For some time you have been using minor triads. Let us now examine the structure of minor scales. Unlike major scales, which have only one form, minor scales have three forms.
The first (and oldest) form is the pure, or natural. By placing your fingers over the eight white keys in the octave A to A, you will be in position to play the natural form of the A minor scale. This is the prototype or pattern for all natural minor scales. Note carefully where the half steps occur. It is this form which determines the key signature of the scale.
If you raise the seventh degree of the scale one half step from G to G# you will have the harmonic form of the scale. This is the form which is usually used in conventional harmonic progressions and harmony exercises, and thus explains why the dominant triad is always major, and why the V7 chord is the same in both major and minor keys. (The raised leading tone is the third of the dominant triad.) In spite of its extensive use, this sharp is not considered part of the key signature but is always written in as an accidental.
The third form of the minor scale is the melodic. In this, not only the seventh degree is raised from G to G#, but also the sixth degree so that F becomes F#. This second sharp is also written as an accidental and never included in the signature. These accidentals, occurring on the sixth and seventh degrees, are not always sharps; in some keys they could be double sharps, and in some keys they would be naturals. Could you identify any of these keys?
The three forms of the minor scale then, along with the major scale, are as follows:
You will notice that between the sixth and seventh degrees of the harmonic scale is a step and a half. This is like the “augmented second” which you encountered in a different context in Club Zara.
As with the major scale, both the pure and the harmonic minors descend just as they ascend, but the melodic minor traditionally descends in the pure form.
The plus sign over the sixth and seventh degrees of the harmonic and melodic forms is the customary indication that a note is to be raised a half step.
If you place your fingers over the tetrachords of any major scale and then make the changes necessary to form a pure minor scale, you will find that you have had to move only three fingers—those over the third, sixth, and seventh degrees—each of which you lowered a half step. When you construct a harmonic scale you need make only two changes, and when you build a melodic scale you change only one note!
To make this structural difference very clear, we will, for the time being, play major and minor scales on the same tonic. For example, we will play C major, then C minor in three forms. We may proceed in dominant, subdominant, or chromatic order, but the major and minor scales will always begin on the same tonic, or keynote.
In theory and harmony courses students frequently are taught to couple minor scales with their relative majors (the tonic three half steps higher) because the key signatures are the same*. For pianists, however, coupling the tonic or parallel minors with their majors not only makes clearer their structural differences but simplifies fingering problems at a later stage.
* They are presented this way in the Appendixd.
When you are sure of the three forms of the minor scale and can isolate 6 and 7 in any form and in any key, you may return to the songs in Chapter 4 and play them in every minor key. Be sure to determine which of the three forms the song is in and play the tetrachord scale in that form. It is a curious fact that although in conventional harmony the harmonic form of the minor scale is regularly used, a significant number of folk-songs and traditional melodies, as well as the works of known composers, are cast in the pure, or natural, form of the minor. To harmonize such melodies one must adopt to a large extent the mode-form implicit in the melody. You will not wish to harmonize the seventh degree of a natural minor scale with a V7 chord, so either leave the tone unharmonized, or consider it an accented passing tone and use a i or iv chord.
Like trees, chords have roots from which they grow. In our Western traditional system of harmony, chords are constructed by the simple device of superimposing a series of thirds over any given tone which is then considered the chord’s root. For example, in your old friend the C major triad, C is the root, E is a third above C, and G is a third above E. We speak of E and G as being the “third” and “fifth” of the triad because they are a third and fifth, respectively, above the root; but the chord is still composed of two thirds—one above the other. In a major triad the lower of the two thirds is major (four half steps) and the upper third is minor (three half steps). In a minor triad the lower third is minor and the upper third is major. In other words, the mode (major or minor) of the triad is determined by the lower third.
In case you are not yet sure of thirds, remember that every third will include three neighboring letter names. The third from C to E includes C, D, E. The third from E to G includes E, F, G. Also, thirds will always be found either on neighboring lines of the staff or in neighboring spaces, so the eye can very easily identify them on the printed page.
