A scale is a sequence of pitches based upon a definite pattern of intervals. All commonly used scales are organized within the framework of an octave, and can be produced at any other octave level. Because of equal temperament tuning (a system of tuning instruments, developed roughly during the time of J.S. Bach, in which the octave is divided into exactly twelve equal semitones), a particular scale may begin on any desired pitch. Scales move continuously in one direction, and the descending pattern of any scale is usually the reverse of the ascending pattern, although there are significant exceptions. Each tone in a scale which contains seven tones or less is usually given a different letter name.

Historically, scales were derived theoretically by the analysis of a large body of music. In other words, the actual music was written before the materials upon which it was based were necessarily fully analyzed. In more recent times this procedure is sometimes reversed, and the basic pitch materials are first organized in a scale or some other form, and a composition based upon this material is then composed. In ancient music the complete pattern of the scale is rarely found in the music, but the scale used as the basis of a particular composition can generally be detected by the presence of certain characteristic melodic formulae. Toward the end of the sixteenth century complete scale passages became more and more a feature of the music ­ notably in instrumental music. The practising of scale patterns has long served as a basis for the development of technique on musical instruments.

More music is probably based upon this scale than upon any other. Most people who have had any contact at all with music are familiar with the singing and the sound of the pattern DO RE MI FA SOL LA TI DO. Let us sing it now. This is the sound of a major scale.

The major scale consists of a sequence of eight pitches. There is an interval of a tone between each of these, except between the third and fourth, and the seventh and eighth pitches which are a semi tone apart. (A tone is an interval equal to the value of two semi tones.) The major scale pattern can be most readily heard and seen by simply sounding the white keys of the keyboard from C to C. Note that there is no black key between E and F, and between B and C, because they are only a semitone apart; there is a black key between each of the others and they are a tone apart. Another way of describing the major scale pattern is simply:- Tone Tone Semitone Tone Tone Tone Semitone. Note that the last note is the octave of the first.

Let us now construct a major scale beginning on a given note - Ab. Here is a suggested step by step procedure:
­ 1) Write up eight notes on successive lines and spaces beginning on Ab.

Figure 16

2) Number each note from 1 to 8 and indicate the semitones by means of a bracket, joining the third and fourth, and seventh and eighth notes.

3) Check each interval. If it is the correct interval, leave it alone; if not, change the upper of the two notes to make the interval correct. A piano keyboard (or a diagram of one) is a useful aid in checking intervals. Remember - ­ any two adjacent keys are a semitone apart; one intervening key between two notes means that the interval is a tone, two intervening keys indicates an interval of a tone and a half.

What is the interval between the first two tones, Ab to B?
A tone and a half.
What do we want, according to our definition of a major scale?
A tone.
How do we achieve this?
Change B to Bb.

Continue this process until the scale is complete. This should be your result. It is called the scale of Ab major.

Figure 17

(If you started on the Ab an octave lower and had all the same pitches an octave lower, it would also be correct.)

As a final check play what you have written on a keyboard or other instrument. It should sound like DO RE MI FA SOL LA TI DO.

Problem: ­ Notate the scale of F# major, using whole notes in the bass clef. Construct this scale by following the above procedure.

Figure 18

Note:­ The numbers and brackets are merely aids in working out the scale, and are not really necessary in the completed notation.

ASSIGNMENT

We have noted in writing and performing major scales that although different ones contain different pitches, they all sound basically the same, and are all, therefore, major scales. What is the one factor that is constant, and that causes the "tune" always to sound the same? Answer:- the pattern of intervals. What is true of a major scale is true of any piece of music. It can be started on any pitch and still come out right if the pattern of intervals is kept unchanged. This phenomenon of reproducing a given piece of music by starting it on a higher or lower pitch than the original is known as TRANSPOSITION. For instance, sing a familiar tune, and then sing it again starting on a different note. If the melody sounded the same in both instances, that was because the distance or interval between each adjacent pitch was kept unchanged. Of course, there will be some difference in the overall effect of each example because of the high or low range in which the melody is performed.

I Tonic - the most important degree in the scale. Its magnetic power tends to draw all other pitches in the scale towards it, and most compositions end on the tonic. Major scales begin and end on the tonic (at the octave), and its name is assigned to the entire scale, e.g.,when the tonic is Ab , the scale is Ab major.

II Supertonic - the degree above the tonic.

III Mediant - comes halfway between the tonic and the second most important degree.

IV Sub dominant - the degree below the second most important note of the scale. Harmonically, it is third in importance, and it occupies the same relative position below the tonic as the dominant (see below) occupies above the tonic.

V Dominant - next in importance to the tonic. Its importance will be more clearly understood when we come to study harmony.

VI Submediant - occupies the same position between the tonic above and the subdominant below as the mediant occupies between the tonic and the dominant.

VII Leading tone - a semitone below the tonic, and has the natural tendency to rise into it.

Figure 19

Always use Roman numerals when referring to degrees of a scale.

Since all music is drawn toward, and ultimately into, the tonic, the tonic is also referred to as the "key-note". Consequently, any music which is built around the notes of any particular major scale is thought of as being in that major key and is labelled according to the key-note. For instance, if a piece revolves around the notes of the Db major scale, it is in Db major. Therefore, for every scale there is a corresponding key.

ASSIGNMENT

We find that melodies based upon particular major scales utilize primarily. just the notes of that scale. Occasionally, tones not belonging to the scale are used to "color" the music. You may have noted in writing a melody in D major that you had to use F# and C# consistently throughout the music. The key signature is a device whereby the accidentals required consistently for music in any particular key are placed at the left hand side of each line of music on the lines or spaces of the notes to be altered, and the performer realizes that throughout the work these accidentals will always be applied to those particular notes. For example, in D major sharps are placed on F and C at the beginning of each line of music, and all F's and C's are played or sung F# and C#.

We know that, in constructing these scales, all except C require one to seven sharps or one to seven flats, but we have also noted that the sharps and flats are never mixed in the same scale. If we examine all the scales containing sharps we find that starting with G major which uses one sharp (F#) we progressively add a sharp to each of the six others until we reach C# major which uses seven sharps. The order in which these sharps were added is F# C# G# D# A# E# B#. Here is how the seven sharps would be notated in the key signature for C# major in the four clefs. For any other key signature using sharps simply drop the unnecessary sharps.

