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    ESSENTIALS OF

    MUSIC THEORY


    ELEMENTARY


    BY

    CARL E. GARDNER

    AUTHOR OF "MUSIC COMPOSITION"


    [Illustration]


    NEW YORK

    CARL FISCHER

    1912




    COPYRIGHT, 1912,

    BY

    CARL FISCHER


    International Copyright Secured




PREFACE.


The primary object in the following pages is to supply the teacher
and student with a text book to accompany the work on instrumental or
vocal technic. Because of the great amount of time required to obtain
proficient technic, text books are often neglected, and, if exhaustive,
are usually ignored. Brevity and conciseness characterize this volume
and it is planned to meet the needs of the busy teacher and student.

Many pupils may not see the importance of some of the following
text, such as the research for theoretical keys; but the results
thus obtained are nothing more than the natural outcome of simple
mathematical reasoning, and are inevitable if the -structure- of the
scales is understood. Pupils should be impressed with the structure of
scales rather than be taught their keys and signatures by rote. Rote
methods have little to recommend them in modern pedagogical ideals,
and are used principally in teaching the young. The lack of knowledge
of the -whys- and -wherefores- of scales, intervals, and chords among
instrumentalists and singers is appalling, and is due partly to the
neglect of text books and partly to incompetent teachers.

A further object of the text is to offer a course of preparation
for the study of harmony, composition, and appreciation. Music
Appreciation, as a cultural course of study, is growing in popularity
and large enthusiastic classes in this subject are to be found in
all our colleges. These students are frequently handicapped by poor
preparation in the essential rudiments of music such as are treated in
this volume.

The author's gratitude for valuable aid, criticisms, and suggestions
is due his wife, Marion Dillon Gardner, and his sisters, Mabel Gardner
Bankart and Lena Gardner Lewis.




CONTENTS.


                                                              PAGE


    PREFACE                                                    iii


    I. RHYTHM: NOTE VALUES: TIME SIGNATURES: TEMPO               3


    II. THEORY OF SCALES (Major and Minor)                      10

            The Major Scale                                     11

            The Minor Scale                                     21


    III. INTERVALS AND CHORD BUILDING                           32

            Inversion of Intervals                              36

            Chord Construction                                  38

            Inversion of Triads                                 40

            The Seventh Chord                                   41

            Inversion of Seventh Chords                         43


    IV. EPITOMIZED ACOUSTICS                                    47


    V. EAR TRAINING                                             51

            The Normal Intervals of the Major Scale             52

            The Normal Intervals of the Minor Scale             55

            Altered Intervals                                   56

            Arpeggio Chords                                     60

            Two Voiced Chords                                   61

            The Four Voiced Chord                               63




ESSENTIALS OF MUSIC THEORY




CHAPTER I.

RHYTHM.


Sound is the effect produced by propagated atmospheric waves which
affect the sense of hearing. Irregular impulses, propagated through the
air, produce noise. Regular impulses produce musical -tone-.

The duration of tone is indicated by symbols called -notes-. Following
is a list of notes:--

    [Double whole note] or [Breve] Double whole note or -Breve-.

    [Semibreve] Whole note or -Semibreve-.

    [Minim] Half note or -Minim-.

    [Quarter] Quarter note or -Crotchet-.

    [Eighth] Eighth note or -Quaver-.

    [Semiquaver] Sixteenth note or -Semiquaver-.

    [Demisemiquaver] Thirty-second note or -Demisemiquaver-.

Occasionally the sixty-fourth note ([Sixty-fourth note]) is used.
Following is a table of the relative value of notes:--

[Illustration]

The -breve- or double whole note is not given in this table as it is
but seldom used. The value of it is twice the whole note, four times
the half, etc.

The whole note is represented by an open oval; the half, by an open
oval and stem; the quarter, by a closed head and stem; the eighth note
is the same as the quarter with a flag; the sixteenth, the same with
two flags; the thirty-second, the same with three flags. As is seen
in the table, the eighth, sixteenth and thirty-second notes are often
grouped when more than one occurs in succession.

Suspension of tone is indicated by symbols called -rests-. Each note
has its equivalent rest. Following is a list of rests:--

    [Double whole rest] Double whole rest.
    [Whole rest] Whole rest.
    [Half rest] Half rest.
    [Quarter rest] Quarter rest.
    [Eighth rest] Eighth rest.
    [Sixteenth rest] Sixteenth rest.
    [Thirty-second rest] Thirty-second rest.

The rate of vibration is called -pitch-. Rapid vibrations produce
"high" (shrill) tones. Slow vibrations produce "low" tones. More
complete information on sound, tone and pitch is given later under the
heading "Acoustics."

The notes are written on the staff which consists of five horizontal
lines together with their spaces. The duration of a tone is determined
by the note used; the pitch, by the note's position on the staff.

A dot placed after a note or rest adds one half its value. A "tie" is
a curved line connecting two notes of the same pitch. Examples of dots
and ties:--

    [Semibreve with Augmentation dot] equals [Semibreve tied with
        Minim] or 1 + 1/2.

    [dotted half note] equals [half note tied to quarter note] or
        1/2 + 1/4.
    [dotted quarter note] equals [quarter note tied to eight note] or
        1/4 + 1/8.
    etc., etc.

A double dot adds one half and one fourth its value, thus:--

    [double dotted whole note] equals [tied notes; tied whole, half
        and quarter note] or 1 + 1/2 + 1/4.
    [double dotted half note] equals [tied notes; tied half, quarter
        and eighth note] or 1/2 + 1/4 + 1/8.
    [double dotted quarter note] equals [music symbol; tied quarter,
        eighth and sixteenth notes] or 1/4 + 1/8 + 1/16.
    etc., etc.

Bars are lines drawn vertically across the staff dividing music into
-measures-. The contents of the measure is determined by the fraction
at the beginning. The denominator of the fraction shows the kind of
notes, and the numerator, the number of that kind contained in a
measure. Each measure must contain the number and kind of notes or
rests designated by the fraction, or their equivalents.

-Artificial groups- are groups of notes played and summed in other
than their fractional value. The most common artificial groups are the
-triplet- and -sextuplet-. A triplet is a group of three notes played
in the time and summed in the value of two of its own kind. A sextuplet
is a group of six notes played in the time and summed in the value of
four of its own kind. A group of five notes is played in the time and
summed in the value of four of its own kind. A group of seven notes
is played in the time and summed in the value of six of its own kind.
Occasionally a group of two notes occurs. This group differs from other
artificial groups inasmuch as it is played more slowly than the tempo
notes. A group of two notes is played in the time and summed in the
value of three of its own kind. Artificial groups are designated by a
curved line over or under the notes with a figure showing the kind of
group, thus:--

    triplet                  [Illustration: [triplet]]
    sextuplet                [Illustration: [sextuplet]]
    group of two notes       [Illustration: [2-note group]]

In "counting" music, it is customary to give as many counts to each
measure as the numerator of the fraction indicates. Each of these
counts is called a -pulse-. Pulses should occur regularly unless
otherwise marked. Irregularities in the occurrence of pulses are
marked in various ways. The -ritardando-, the -hold- ([Fermata]),
and the -accelerando- are the principal marks of irregularities. The
-ritardando- (abbreviated -ritard.- or -rit.-) means to lessen the
speed, the -accelerando- (abbreviated -accel.-) to quicken the speed,
and the -hold- ([Fermata]) to hold the note, over or under which it is
placed, as long as musical taste dictates.