It is absolutely essential that you learn to compute intervals by both methods—letter names and half steps. The letter names will tell you of what kind the interval is—second, third, and so on; the number of half steps will tell you its quality—major, minor, augmented, and so on. You must also learn to compute intervals from the upper note down as well as from the lower note up.
Bear in mind that although you include the first note when you calculate intervals by letter names, you must not include the first note when you count half steps because what you are calculating here is the distance from the first note. Thus, in counting the half steps between C and E, you would proceed as follows:
In counting downward from E reverse the process:
Identify the thirds in Figure 55, indicating those which are major with a capital M and those which are minor with a small m.
We have been playing our tonic triads in what is known as root position, that is, with the root positioned as the lowest tone. The three notes of our triad can, however, be regrouped so that a tone other than the root appears at the bottom while the root appears in one of the other parts, or voices. Such rearrangements are called inversions. In piano music there are many occasions when the left hand plays the root in the bass (lowest part) while the right hand plays the three notes of the triad in various groupings. We cannot properly call these groupings inversions as long as the root remains in the bass, so we call them positions. For the present it will be convenient to call them “positions” even when the left hand is tacit (not playing).
Notice that whatever its location, C is always the root. Likewise, E is always the third, and G is always the fifth. Can you identify the root, third, and fifth in each of the chords in Figure 57? This rule will help you:
When all the tones of a chord are arranged so that they form a series of thirds, the lowest tone will be the root.
To help get the feel of these positions with ease and precision, do the following warmup:
Play this through all major and minor keys until you can do it easily hands alone, then play the hands together. Play it also in subdornant and chromatic order.
If you build a major triad on C (C E G) and then build another major triad on the subdominant (F A C), you will observe that while two of the notes in each triad are different, one is the same—C. This is called the common tone because the two chords have this tone in common. One of the first rules of chord progression the student of harmony learns is to keep the common tone in the same voice and move the other voices to the nearest chord tones. When both I chord and IV chord are in first position, the two C’s are an octave apart, but since C is the common tone it must remain in the voice in which it first appeared (the lowest). To accomplish this we must regroup the notes of the F chord so that C can remain in the lowest voice. The two other notes of the C triad move to the two other notes of the F triad which are nearest them, E moving to F and G moving to A. This will explain the progression I IV I which you learned by rote in Chapter III.
This applies, of course, to all positions of the triads. If we begin with a second position of the C triad, the common tone will still be C, only now it will be in the highest part, or soprano, and there it must remain. If we begin with a C triad in third position, the common tone will be in the alto (middle part) and must stay fixed like the filling in a sandwich. You will notice that the position of the second chord is never the same as that of the first chord.
Play I IV I starting from all three positions, in all major and minor keys, and with each hand separately. CAUTION! Remember that when the tonic triad is minor the subdominant triad will also be minor, but before you automatically lower the middle tone of the iv chord a half-step, ask yourself which tone really is the third.
Let us now examine the progression I V (not V7).
Here the common tone, G, is in the same octave for both chords, but is not in the same voice. In the I chord the G is in the soprano, while in the V chord it is in the tenor (lowest of the three upper parts). We solve the problem by retaining the common tone in the soprano of both chords and moving the two lower voices of the I chord to the nearest tones of the V chord.
You will quickly notice that the V chord in this position (second) is very like the V7 chord which you have been playing by rote, but do not confuse them. In the V7 the third of the I chord moves upward, while in the V the third moves downward.
The progressions V I and IV I are often heard at the end of compositions or sections of compositions and are then called cadences. A cadence is a conventional chord progression at the end of a piece or section, designed to give a feeling of rest or finality. The progression IV I is called a plagal or church cadence, and you hear it in the “A-men” at the conclusion of hymn tunes. V I is an authentic cadence, and the combination I IV (I) V I is known as a complete cadence.
Here are the three cadences with the upper voices in all three positions. They are written in major mode but you should play them in minor mode as well and in all keys. Just remember that although the subdominant chord follows the mode of the tonic, the dominant chord is always major. The connecting lines indicate common tones and should not be read as ties.