Figure 20

The same procedure as we have just used can be applied to flat keys. The order in which the flats occur is Bb Eb Ab Db Gb Cb Fb, exactly the reverse order of the sharps. Therefore, the key signature for Cb major requiring all seven flats is written as follows in the four clefs:

Figure 21

ASSIGNMENT

We have now discovered all the major key signatures by constructing major scales on each of the twelve semitones of the octave. Let us now seek another relationship which might exist between all major scales. Examine the following C major scale and see if you can discover any aspect of its construction other than the basic tone semi tone relationship.

Figure 22

Did you notice that if you split the scale right down the middle the top half is identical in construction to the bottom half? Therefore, the top half of C major might be considered the bottom half of G major. What accidentals are required for G major? (One sharp - F#.) Let us now examine the G major scale in the same way. We find that its top half is the bottom half of the D major scale. What accidentals are needed in D major? (Two sharps - F# C#) We can follow this same process straight through all the sharp and flat keys. The fifth degree of each major scale is the first degree of another major scale requiring one additional sharp. This phenomenon is known as the Cycle of Fifths and can be charted as follows:

Figure 23

If you know your order of sharps and flats, this chart might be useful as a memory guide to your major key signatures. However, its main purpose is to demonstrate the underlying theoretical relationship between all major keys. It also serves to point out the relative closeness of relationship between various keys. This will become clearer in the study of harmony, and particularly in connection with modulation.

ASSIGNMENT

Figure 24

What adjectives might you use to describe this tune? (Possible answers:- sad, haunting, melancholy, exotic.) Does it sound like a major key melody? No. The reason is, of course, that it is not based upon the major mode. (We shall use the term "mode" from now on to describe the specific tonal resource of a particular piece of music.) Although perhaps the majority of melodies with which you are familiar are based upon the major mode, a considerable amount of music is based upon other modes.

If this work (Fig. 24) were in the major mode, " according to its key signature it would be in C major. However, what pitch is in fact its tonic? (The last note would be the most significant clue.) Its tonic is A, the submediant of C major, and this piece is based upon the Minor mode. Let us sing the scale of the Minor mode from A to A, all pitches natural. Perhaps it would be simpler to sing the syllables "la" to "la". This is called the Natural Minor Scale, and the best description of its sound would be to say that it is like singing "la ti do re mi fa sol la " . (Remember that the Major scale is the sound from "do" to "do".) Sing some minor scales starting on different pitches:- B, Bb, D. (Use the "solfa" syllables.) Although, of course, this scale has its own tone semi tone structure, we shall not concern ourselves with it as we did with the major scale. Which degree of this scale contrasts it most significantly with the major scale? The third degree, just a semi tone above the second, seems to be predominant in creating the characteristic minor quality of sound.

Relative and Tonic Minor
We have derived the minor scale from a given major scale, and, because of this, it is very closely related in sound to that major scale. It has the same key signature, and therefore utilizes all the same pitches. A minor scale with that type of relationship to a given major scale is called a Relative Minor scale; it has the same key signature and begins on the submediant degree of the major scale. What is the relative minor of E major? C# minor. Let us sing it, starting on C#, but singing the syllable names ("la" to "la").

Let us now sing a minor scale starting on the same pitch as our given major (E) using the syllable names. What actual pitches did we sing? E F# G A B C D E. Therefore, this minor scale has a key signature of one sharp. A minor scale which starts on the same pitch as a given major is called the Tonic Minor. In this case the Tonic Minor of E major is E minor. What must be done to a major key signature to change it to its Tonic Minor key signature? Add three flats. A Tonic Minor scale, then, starts on the same note as a given major, but its key signature contains three additional flats. (Flats cancel out existing sharps, so if we add three flats to a key signature consisting of one sharp, we end up with a key signature of two flats).

Summary:  ­ A Relative Minor scale has the same key signature as a given major scale but starts on the submediant degree of the major scale. A Tonic Minor scale starts on the same pitch as a given major scale but requires three additional flats in the key signature.

ASSIGNMENT

Harmonic and Melodic Minor Scales
What is the most critical difference between the natural minor scale we have been singing, and a major scale? The lack of a leading tone. This lack will become much more obvious when we get into the study of harmony. The fact that the seventh degree of the minor scale did not lead into the tonic by a semi tone step also troubled many composers of the past. They overcame this problem by a habit of altering the seventh degree of the scale and making it into a true leading tone. Let us write the A minor scale, and then alter the seventh degree to make it into a true leading tone. This is done by raising the natural to a sharp. (If the seventh degree were a flat, we would make it natural. If it were a sharp, we would make it a double sharp.)(Click for example)

Figure 25

Sing this scale now. In this form it was now apparently suitable for harmonization, but melodically it was still unsatisfactory to the tastes of many composers. Where is the problem now? The unusual interval between the sixth and seventh degrees (augmented second). Therefore, another form of minor had to evolve for melodic purposes. Raise the sixth as well as the seventh degree of the A minor scale and sing it now. This is called the Melodic Minor scale. In its descending form it is converted back to the natural minor since the raised leading tone is no longer necessary. We always write melodic minor scales in ascending and descending forms, as follows:(Click for example)

Figure 26

It should be noted that in the music of Bach, Beethoven, and others, the harmonic minor scale was on occasion used melodically, and this particular sound is actually favored in certain parts of the world.

Summary:
­ The natural minor scale sounds like singing from "la" to "la", or the playing of the white keys from A to A, or performing from submediant to submediant of any major key. To form the Harmonic Minor, simply raise the leading tone one semitone.

To form the Melodic Minor, raise the sixth and seventh degrees ascending by one semitone, and convert the descending form of the scale back to natural minor.

We have not bothered about the interval pattern of the various minor scales, but, of course, each scale has its own specific pattern of intervals which the student can discover for himself or herself.

Here are all minor key signatures as charted on the cycle of fifths:

Figure 27

ASSIGNMENT

Figure 28

Upon what scale or mode is this melody based? It is pentatonic, using only five notes. It is neither major nor minor, but resembles minor more closely because of the last note and some of the melodic patterns it contains.

