determine what (relatively) simple chords may be
substituted for more complex chords and what extensions may be added
to chords while remaining harmonically correct. Most importantly, you
will be able to figure out what notes to leave out when playing a
chord.
All the examples will be written in the keys of C Major and A Minor.
Students are strongly encouraged to examine results in other keys.
It's important when learning the material that you work through the
exercises yourself without first looking at the completed exercises.
I developed this material as a way of learning it myself. I don't
intend it to be a list of prescriptions; merely as a way to take a
simple concept as far as I can for the background of interested
players.
This set of lessons is divided into several parts. Each part except
the first builds upon material developed in the previous lesson. My
plans for the set include the following:
==> Part 1. Preliminaries and an introduction to chord construction
Part 2. 4-, 5-, 6-, and 7-note chords; naming chords
Part 3. What to leave out while retaining chord identity
Do the exercises!
Development of Chords from Scale Tones in Thirds
Part 1. Preliminaries and an Introduction to Chord Construction
Section 1.1. Definitions and notation conventions.
First, let's define some terms.
An "interval" is the distance between two tones. There are five
qualities of intervals; their names are perfect, major, minor,
diminished, and augmented. These qualities of intervals are defined
as follows:
o Perfect interval: an interval which, when inverted, becomes
another perfect interval (a self-referential definition if ever
I heard one). E.g., C-F is a perfect 4th, F-C is a perfect 5th;
C1-F2 is a perfect 11th (where the 1 and 2 mean that the C and F
are in different octaves), C2-F2 is a perfect 4th, F2-C3 is a
perfect 5th; and so on.
o Major: an interval other than a perfect interval that appears in
a major scale.
o Minor: an interval that does not appear in a major scale.
o Augmented: a raised perfect or major interval.
o Diminished: a lowered perfect or minor interval.
In defining major and minor scales, the intervals between adjacent
notes in the scale are sometimes called "half step" and "whole step",
or, equivalently, "semitone" and "whole tone".
o Semitone: the interval between the notes of two adjacent keys on
the piano, or two adjacent frets on the guitar. Also called a
"minor 2nd" or "half step". Example: C-Db. [b is used to denote
flat]
o Whole tone: the interval between a key and the key next to the
adjacent key on the piano [two keys away], or at two frets' apart
on the guitar. Also called a "major 2nd" or "whole step".
Example: C-D.
I will use the following conventions in my notation:
o M: major interval, scale, or chord
o m: minor interval, scale, or chord
o b: the "flat" symbol, i.e., the specified note is lowered by one
semitone. Example: Bb is a semitone lower than B.
o #: the "sharp" symbol, i.e., the specified note is raised by one
semitone. Example: G# is a semitone higher than G.
o nat: used to indicate that a note is neither sharped nor flatted
(usual music notation uses a sort of L7 symbol that I can't
reproduce at the computer keyboard).
o upper case Roman numeral: a major-, dominant-, or augmented-
family chord. The number refers to the degree of the scale on
which a chord is built. Example: I indicates the major chord
built on the first degree of a scale (e.g., C in the key of C).
o lower case Roman numeral: a minor-, half-diminished-, or
diminished-family chord. The number refers to the degree of the
scale on which a chord is built. Example: vi indicates the
minor chord built on the sixth degree of a scale (e.g., Am in
the key of C).
Section 1.2. The Major and Minor Scales
1.2.1. The Major Scale.
The major scale is defined as an 8-tone scale comprising the set of
intervals (in terms of whole- and half-steps). The intervals are:
whole whole half whole whole whole half
The C Major scale is:
C D E F G A B C
1.2.2. The Natural Minor Scale.
The natural minor scale is defined as an 8-tone scale containing the
same notes as its relative major scale, but starting on the 6th scale
degree of its relative major scale; also known as the Aeolian mode.
The relative minor of C Major is A Minor, and its intervals are:
whole half whole whole half whole whole
The A natural minor scale is:
A B C D E F G A
1.2.3. The Harmonic Minor Scale.
Similar to the natural minor scale but with a raised 7th scale degree.