By now you are familiar with a number of intervals—half steps, whole steps, thirds, fifths, octaves. Reference has been made to “augmented seconds,” and “sevenths” in connection with dominant sevenths. Let us complete our knowledge and formalize it in the schedule at Figure 63.
You will have noticed that a number of qualitative words are used to describe the intervals: perfect, major, minor, augmented, diminished. It is frequently said that the major intervals are those which occur in the major scale—counting up from the tonic; the minor intervals are those which occur in the minor scale; the perfect intervals are those which occur in both major and minor scales. Immediately you are going to point out that this is all very well for 1, 4, and 5—they do occur in both modes. But what about 2? It also is the same in both modes, yet it is called a “major second.” This apparent inconsistency will soon be explained.
The trouble with using this method to establish the identity of intervals is that one always is obliged to consider the lower tone in the interval as the tonic of the key whereas, in context, the lower tone could be any degree of the scale. Moreover, counting up becomes the sole means of identifying an interval and the very needful ability to reckon downward is not acquired.
Because of these limitations we shall stress learning intervals by the number of half steps they include, and in this way all intervals will be easily recognizable regardless of context, position in octave, or whether they are calculated upward or downward.
The perfect intervals retain their quality of “perfection” when they are inverted. (Bear in mind that “inverted” only means “upside down.”) Thus, a perfect fourth inverts to a perfect fifth; a perfect prime inverts to a perfect octave. On the other hand, major intervals invert to minor, and minor intervals invert to major. This is the reason the interval between 1 and 2 is a major second even though it occurs in both modes: it inverts to a minor seventh.
Perhaps you have noticed that the interval and its inversion always total nine. A third inverts to a sixth, a fifth to a fourth, and so on.
Also, you may have noticed in the interval schedule the inclusion of two intervals, each with six half steps. This is really a matter of spelling. If you count six half steps up from C (see Figure 54) you will arrive at the lowest of the group of three black keys, but, out of context, who is to say if this key is F sharp or G flat? If it is a G flat it will be a fifth up from C because the interval C—G embraces five letter names—C, D, E, F, G. If, however, the key is F sharp the interval will be only a fourth because it embraces only four letter names. In the case of the fifth, the interval is one half step smaller than the perfect fifth, so we say it is diminished. In the case of the fourth, the interval is one half step larger than the perfect fourth, therefore we say it is augmented.
All perfect and major intervals may be augmented by adding one half step. All perfect and minor intervals may be diminished by subtracting one half step.
Figure 64 will make this clear.
Major intervals may not become diminished; minor intervals may not become augmented. Augmented intervals may become doubly augmented by adding another half step; diminished intervals may be doubly diminished by subtracting another half step.
In identifying intervals, follow these steps:
- Determine the kind of interval by counting letter names.
- Determine its quality by counting half steps as described in Figure 54.
- Consult the interval schedule. If the number of letter names and of half steps match a listed interval, you have your answer.
- If the interval includes one more half step than given in the schedule for the perfect or major interval with the same number of letter names, the interval will be augmented.
- If the interval includes one less half step than given in the schedule for the perfect or minor interval with the same number of letter names, the interval will be diminished.
For example, C to A sharp is a sixth because it includes six letters, but it has ten half steps—one more than the major sixth in the schedule—therefore it must be an augmented sixth. C sharp to A flat also has six letters but only seven half steps—one less than the minor sixth—therefore it has to be a diminished sixth, even though it sounds like a perfect fifth and has the same number of half steps.
What are the following intervals?
Above and below each note in Figure 66 write notes at the designated intervals. M = major, m = minor, P = perfect, A= augmented, DA = doubly augmented, d = diminished, dd = doubly diminished.
So far we have been harmonizing melodies with chords (block chords in which the tones are struck together). Some melodies, particularly those in triple or compound meters, lend themselves to a type of accompaniment in which the tones of the chord are played individually or in different combinations (broken chords). Thus, the chord progression I V7 I could be played in the following ways:
DID YOU EVER SEE A LASSIE?
