Scale of G
Scale of D
Scale of A
Scale of E
Scale of B
Scale of F#
Scale of C#
Scale of F
Scale of Bb
Scale of Eb
Scale of Ab
Scale of Db
Scale of Gb
Scale of Cb
THE MINOR SCALE.
There are two forms of minor scales, harmonic and melodic, both differing in construction from the major form.
The minor key having no sharps or flats in the signature is a. Starting at a and sounding the seven white keys in order to the right produces a form of scale with whole steps between 1 and 2, 3 and 4, 4 and 5, 6 and 7, 7 and 8, and half steps between 2 and 3 and between 5 and 6. This scale is unsatisfactory to the ear as its subtonic is not a leading tone. The effect of a leading tone should be urgent, restless, and demand its tonic in order to obtain a restful effect. This urgent effect can only be obtained by the subtonic being one half step below the tonic. This may be obtained by simply raising the seventh one semi-tone in the above scale formation and thus is produced the so-called harmonic minor scale.
The symbols for raising a note are the sharp (#), the double sharp (x), and the cancel (⊄) (also called natural) when placed before a note that has been previously affected by a flat. The symbols for lowering a note are the flat (b), the double flat (bb), and the cancel when placed before a note that has been previously affected by a sharp. By these statements it can be seen that the cancel (⊄) is both a lowering and a raising symbol. The cancel lowers a tone when it cancels a sharp and raises a tone when it cancels a flat.
The harmonic minor scale is formed by whole steps between 1 and 2,—3 and 4,—4 and 5,—half steps between 2 and 3,—5 and 6,—7 and 8, and an interval of one and one-half steps (called an augmented step) between 6 and 7. In demonstrating the minor keys, a curved line will be used to connect those figures representing tones one half step apart and a bracket to connect those figures representing tones an augmented step apart.
The key of a minor (harmonic form) is as follows:—
a b c d e f g# a
1 2 3∼4 5∼6∪7∼ 8
The student will notice that this scale has one sharp (g). Nevertheless, the a minor is the minor key which has neither sharps nor flats in its signature. The raised seventh of all minor keys is never present in the signature, but appears as accidental.
When a sharp, double sharp, flat, double flat or cancel, which is not present in the signature, is placed before a note, it is called an accidental. If the raised seventh were present in the signature, uniform signatures in the minor would be impossible. It may also be remarked here that the seventh is not always raised during the course of a composition and is necessarily raised only when the composer desires the listener's ear to come at rest on the tonic, in which case the tonic must be preceded by the raised seventh, if the subtonic precedes the tonic in the melody or harmony.
The same rules (pages 13 and 15) used in the major for finding the key having the next number of sharps and the key having the next number of flats are applicable in the minor. The order of the letters in both the sharp and flat signatures is the same in the minor as in the major.
A minor has no sharps, the fifth of a is e and has one sharp (f):—
e f# g a b c d# e
1 2∼3 4 5∼6∪7∼8
The fifth of e is b and has two sharps (f and c):—
b c# d e f# g a# b
1 2∼3 4 5∼6∪7∼8
The fifth of b is f# and has three sharps (f, c and g):—
f# g# a b c# d e# f#
1 2∼3 4 5∼6∪7∼8
The fifth of f# is c# and has four sharps (f, c, g and d):—
c# d# e f# g# a b# c#
1 2∼3 4 5∼6∪7∼8
The fifth of c# is g# and has five sharps (f, c, g, d and a):—
g# a# b c# d# e fx g#
1 2∼3 4 5∼6∪7∼8
The student will notice that in this key, f is double sharped. F is sharped in the signature, but because the subtonic requires raising, f demands a double sharp.
The fifth of g# is d# and has six sharps (f, c, g, d, a and e):—
d# e# f# g# a# b cx d#
1 2∼3 4 5∼6∪7∼8
The fifth of d# is a# and has seven sharps (f, c, g, d, a, e and b):—
a# b# c# d# e# f# gx a#
1 2∼ 3 4 5∼6∪7 ∼8
The minor keys having more than seven sharps should be found by the student and submitted to the teacher for correction. For the sake of brevity, they are not given here, but the student should be thoroughly capable, by this time, of finding them all.