A great deal of music, particularly folk music, is based upon the pentatonic scale. Its main characteristic is that it contains only five different pitches within the octave of a major or minor scale, and has no semitones. You can derive it by removing "fa" and "ti" (subdominant and leading tone in the major, or supertonic and submediant in the natural minor), or simply by playing on the black keys of the keyboard.

Figure 29

Find the keynote, and then write the scale upon which this tune is based. {D E F G A B C D.) Is it major or minor? Neither. The pattern of intervals is that formed by playing the white keys from D to D or by singing from "re" to "re". This is called the Dorian mode and its use goes back to the Middle Ages. The body of music known as Gregorian Chant is based upon the church modes, one of which is the Dorian. Some folk songs are also based upon this mode and many contemporary composers are again using it in their works. Like any other scale or mode, it can of course be transposed to any key. There are two basic ways of doing this: one is to analyze the interval structure of the Dorian mode (T S T T T S T) and simply apply it to any key note; the other is to determine the new key signature by considering the key-note of the Dorian key as the "re" of a major key ( remember that the Dorian mode is a scale from "re" to "re").

For example, write the A Dorian mode. If we construct it on the interval pattern we get A B C D E F# G A. If we think of A as "re" in a major key, we have the key signature of G major, one sharp, and arrive at the same answer.(Click for example)

Figure 30

Let us sing and play these in different keys. Note that Dorian has a slight resemblance to minor, Lydian and Mixolydian have qualities of the major mode, whereas Phrygian seems to be a law unto itself. An interesting theoretical point to note about Phrygian is that it has the same interval pattern in its ascending form as the descending form of major, and vice versa. Another way of:saying this is that the Phrygian mode is the inversion of the major mode.

A scale form which is strictly a development of recent times is the Whole Tone Scale. This scale contains only six notes within the octave, with an interval pattern of T T T T T. Music based upon this scale has a unique exotic quality, and, because there are no semitones, there is no strong feeling of tonic. There are only two tonal possiblities for this scale - one starting on C and one starting on Db. If you start on any other pitch, you have simply a variant of one of these two scales. Sing and play whole tone scales starting on different notes.

A Chromatic scale is the sequence of the twelve semitones within the octave. There is only one tonal possibility for this. This particular scale is not really usable as the basis of a musical composition. However, a modern technique in musical composition utilizes the twelve semitones of the octave in a particular manner as a basis for composition. This is called the "twelve tone" or "dodecaphonic" ­ style of composition, and this kind of music is often referred to as "atonal" music.

Simply stated, this is how the technique operates: the composer arranges the twelve tones of the octave in a sequence of his own invention. He then proceeds to use this "tone row", the main rule being that the notes must always be presented in sequence, although they can be in any rhythmic pattern, and can occur either consecutively or simultaneously. The resultant music differs radically in effect from all music of the past. Further information on this technique, invented by Schoenberg, can be found in any number of books and articles on music.

The above are the principal scale patterns used as the foundation for most western music of the last fifteen hundred years. Of course, new "synthetic" scales can be, and are, sometimes invented by composers who are seeking out new sonorities.

The important point to remember is that the scale itself is not significant except for its use in actual music, and that any scale may be transposed to any tonality with a minimal sacrifice of its intrinsic quality.

As an illustration of the variety of sound possible in music, here are ten scales,. all in the tonality of D. It may be interesting to test a group in their ability to identify these various modes by ear. However, remember that this is not any easy task for the inexperienced ear.

Figure 31

ASSIGNMENT

Index

An interval is the distance in sound between any two pitches sounded concurrently or consecutively. Each different musical interval has its own name for purposes of identification, although as in the naming of single pitches more than one name is possible for each interval. Here are two intervals with which you are already familiar (Click for example) :

Figure 32

They are the perfect octave and the semitone. The perfect octave produces a sound as pure as it is possible for two notes to produce without actually being the same note (unison). In other words, when you hear an octave, it is almost like hearing one note, but not quite: Therefore, the octave would have to be considered our most consonant interval. The semitone , on the other hand, is a sort of paradox, as the distance between the two notes is as small as is possible in conventional tuning, but when the two notes are sounded simultaneously, extreme dissonance is produced which implies that in sound the two notes are actually very far apart.

Each interval can be reproduced from any pitch provided that the distance between the two notes (i.e. the number of semitones) remains constant. Here are some other examples of octaves and semitones (Click for example) :

Figure 33

Let us see if we can identify a few intervals by ear. This is the sound of the perfect fifth, perfect fourth, and major third (Click for example) :

Figure 34

(Play various perfect octaves, perfect fifths, perfect fourths, and major thirds for ear training. (Click for examples - Perfect 5th, Major 3rd, Octave, Perfect 4th, Octave, Perfect 5th, Perfect 5th, Perfect 4th) Do them consecutively, playing the lower note first; but it often helps if the lower note is sustained while the second note is being sounded. As much practice of this type as is possible is desirable.)

The Naming of Intervals
As you probably have noticed, the name of each interval has two parts, e.g. Perfect Fifth. The first part is called the quality and the second part is called the number. Let us now find out how we apply names to intervals. We will deal first with the number.

Procedure:  ­ To determine the number of any interval count each successive letter name from the lower note to the higher, including both. If the number is greater than an octave, the interval might be called Compound, e.g. octave plus third is compound third, or it might be called the actual number (a tenth). Ninths are usually called ninths, although intervals greater in number are often called compound.

Figure 35

Now we will investigate how we name the quality of the intervals. There are five possible names in use for intervals - major, minor, perfect, diminished, augmented. 2nds, 3rds, 6ths and 7ths may be major, minor, diminished or augmented. Unisons, 4ths, 5ths, and octaves may be perfect, diminished or augmented. The relationship between these various names is as follows:

Therefore, to determine the quality name of an interval, we must first decide whether the interval is major or perfect, and then work from there.

Rule:- If the top note is a part of a major scale of which the bottom note is the tonic, then the interval is major or perfect according to its number. Your first step must be then to look at the lower note of an interval and think of it as the tonic of a major scale.

Figure 36

Before one can do intervals with any degree of fluency, one must be extremely familiar with all major key signatures. Always associate the names of intervals with their sounds, and practise hearing intervals as much as possible.

Figure 37

(From now on we will utilize the following symbols to denote the quality of intervals:­ + major, - minor, P perfect, ° diminished, x augmented.)