This occurrence of pulses is called -rhythm-. The most common rhythms
are =4/4= or [common time], =3/4=, =2/4=, =6/8=,
=3/8=, =4/8=, =9/8=, =12/8=, and =2/2= or [2/2
alla breve] also called -alla breve-. Other rhythms not so common are
=6/4=, =8/4=, =1/4=, =2/8=, =1/2=, =6/2=,
=3/2=, =4/2=, =3/16=, and less often =1/1=,
=5/4=, =5/8= and =5/16=, etc.

On the first pulse of all kinds of rhythm is a primary accent called
-thesis-. Secondary accents, called -arsis-, occur in =4/4= on
the third count and in =6/8= on the fourth count. These natural
accents give a "swing" to the music. They can only be displaced or
overshadowed by artificial accents which are designated in various
ways. The most common artificial accents are the -forzando- (designated
thus: [Sforzato], [Marcato], or -fz-), meaning a sudden strong
accent to the note or chord over or under which it is placed; the
-rinforzando- (which is not quite so marked as the -forzando-); the
-staccato- (designated by a dot placed over or under the note or
chord) which makes the note thus indicated short and crisp, and the
-syncopation-, which is a form of rhythm displacing the natural accent
by the note's entrance on an unaccented part of the measure and its
sustentation through the pulse.

The rapidity of the occurrence of pulses is called -tempo-, which is
indicated at the beginning of a movement by Italian words usually, the
most common of which are as follows:--

-Grave-, slow and solemn (the slowest tempo).

-Largo-, slow, a trifle faster than Grave.

-Larghetto-, a trifle faster than Largo.

-Adagio-, a trifle faster than Larghetto.

-Lento-, slow.

-Andante-, moderately slow.

-Andantino-, translated literally means slower than Andante, but it is
more often used incorrectly meaning faster than Andante.

-Moderato-, moderate; the mediate between fast and slow.

-Allegretto-, cheerful.

-Allegro-, quick.

-Vivo-, quick.

-Presto-, very quick.

To many of the above words is added the ending -issimo- which gives
the word to which it is added its superlative degree. Other terms are
oftentimes combined with the above words to characterize the movement.
Every pupil should have a dictionary of musical terms for constant
reference.

The majority of piano students have an absolute disregard for note
values and tempo marks which are so important that the pupils fail to
gain any good results from their study unless they understand and pay
strict attention to these marks. The incompetency of so many teachers
is somewhat responsible for this state of affairs, but the majority of
piano studies and methods is more largely responsible. In second grade
studies, there are many which, if written in a judicious manner, would
be excellent second grade work, but when played as they are written
and as their tempo mark demands require a virtuoso to execute them
correctly. These studies have led pupils to playing -allegro- movements
in -largo- tempo. At the end of a week's practice a -moderato tempo-
may be the result. Continued enforced disregard can produce nothing
but habitual disregard for tempo marks. The teacher should constantly
remind the student of these facts and, in as far as possible, omit such
studies as cannot be played -a tempo-. Many exercises may be rewritten
in a playable manner by the teacher, who, by so doing, would not only
impress the pupil with the importance of tempo marks, but would develop
his ability to read from manuscript, an ability which, unfortunately,
is almost universally lacking in pupils.


EXERCISES.

ORAL AND WRITTEN.

1. What is sound?

2. Tell the difference between noise and musical tone.

3. What is a note?

4. Describe the most common notes.

5. Write a table of the relative value of notes commencing with the
whole note.

6. Write a table of the relative value of notes commencing with the
dotted half note.

7. Describe the rests.

8. For each dotted note, show its equivalent by two tied notes.

9. For each double dotted note, show its equivalent by three tied notes.

10. Describe measure and bar and upon what the measure's contents
depends.

11. Describe the manner of counting the different rhythms.

12. Name the marks that designate irregularities in rhythms and
describe the character of each mark.

13. Describe the natural accent.

14. Name the most common artificial accents and describe the character
of each.

15. What is meant by tempo?

16. Name and define twelve different tempo marks.

17. Explain and notate artificial groups.




CHAPTER II.

THEORY OF SCALES.

MAJOR AND MINOR.


As stated before (Chapter I, page 4), the rate of vibration
is called pitch. Tones vibrating an equal number of times produce
an -unison- which is a perfect concordance and is pleasant to the
ear. Equally as pleasant to the ear is the ratio of two vibrations
against one. A tone vibrating twice as fast as a given tone is called
the given tone's -octave-. Between these two tones many tones may be
found. For example, suppose a note vibrating two hundred times in a
second, its octave would vibrate four hundred times in the second.
Between these two tones there would be (avoiding fractions which would
produce more different pitches) one hundred and ninety-nine tones of
different pitch. The ear is incapable of locating all these tones and
modern custom has divided all octaves into twelve parts, each part
being called a half step or, literally incorrect, a -semi-tone-. These
semi-tones sounded successively upwards or downwards from any tone to
its octave produce the -chromatic scale-.[A]

[Footnote A: All references to scales, intervals and enharmonic changes
treat of the tempered scale.]

A -diatonic scale- is a progression from any tone to its octave in
which certain chromatic steps are omitted. In modern music there are
three forms of diatonic scales, called:--

    1. Major.
    2. Harmonic minor.
    3. Melodic minor.

All three forms have eight tones, the eighth being the octave of the
first and is given the same name.

The tones of the diatonic scale are named in four different ways:--

    1. by numerals (Arabic and Roman),
    2. by the first seven letters of the alphabet,
    3. by the Italian syllables (-do-, -re-, -mi-, -fa-, -sol-,
        -la-, -si-,) and
    4. by the theory names (-tonic-, -supertonic-, -mediant-,
       -subdominant-, -dominant-, -submediant- and -subtonic-).

-The key-tone is the tone from which a diatonic scale is built.-

The numerical system is a movable system which means that 1 is always
the key-tone. The theory name system is a movable system, the tonic
being always the key-tone or 1. The alphabet system is a fixed system
which means that a letter is always the same tone or its octave. The
Italian system is treated as both a fixed system and a movable system.
This book treats of the movable -do-, -do- always being the key-tone, 1
and tonic.


THE MAJOR SCALE.

A major scale is a progression from any tone to its octave in which
chromatic steps are omitted between 1 and 2,--2 and 3,--4 and 5,--5 and
6,--6 and 7; from 3 to 4 and from 7 to 8 half steps are made.

Following is a diagram of a two octave keyboard:--

[Illustration:

      C♯  D♯  F♯  G♯  A♯   C♯  D♯  F♯  G♯  A♯
      or  or  or  or  or   or  or  or  or  or
      D♭ E♭  G♭ A♭  B♭ D♭  E♭ G♭ A♭  B♭

      C   D   E  F  G  A  B  C  D   E   F  G  A  B  C
]

The keyboard shows white and black keys. The black keys are in groups
of two and three. As can be seen in the diagram, the white key next
to the left of the group of two black keys is -c-. The white keys in
order to the right of -c- are respectively -d-, -e-, -f-, -g-, -a- and
-b-. Following -b- is a repetition of -c- at the distance of an octave.
Notice that between -e- and -f- there is no black key as is also the
case between -b- and -c-. In these two cases, where no black key
separates the white keys, the white keys are one semi-tone apart. Two
white keys separated by a black key are one whole step apart. A black
key is at the distance of a semi-tone from an adjoining white key. The
black keys derive their letter names from the white keys. A black key
is named from either of the white keys between which it is situated.
The black key between -c- and -d- is named -c sharp- (♯) or -d flat-
(♭).