These two songs embody a common feature in their melody structure: the second half of each first phrase defines the dominant triad in its root position. The third and fourth measures of Where, Oh Where Has My Little Dog Gone? give the complete chord while the third full measure of Did You Ever See A Lassie? has only the root and fifth of the V chord, but the force of the dominant harmony is the same.
Ten Little Indians shares this same feature—indeed, the entire tune is composed of only the tones of the I and V triads in their root positions. Write the melody in any key you wish, and then add the appropriate harmonies.
Use the new accompaniment patterns to harmonize the melodies in this chapter, then use them to accompany earlier songs and melodies when appropriate.
Warmup XIV is designed to give individual fingers greater speed and precision in articulation. In each measure hold down silently all the keys except the one which is being reiterated. Play both A and B in all major and minor keys, first with the right hand, then with the left.
Warmup XV is another way in which to get the feel of the octave span. We call this inch worming, and if you have ever observed an inch worm’s method of locomotion you will understand why. Be sure to bring your thumb up to your little finger before stretching your hand out for the next octave on the way up, and on the way down be sure to bring your little finger to your thumb. This is one time when you should not keep your hand framed to the octave. Play the warmup on all keys.
Until now your attention has been fully engaged by the complexities of fingers, keys, and chords. Nothing has been said about expression, and expression marks have been kept to a minimum in your music. But some magic beyond accurately ordered sounds is needed to make music come to life, and that magic is interpretation.
Since all music disciplines (together with other art forms) are governed by certain general esthetic values, let us restrict our discussion of interpretation to a few practical procedures which relate particularly to piano playing.
1. A good effect is frequently produced by playing louder as a melodic line ascends, and softer as it descends. (Beware of increasing speed during a crescendo.)
2. Unless the left hand has the melody, it should play a little softer than the right hand. If the right hand is playing mezzo forte (mf) the left hand should be playing mezzo piano (mp). If a crescendo occurs in the right-hand part the left hand should also make a crescendo, but always one degree softer.
3. It is usually good practice to taper off or “lighten up” dynamically at the end of a phrase, even when the last note under the slur falls on an accented beat. Slavish accenting of first beats is fine for beginning pianists and marching bands but gives a “squared off” effect to music, while “shaping” the phrase makes the music more supple and eloquent. The same principle applies to two notes slurred together: the second should usually be softer than the first.
4. Generally, a broadening out or slight ritard at the end of a composition gives an effect of importance and finality, and, if you are careful not to overdo it, you may hesitate just the least bit before the final note or chord.
5. Think of the music as being horizontal rather than vertical, even in a forest of chords. In good music every note, chord, phrase, or section grows logically out of what preceded it and leads logically to what follows.
6. If a phrase or motif is repeated one or more times, try to vary each repetition. You can do this by playing softer or louder, or modifying the time (rubato), or varying the touch (staccato, legato, portato), or bringing out an inner voice—to mention a few possibilities.
7. Never “punch,” “poke,” or “hit” the keys. Try to achieve a “singing tone” by transferring the weight of your fingers, hand, and arm from one key to the next in a lateral and gliding motion.
8. Always be aware of left-hand interest. Even when the bass part has a strictly accompanying task, a good composer will try to build in elements of counterpoint, imitation, or rhythmic variety. When you discover such details, highlight them. This is another way in which repetitions may be varied.
9. Bring out the highest notes of right-hand chords.
10. Observe phrasing carefully. If it is not indicated in your music, work it out for yourself. Remember that a phrase is more than a collection of notes which are to be played legato: it is a musical utterance with a specific meaning to communicate. Singing the melody will help you feel where the musical commas and periods should go. Do not feel bound by the phrasing given in your music, but if you change it make sure you have a good reason for doing so. Sometimes the meaning of the music is so clear that no phrasing marks are necessary.
Be careful to make an actual (though slight) break between phrases. Do not content yourself with a mere twitch of the wrist which looks effective but permits the last note of one phrase to connect with the first note of the next phrase. You must actually lift your fingers from the keys for an instant and “let the air in.” (Sometimes, of course, the last note of one phrase is the first note of the next.)