Play the intervals (Figure 37) and check their sounds.(Click for example) Actually, we have only three sounds, although we have six names. This phenomenon of enharmonic names, whether applied to single pitches or to intervals, is not new to us.

Figure 38

The above are listed according to size. Let us now reorganize these intervals according to relative consonance and dissonance. In order to determine this aurally, the two notes should be played or sung simultaneously. There is a scientific method of explaining relative consonance and dissonance, but we will do it more or less strictly on aural intuition, and this is what we come up with:­

Figure 39

Awareness of the sound of intervals in this manner may help us in hearing and identifying different intervals. Two things to be particularly aware of are whether the pitches are close together or far apart, and whether they are relatively consonant or dissonant.

Besides being able to name intervals, we also have to write intervals above and below given notes. Writing an interval above a given note should not present any difficulty if one can name intervals properly. However, writing an interval below a given note does bring up a few problems in procedure because of the fact that all intervals are named by checking the number and quality from bottom to top. In other words, the bottom note is your main reference note.

Notice that a flat added to the lower note increases an interval, and an added sharp would reduce the interval. Practice writing intervals below given notes.

Inversion of intervals.
Sing a P5 above E. Now sing a P4 below E. What have you discovered? In both cases the note B (an octave apart) produced the correct interval. The second interval you sang is called the inversion of the first one, and vice versa. Each interval has its corresponding inversion. For instance, C up to E is a major third. Drop the E an octave below C and you have a minor sixth. Similarly, what is an augmented 2nd above C? D#. When the D# is dropped below the C, a diminished 7th is now produced.

From these observations what general conclusions can we arrive at regarding inversions?
1) An inversion of an interval is simply the completion of an interval to its octave.
2) The number of an interval plus the number of its inversion adds up to nine. In order to determine the number of the inversion of an interval, subtract the number of the interval from nine.
3) Major intervals become minor, and vice versa.
4)Perfect intervals remain perfect.
5)Diminished intervals become augmented, and vice versa.

Knowledge of inversions is particularly useful in writing intervals below a given note, because the correct note can now be found simply by writing the inversion of the required interval above the given note and then transposing it down an octave. This knowledge becomes particularly pertinent when we come to deal with triads. In making the inversion of a compound interval consider it in its simple form; in writing it, complete the interval to two octaves.

ASSIGNMENT

A less practical, although perhaps more meaningful, means of measuring intervals is simply by counting the number of semitones within each interval.

   Twelve  -  perfect octave, augmented seventh, diminished ninth.
   Eleven  -  major seventh, diminished octave.
   Ten     -  minor seventh, augmented sixth.
   Nine    -  major sixth, diminished seventh.
   Eight   -  minor sixth, augmented fifth.
   Seven   -  perfect fifth, diminished sixth.  
   Six     -  diminished fifth, augmented fourth.
   Five    -  perfect fourth, augmented third.
   Four    -  major third, diminished fourth.
   Three   -  minor third, augmented second. 
   Two     -  major second, diminished third.
   One     -  minor second, augmented unison.

This is useful information for the student, but the preceding method of writing and naming intervals is felt to be the more practical one. However, it is the end result (fluency) that counts, and any means that assists in attaining this goal is acceptable.

Index

The simultaneous sounding of any number of tones constitute what we might consider a chord. Sometimes a chord may be "broken" so that the pitches are sounded consecutively, but the chord feeling remains, either because each tone is sustained against the succeeding one, or because of the familiarity of this combination of tones as a chord sound. The purpose of chords is to provide a fullness to the musical tex­ ture, often as accompaniment to a melodic line, and also to provide a certain tonal direction to the music. As we get into our study of harmony these concepts will become clearer.

The TRIAD is the most familiar and most basic chord form. It consists of three pitches, each one the interval of a third higher than its predecessor. The fundamental note of the triad is called the ROOT, the third above it is called the THIRD, and the fifth above it is called the FIFTH (makes sense). Therefore, we can also analyze the structure of a triad by saying that it comprises an interval of a third and an interval of a fifth above a given root.

Let us sing a triad. One group will sing C, one will sing E, and one will sing G. (Sing it on "ah" and stress a good clear ringing major triad.)Click here for example This is what we sang:

Figure 40

Figure 41

Figure 42

Figure 43

Triads are named according to the root and the kind. For example, we have just sung and written a C major, C minor, C diminished, and C augmented triad.

Let us practice writing a few triads: F# °, Ab +, B x, Bb -. Simply follow the proceedure of writing the proper third and fifth above the root.

Figure 44

Inversions of Triads
Let us first write and then sing a D+ triad. (Have male voices all sing lowest part, or if there is a fairly equal mixture of men and women, divide both groups into three parts.)

(a) Figure 45

We have just sung this major triad in what is known as ROOT POSITION. This means that the lowest voice is the root of the triad. Its effect is one of clarity and strength. We will continue to sing this same D+ triad, but change the voicing so that the third, and then the fifth become the lowest parts.

(b) (c) Figure 46

All triads of all kinds can be inverted in this manner. For convenience, a numbering is often used to specify the inversion.

Root position is 5/3, a fifth and third above the bass. First inversion is 6/3, a sixth and third above the bass. Second inversion is 6/4, a sixth and fourth above the bass.

Figure 47

NOTE: In order to determine the root and the kind of any given triad in first or second inversion, it is desirable to first re-juggle the pitches to revert the triad to root position, and then analyze.

ASSIGNMENT

Triads in relation to major keys
So far, we have been considering triads. as isolated entities. In the study and practice of harmony, we tend to think more of triads as they relate to a given key. Let us consider the variety of triads available to any major key. For purposes of convenience, we will work in C major, but what is true of C major is true of all major keys.

Write the scale of C major in the treble clef, and then write a triad in root position above each degree of this scale using only the pitches of C major.

We will divide in three parts and sing what we have written.(Click for example) Now we will number each degree with the proper Roman Numeral and analyze each triad to determine its kind.

Figure 48

Therefore, in any major key, there are three major, three minor, and one diminished triad which will be used as the primary materials for harmonization. The most important or key triads are the major ones, I, V, and IV. The others are secondary, but are useful for additional color and variety.

We must become extremely fluent in identifying and writing triads on any degree of any key. We will limit our work at this point strictly to major keys. Remember, what­ ever the major key, vii° is always diminished, ii is always minor, etc.