Starting at -c- and sounding the white keys in order to the right as
far as the octave produces the ascending major scale of -C-; sounding
in order to the left produces the descending major scale of -C-. Notice
that no black keys are necessary in the case of the -C- major scale,
the whole and half steps being in their proper places; namely, whole
steps between 1 and 2, 2 and 3, 4 and 5, 5 and 6, 6 and 7, and half
steps between 3 and 4 and between 7 and 8. The student must constantly
keep in mind the order of whole and half steps in all scales explained.
In each scale explained the letters will be numbered and a curved line
will connect those figures representing tones one half step apart.

All major keys except -C- major require one or more black keys. The
number of sharps or flats required for a key is placed at the beginning
of the staff and this is called the signature.

A sharp (♯) placed before a note raises the tone one half step and a
flat (♭) lowers a tone one half step.

The sharp keys will be considered first and a sharp major scale will be
built from each of the twelve tones.

       *       *       *       *       *

=Rule 1. The Fifth of a Scale is the Tonic (or 1) of the Scale having
the next Number of Sharps.=

-C- has no sharps, the fifth of -C- is -g- and therefore by following
the rule, we find that -G- has one sharp. The scale of -G- is as
follows:--

    G a b c d e f♯ G
    1 2 3⌣4 5 6 7⌣8

Notice that the seventh of the scale is a black key.

The fifth of -G- is -d- and has two sharps:--

    D e f♯ g a b c♯ D
    1 2 3⌣4 5 6 7⌣8

Notice that -f- remains sharped and the added sharp is the seventh of
the scale. This is always the case, the added sharp is the seventh of
the new scale.

The fifth of -D- is -a- and has three sharps:--

    A b c♯ d e f♯ g♯ A
    1 2 3⌣4 5 6 7⌣8

The fifth of -A- is -e- and has four sharps:--

    E f♯ g♯ a b c♯ d♯ E
    1 2 3⌣4 5 6 7⌣8

The fifth of -E- is -b- and has five sharps:--

    B c♯ d♯ e f♯ g♯ a♯ B
    1 2 3⌣4 5 6 7⌣8

The fifth of -B- is -f-♯ and has six sharps:--

    F♯ g♯ a♯ b c♯ d♯ e♯ F♯
    1 2 3⌣4 5 6 7⌣8

Notice that -e-♯ is not a black key but the white key which has been
previously considered as -f-. It must be called -e-♯ to retain the
alphabetical order.

The fifth of -F-♯ is -c-♯ and has seven sharps:--

    C♯ d♯ e♯ f♯ g♯ a♯ b♯ C♯
    1 2 3⌣4 5 6 7⌣8

In this scale all the notes are sharped. The -b-♯ as well as the -e-♯
is a white key.

The fifth of -C-♯ is -g-♯ and has eight sharps. This key necessitates
one double sharp and -f- is double sharped. The double sharps are
added in the same order that the single sharps are. The double sharp
(designated thus: =x=) raises a tone one whole step.

    G♯ a♯ b♯ c♯ d♯ e♯ f=x= G♯
    1 2 3⌣4 5 6 7⌣8

The fifth of -G-♯ is -d-♯ and has nine sharps (two double sharps, -f-
and -c-):--

    D♯ e♯ f=x= g♯ a♯ b♯ c=x= D♯
    1 2 3⌣4 5 6 7⌣8

The fifth of -D-♯ is -a-♯ and has ten sharps (three double sharps, -f-,
-c- and -g-):--

    A♯ b♯ c=x= d♯ e♯ f=x= g=x= A♯
    1 2 3⌣4 5 6 7⌣8

The fifth of -A-♯ is -e-♯ and has eleven sharps (four double sharps,
-f-, -c-, -g- and -d-):--

    E♯ f=x= g=x= a♯ b♯ c=x= d=x= E♯
    1 2 3⌣4 5 6 7⌣8

The fifth of -E-♯ is -b-♯ and has twelve sharps (five double sharps,
-f-, -c-, -g-, -d- and -a-):--

    B♯ c=x= d=x= e♯ f=x= g=x= a=x= B♯
    1 2 3⌣4 5 6 7⌣8

-B-♯ has taken us back to our starting key called by a different name.

All twelve keys have now been named with their sharp signatures. To
continue counting five would take us over the same keys called by
different names. The student is advised to do a little of this for
mental discipline. If this is done beyond fourteen sharps, it will be
necessary to add triple sharps. Of course, triple sharps are never used
in musical notation and such a research would be entirely arithmetical.

The order of the letters in the sharp signature which follows should be
committed to memory:--

    F C G D A E B.

All keys having one double sharp or more would be difficult to read,
and so instead of using the sharp signatures on such keys, the flat
signatures are used. All twelve keys with their flat signatures will
now be given.

       *       *       *       *       *

=Rule 2. The Fourth of a Scale is the Tonic of the Scale having the
Next Number of Flats.=

-C- has no flats; the fourth of -C- is -f-; therefore, by following the
rule, we find that -F- has one flat:--

    F g a b♭ c d e F
    1 2 3⌣4 5 6 7⌣8

Notice the fourth of the scale is a black key.

The fourth of -F- is -b-♭ and has two flats:--

    B♭ c d e♭ f g a B♭
    1 2 3⌣4 5 6 7⌣8

Notice that the -b- remains flat and that the added flat is the fourth
of the scale. This is always the case--the added flat is the fourth of
the new scale.

The fourth of -B-♭ is -e-♭ and has three flats:--

    E♭ f g a♭ b♭ c d E♭
    1 2 3⌣4 5 6 7⌣8

The fourth of -E-♭ is -a-♭ and has four flats:--

    A♭ b♭ c d♭ e♭ f g A♭
    1 2 3⌣4 5 6 7⌣8

The fourth of -A-♭ is -d-♭ and has five flats:--

    D♭ e♭ f g♭ a♭ b♭ c D♭
    1 2 3⌣4 5 6 7⌣8

The fourth of -D-♭ is -g-♭ and has six flats:--

    G♭ a♭ b♭ c♭ d♭ e♭ f G♭
    1 2 3⌣4 5 6 7⌣8

The fourth of G♭ is -c-♭ and has seven flats:--

    C♭ d♭ e♭ f♭ g♭ a♭ b♭ C♭
    1 2 3⌣4 5 6 7⌣8

The fourth of -C-♭ is -f-♭ and has eight flats. This key necessitates
one double flat and -b- has the double flat. The double flats are
added in the same order that the single flats are. The double flat
(designated: ♭♭) lowers a tone one whole step.