A minor has no flats, the fourth of a is d and has one flat (b):—
d e f g a bb c# d
1 2∼3 4 5∼6∪7∼8
The fourth of d is g and has two flats (b and e):—
g a bb c d eb f# g
1 2∼3 4 5∼6∪7∼8
The fourth of g is c and has three flats (b, e and a):—
c d eb f g ab b⊄ c
1 2∼3 4 5∼6∪7∼8
The student will notice a contradiction in the above scale; it is stated that c has three flats and in the example, b is cancelled. This cancel, however, appears as accidental (the raised seventh) and must be a flat in the signature.
The fourth of c is f and has four flats (b, e, a and d):—
f g ab bb c db e⊄ f
1 2∼3 4 5∼6∪7∼8
The fourth of f is bb and has five flats (b, e, a, d and g):—
bb c db eb f gb a⊄ bb
1 2∼3 4 5∼ 6∪7∼8
The fourth of bb is eb and has six flats (b, e, a, d, g and c):—
eb f gb ab bb cb d⊄ eb
1 2∼3 4 5∼6∪7∼8
The fourth of eb is ab and has seven flats (b, e, a, d, g, c and f):—
ab bb cb db eb fb g⊄ ab
1 2∼ 3 4 5∼6∪7∼ 8
The student should find the minor keys having more than seven flats.
The harmonic minor scale is awkward in formation on account of the augmented second step between steps six and seven. All augmented intervals sound harsh and are difficult to sing tunefully. For this reason, another form of minor scale is sometimes used which eliminates the augmented second step. This form is called melodic minor and is used, as its name implies, only for melodic purposes. It defies harmonization for the obvious reason that its ascending form differs from its descending form.
The melodic minor scale has the sixth as well as the seventh raised by accidental in ascending, but in descending, both the sixth and seventh are restored. The ascending form has whole steps between 1 and 2,—3 and 4,—4 and 5,—5 and 6,—6 and 7, and half steps between 2 and 3 and between 7 and 8. The descending form has its half steps between 6 and 5 and between 3 and 2. Notice that the descending form is as its signature dictates.
raised raised
Ascending: —1 2∼3 4 5 6 7∼ 8
Descending:—8 7 6∼5 4 3∼2 1
The ascending form of the melodic minor is nearly the same as the major scale, and for this reason it is best not to retain the raised sixth and seventh in descending. The subtonic in a descending scale does not lead (progress) to the tonic and therefore need not necessarily be situated one half step below the tonic.
Any minor key is called the relative of the major key having the same signature; therefore, the relative minor of C major is a[C] as they both have neither sharps nor flats.
Rule 5. The Relative Minor is found on the Sixth of the Major Scale.
Rule 6. The Relative Major is found on the Third of the Minor Scale.
Some writers have called the relative minor parallel minor, using relative and parallel synonymously. This is a usage to be regretted as it causes considerable confusion. By most writers, the parallel minor is treated as the scale commencing on the same key-note as the major and will thus be treated in this book, therefore:—
the relative minor of C is a;
the parallel minor of C is c.
The parallel minor scale has three more flats or three less sharps in its signature than the major scale. In other words, by lowering steps 3, 6 and 7 of the major scale one semi-tone, the signature of the parallel minor is obtained.
The notation in the treble clef of all the minor scales (harmonic and melodic) follows:—
| Scale of a Harmonic | |
| Scale of a Melodic |
|
| Scale of e Harmonic |
|
| Scale of e Melodic |
|
| Scale of b Harmonic |
|
| Scale of b Melodic |
|
| Scale of f# Harmonic |
|
| Scale of f# Melodic |
EXERCISES
ORAL AND WRITTEN
1. Into how many parts does modern custom divide an octave?
2. What is each part called?
3. What is the difference between a chromatic scale and a diatonic scale?
4. How many forms of diatonic scales are there and what are their names?
5. Name and define the four ways in which the tones of the diatonic scales are named.
6. What is the key-tone?
7. Describe the movable and fixed systems.
8. Describe the major scale.
9. Describe the effect of a sharp; of a double sharp; of a flat; of a double flat; of a cancel.
10. State the rule for finding the key having the next number of sharps and the rule for finding the key having the next number of flats.