ASSIGNMENT

Index

Part Three
HARMONY

In a sense, we have already been dealing with harmony, since one concept of the meaning of the word "harmony" suggests the blending together of different sounds. In writing and singing triads (and intervals) we have already produced harmony in this sense. However, there is also another aspect to the meaning of the word, and it is with this aspect that we shall be mainly concerned at this point.

What is being referred to here is the relationship or interconnection between chords. When we think of the musical effect of a series of chords, we are thinking of CHORD PROGRESSION. With every melody there is a logical sequence of chords which can be used to enhance and accompany it. Sometimes there are several possibilities available in choice of chords which would be almost equally effective, and that is why harmony is to some extent a creative endeavour. However, if one's objectives are clear, then there are certain rules of style and logic which can be used as guides in the choice of chords.

In order to successfully harmonize melodies, you must be thoroughly familiar with the tone materials with which you will be working. What are these materials?

1.   Theoretically, any kind of chord, or series of chords is available. However, for the present, we will limit our­ selves to all the diatonic triads within any given key and mode. You should know which ones are major, minor, diminished and augmented.
2.   You must now also think of the music in connection with a definite medium. You must know if you are harmonizing for voices, or for any particular instrument or combination of instruments. The same chord progression written for different media would in each case be handled differently, and would produce a somewhat different result. The manner, or the medium, in which harmonies are presented, is often thought of as the TEXTURE of the music.

Let us illustrate this point together. We will select a chord progression, and write it and perform it in different media. We will work in C major, although what is done in C major, theoretically, could be transposed to any other key. However, certain limitations now begin to affect our choice of key: primarily, these are the limitations of range and technical flexibility of any given voice or instrument.

The progression of triads that we will write and perform in C major is as follows: I vi ii V I. We will arrange this chord progression for (a) the autoharp (b)the piano and (c) four voices.

The Autoharp
On this instrument the names of certain chords are written directly on each of the bars that are used in playing the instrument. When a particular bar is depressed it blocks off or dampens all the strings that are not needed for that chord, and when one strums the instrument only the notes required for the selected chord will sound. What is it then that we must know in order to play our progression on the auto­ harp?

You will notice that in playing the progression on the auto­harp we do not concern ourselves with individual notes since this is taken care of for us by the instrument. However, we should decide on a particular rhythm, and know how long we wish to retain each chord.
For example, in 4/4 time we might play it like this:(Click for example)

Figure 49

What other possibilities can you suggest? (try different ones) Of course, this would depend upon the type of song you were accompanying, its rhythm and its metre.

Harmonic Rhythm
This term refers to the relative duration of chords within a given piece of music. If, as in Bach chorales, the chord changes are frequent (more or less one on every beat), then the harmonic rhythm is considered to be fast. If, on the other hand, the tendency is to retain the same chord over one or several measures as in many simple folk songs, then the harmonic rhythm is considered to be slow. Harmonic rhythm usually affects the character of music and more tension is generally built up with a fast harmonic rhythm. (Comparisons might be made in this regard between the music of Bach or Handel, and that of Mozart or Haydn.)

It is possible to play our chord progression on the guitar similarly to the way in which we played it on the autoharp. Many guitarists do not concern themselves with individual notes or chords, but simply with the finger pattern they must use for each chord. Therefore, the music for the guitarist is often written in the form of a tablature, showing the placement of the fingers on the frets and strings for each chord. The vertical lines represent the strings, the horizontal lines represent the frets, and the black dots represent the fingers. The selection of harmonic rhythm would be the same as that of the autoharp.

Figure 50

The Piano
To write this chord progression for piano, what additional information must we have?

Answer: We must know the exact notes needed for each triad, and we must notate them properly for the pianist. Let us plan to write this progression as the left hand accompaniment to a waltz. First, we will write out the chords in full on a staff so that we will know what notes are required. Write each triad in root position.

Figure 51

Now let us write this progression for the left hand in a waltz accompaniment pattern -- oom pa pa ; 3/4 time. The usual way to do this is to strike the root of the chord in octaves on the first beat of each measure, and the complete triad on the second and third beats. In writing out the second and third beats, don't worry about writing the triad in root position since the feeling of the octave on the first beat will carry through the bar. Therefore, our concern should be in making the triads connect as smoothly as possible, one to the other. (Work this out on the blackboard together with the class). Most logical result: ­­

Figure 52

Of course, it is possible on the piano to set this harmonic progression to an infinite variety of textures and rhythms.

Let us now write this same harmonic progression for the standard mixed voice choir -- soprano, alto, tenor, . and bass.

Before we can do this, we must be familiar with the workable ranges of these voices. Here they are:

Figure 53

In writing four part harmony for the mixed choir, we will use only treble and bass clef, putting the soprano and alto on the treble and the tenor and bass on the bass staff. Often these parts are placed on four separate staves (the tenor part being written an octave higher than it actually sounds in the treble clef). However, it is easier for beginners to read the harmony on two rather than four staves. One way of writing our progression for SATB is as follows:Click for example

Figure 54

Let us divide into four parts and sing this example to the syllable "ah". Once the class has learned to sing this progression properly, the teacher can play many popular melodies such as "Heart and Soul", "Blue Moon" and "These Foolish Things" on the piano or other instrument to the accompaniment of this chorus).

Now study the music and point out any features you notice.

We will pursue our study of harmony for the present in the SATB medium. We will be concerned with discovering, writing, and singing various harmonic progressions using the diatonic chords in a given key. Because it is necessary for the beginner to have definite limitations within which to work, we will impose some rules. These rules are, in a sense, arbitrary, but they are justified by being part of a long tradition of choral music.

This term refers to the individual movement or melodic character of each volce as considered in isolation as well as in relation to the other voices. Here are rules to follow for the present:

1.   Make each voice as smooth as possible avoiding any unnecessary leaps. Wherever possible, keep the common tone between two adjacent chords in the same voice.

2.   Consider the relationship between soprano and bass parts. There should be a fair amount of contrary motion (when one part moves up, the other part moves down and vice versa), some oblique motion (one part stands still while the other moves) and occasional similar motion.