    F♭ g♭ a♭ b♭♭ c♭ d♭ e♭ F♭
    1 2 3⌣4 5 6 7⌣8

The fourth of -F-♭ is -b-♭♭ and has nine flats (two double
flats, -b-♭♭and -e-♭♭):--

    B♭♭ c♭ d♭ e♭♭ f♭ g♭ a♭ B♭♭
    1 2 3⌣4 5 6 7⌣8

The fourth of -B-♭♭ is -e-♭♭ and has ten flats (three double flats,
-b-♭♭, -e-♭♭ and -a-♭♭):--

    E♭♭ f♭ g♭ a♭♭ b♭♭ c♭ d♭ E♭♭
    1 2 3⌣4 5 6 7⌣8

The fourth of -E-♭♭ is -a-♭♭ and has eleven flats (four double
flats, -b-♭♭, -e-♭♭, -a-♭♭ and -d-♭♭):--

    A♭♭ b♭♭ c♭ d♭♭ e♭♭ f♭ g♭ A♭♭
    1 2 3⌣4 5 6 7⌣8

The fourth of -A-♭♭ is -d-♭♭ and has twelve flats (five double
flats, -b-♭♭, -e-♭♭, -a-♭♭, -d-♭♭, and -g-♭♭):--

    D♭♭ e♭♭ f♭ g♭♭ a♭♭ b♭♭, c♭ D♭♭
    1 2 3⌣4 5 6 7⌣8

D♭♭ has taken us back to our starting key called by a different name as
was the case when we had twelve sharps. To continue counting four would
take us over the same keys called by different names. As was advised in
the sharp keys, this research should be continued by the student. If
more than fourteen flats are considered, it will be necessary to add
triple flats.

The order of the letters in the flat signature which follows should be
committed to memory:--

    B E A D G C F.

By comparing the order of the letters in the flat signature with that
of the sharp signature, it will be seen that the order of the letters
in the flat signature is that of the sharp signature reversed.

Notice that each key has more than one name; for example, the white key
next to the left of the group of two black keys has been called -C-,
-D-♭♭ and -B-♯.

       *       *       *       *       *

=Rule 3. An Enharmonic Change is the Change of a Name of a Tone
without Altering its Pitch.=

Two or more scales played from the same pitched tone but called by
different names are called -enharmonic scales-. In practice,[B] fifteen
major scales are used, three of which are enharmonic scales. Following
is a list of the major scales used in practice together with their
signatures:--

    C       no sharps or flats
    G       1 sharp (f)
    D       2 sharps (f and c)
    A       3 sharps (f, c and g)
    E       4 sharps (f, c, g and d)
    F 1 flat (b)
    B♭ 2 flats (b and e)
    E♭ 3 flats (b, e and a)
    A♭ 4 flats (b, e, a and d)
    D♭ 5 flats (b, e, a, d and g). enharmonical to C♯.
    G♭ 6 flats (b, e, a, d, g and c) enharmonical to F♯
    C♭ 7 flats (b, e, a, d, g, c and f) enharmonical to B
    B 5 sharps (f, c, g, d and a)
    F♯ 6 sharps (f, c, g, d, a and e)
    C♯ 7 sharps (f, c, g, d, a, e and b)

[Footnote B: Theoretical keys appear in many compositions during
transitions, but they are not obvious (except by analysis) because of
the fact that their signatures do not appear.]

The enharmonic keys used in practice are:--

    B (five sharps) enharmonical to C♭ (seven flats)
    F♯ (six sharps) enharmonical to G♭ (six flats)
    C♯ (seven sharps) enharmonical to D♭ (five flats)

       *       *       *       *       *

=Rule 4. The Sum of the Enharmonic Flat and Sharp Signatures is
Twelve.= Notice that this is true in the above three keys.

By this rule the theoretical keys (that is, those having more than
seven sharps or flats) are easily found. For example:--to find the key
having eleven flats; the key having eleven flats is enharmonical to the
key having one sharp (11 + 1 = 12). -G- has one sharp and changing its
name to the enharmonic flat key, we obtain -A-♭♭ which, therefore, has
eleven flats. This process should be applied to all keys having eight
to eleven sharps and eight to eleven flats inclusive.

As stated in the first chapter, the pitch of a tone is determined
by the note's position on the staff. The staff of five lines with
its spaces allows of the designation of but one octave and one step,
whereas, in modern music, there is need of the notation of at least six
octaves. This necessitates the use of symbols called -clefs-, and lines
added to the staff called ledger lines. Ledger lines are short lines
parallel to the staff lines added above or below the staff lines.

There are three clefs:--

    1. the treble or G clef, 2. the tenor, movable or C clef, 3. the
    bass or F clef.

Middle -C- is the tone which all voices can sing. It is in the lower
register of the high female voice and in the upper register of the low
male voice.

The -G clef- (treble clef) is for high voices or instruments. Its
symbol shows the position of the -G- next above middle -C- thus:
[Illustration: [music]]. Middle -C- is found on the first ledger line
below the -G clef-, thus: [Illustration: [music]].

The -tenor- or -C clef- is for the use of medium voices or instruments.
Its symbol shows the position of middle -C-. This clef is movable
and may place middle -C- on any line or space of the staff. Its most
common position is on the third line of the staff, thus: [Illustration:
[music]]. It is not uncommon, however, to find it on the second or
fourth line. For vocal music it is often written in the third space.

The -bass- or -F clef- is for the use of low voices or instruments.
Its symbol shows the position of the -F- next below middle -C-, thus:
[Illustration: [music]]. Middle -C- is found on the first ledger line
above the -F clef-, thus: [Illustration: [music]].

The notation in the treble clef of all the major scales used in
practice follows:

    Scale of C [Illustration: [Music: C d e f g a b C]]

    Scale of G [Illustration: [Music: G a b c d e f♯ G]]

    Scale of D [Illustration: [Music: D e f♯ g a b c♯ D]]

    Scale of A [Illustration: [Music: A b c♯ d e f♯ g♯ A]]

    Scale of E [Illustration: [Music: E f♯ g♯ a b c♯ d♯ E]]

    Scale of B [Illustration: [Music: B c♯ d♯ e f♯ g♯ a♯ B]]

    Scale of F♯ [Illustration: [Music: F♯ g♯ a♯ b c♯ d♯
      e♯ F♯]]

    Scale of C♯ [Illustration:
      [Music: C♯ d♯ e♯ f♯ g♯ a♯ b♯ C♯]]

    Scale of F [Illustration: [Music: F g a b♭ d e F]]

    Scale of B♭ [Illustration: [Music: B♭ c d e♭ f g a B♭]]

    Scale of E♭ [Illustration: [Music: E♭ f g a♭ b♭ c d E♭]]

    Scale of A♭ [Illustration: [Music: A♭ b♭ c d♭ e♭ f g A♭]]

    Scale of D♭ [Illustration: [Music: D♭ e♭ f g♭ a♭ b♭ c
        D♭]]

    Scale of G♭ [Illustration: [Music: G♭ a♭ b♭ c♭ d♭ e
       f G♭]]

    Scale of C♭ [Illustration: [Music: C♭ d♭ e♭ f♭ g♭
       a♭ b♭ C♭]]


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- (♭), the -double flat- (♭♭), 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 b♭ 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 b♭ c d e♭ 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 e♭ f g a♭ 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 a♭ b♭ c d♭ e♮ f
    1 2⌣3   4  5⌣6 ⌴7 ⌣8

The fourth of -f- is -b-♭ and has five flats (-b-, -e-, -a-, -d- and
-g-):--

    b♭ c d♭ e♭ f g♭ a♮ b♭
    1  2⌣ 3  4   5⌣6  ⌴7⌣8

The fourth of -b-♭ is -e-♭ and has six flats (-b-, -e-, -a-, -d-, -g-
and -c-):--

    e♭ f g♭ a♭ b♭ c♭ d♮ e♭
    1   2⌣3  4   5⌣ 6⌴  7 ⌣8

The fourth of -e-♭ is -a-♭ and has seven flats (-b-, -e-, -a-, -d-,
-g-, -c- and -f-):--

    a♭ b♭ c♭ d♭ e♭ f♭ g♮ a♭
    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.