The component intervals are:
whole half whole whole half m3 half
The A harmonic minor scale is:
A B C D E F G# A
1.2.4. The Melodic Minor Scale.
Similar to the natural minor scale but with a raised 6th and a raised
7th when ascending; identical to the natural minor scale when played
descending. The component intervals are:
whole half whole whole whole whole half
The ascending A melodic minor scale is:
A B C D E F# G# A
Other definitions and conventions will be introduced as needed.
Section 1.3. Elementary Chord Construction From Tertiary Harmony.
One can develop a useful set of chords by stacking notes from the
scale. For the purposes of this set of lessons I will stack thirds.
I will start with, say, a C major scale; over that I will place the
same scale but starting with the 3rd scale degree (E); over that I
will place the same scale starting with the 5th scale degree (G). The
harmony deriving from stacking alternate scale tones is called
"tertiary harmony".
The harmonized scales in C and its relative minors are:
C major:
G A B C D E F G - fifth above root
E F G A B C D E - third above root
C D E F G A B C - root of chord
A natural minor:
E F G A B C D E - fifth above root
C D E F G A B C - third above root
A B C D E F G A - root of chord
A harmonic minor:
E F G# A B C D E - fifth above root
C D E F G# A B C - third above root
A B C D E F G# A - root of chord
A melodic minor:
E F# G# A B C D E - fifth above root
C D E F# G# A B C - third above root
A B C D E F# G# A - root of chord
If we examine the intervals contained in these stacks of notes, we'll
discover that there are only a few distinct sets of relationships.
Listed with the bottommost interval first, these are:
o M3 m3 - defined as a "major" chord, e.g., C-E-G. It's called a
"major" chord because the chord built upon the tonic of the
major scale is of this type. (Warning - another kind of chord
containing the intervals M3 m3 on the bottom is called a
"dominant" chord. Dominant chords are not distinguishable from
major chords in three-note chords, but are distinguishable in
chords having four or more notes. See part 2 for more
information.)
o m3 M3 - defined as a "minor" chord, e.g., A-C-E. It's called a
"minor" chord because the chord built upon the tonic of the
minor scale is of this type.
o m3 m3 - ambiguous, either diminished or half-diminished, e.g.,
B-D-F. This chord will divide in unambiguous ways starting with
4-note chords in Part 2.
o M3 M3 - defined as an "augmented" chord, e.g., C-E-G#.
These interval patterns, along with one or two others, will serve as
the basis for a chord classification system to be introduced in Part
2.
Do the exercises!
Exercise 1. Table of Intervals.
Create a table of intervals for all note pairs between unison and
two octaves. Format the table so that one column reflects the
number of semitones between the note pair and another column
shows the name of the interval. You may include any other
information that you find useful, such as the sequence of major
and minor thirds that make up a particular interval, or examples
of the interval. Information developed in this table will be
used later to assist in the naming of chords.
As an example, here are the first few lines from such a table.
_____________________________________________________
Example for Exercise 1. Intervals
_____________________________________________________
Semitones Interval Thirds Example
0 d2 unison
1 m2 1 semitone C-C#, C-Db
2 M2 2 semitones C-D
3 m3 m3 C-D#, C-Eb
4 M3, d4 M3 C-E
5 P4 M3 + 1 semitone C-F
...
_____________________________________________________
Exercise 2. Three-note chords in C major and A minor.
Create a table of three-note chords based on alternate
notes taken from the various major and minor scale
types for the key of C major/A minor. Format the table
so that the root of the chord is on the bottom, the
third is in the middle, and the fifth is on the top,
leaving spaces between each line and between each
column. In the intermediate lines and columns, indi-
cate whether the accompanying interval is a major third
or a minor third. Below the intervals column, place
the generic symbol for the type of chord based on the
interval. These chords comprise the family of major
and minor chords.
Notice that the intervals contained in each chord are
unique to the position of the chord within the scale,
and that the same chord type appears at the same posi-
tion within each key.
As an example, here is the table for C major.