3.   Consecutive perfect fifths or perfect octaves between any two parts are strictly forbidden.

4.   The leading note should rise to tonic.

Spacing
This refers to the interval between any two adjacent parts: soprano and alto, alto and tenor, tenor and bass. An interval greater than an octave is allowed only between tenor and bass. The interval between the other parts should always be an octave or less.

Doubling
For the present, always double the root of the chord. Occasionally, it will be found necessary to triple the root of a chord and omit the fifth. This is acceptable. The third of a triad may never be omitted.

These are our basic rules, although many exceptions will gradually be introduced.

ASSIGNMENT

1)   Transpose the choral setting of our harmonic progression to the keys of E major and A major. Sing and discuss these transpositions with regard to any alteration of effect, or any special problems they present.

The Cadence
Certain chord progressions come to be recognized as standard formulas or cliches through constant and recurring use in similar musical situations. When one hears a progression of this type, one immediately receives clues regarding the structure and direction of the music. 0f course, when these formulas are deliberately used in an unpredictable manner, then the opposite effect, usually one of surprise, is created.

The most common of these musical cliches are called CADENCES or cadential formulas. Cadences serve the dual purpose of establishing a key or tonality, and of clarifying the form or structure of the music. When cadences are deliberately avoided, both form and tonality become less clearly defined, and the music assumes a quality of vagueness. A cadence operates much in the same manner as punctuation, signifying the end of a phrase or section in the music, and the degree of finality inherent in this ending; it also emphasizes the key or tonal region reached at this point.

Most cadence formulas consist of two chords, but a third chord is often necessary in order to clearly establish the key. This will be made evident when we study our first cadence.

The Perfect Cadence
This cadence would be similar to a period in language punctuation in that it generally denotes a full stop. The basic chord progression for the perfect cadence is V 5/3 to I 5/3. This is what it looks like in four parts. (Sing it on "ah" and play it on the piano.)(Click for example)

Figure 55

1.   The two chords in isolation do not definitely establish the tonality. This same progression might be thought of as I to IV in G major. In order to clearly establish C major, a third chord is necessary to precede this. The chord must have an F natural in it (ii, IV, or vii°).(Click for example)

Figure 56

2.   This example contains the root of the I chord in the soprano. It would still be a perfect cadence if the 3rd and 5th were used, but the strength and finality of the cadence would be weakened. Let us write and sing examples of perfect cadences with either root, 3rd, or 5th of the tonic chord in the soprano in various keys.

By and large, the part writing in each example will be the same. That is, the root, third and fifth of the V chord will always logically progress to the same note of the I chord. This follows the basic rule of moving as smoothly as possible within any given part, and keeping the common tone wherever possible. Also of having leading tone go to tonic. However, if the fifth of V is made to go to the root (rather than the third) of I, you will notice that we end up with a I chord containing three roots and a third. This is quite acceptable and for the time being we will consider it the only solution to this problem.(Click for example)

Figure 57

Plagal Cadence
This cadence is not really a cadence at all, but rather a conventional musical tag, or epilogue. It is always preceded by a perfect cadence, and is the familiar "amen" found at the end of hymn tunes. Its formula is IV 5/3 to I 5/3.(Click for example)

Figure 58

1)   Because it is preceded by a Perfect Cadence, tonality is clearly established. Otherwise it might also be considered as I to V in F major.

2)  Conventionally, the root of I is in the soprano. Write examples in various keys with the third and fifth of I in the soprano, and note the difference in effect.

3)   Regardless of what note is given to the soprano, the voice leading never varies in this cadence formula.

Let us now write a Perfect Cadence followed by a Plagal Cadence in Eb major. Remember that all chords should be in root position.(Click for example)

Figure 59

Sometimes, in writing this progression (V I IV I) the IV - I is thought of purely as an extension of the Perfect cadence, and when the bass first sounds the root of the I chord, it sustains it right to the end. This puts the IV chord in second inversion, and its fifth is the conventional note to double in this case. ­­

Figure 60

Note that the basic effect of this prognession is a sustained tonic chord with a slight ornamentation in the two inner voices. A sustained bass note which contains chord changes above it is called a PEDAL tone. It is a common device found particularly at the end of many organ works in which an organ pedal is sustained for several measures, and changing harmonies occur above it.

As the perfect cadence resembles the period in punctuation, the half cadence resembles the comma. It is, therefore, to be found at the end of various phrases in a musical work, but never at the end of a section. Frequently, one finds a musical sentence made up of two balanced phrases. This is often referred to as question and answer, or thesis and antithesis. In this case, the first phrase would normally end with a half cadence and the second phrase with a perfect cadence.(Click for example)

Figure 61

There are various forms possible of the half cadence, but in all cases the second of the two chords is a V 5/3. It can be preceded by any triad (iii or vii° would be rarely used) in either root position or first inversion.

Let us write various possibilities of half cadences in A major.

Figure 62

Figure 63

Figure 64

This new cadence formula is called "deceptive" for obvious reasons. It starts out like a perfect cadence but its course is altered. Thus, we are deceived or thwarted. V 5/3 to vi 5/3 is the standard formula for this cadence. Note . the unusual doubling in the vi chord. What is it? (Third). Why do you think it is used? The third of the vi chord is the tonic note of the key, and thus the same note is doubled as would have been doubled if this chord was a I. Thus the ambiguity of this particular cadence is re-inforced. Actually, any chord other than I following V 5/3 would be considered deceptive. However, V to vi is the only form we will consider here. In some ways the word cadence in connection with this formula is a misnomer since it is more likely to occur in the middle rather than at the end of phrases.

Write deceptive cadences in various keys.

Ear Training

1.   Identify the following cadences:

Figure 65

2.   After being informed of the key, write these cadences. (Supplement these examples with others.)

ASSIGNMENT

Above the given bass notes, write the cadences they suggest. Name each cadence, and fully number each triad with the Roman numeral for degree and Arabic numerals for inversion.