[Footnote C: Capital letters are used to designate major keys and small
letters to designate minor keys.]

       *       *       *       *       *

=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 [Illustration: [music]]
    Harmonic

    Scale of a [Illustration: [music]]
    Melodic

    Scale of e [Illustration: [music]]
    Harmonic

    Scale of e [Illustration: [music]]
    Melodic

    Scale of b [Illustration: [music]]
    Harmonic

    Scale of b [Illustration: [music]]
    Melodic

    Scale of f♯ [Illustration: [music]]
    Harmonic

    Scale of f♯ [Illustration: [music]]
    Melodic

    Scale of c♯ [Illustration: [music]]
    Harmonic

    Scale of c♯ [Illustration: [music]]
    Melodic

    Scale of g♯ [Illustration: [music]]
    Harmonic

    Scale of g♯ [Illustration: [music]]
    Melodic

    Scale of d♯ [Illustration: [music]]
    Harmonic

    Scale of d♯ [Illustration: [music]]
    Melodic

    Scale of a♯ [Illustration: [music]]
    Harmonic

    Scale of a♯ [Illustration: [music]]
    Melodic

    Scale of d [Illustration: [music]]
    Harmonic

    Scale of d [Illustration: [music]]
    Melodic

    Scale of g [Illustration: [music]]
    Harmonic

    Scale of g [Illustration: [music]]
    Melodic

    Scale of c [Illustration: [music]]
    Harmonic

    Scale of c [Illustration: [music]]
    Melodic

    Scale of f [Illustration: [music]]
    Harmonic

    Scale of f [Illustration: [music]]
    Melodic

    Scale of b♭ [Illustration: [music]]
    Harmonic

    Scale of b♭ [Illustration: [music]]
    Melodic

    Scale of e♭ [Illustration: [music]]
    Harmonic

    Scale of e♭ [Illustration: [music]]
    Melodic

    Scale of a♭ [Illustration: [music]]
    Harmonic

    Scale of a♭ [Illustration: [music]]
    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:
[Illustration: [music]] -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

[Illustration: [music]

    prime. 2nd 3rd 4th 5th 6th 7th 8th    or 9th
                                   octave
]

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:--

[Illustration: [music]

    perf. maj. maj. perf. perf. maj. maj. perf.  maj.
    prime 2nd  3rd  4th   5th   6th  7th  octave 9th
]

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:--

[Illustration: minor 2nd

[music]]

[Illustration: minor 3rd

[music]]

[Illustration: minor 6th

[music]]

[Illustration: minor 7th

[music]]

etc.

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:--

[Illustration:

    dim.  dim.  dim.  dim.  dim.  dim.  dim.
    2nd   3rd   4th   5th   6th   7th   8th

    [music]

    etc.
]

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
[Illustration: [music]].

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:--

[Illustration:

    aug.   aug.  aug.  aug.  aug.  aug.  aug. aug.
    prime  2nd   3rd   4th   5th   6th   7th  8th

    [music]

    etc.
]

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 [Illustration:
[music]] is [Illustration: [music]]. 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:--

         major                  minor
           2nd                   7th

    [Illustration: [music]]   inverts to  [Illustration: [music]]

         major                  minor
           6th                   3rd

    [Illustration: [music]]     "      "  [Illustration: [music]]

          perf.                  perf.
          prime                  8th

    [Illustration: [music]]     "     "   [Illustration: [music]]

          perf.                 perf.
          4th                   5th

    [Illustration: [music]]     "     "  [Illustration: [music]]

          aug.                  dim.
          4th                   5th

    [Illustration: [music]]     "     "  [Illustration: [music]]

          dim.                  aug.
          7th                   2nd

    [Illustration: [music]]     "     "  [Illustration: [music]]

             etc.                    etc.

The -prime- is also called an -unison-, but in speaking of intervals,
it should always be called a -prime-. Correctly speaking, a -perfect
prime- is not an interval, but in the theory of music it is so called.
There is good reason for making this error, but none for calling a
-diminished prime- a -diminished unison-. Notice that the -diminished
second- as well as the -perfect prime- is an -unison-.

Intervals are considered both harmonically and melodically, or in other
words, both when sounded together or separately. In either case, the
lower note is the one from which to determine the interval.


CHORD CONSTRUCTION.

A chord is a combination of two or more tones sounded simultaneously.
All chords are constructed in -thirds-. The -fundamental tone- of a
chord is the tone on which the chord is constructed.

A chord of three tones is a -triad- which consists of a -fundamental-
together with its third and its fifth. Triads are divided into two
classes, -independent- and -dependent-. The independent triads have no
dissonant intervals and may end a composition. The dependent chords
have one or more dissonant intervals and are "restless" chords and
demand another chord to follow. The progression of a dependent chord
to an independent chord, thereby obtaining a restful effect, is called
-resolution-.

There are two kinds of independent triads, -major- and -minor-. -A
major triad- consists of the fundamental, the -major- third, and the
-perfect- fifth. Example:-- [Illustration: [music]]. -A minor triad-
consists of the fundamental, the -minor- third, and the -perfect-
fifth. Example:-- [Illustration: [music]].

If the fifth of a triad is augmented or diminished, the triad is a
dependent. Dependent triads are found constructed on the subtonic of
major keys; on the subtonic, supertonic, and mediant of the minor keys.
The triad on the mediant of the minor key is an augmented triad and the
first three mentioned triads are diminished.

Music written for four voices necessitates the doubling of one of the
factors of the triad. Any factor of the triad may be doubled. The
factor most frequently doubled is the fundamental in the octave, double
octave, or unison. The four voices are -soprano- (high female voice),
-alto- (low female voice), -tenor- (high male voice), and -bass- (low
male voice). Chords are figured under the bass by Roman numerals. Large
numerals designate -major- triads; small numerals designate -minor-
triads; large numerals with the mark (´) affixed designate augmented
triads; small numerals with a cipher affixed designate diminished
triads. The notation of the triads on each degree of the major and
minor scales follows:--

[Illustration:

    Major Mode.       N.B.

[music]

    I     II  III  IV  V  VI  VII°
]

[Illustration:

Minor Mode.

[music]

    I  IIº  III´  IV  V  VI  VIIº
]

N. B. Although doubled in the above examples, the fundamental of the
subtonic triad is seldom doubled in four voice writing and if doubled,
a bad progression results in many cases.

The three upper voices in a fundamental chord may be arranged in a
different manner:--

With the fundamental in the soprano:-- [Illustration: [music C I]].


With the third in the soprano:-- [Illustration: [music C I]].

If the three upper voices do not exceed the compass of an octave, the
chord is said to be in "close position." If the three upper voices
exceed the compass of an octave, the chord is said to be in "open
position."


INVERSION OF TRIADS.

A chord is inverted when a factor other than the fundamental is in the
bass. The first inversion of the triad is where we have the third in
the bass. It is called the chord of the sixth, because the fundamental
is the sixth of the bass. This chord is figured by a small Arabic
figure (⁶) over the bass note, the Roman numeral under the bass showing
the fundamental. Examples:--

[Illustration: doubled fundamental

[music: C I]]

[Illustration: doubled third

[music: C I]]

The second inversion of the triad has the fifth in the bass and is
called the chord of the sixth and fourth, or six-four chord, because
the fundamental is the fourth of the bass. This chord is figured by the
small Arabic figures (6/4) over the bass, the Roman numeral under the
bass showing the fundamental. Examples:--

[Illustration:

    doubled
    fifth

[music: 6/4, C I]]

[Illustration:

    doubled
    fundamental

[music: 6/4, C I]]


THE SEVENTH CHORD.