11. Write on the staff, using the treble clef, all the major keys to eleven sharps and eleven flats. Write several scales (the teacher deciding the number) using the bass and tenor clefs. (Show by curved line those notes situated one semi-tone apart.)
12. What is the order of the letters in the sharp signature? In the flat signature?
13. What is meant by enharmonic?
14. What are the enharmonic scales used in practice?
15. Give enharmonic letter names for each of the twelve keys.
16. What is the sum of sharp and flat signatures of enharmonic keys?
17. By the use of this enharmonic sum, find all the theoretical keys.
18. What is the construction of the harmonic minor scale? Of the melodic minor?
19. Write on the staff all the minor scales (both harmonic and melodic) to eleven sharps and eleven flats, letting the teacher determine which clef or clefs to use.
20. What is the reason for raising the seventh in harmonic minor?
21. What is the reason for raising the sixth in melodic minor?
22. Why does the descending form of melodic minor differ from the ascending form?
23. Why does not the raised sixth or seventh appear in the signature?
24. What is an accidental?
25. What is the relative minor and how is it found?
26. What is the parallel minor and how does its signature differ from its parallel major?
N. B. Before proceeding to the next chapter all these exercises should be properly answered and corrected by the teacher.
CHAPTER III.
INTERVALS AND INTRODUCTION TO CHORD BUILDING.
An interval is the distance between two tones; intervals are named by the ordinals. The number of letters comprised in the notation of two tones determines the ordinal name of the interval. Example: c to d is an interval of a second because two letters are comprised. It makes no difference whether or not either or both of the above tones is affected by an accidental, the interval still comprises two letters and is a second.
Reckoning from the tonic of the major scale to each degree of the scale produces the following intervals:—
8th or prime. 2nd 3rd 4th 5th 6th 7th octave 9th
The interval of the ninth is often called a second, the octave not being considered.
These intervals are the normal intervals of the major scale. These normal intervals are qualified in two ways. The prime, fourth, fifth and octave are called perfect. The second, third, sixth and seventh are called major; thus:—
All intervals should be reckoned from the lower note, which is considered a major key-note. If the upper note is in the major scale of the lower note, the interval is normal; that is, either perfect or major. If the upper note is not in the major scale of the lower note, the interval is a derivative interval. The derivative intervals are called minor, diminished and augmented.
A minor interval is derived from a major interval and is one semi-tone smaller. By lowering the upper tone of any major interval one half step or by raising the lower tone of any major interval one half step (not altering the letter name in either case) a minor interval is formed, thus:—
A diminished interval is one half step smaller than a minor or a perfect interval. By lowering the upper tone of any minor or any perfect interval one half step, or by raising the lower tone of any minor or any perfect interval one half step (not altering the letter name in either case) a diminished interval is formed, thus:—
The tones of the diminished second are the same pitch, but must be called a second because two letters are comprised. The diminished prime is possible melodically, but harmonically, only in theory. It is .
An augmented interval is one half step larger than a major or a perfect interval. By raising the upper tone of any major or perfect interval one half step, or by lowering the lower tone of any major or perfect interval one half step (not altering the letter name in either case) an augmented interval is formed, thus:—
Notice that the tones of the augmented seventh are the same pitch, but must be called a seventh as seven letters are comprised.