Index

Normal Progression

In harmonizing a melody, our primary concerns are with the selecton of chords and with the frequency of chord change: Chord progression and harmonic rhythm. In Bach chorales the harmonic rhythm is relatively fast; generally one chord change on each beat of the measure. McHose (McHose, A. I., Basic Principles of the Technique of 18th and 19th Century Composition, Appleton-Century Crofts, Inc., 1951), through an exhaustive study of Bach chorales has arrived at a basic concept of chord progression which he refers to as "normal progression." Although exceptions are many, normal progression implies the usual chord choices one would find in Bach chorales. The following diagram is a simple depiction of normal progression:

                ---------------------->
                I   iii   IV   V    I
                          ii   vii°
                     
                     Figure 66

This diagram indicates which chord would normally follow any other given chord. However, this does not suggest that each chord must be used, but only that the direction as indicated by the arrow be maintained. In other words VI to V to I would be normal progression. I ii V I would also be normal progression. V IV vi would not be normal progression. Note that the supertonic and subdominant triads, as well as the dominant and leading tone triads are interchangeable in function. If both are used consecutively, it is usually stronger to go from IV to ii (rather than vice versa) and from Vll° to V.

Of course, if we analyze a large number of Bach chorales, we will probably detect many instances when normal progression is not followed, but until a student attains a certain proficiency and a certain insight into this style, he should utilize normal progression in every instance where it is possible to do so. Occasionally a situation arises where it is impossible, and then, of course, it cannot be used.

That having been said, three common exceptions can now be given:
1)   V - vi is acceptable, since this is our familiar deceptive cadence. (Remember the unusual doubling.)
2)   vi - iii is a common exception and may be used, although sparingly.
3)   iii vi and ii should not go directly to I.

Other points to note:

iii is relatively infrequently used.
vii° should always be in first inversion, and the bass (3rd of the chord) is the note normally doubled. Some­ times the 5th may be doubled, but the leading tone is never doubled.

Summary of basic rules of chord progression in Bach Chorale style.

I can go to any chord
ii usually to V, sometimes to vii°
iii usually to VI; could also go to IV or V
IV could progress equally to V ii or I
V usually to I, sometimes to vi
vi could go to IV ii V or vii° sometimes to iii
vii° usually goes to I; sometimes to V

Let us now harmonize together a phrase from a chorale so that we may discover a suitable working procedure.

Here is the melody. Sing it over. Remember that although there are many rules and guidelines which help us in harmonization, our ultimate criterion is always the sound. Usually it is possible to come up with more than one correct result, and, therefore, within certain limitations, one's choice of chords is purely subjective and should be based upon what one eventually will be able to identify as typically Bach style.

Figure 67

Suggestions for procedure in the harmonization of this tune:

1)   Note the key - A major. Know the chords in this key.
2)  Check the cadence at the end. It looks like a Perfect cadence with the third in the soprano of the tonic chord. A deceptive cadence wouldn't work because it would produce parallel fifths between bass and soprano.
3)   Proceed with chord selection. Two things should be borne mind in this work: (a) try for strong normal progression; (b) try for an interesting bass line. The expression an "interesting bass line" is a somewhat abstract concept. It should be interesting mainly in relation to the soprano and the two parts together should sound quite well. Generally speaking, the bass line should have a certain melodic interest, with some leaps, some stepwise motion, and a pleasing contour. It is frequently desirable to have some contrary motion with the soprano (that is when the soprano moves downward, the. bass moves upward, and vice versa), and some conjunct motion (both parts moving in the same direction).
Here are all the chord possibilities for this tune:
­­

Figure 68

These seem like quite a few rules to remember, and they are, but with practice and correction, they will become more or less second nature. Your primary guides are the models themselves -- the Bach Chorales, and each of the phrases you will be given to harmonize will be actual phrases that Bach has harmonized. After you have completed each exercise to the best of your ability, comparisons should be made with the original. Differences, of course, should be noted, but it is not to be expected that your harmonization will be the same as Bach's.

This is the Bach harmonization (modified to eliminate some elements not yet covered) of the phrase you have completed. It is the first phrase of the Chorale "Nun lob; mein Seel', den Herren".

Figure 69

Note these points:
1)   Normal progression is followed with the exception of vi - iii which has been noted as a common exception.
2)   The first two chords of the second last measure are both in first inversion (root position would have produced parallel octaves).
3)   The root of the chord is doubled in each case, except in the ii 6/3 chord where the fifth is doubled. Note that the doubling of the root here would have produced parallel octaves between tenor and soprano.
4)   There is some contrary motion, between soprano and bass, some similar motion, and some oblique motion. The bass is an interesting melody in its own right. Play, sing, and listen carefully to this harmonization.

ASSIGNMENT

Harmonize the following phrases in four parts. Note that this is the same melody as the one we have just done but in a different key and a different metre. Bach does not harmonize it exactly the same as the previous one; therefore, approach it from a fresh view point.

From this point on, the work of individual students will have to be corrected for technical errors as well as stylistic weaknesses. However, the Bach models should still be studied by the class after each harmonization is completed and retained for reference. This is Bach's harmonization of "O Herre Gott, dein gottlich Wort." (Slightiy modified.)(Click for example))

Figure 70

It is not always necessary to decide upon a complete chord progression and bass line for a phrase before doing the inner parts. Sometimes, it might be found desirable to sketch in a few bass notes and then complete the inner parts, and continue along in this manner. Often, it is desirable to do the cadences first and then go back to the beginning. Any method which produces good results is acceptable.

Voice Leading

We have been discussing harmonization so far primarily in terms of chord progression. The selection of an effective chord progression is fundamental, and at this stage of development a harmonization should mainly be the result of a methodical selection of chords. However, the importance of a good bass line has already been indicated, and this to some extent will determine chord choices. As one becomes more adept at this work, the melodic interest in the bass, as well as the innier parts, will more and more become determining factors of the harmony, although the underlying chord structure will always predominate. The main point here relates to a method of working. At this stage in the study of harmony, the student will first work out the chord progression, and then consider the voice leading (melodic interest of the individual parts). Sometimes an inversion or a chord will be changed to improve the voice leading. At a more advanced stage, the voice leading and the chord progression will be determined more or less simultaneously, with both factors having equal importance.

The following is an example taken from Piston (Piston, Walter, Harmony ( 3rd edition), W.W. Norton and Co. Co.Inc., 1962, Page 22 and 28.) illustrating how voice leading might be improved in a predetermined harmonic progression. The basic approach is to open up the spacing between the parts. Note changes in doublings and instances where the common tone is not kept in the same voice in successive chords.(Click for example))

Figure 71

The dominant seventh chord.