The seventh chord is obtained by adding the seventh of the fundamental
to any triad. A seventh chord may be formed on each degree of the major
and minor scales. It is figured with the Roman numerals below the bass
and a small Arabic (7) over the bass. The notation of the seventh
chords on each degree of the major and minor scales follows:--

[Illustration:

C Major.

[music]

    7 7 7 7 7 7 7

    I  II  III  IV  V  VI  VII°
]

[Illustration:

a̲ Minor.

[music]

    7 7 7 7 7 7 7

    I  II° III´ IV  V  VI  VII°
]

The tonic and subdominant of the major mode and the submediant of the
minor mode are formed with the major triad and the major seventh. The
dominant seventh in both modes is formed with the major triad and the
minor seventh. The seventh chords on II, III and VI in the major mode
and on IV of the minor mode are formed with the minor triad and the
minor seventh. The seventh chords on VII° in the major mode and on II°
in the minor mode are formed with the diminished triad and the minor
seventh. The subtonic seventh chord in the minor mode is called the
-diminished seventh- and is formed with the diminished triad and the
diminished seventh. The seventh chord on the mediant in the minor mode
is formed with the augmented triad and the major seventh. The seventh
chord on the tonic of the minor mode is formed with the minor triad
and the major seventh. In four voice writing, all the seventh chords
with the exception of those on the subtonic of both modes are often
written without the fifth and with the doubled fundamental. All seventh
chords are dependent chords and their natural resolution is to the
chord the fundamental of which is situated a fourth above or a fifth
below the fundamental of the seventh chord. This progression is called
"cadencing resolution." The subtonic seventh chord of both modes may
also naturally resolve to the tonic. The most important seventh chord
is the dominant, which resolves to the tonic. This progression is
called the -authentic close-. Another method of ending a composition is
by the -plagal close- which is a progression from subdominant harmony
(triad) to tonic. The plagal close is preceded by the authentic close
and is also called the -after cadence- and the -Amen cadence-.


INVERSION OF SEVENTH CHORDS.

The first inversion of the seventh chord is called the chord of the
fifth and sixth (six-five chord). The chord is figured by the Roman
numeral below the bass note and the Arabic figures (6/5) over the bass.
Example:--

[Illustration:

[music]

    6/5

    C  V⁷
]

The second inversion is called the chord of the third, fourth and
sixth (four-three chord). It is figured by the Roman numeral below the
bass and the Arabic figures (6/4/3) or simply (4/3) over the bass.
Example:--

[Illustration: [Music]

    4/3

    C V⁷
]

The third inversion is called the chord of the second and fourth. It
is figured by the Roman numeral below the bass and the Arabic figures
(6/4/2), (4/2), or simply (2) over the bass. Example:--

[Illustration: [Music]

    2

    C V⁷
]

The study of chord progression, altered chords, melody writing, passing
tones, etc., belongs properly to the study of harmony and counterpoint
which is not the subject of this volume.


EXERCISES.

    ORAL AND WRITTEN.

    1. What is an interval?

    2. How are intervals named?

    3. What are the normal intervals?

    4. Qualify the derived intervals; from what is each derived?

    5. Notate all the normal intervals and all the derivative intervals.

    6. Name all the following intervals:--

    (Accidentals affect only those notes before which they are placed.)

[Illustration:

[music]]

    7. Name several intervals (the teacher to determine the number)
    having different names, but sounding the same.

    8. What is the sum of inversions?

    9. State the manner in which qualifications invert.

    10. Invert all the intervals given in exercise 6.

    11. Notate and figure all the triads in several different major and
    minor keys. Which are dependent and which are independent?

    12. Describe a dependent triad.

    13. Describe open and close position.

    14. Notate and figure several sixth chords; several six-four
    chords.

    15. Notate and figure all the seventh chords in several different
    major and minor keys.

    16. What is the most important seventh chord?

    17. What is the authentic close?

    18. What is the plagal close?

    19. Notate and figure a six-five dominant chord in several major
    and minor keys.

    20. Notate and figure a four-three dominant chord in several major
    and minor keys.

    21. Notate and figure a four-two dominant chord in several major
    and minor keys.

    22. Choose some standard chorals (Bach's are advised) and analyze
    the chords therein.




CHAPTER IV.

EPITOMIZED ACOUSTICS.


The science of sound, including its cause and effect and the manner,
velocity, and intensity of its conveyance through different media, is
called -acoustics-.

The medium through which sound is most commonly propagated is air.
Through this medium, at a temperature of 32° Fahrenheit, sound travels
at a rate of 1090 feet per second. The quality and intensity of sound
do not alter the rate of speed. If this were not true, ensemble music
would be impossible. Intensity of sound is greater in condensed air;
velocity of sound is greater in a warm temperature.

Many experiments have been made to determine the velocity of sound,
the most reliable of which vary not over seven feet per second. The
average of six of the best experiments, made in the early part of the
nineteenth century, is 1089.7 feet per second at 32° Fahrenheit. Ten
hundred and ninety feet per second is the rate now generally adopted.

Wind and temperature are the only circumstances affecting the velocity
of sound in the air to any extent. Sound travels about four times
faster through water than through air, and about ten times faster
through solids such as metals and wood than through air. A sudden
displacement of the molecules of a medium produces sound which travels
in waves at an equal velocity in all directions. An idea of the manner
in which sound waves travel may be obtained by throwing a stone in
water; small waves are propagated from the point of impact which, if
the water be still, spread equally in all directions, but if it be
running water, the waves extend a greater distance down stream than up
stream. The effect of wind on sound waves may be compared to the effect
of running water on the waves propagated by the impact of the stone.

Musical tone is produced by regular vibrations; noise by irregular
vibrations. The tones of the tempered chromatic scale have the
following number of vibrations per second:--

    Middle   c               258.6
        c♯ or d♭  274.0
             d               290.3
        d♯ or e♭  307.6
             e               325.9
             f               345.2
        f♯ or g♭  365.8
             g               387.5
        g♯ or a♭  410.5
             a               435.0
        a♯ or b♭  460.8
             b               488.2

The preceding figures represent the vibrations of the "International
Pitch" which was adopted by the Piano Manufacturers' meeting in 1891.
A is the standard pitch having 435 double vibrations per second at a
temperature of 68° Fahrenheit. Many pitches have prevailed in different
countries at different times. At the time of Handel and Mozart, the
pitch was lower (422.5 and 421.6). England has had the pitch run as
high 454.7 and the United States as high as 460.8.

Sounds vibrating below a certain number lose the character of musical
tones as do those vibrating above a certain number. Great discrepancies
of opinion exist among theorists on this subject. Savart claims the
lowest audible sound has eight vibrations per second; Helmholtz claims
that there is no definite pitch of sounds having less than forty
vibrations per second; Herr Appum claims to hear fifteen vibrations
by the use of specially loaded tongues in reed pipes. He claims the
character of tone commences at twenty vibrations, but the musical
character of bass tones does not exist until frequencies exceed
twenty-four vibrations per second. On the subject of the audibility
of acute sounds, opinions are advanced ranging from 6,400 to 36,000
vibrations per second.

