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.)

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

Middlec258.6
c# ordb274.
d290.3
d# oreb307.6
e325.9
f345.2
f# orgb365.8
g387.5
g# orab410.5
a435.
a# orbb460.8
b488.2

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:—

BassE81.5D290.3
BaritoneF86.3F#365.8
TenorA108.7A435.0
ContraltoE163.0F690.5
Mezzo SopranoF172.6A870.0
SopranoA217.5C1034.6

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

Ratio of Intervals:—

Octave1— 2
Perfect fifth2— 3
Perfect fourth3— 4
Major third4— 5
Minor third5— 6
Major sixth3— 5
Minor sixth5— 8
Major second8— 9
Minor second15—16
Major seventh8—15
Minor seventh9—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.

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:—

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.

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:—

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2. All intervals of the major scale:—

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3. Tonic triad:—

[[Midi]] [[audio/mpeg]] [[XML]]

4. Perfect fifth:—

[[Midi]] [[audio/mpeg]] [[XML]]

The teacher should use various rhythms besides those given.

5. Perfect fourth:—

[[Midi]] [[audio/mpeg]] [[XML]]

6. Perfect octave:—

[[Midi]] [[audio/mpeg]] [[XML]]

7. Perfect intervals combined:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

9. Minor sixth:—

[[Midi]] [[audio/mpeg]] [[XML]]

10. Major sixth:—

[[Midi]] [[audio/mpeg]] [[XML]]

11. Minor third:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

2. Harmonic minor scale:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

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

[[Midi]] [[audio/mpeg]] [[XML]]

6. The minor triad:—

[[Midi]] [[audio/mpeg]] [[XML]]

7. The minor third:—

[[Midi]] [[audio/mpeg]] [[XML]]

8. The major sixth:—

[[Midi]] [[audio/mpeg]] [[XML]]

9. The minor sixth:—

[[Midi]] [[audio/mpeg]] [[XML]]

10. The major third:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

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

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

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

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

Exercise:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

2. The minor triad:—

[[Midi]] [[audio/mpeg]] [[XML]]

3. The diminished triad:—

[[Midi]] [[audio/mpeg]] [[XML]]

4. The augmented triad:—

[[Midi]] [[audio/mpeg]] [[XML]]

5. The dominant seventh chord:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

7. The subtonic seventh chord:—

[[Midi]] [[audio/mpeg]] [[XML]]

8. The diminished seventh chord:—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

2. Sixths:—

[[Midi]] [[audio/mpeg]] [[XML]]

3. Mixed intervals and rhythm (contrapuntal):—

[[Midi]] [[audio/mpeg]] [[XML]]

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:—

[[Midi]] [[audio/mpeg]] [[XML]]

2. The primary and secondary triads:—

[[Midi]] [[audio/mpeg]] [[XML]]

3. Introducing the dominant seventh chord:—

[[Midi]] [[audio/mpeg]] [[XML]]

[[Midi]] [[audio/mpeg]] [[XML]]

4. Introducing the dominant and secondary seventh chords:—

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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.