_____________________________________________________________________________
Example for Exercise 2. Triad chord stacks, key of C major
_____________________________________________________________________________
1 2 3 4 5 6 7 Degree
Chord tone
_____________________________________________________________________________
G A B C D E F fifth
m3 M3 M3 m3 m3 M3 m3
E F G A B C D third
M3 m3 m3 M3 M3 m3 m3
C D E F G A B root
_____________________________________________________________________________
I ii iii IV V vi vii symbol
_____________________________________________________________________________
Exercise 3. Three-note chords in all keys.
Create a key-independent abstract of the information
developed in exercise 2 (three-note chords). The
columns of the table should contain:
o the chord type and name
o the interval between the root and third,
expressed as a major or minor third
o the interval between the third and fifth,
expressed as a major or minor third
o the number of semitones between the root and
third (see exercise 1)
o the number of semitones between the root and
fifth (see exercise 1)
o the scale degrees on which this type of chord
occurs for each scale type (see exercise 2)
Here is an example of such a table.
____________________________________________________________________________
Example for Exercise 3. Naturally-occurring triads, grouped by chord type
____________________________________________________________________________
Chord type Intervals Semitones Scale source
and chord name 3rd 5th M nm hm mm
____________________________________________________________________________
Major
major M3 m3 4 7 1 4 5 3 6 7 5 6 4 5
Minor
minor m3 M3 3 7 2 3 6 1 4 5 1 4 1 2
Diminished or
half-diminished
(ambiguous)
m3 m3 3 6 7 2 2 7 6 7
Augmented
aug M3 M3 4 8 3 3
____________________________________________________________________________
Interval Meaning
M3 major 3rd, 4 semitones
m3 minor 3rd, 3 semitones
Semitones Meaning
(#) number of semitones of chord tone above chord root
Scale source Meaning
M major scale
nm natural minor scale
mm melodic minor scale
hm harmonic minor scale
Note: if one desires a more conventional notation in
the "Semitones" column, replace the numbers by the
corresponding interval names from Table 1.
******************************************
******************************************
******************************************
*** ***
*** STOP! ***
*** ***
*** Answers to exercises appear below. ***
*** Do the exercises before peeking. ***
*** ***
******************************************
******************************************
******************************************
____________________________________________________________
Solution to Exercise 1. Intervals
____________________________________________________________
Semitones Interval Thirds Example
0 d2 unison
1 m2 1 semitone C-C#, C-Db
2 M2 2 semitones C-D
3 m3 m3 C-D#, C-Eb
4 M3, d4 M3 C-E
5 P4 M3 + 1 semitone C-F
6 a4, d5 m3 m3 C-F#, C-Gb
7 P5 M3 m3 C-G
8 a5, m6 M3 M3 C-G#, C-Ab
9 M6, d7 m3 m3 m3 C-A, C-Bbb
10 m7 M3 m3 m3 C-A#, C-Bb
11 M7 M3 M3 m3 C-B
12 P8 M3 M3 M3 C1-C2
13 a8, m9 M3 m3 m3 m3 C1-Db2
14 M9 M3 M3 m3 m3 C1-D2
15 m10 M3 M3 M3 m3 C1-D#2, C1-Eb2
16 M10, d11 M3 M3 M3 M3 C1-E2
17 P11 M3 M3 m3 m3 m3 C1-F2
18 a11, d12 M3 M3 M3 m3 m3 C1-A#2, C1-Gb2
19 P12 M3 M3 M3 M3 m3 C1-G2
20 a12, m13 M3 M3 M3 M3 M3 C1-G#2, C1-Ab2
21 M13 M3 M3 M3 m3 m3 m3 C1-A2
22 m14 M3 M3 M3 M3 m3 m3 C1-Bb2
23 M14 M3 M3 M3 M3 M3 m3 C1-B2
24 p15 M3 M3 M3 M3 M3 M3 C1-C2
____________________________________________________________
Notes:
1. In the "Thirds" column, note that M3 M3 M3 = m3 m3 m3
m3; therefore one can substitute four minor thirds for
three major thirds in any interval with no change in
the total size of the interval.
2. In the "Example" column, if note names have numbers
appended, the numbers refer to the relative octave in
which the notes appear.
Definitions:
Perfect interval: an interval which, when inverted, becomes
another perfect interval (a self-referential definition
if ever I heard one). E.g., C-F is a perfect 4th, F-C
is a perfect 5th; C1-F2 is a perfect 11th, C2-F2 is a
perfect 4th, F2-C3 is a perfect 5th; and so on.
Augmented: a raised perfect or major interval.
Diminished: a lowered perfect or minor interval.
Major: an interval other than a perfect interval that
appears in a major scale.
Minor: an interval that does not appear in a major scale.
Solutions to Exercise 2. Triad Chord Stacks
___________________________________________________________________________
Exercise 2a. Triad chord stacks, key of C major
___________________________________________________________________________
1 2 3 4 5 6 7 Degree
Chord tone
___________________________________________________________________________
G A B C D E F fifth
m3 M3 M3 m3 m3 M3 m3
E F G A B C D third
M3 m3 m3 M3 M3 m3 m3
C D E F G A B root
___________________________________________________________________________
I ii iii IV V vi vii symbol
___________________________________________________________________________
___________________________________________________________________________
Exercise 2b. Triad chord stacks, key of A natural minor
___________________________________________________________________________
1 2 3 4 5 6 7 Degree
Chord tone
___________________________________________________________________________
E F G A B C D fifth
M3 m3 m3 M3 M3 m3 m3
C D E F G A B third
m3 m3 M3 m3 m3 M3 M3
A B C D E F G root
___________________________________________________________________________
i ii III iv v VI VII symbol
___________________________________________________________________________
___________________________________________________________________________
Exercise 2c. Triad chord stacks, key of A harmonic minor
___________________________________________________________________________
1 2 3 4 5 6 7 Degree
Chord tone
___________________________________________________________________________
E F G# A B C D fifth
M3 m3 M3 M3 m3 m3 m3
C D E F G# A B third
m3 m3 M3 m3 M3 M3 m3
A B C D E F G# root
___________________________________________________________________________
i ii III iv v VI VII symbol
___________________________________________________________________________
___________________________________________________________________________
Exercise 2d. Triad chord stacks, key of A melodic minor
___________________________________________________________________________
1 2 3 4 5 6 7 Degree
Chord tone
___________________________________________________________________________
E F# G# A B C D fifth
M3 M3 M3 m3 m3 m3 m3
C D E F# G# A B third
m3 m3 M3 M3 M3 m3 m3
A B C D E F# G# root
___________________________________________________________________________
i ii III iv v VI VII symbol
___________________________________________________________________________
_______________________________________________________________________________
Solution to Exercise 3. Naturally-occurring triads, grouped by chord type
_______________________________________________________________________________
Chord type Intervals Semitones Scale source
and chord name 3 5 M nm hm mm
_______________________________________________________________________________
Major
major M3 m3 4 7 1 4 5 3 6 7 5 6 4 5
Minor
minor m3 M3 3 7 2 3 6 1 4 5 1 4 1 2
Diminished or
half-diminished
(ambiguous)
m3 m3 3 6 7 2 2 7 6 7
Augmented
aug M3 M3 4 8 3 3
_______________________________________________________________________________
Column key for this table
Interval Meaning
M3 major 3rd, 4 semitones
m3 minor 3rd, 3 semitones
Semitones Meaning
(#) number of semitones of chord tone above chord root
Scale source Meaning
M major scale
nm natural minor scale
mm melodic minor scale
hm harmonic minor scale
Note: if one desires a more conventional notation in the
"Semitones" column, replace the numbers by the corresponding
interval names from Table 1.
==============================================================================
FUTURE LESSONS
--------------
No Name Style Level Instructor
9 Right hand Left hand technique Technique B Tim Fullerton
10 How Chords work Theory B Tim Fullerton
11 Right and Left hand techniques theory (etc.) b Tim Fullerton
12 Modes Theory I Dave Good
13 Octaves Theory B Bill Quinn
==============================================================================
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