11. Finally, plan your dynamics in advance. Never depend on the inspiration of the moment. If, after careful planning, you do feel inspired, your performance will be all the better. Plotting the dynamic scheme of any but the simplest music requires thought and experimentation. When you have decided what you want, mark it in your music.
Most important of all, listen to your playing, not just for correct notes and time, but for music—beautiful music!
from his notebook
LITTLE ETUDE NO. I
MINUET IN G MAJOR, NO. II
Here is some more boogie; the pattern is the one you had in Chapter 4 but now it employs IV and V chords as well as I chords in root position.
By this time you should feel at ease with unaccented passing tones, so let us add another of the non-harmonic tones to our resources—the chromatic lower neighbor. This, you may recall from Chapter 5, is the half step below any scale degree, always moving away from and returning to the same tone. It may be a normal degree of the scale as at b in Figures 68 and 69, or it may be chromatically raised (sharped or naturaled), as at a.
Complete the beginnings, adding measures to make two four-measure phrases. Then improvise your own creations in a predetermined number of measures and phrases.
In Chapter 4 you had a piece called Pentatonia which was based on the pentatonic scale. The pentatonic scale is like a major scale with the 4 and 7 omitted, or a pure minor scale with 2 and 6 omitted. It can be played entirely on the five black keys, or any others so long as the same interval structure is preserved. You may start the scale on any of the five tones. There is no true major or minor, but certain combinations give modal effects.
Carry on from the model beginnings, then improvise your own melodies. These may be divided between the hands or played by the two hands simultaneously. Do not harmonize them.
A charming effect can be had by group improvisation on the pentatonic scale. To do this, everyone plays at random on the black keys of the two highest octaves, as softly as possible. You will hear a Westerner’s idea of pagoda bells.
- Play a note on the piano and sing a pitch a major third higher; a minor third higher; a major third lower; a minor third lower.
- Identify major and minor thirds as someone else plays them.
- Identify the positions of triads as someone else plays them.
- Identify plagal, authentic, and complete cadences as someone plays them. Tell whether they are major or minor and which chord tone is in the soprano.
- Identify the three forms of the minor scale as someone plays them up and down one octave.
- Play the first five tones of a minor scale and sing 6, 7, 8, in the pure form. Do the same in the harmonic and melodic forms—always after playing the first five tones.
- As someone plays the five tones of a pentatonic scale (on black keys), identify the first note.
- As your teacher, or a fellow student, plays songs and pieces—sometimes observing phrasing, sometimes not—raise your hand when you hear a phrase properly played.
- Play and harmonize Oh, Dear, What Can the Matter Be? and Ten Little Indians by ear, and in all keys.
- Define the following terms:
pure minor scale; natural minor scale; harmonic minor scale; melodic minor scale; tonic minor scale; parallel minor scale; relative minor scale; root; major third; minor third; root position; doubled; tacit; first position; second position; third position; third (of triad); fifth (of triad); soprano; common tone; alto; cadence; authentic cadence; plagal cadence; complete cadence; voice leading; block chords; triple time; compound time; diminished; augmented; perfect (of an interval); inverted; broken chords; accidental.
- Play notes a major third and a minor third above and below each key in an octave.
- Construct major and minor triads in first, second, and third position on all keys in an octave.
- Play major triads in second and third positions and lower the thirds to make them minor.
- Play I IV I through all major and minor keys with three voices in the right hand and the roots (single notes) in the left hand.
- Play I V I, then I V7 I (the right hand starting with first position only) in all major and minor keys, and compare the progression of parts.
- Play I V I through all major and minor keys in all positions and in all orders with the right hand alone.
- Proceed as above and double the roots in the bass (left hand).
- Select an appropriate accompaniment for Silent Night and transpose the song into all keys.
- Harmonize Oh Dear, What Can the Matter Be? first with block chords, then with broken chords, and transpose into all keys.
- Construct minor tetrachord scales in three forms on every note in an octave. First play the scales up and down one octave, then two or more octaves.
- Write Ten Little Indians—melody and chords—in any key but C major.