The limits of the human voices are tabulated below:--

    Bass          E   81.5     D        290.3
    Baritone      F   86.3     F♯       365.8
    Tenor         A  108.7     A        435.0
    Contralto     E  163.0     F        690.5
    Mezzo Soprano F  172.6     A        870.0
    Soprano       A  217.5     C       1034.6

Occasionally there are exceptional voices having a wider range than the
above scale indicates.

Ratio of Intervals:--

    Octave           1-- 2
    Perfect fifth    2-- 3
    Perfect fourth   3-- 4
    Major third      4-- 5
    Minor third      5-- 6
    Major sixth      3-- 5
    Minor sixth      5-- 8
    Major second     8-- 9
    Minor second    15--16
    Major seventh    8--15
    Minor seventh    9--16

Each tone generates "over tones" called -harmonics-. These -harmonics-
are the octave, the twelfth (perfect fifth), the seventeenth (major
third), the twenty-first (minor seventh) and the twenty-third (minor
ninth). Other -harmonics- than the above exist but are not used at the
present time in chord construction. The old theorists treated chords of
the eleventh and thirteenth, but modern theorists treat these intervals
as suspensions, anticipations, etc.[D] The origin of chord construction
may be seen from these -harmonics-. These over tones, generated from
a fundamental, are the pure (untempered) intervals. The tempered
intervals, with the exception of the octave, are slightly out of tune
but not enough so to shock the ear.

[Footnote D: Composers of the present day often use these intervals as
chord factors.]

The pure (untempered) scale of C has the following number of vibrations
per second:--

    Middle c    261.0
           d    293.6
           e    326.2
           f    348.0
           g    391.5
           a    435.0
           b    489.3

An entire volume would be necessary to explain completely the science
of acoustics. All ambitious students should consult books on acoustics.
The author recommends the books on sound by the following writers:--

    Appum
    G. B. Airy
    Pietro Blaserno
    Helmholtz
    Pole
    Benjamin Peirce
    Rodolphe Radau
    Savart
    Tyndall
    J. August Zahn




CHAPTER V.

EAR TRAINING.


A person with an untrained ear can appreciate music comparatively
little, even though he is well educated in the theory of music.
Absolute pitch is the ability to recognize and intonate any tone
indicated. Very few persons possess naturally absolute pitch, but it
may be acquired by a systematic study of ear training. Relative pitch
is the ability to recognize a tone by comparison with a known tone.
Advancement in relative pitch eventually leads to the attainment of
absolute pitch.

In practicing ear training, only a few minutes at a sitting are
advised. Too much time at once devoted to this practice tires the ear
and does more harm than good. On the other hand, these sittings should
be many each day. Students who do not have a teacher daily, should
have a member of the household play the exercises in ear training and
correct the mistakes. If no member of the household is musical, the
student should co-operate with another student.

Each exercise should be thoroughly learned before proceeding to the
following exercise. It may be recognizable and properly intonated
at once or it may take several sittings. The pitch A at 435 is the
standard for orchestral tuning and is recommended to the student for a
fundamental. The student should carry upon his person a tuning fork of
this pitch and sound it as often as an opportunity permits, and thereby
fix this fundamental thoroughly in his mind. Eventually, the student
will recognize this pitch whenever he hears it. Other tones will be
recognizable by comparison with this fundamental. Any other pitch for a
fundamental may be chosen with equally good results. The argument is in
favor of A because of its use in orchestral tuning.

Many systems of ear training, which produce the desired results, exist.
The following system has been found the most satisfactory by the
author.[E] Deviations from and additions to this system do no harm and
are advisable in certain individual cases.

[Footnote E: Many young children find difficulty in intonating small
intervals and it is necessary in such cases to commence with large
intervals and work toward the smaller intervals.]

The Italian syllables should be used in singing the exercises.
Movable -do- is advised. Any instrument may be used by the teacher or
co-operator to play the exercises. For low voices, the exercises should
be played two octaves lower than indicated, and for medium voices, one
octave lower. The teacher, after having played an exercise, should
explain the interval or intervals therein. The student should sing the
exercise first with the instrument and then without. Each exercise
should be faultlessly intonated before proceeding to the following
exercise.


GROUP I.

    THE NORMAL INTERVALS OF THE MAJOR SCALE.

1. Major scale:--

[Illustration: [music]]

2. All intervals of the major scale:--

[Illustration: [music]]

3. Tonic triad:--

[Illustration: [music]]

4. Perfect fifth:--

[music]

The teacher should use various rhythms besides those given.

5. Perfect fourth:--

[Illustration: a)

[music]]

[Illustration: b)

[music]]

6. Perfect octave:--

[Illustration: a)

[music]]

[Illustration: b)

[music]]

7. Perfect intervals combined:--

[music]

The teacher should combine these intervals in various ways and in
several different rhythms and the student should notate the exercise.
The teacher may also choose some melody free from accidentals and play
it slowly while the student notates. Such practice accomplishes a
two-fold result, ability to notate rhythm as well as intervals.

8. Major third:--

[Illustration: [music]]

9. Minor sixth:--

[Illustration: [music]]

10. Major sixth:--

[Illustration: [music]]

11. Minor third:--

[Illustration: [music]]

The first measure of exercise 2 contains the major second which is an
easily recognized interval. The last measure but two of exercise 2
contains the major seventh. This interval is ordinarily a difficult
interval to intonate but coming as it does in exercise 2 it is easy
to intonate because of the ascending scale on the second half of the
measures. For the present, it is not advisable to practice the major
seventh except in some such sequence as exercise 2. All other normal
intervals may be practiced separately and in combinations. After the
student has become thoroughly proficient in recognizing and properly
intonating all the intervals in group I, he may proceed to the
intervals of the minor scale found in group II.


GROUP II.

    THE NORMAL INTERVALS OF THE MINOR SCALE.

1. Melodic minor scale:--

[Illustration: [music]]

2. Harmonic minor scale:--

[Illustration: [music]]

If the student finds difficulty in singing the harmonic form with the
awkward augmented step, the singing of this scale may be postponed.

3. Play alternatively the major scale and both forms of minor, and
require the student to distinguish between them. Do not proceed until
the student is capable of recognizing and distinguishing between all
diatonic scales.

    4. All intervals of the harmonic minor scale:--

    [Illustration: [music]]

    5. All intervals of the ascending melodic minor scale:--

    [Illustration: [music]]


    6. The minor triad:--

    [Illustration: [music]]

    7. The minor third:--

    [Illustration: [music]]

    8. The major sixth:--

    [Illustration: [music]]

    9. The minor sixth:--

    [Illustration: [music]]

    10. The major third:--

    [Illustration: [music]]

11. Play slowly several minor melodies free from modulations and
require the student to notate.

The teacher should now play melodies in which are transitions from
major to -parallel- minor and -vice versa-. Great familiarity with the
normal intervals is necessary before studying altered intervals. It is
hoped that the major seventh may now be properly intonated.

The student may experience considerable difficulty with the following
group, in which case it is advisable to postpone this group until
the ear is more thoroughly trained. The object of its following the
normal intervals is to fix firmly the fundamental and all intervals by
comparison with this fundamental.


GROUP III.

    ALTERED INTERVALS.