The following intervals are the same in sound, but not in name:—
| perfect prime | sounds | the | same | as | diminished 2nd |
| augmented prime | " | " | " | " | minor 2nd |
| diminished prime | " | " | " | " | minor 2nd |
| major 2nd | " | " | " | " | diminished 3rd |
| minor 3rd | " | " | " | " | augmented 2nd |
| major 3rd | " | " | " | " | diminished 4th |
| perfect 4th | " | " | " | " | augmented 3rd |
| augmented 4th | " | " | " | " | diminished 5th |
| perfect 5th | " | " | " | " | diminished 6th |
| minor 6th | " | " | " | " | augmented 5th |
| major 6th | " | " | " | " | diminished 7th |
| minor 7th | " | " | " | " | augmented 6th |
| major 7th | " | " | " | " | diminished 8th |
| perfect octave | " | " | " | " | augmented 7th |
From the preceding list the following rule is apparent:—
Rule 7. By Changing Enharmonically Either or Both of the Tones of an Interval, a Different Interval is Obtained Which Sounds the Same as the Original Interval.
The distance in semi-tones of all the intervals to an octave is as follows:—
| prime | = | unison | comprises | 1 | letter |
| augmented prime | = 1 | semi-tone | " | 1 | " |
| diminished 2nd | = | unison | " | 2 | letters |
| minor 2nd | = 1 | semi-tone | " | 2 | " |
| major 2nd | = 2 | semi-tones | " | 2 | " |
| augmented 2nd | = 3 | " | " | 2 | " |
| diminished 3rd | = 2 | " | " | 3 | " |
| minor 3rd | = 3 | " | " | 3 | " |
| major 3rd | = 4 | " | " | 3 | " |
| augmented 3rd | = 5 | " | " | 3 | " |
| diminished 4th | = 4 | " | " | 4 | " |
| perfect 4th | = 5 | " | " | 4 | " |
| augmented 4th | = 6 | " | " | 4 | " |
| diminished 5th | = 6 | " | " | 5 | " |
| perfect 5th | = 7 | " | " | 5 | " |
| augmented 5th | = 8 | " | " | 5 | " |
| diminished 6th | = 7 | " | " | 6 | " |
| minor 6th | = 8 | " | " | 6 | " |
| major 6th | = 9 | " | " | 6 | " |
| augmented 6th | = 10 | " | " | 6 | " |
| diminished 7th | = 9 | " | " | 7 | " |
| minor 7th | = 10 | " | " | 7 | " |
| major 7th | = 11 | " | " | 7 | " |
| augmented 7th | = 12 | " | " | 7 | " |
| diminished 8th | = 11 | " | " | 8 | " |
| perfect 8th | = 12 | " | " | 8 | " |
A quicker and better method of determining an interval than by committing to memory the above table is to consider the lower note the tonic of the major scale. If the upper note is in the major scale of the lower note, the interval is normal (major or perfect). After a little practice the number of letters in an interval can be determined at a glance. If the upper note is not in the major scale of the lower note the interval is derivative and is determined by the information heretofore given.
INVERSION OF INTERVALS.
Intervals are said to be inverted when the lower note of the original interval is placed an octave higher, thereby becoming the upper note of the interval thus formed. Example: the inversion of is . The same letters are in both intervals, but the first interval is a third and the inverted interval is a sixth.
Rule 8. The Sum of an Interval and Its Inversion is Nine.
The above rule, therefore, gives the following inversions:—
| a prime | inverts | to | an octave | (1 + 8 = 9) |
| a second | " | " | a seventh | (2 + 7 = 9) |
| a third | " | " | a sixth | (3 + 6 = 9) |
| a fourth | " | " | a fifth | (4 + 5 = 9) |
| a fifth | " | " | a fourth | (5 + 4 = 9) |
| a sixth | " | " | a third | (6 + 3 = 9) |
| a seventh | " | " | a second | (7 + 2 =9) |
| an octave | " | " | a prime | (8 + 1 = 9) |
To find to what intervals ninths, tenths, elevenths, twelfths, etc., invert, consider them respectively as seconds, thirds, fourths, fifths, etc., and consider the lower note placed two octaves higher instead of one octave.
Qualifications invert in the following manner:—
| major | intervals | invert | to | minor | intervals |
| minor | " | " | " | major | " |
| perfect | " | " | " | perfect | " |
| diminished | " | " | " | augmented | " |
| augmented | " | " | " | diminished | " |
By the use of the above table and rule 8, all inversions may be determined. Examples:—