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

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:--

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:--

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:--

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:--

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

major minor 6th 3rd

perf. perf. prime 8th

perf. perf. 4th 5th

aug. dim. 4th 5th

dim. aug. 7th 2nd

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:--

I II III IV V VI VII° ]

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:--

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:--

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:--

7 7 7 7 7 7 7

I II III IV V VI VII° ]

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:--

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:--

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:--

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.)

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.