    1. The chromatic
    scale:--

    [Illustration: [music]]

In syllabicating the chromatic scale or any of the altered intervals,
the syllable -ah- may be used on each tone. To those wishing to adhere
to the Italian syllables, the tonic sol-fa syllables, invented by Miss
Sarah Ann Glover, may be used which are as follows: ascending chromatic
scale---doh-, -de-, -ray-, -re-, -me-, -fah-, -fe-, -soh-, -se-, -lah-,
-le-, -te-, -doh;- descending chromatic scale---doh-, -te-, -ta-,
-lah-, -la-, -soh-, -sa-, -fah-, -me-, -ma-, -ray-, -ra-, -doh-. Miss
Glover changed the spelling of the Italian syllables to coincide with
the English pronunciation. She also changed the subtonic from -si- to
-te-.

2. -The augmented fourth- is found as a scale interval between the
fourth and seventh steps of the major scale (-fah- to -te-). It is more
difficult to conceive and intonate properly the augmented fourth when
it is constructed upon the tonic. In order to make this interval less
difficult, the following exercise contains the intermediate scale steps
previous to the skip of an augmented fourth:--

[Illustration: [music]]

It may be necessary with some students to interpolate some or all the
intermediate steps previous to skips to all altered intervals. It may
not be amiss to state here that in correct melody writing augmented and
diminished intervals are usually avoided. Singers almost invariably
intonate them out of tune. When these intervals exist as constituent
parts of an arpeggio chord progression, they are comparatively easy.
As altered intervals these skips are given to instruments (instruments
being capable of properly intonating all skips) when a dramatic effect
is desired.

3. -The diminished fifth- is analogous to the augmented fourth. Being
the inversion of the augmented fourth, it is found as a scale interval
between the seventh and fourth steps (-te- to -fah-). The following
exercise contains the diminished fifth built upon the tonic:--

[Illustration: [music]]

4. -The augmented fifth- is analogous to the minor sixth. It is found
as a scale interval between the third and seventh steps of the harmonic
minor scale and ascending melodic minor scale. The following exercise
contains the augmented fifth built upon the major tonic:--

[Illustration: [music]]

5. -The diminished fourth- is the inversion of the augmented fifth and
is analogous to the major third:--

[Illustration: [music]]

6. -The augmented second- is analogous to the minor third. It is found
as a scale interval between the sixth and seventh steps of the harmonic
minor. The following exercise contains the augmented second built upon
the major tonic:--

[Illustration: [music]]

7. -The diminished seventh- is the inversion of the augmented second
and is analogous to the major sixth:--

[Illustration: [music]]

8. -The augmented third- is analogous to the perfect fourth. This
interval is found in the altered minor triad between the third and the
raised fifth of the triad. Derivation of the augmented third:--

[Illustration: [music]]

E II]

Exercise:--

[Illustration: [music]]

9. -The augmented sixth- is analogous to the minor seventh. This
interval is found in the augmented sixth chord. Origin of the augmented
sixth chord:--

[Illustration: [music]]

E II]

-The diminished sixth-, which is the inversion of the augmented third
and analogous to the perfect fifth, is not used melodically. -The
diminished third-, which is the inversion of the augmented sixth and
analogous to the major second, is but seldom used melodically.

If the singing of the harmonic minor scale has been postponed, it
should now be practiced. Exercises containing the major seventh may now
be given.


GROUP IV.

    ARPEGGIO CHORDS.

If the fundamental is not yet fixed in the student's mind exercises
pertaining to groups I and II should be given before proceeding. The
intervals already given should be expanded, the major second to a major
ninth, the major third to a major tenth, etc., etc.

In practicing the following exercises, the student should name the
intervals between consecutive notes and between each note of the chord
and the fundamental.

    1. The major triad:--

    [Illustration: [music]]

    2. The minor triad:--

    [Illustration: [music]]

    3. The diminished triad:--

    [Illustration: [music]]

    4. The augmented triad:--

    [Illustration: [music]]

    A III'

    5. The dominant seventh chord:--

    [Illustration: [music]]

The teacher should explain the dominant seventh chord and its
resolution. Also give exercises on the skip of a minor seventh.

6. The supertonic seventh chord:--

[Illustration: [music]]

7. The subtonic seventh chord:--

[Illustration: [music]]

8. The diminished seventh chord:--

[Illustration: [music]]

A great many exercises on these chords should be given together with
the natural resolution of the dependent chords. Exercises on the
inversions of these chords may be given when the student has obtained
proficiency on the fundamental position. The inversions may be found in
Chapter III.


GROUP V.

    TWO VOICED CHORDS.

The student should name the interval that one voice forms with the
other. The upper melody should then be sung as the exercise is played.
Repeat the exercise, the student singing the lower melody this time. If
the student experiences difficulty in naming the intervals, the chords
should be played in arpeggio style.

1. Thirds:--

[Illustration: [music]]

2. Sixths:--

[Illustration: [music]]

3. Mixed intervals and rhythm (contrapuntal):--

[Illustration: Cherubini

[music]]


GROUP VI.

    THE FOUR VOICED CHORD.

It becomes necessary to use organ or piano for this group. The student
should name the kind of chord and sing the upper voice.

    1. The primary triads:--

    [Illustration: [music]]

2. The primary and secondary triads:--

[Illustration: [music]]

3. Introducing the dominant seventh chord:--

[Illustration: (a)

[music]]

[Illustration: (b)

[music]]

4. Introducing the dominant and secondary seventh chords:--

[Illustration: (a)

[music]]

[Illustration: (b)

[music]]

The chorals that were chosen for analysis in Chapter III should now
be played for ear training. The teacher's judgment is very necessary
in deciding the limitations of each individual student. At the
proper time modulations may be made. Before the student may be called
proficient, he must be capable of instantly recognizing and properly
intonating any and all chords sounded.




Can You Compose Music?


In this, the latest of methods for the study of Harmony, the author,
Carl E. Gardner, presents a system of training which, both in purpose
and plan, provides, what up to the present has frequently been hinted
at, but never practically accomplished--a "direct" method for the
teaching of music composition.

In the writing of the work the student's practical development has been
uppermost in the author's mind and to this end he has provided not a
mere treatise on musical grammar, along conventional, hackneyed lines,
but a new method which will -allow and encourage the student to compose
as he advances and develops-.

    MUSIC COMPOSITION

    A NEW METHOD OF HARMONY

    BY

    CARL E. GARDNER

    Author of "Essentials of Music Theory."

    Price, $1.50

       *       *       *       *       *


OPINION OF THE PRESS

"His work--numbering 161 small pages--is of necessity simply an
abbreviated affair. It is soundly done, the work of a man who knows his
subject through and through, and it is capitally written."

    MUSICAL AMERICA.

"The author calls his method the direct method, in that he makes the
pupil begin to compose from the beginning instead of after a long and
tedious course of technical rules. Time will tell whether this new
method will make better composers than the old way or not, but the
new method will certainly make the way of the pupil less thorny. The
average student will probably enjoy learning composition according to
the method by Carl E. Gardner. There is no reason whatever why this
method should not be as useful as the long established methods of
Jadassohn, Prout, Richter and others who believe in keeping the pupil's
nose to the grindstone for several years before furnishing him with
wings."

    MUSICAL COURIER.

"'Music Composition,' a 'new method of harmony,' by Carl E. Gardner,
published by Carl Fischer, New York, is a meritorious text book which
seeks to combine, in efficient manner, the teaching of simple forms
with the customary guidance in chord connection. The abandoning of the
isolated manner in which harmony is generally taught and the stimulus
of life it undoubtedly receives by joining to it symmetry, rhythm and
melody, is undeniably a progress."

    CANADIAN JOURNAL OF MUSIC.

            PUBLISHED BY
            CARL FISCHER
    BOSTON    NEW YORK    CHICAGO