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Previous page:

Development of Chords from Scale Tones in Thirds

Next page:
Previous page: Right and Left Hand Technique Next page: lw9 - Right hand Left hand technique - cont.

Lessons of The Week was a series of guitar lessons circulated in "News", in the pre-web days of the Internet. 29 lessons were written before it died out, and I happende to write the first three. They represent a little bit of internet history, as they may have been the first guitar lessons written for the internet.

The lessons were all written in txt format - they were written around the same time as Tim Berners Lee were sitting in Switzerland specifing the first version of html. I have converted them to html, and may have added a few links from the lessons.

Lesson: 8

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:

  • 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.
  • 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.
  • Augmented: a raised perfect or major interval.
  • 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".

  • 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]
  • 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:

  • M: major interval, scale, or chord
  • m: minor interval, scale, or chord
  • b: the "flat" symbol, i.e., the specified note is lowered by one semitone. Example: Bb is a semitone lower than B.
  • #: the "sharp" symbol, i.e., the specified note is raised by one semitone. Example: G# is a semitone higher than G.
  • 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).
  • 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).
  • 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:

  • 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.)
  • 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.
  • 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.
  • 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

Section 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, indicate 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 position 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 ex. 1)
  • o
  • the number of semitones between the root and fifth (see ex. 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.


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