Triads are not the only available tonal material which can be used in the harmonization of melodies. By super imposing an additional third on the two existing thirds of the triad, we form a new type of chord called the seventh chord. This is a thicker, much fuller sounding chord. Any triad can be converted into a seventh chord. The most common and important of these seventh chords is the Dominant Seventh, the seventh chord constructed on the dominant degree of a scale. Here is the dominant seventh of C major. It is frequently familiarly known as G7.

Figure 72

This chord is used functionally in exactly the same way as the ordinary dominant triad. In four part writing, the problem of doubling does not usually exist since this is a four note chord. However, occasionally the fifth might be dropped, and then the root would be the logical note to double. The basic rules of voice leading in the use of V7 are that the seventh always descends by step, and the leading note, of course, rises by step to tonic. This is V7 to I in C major (a Perfect Cadence).(Click for example))

Figure 73

The Dominant Seventh chord can be written in any inversion. The following are the four inversions of the chord with their full numberings. (Remember that the numbers simply indicate the intervals above the bass.) The full numberings are not used in actual practice and the bracketed numbers indicate the ones which are omitted.

Figure 74

Here are the V7 to I progression in first, second, and third inversion. Notice the voice leading. Notice the necessity for I 6/3 in the third inversion; this is a very useful progression. Play and sing these examples and listen carefully to the sounds.(Click for example))

Figure 75

ASSIGNMENT

Index

(c) KEYBOARD HARMONIZATION OF SIMPLE SONGS
We will now turn to a form of harmonization which will necessitate a different approach from the Bach chorale style -- keyboard chording of simple song. In this work, intuition and ear, rather than rule will be of primary importance. However, a theoretical understanding of the processes should not be dismissed if the most effective results are to be achieved. (In conjunction with, or in lieu of a keyboard, a guitar, or an autoharp with a sufficient number of chords on it can be used.) The harmonic rhythm in most songs is much slower than in the Bach chorales, and frequently one chord will last for several measures. Relatively few chords will satisfactorily harmonize many songs -- I, IV, and V or V7 are often all that are needed.

Figure 76

Procedure:
1)   Sing the melody.
2)  Determine the key (Eb +) and write out I IV V and V 7 in block form in bass clef. (To play a thick chordal accompaniment is not always musically very satisfactory, but the primary purpose of this exercise is chord selection and the chords should be very easy to perform.)
3)    Work out the chord selection together with the class primarily by a trial and error method. Play little bits of the melody and find out what chords are needed as you go along. Strike one chord per measure even if it is a repetition.

The most logical chord choice for this piece is as indicated in Figure 76.

Analysis of Result:
1)   Only the three chords I IV and V7 were needed
2)   Frequently the melody provides clues by outlining parts of chords or complete chords (bar 1, bar 9, bar 13).
3)   There are several instances of notes which do not belong to a chord, but which nevertheless fit in well with a particular chord. These are called "non-chord tones" and are very important to our study of harmony. These non-chord tones serve to keep the music moving, add tension to it, and color the harmony. (Examples: D in bar 3, C in bar 4, G in bar 5, etc.) We will study the various types in more detail later, but for the time being be aware of their existence. It is these non-chord tones, particularly when they are located on strong beats, which make it most difficult to determine the correct chord, and it is in these instances that one's ear must be most keenly attuned to what is needed.

Do the following tune together with the class. They probably will have great difficulty in discovering the key to the correct harmonization of this, which is the fact that the first beat of practically every measure is a non-chord tone. When this is discovered the chord progressions turns out to be very simple. Without the proper harmonization the basic rhythmic drive of this tune would be lacking.

Figure 77

ASSIGNMENT

In the same manner harmonize the following tune (or tunes). It might be advisable to give out several tunes to a group, with different people doing different ones, or allow students to select their own from a song book. For the time being stick to major keys, and only material that can be harmonized simply with three or four chords. The students should play their own solutions at the keyboard, and suggestions for improvement, if necessary, be made by teacher and fellow students. Suggested tunes for harmonization. ­­

Figure 78

Index

(d) NON-CHORD TONES
We have already noted the presence of non-chord tones in the songs we have been harmonizing. Let us now take a closer look at them to see if we can discover the different varieties of non-chord tone. All the circled notes in the following example are non Chord tones.

Figure 79

Figure 80

Click to play example

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Click to play example

Figure 81

Possibilities for the use of non-chord tones in harmonization are as follows:
1.   An unaccented passing tone may be inserted when any given voice leaps a third.
2.   A suspension may be inserted when any given voice decends by step.
3.   A neighboring tone may be inserted when any given voice repeats a tone on two consecutive beats.
4.   An anticipation may be used when any given voice descends by step.
5.   An escape tone may be inserted when a part descends by step.
6.  An appogiatura might be inserted anywhere if it sounds effective.
7.   A changing tone might be used when a voice stands still or leaps a third.

For the time being use only unaccented passing tones, suspensions, and neighboring tones when harmonizing a melody. However, consider that any of these non-chord tones might be present in the given melody. Add the non-chord tones after the harmonization has been completed, and use them judiciously; do not overuse them. Generally speaking, non-chord tones serve to keep the music moving, and in chorale style are usually in the form of eighth notes. They also serve to spice up the harmony and to create points of tension. They are usually sprinkled throughout the texture, occuring at various points in all the voices. However, sometimes they are used more or less continuously in a single voice, mainly the bass. Suspensions are numbered according to the interval between the dissonance and the bass (see figure 81, number 2: 4-3 sus, 7-6 sus, 9-8 sus). The 4 - 3 is the most effective suspension. Avoid doubling the suspended note. The 9 - 8 is the next most effective suspension. The suspended note in this case is obviously doubled in the bass.

ASSIGNMENT

Harmonize the following chorale phrase in four parts. The dominant seventh chord may be used, as well as all triads in either root position or first inversion.. Also try to use some passing tones in alto, tenor, or bass parts. (Give the students the melody from the following four part harmonization.)
Bach's solution of #54 "Lobt Gott, ihr Christen, allzugleich."(Click to play example). ­­

Figure 81a

Dictation
The following dictation exercises should be given periodically interspersed with the other work. Intensive training in ear training and sight singing are beyond the

Index

Part Four
ARRANGING (Studies in texture)

Index

Part Five
MELODY WRITING (Studies in form)

Index