Transcriber Note:

The natural sign inline does not appear in ADE readers, so I have used the ⊄ mathematical symbol instead. Flats and sharps have been created for the ADE reader for inline use. Mathematical symbols have been used instead of the ties and square shapes as shown in the book.

ESSENTIALS OF
MUSIC THEORY
ELEMENTARY

CARL E. GARDNER

AUTHOR OF "MUSIC COMPOSITION"

NEW YORK

CARL FISCHER

1912

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:—

or	Double whole note or Breve.
	Whole note or Semibreve.
	Half note or Minim.
	Quarter note or Crotchet.
	Eighth note or Quaver.
	Sixteenth note or Semiquaver.
	Thirty-second note or Demisemiquaver.

Occasionally the sixty-fourth note

is used. Following is a table of the relative value of notes:—

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.
	Whole rest.
	Half rest.
	Quarter rest.
	Eighth rest.
	Sixteenth 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:—

equals

or 1 + 12.

equals

or 12 + 14.

equals

or 14 + 18.
etc., etc.

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

equals

or 1 + 12 + 14.

equals

or 12 + 14 + 18.

equals

or 14 + 18 + 116.
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

sextuplet

group of two notes

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 (

), 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 (

) 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 44 or

, 34, 24, 68, 38, 48, 98, 128, and 22 or

also called alla breve. Other rhythms not so common are 64, 84, 14, 28, 12, 62, 32, 42, 316, and less often 11, 54, 58 and 516, etc.

On the first pulse of all kinds of rhythm is a primary accent called thesis. Secondary accents, called arsis, occur in 44 on the third count and in 68 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: >, ^, 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]

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:—

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 (b).

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 (b) 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# fx 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# fx g# a# b# cx 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# cx d# e# fx gx 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# fx gx a# b# cx dx 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# cx dx e# fx gx ax 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 bb 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 bb and has two flats:—

Bb c d eb f g a Bb
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 Bb is eb and has three flats:—

Eb f g ab bb c d Eb
1 2 3∼4 5 6 7∼ 8

The fourth of Eb is ab and has four flats:—

Ab bb c db eb f g Ab
1 2 3∼4 5 6 7∼8

The fourth of Ab is db and has five flats:—

Db eb f gb ab bb c Db
1 2 3∼4 5 6 7∼ 8

The fourth of Db is gb and has six flats:—

Gb ab bb cb db eb f Gb
1 2 3∼4 5 6 7∼8

The fourth of Gb is cb and has seven flats:—

Cb db eb fb gb ab bb Cb
1 2 3∼4 5 6 7∼ 8

The fourth of Cb is fb 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: bb) lowers a tone one whole step.

Fb gb ab bbb cb db eb Fb
1 2 3∼4 5 6 7∼ 8

The fourth of Fb is bbb and has nine flats (two double flats, bbb and ebb):—

Bbb cb db ebb fb gb ab Bbb
1 2 3∼ 4 5 6 7∼ 8

The fourth of Bbb is ebb and has ten flats (three double flats, bbb, ebb and abb):—

Ebb fb gb abb bbb cb db Ebb
1 2 3∼4 5 6 7 ∼ 8

The fourth of Ebb is abb and has eleven flats (four double flats, bbb, ebb, abb and dbb):—

Abb bbb cb dbb ebb fb gb Abb
1 2 3∼ 4 5 6 7∼ 8

The fourth of Abb is dbb and has twelve flats (five double flats, bbb, ebb, abb, dbb, and gbb):—

Dbb ebb fb gbb abb bbb, cb Dbb
1 2 3∼4 5 6 7 ∼ 8

Dbb 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, Dbb 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)
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)

F 1 flat (b)
Bb 2 flats (b and e)
Eb 3 flats (b, e and a)
Ab 4 flats (b, e, a and d)
Db 5 flats (b, e, a, d and g) enharmonical to C#
Gb 6 flats (b, e, a, d, g and c) enharmonical to F#
Cb 7 flats (b, e, a, d, g, c and f) enharmonical to B

The enharmonic keys used in practice are:—

B (five sharps) enharmonical to Cb (seven flats)
F# (six sharps) enharmonical to Gb (six flats)
C# (seven sharps) enharmonical to Db (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 Abb 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:

. Middle C is found on the first ledger line below the G clef, thus:

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: