When you play a IV chord you extend the five-finger position upward by one scale degree. When you play a V7 chord you extend the five-finger position downward one scale degree. These two additional notes in combination with the notes of the five-finger position give all the degrees of a major scale —the IV chord providing the sixth degree, the V7 providing the seventh. By putting the seventh degree up an octave so that it follows the sixth, and repeating the first degree, or tonic, an octave higher, we have a complete scale from 1 to 8. This also applies to minor scales but we will consider them later.
Dividing these eight tones into four lower and four higher tones gives us two tetrachords, which are not chords in the usual sense at all, but a diatonic series of four notes each. If you examine these tetrachords you will observe that they are structurally alike: whole steps between each of the lower three tones, but only a half step between the third and fourth tones. In other words, the half steps in a major scale occur between the third and fourth, and between the seventh and eighth degrees.
For a while we will play our scales in tetrachords —the lower tetrachord with the left hand, the upper tetrachord with the right hand. There are several reasons for doing this. One is that the fingering is the same for all scales, thus eliminating fingering problems. Another is that by having a different finger for every key you can see the entire scale spread out like a map under your hands before you begin to play. You will not use your thumbs, so fold them under and close to the palms of each hand and keep them there while you play.
When you are able to organize and play easily the eight tones of every major scale, try extending the scales over two octaves. To do this, simply strike the highest tonic of the first octave once with the fifth finger of the right hand, then again with the fifth finger of the left hand, and continue up the second octave, thus:
You have already learned that the first degree of any scale is called the tonic. Let us now consider the names of the other scale degrees. They are easy to remember because they are descriptive. After the tonic, the most important scale degree (for acoustical, theoretical, and philosophical reasons which we need not explore here) is the fifth —the dominant —hence its name. If we descend five degrees from the tonic, we arrive at the fourth degree of the scale, and since it is five degrees under the tonic it is called the subdominant. Halfway between the tonic and the dominant is the third —the mediant (note the similar words with similar meanings: medium, median, mediate), and halfway between the tonic and subdominant is an underneath third —the sixth degree of the scale —which is called the submediant. This leaves only two scale degrees unnamed: the second and seventh. The second is called the supertonic, which is self-explanatory; the seventh is called the leading tone because of its strong tendency to move or “lead” to the tonic. There is one circumstance in which the seventh is not called the leading tone and this is when, as in the pure, or natural, form of the minor (see Chapter 6), the seventh degree is a whole step below the tonic instead of a half step. In this case it is called the subtonic. This distinction is frequently ignored, however, and more often than not you will hear the seventh degrees of all scales referred to as leading tones.
In order, they are:
| Scale degree | Name |
| 1st | TONIC |
| 2nd | SUPERTONIC |
| 3rd | MEDIANT |
| 4th | SUBDOMINANT |
| 5th | DOMINANT |
| 6th | SUBMEDIANT |
| 7th | LEADING TONE (or SUBTONIC) |
| 8th | TONIC |
The First Noel is a two-hand melody which incorporates an ascending major scale. Say the scale-degree numbers as you play it, and transpose it into every major key.
THE FIRST NOEL
Traditional English Carol
Another song which incorporates a major scale is to the World. Play this by ear in several keys. Start on black keys as well as white keys, and of course, play it tetrachord-wise.
The twelve-tone “row” concept, or numerically planned sequences of pitches, is not new; it has been the practice of bellringers, particularly in England, for hundreds of years. The permutations, or sequential combinations of which any given number of bells is capable, are called changes. It has been estimated that twelve bells could produce 479,001,600 changes which would take 37 years and 355 days to ring.
Without embarking on any such formidable venture, we can still enjoy a taste of change-ringing by playing some of the changes on the keyboard, and what better way really to learn scales?
Place your fingers, tetrachord-fashion, over the C-major scale. The first four changes, in music notation, are shown in Figure 44. After you have played all the changes in C major, play them in other major keys. Say, or sing, the scale-degree numbers as you play.
HUNTING UP AND COURSING DOWN
The title “Hunting Up and Coursing Down” refers to the way in which the “lead” or highest bell moves one position further away from first place in each change. This is “hunting up.” “Coursing down” is the return journey to first place.
If you look back over the first eight warmups, you will notice that they all begin with either a five-finger position or a triad. In both cases the lowest note was the tonic and the highest note the dominant. You have played these warmups in what we called chromatic order, that is, ascending or descending a half step for each repetition. Now let us try them in a new way. Taking the highest tone of the five-finger position or triad (the dominant) as the new tonic of a new key, Warmup I would progress like this:
This is called playing in dominant order, and if you continue through enough keys in this way you eventually will return to the key from which you started. You may also do this in subdominant order. Progressing through all the keys in this manner is known as making the circle —or cycle —of fifths; or, if in subdominant order, making the circle of fourths. When you examine Figure 46 you will observe that as you move to the right, or clockwise, from C, each new key will have an additional sharp or one less flat. As you move to the left from C, each key will have an additional flat or one less sharp. The three pairs of bracketed keys represent a continuation of the same pattern—the sharp keys moving clockwise around to C#, which has seven sharps, and the flat keys moving counterclockwise around to C♭, which has seven flats. It so happens that for each of these pairs the pitch is the same (on the piano) although the “spelling” is different. To such a relationship we give the term enharmonic.
Play all the warmups in dominant order, then in subdominant order. Warmup XII is given here in dominant order, but can be played as well in subdominant order or up or down chromatically.
WARMUP XII
Harmonizing Each Degree of the Scale
Now that our melodies are covering a wider range, let us consider how each scale degree in the octave may be harmonized, using those chords of which the melody tone is part.
Three Melodies to Harmonize and Transpose
Fingering. The increased range of the melodies and songs will mean hand shifts and fingering adjustments. Since fingerings differ in different keys, only general provisions can be made. For instance, always begin with a finger which will allow you to make maximum use of all your other fingers. Don’t start a downward passage in the left hand with your fifth finger. If you find yourself in danger of running out of fingers don’t wait until the emergency is upon you: put two or three fingers over or pass your thumb under (depending on direction), or use a contraction which, when “let out,” will give you a greater reach.
Transposition. To transpose the new songs, follow these steps which are basically the same as those you have been following.
- Play melody alone in original key, singing scale-degree numbers.
- Determine which scale degree the melody begins on.
- Choose new key.
- Play tetrachord scale up and down one octave in new key, saying scale-degree numbers aloud, and paying particular attention to the location of 6 and 7. (Remember, it is easier to find 7 by counting down one half step from 8 than by counting up seven degrees from 1.)
- Locate first note. (Same scale degree in new key as it was in original key.)
- Play melody with one hand alone.
- Harmonize melody.
THEME FROM THE “NEW WORLD” SYMPHONY
Antonin Dvorak
1841-1904
OLD BLACK JOE
Stephen C. Foster
SWANEE RIVER
Stephen C. Foster
In the chord guide (Figure 47) only those chords which included the melody tone were given. The same principle in reverse has governed your improvisation: you have been limited to using only chord tones in your melodies.
Many beautiful melodies found in music literature are fashioned from these tones alone, but such a practice is necessarily restrictive. If you look back over the melodies and songs which you have been harmonizing, you will find that in general only chord tones were accompanied by a chord, on strong beats. However, many non-chord tones occurred between the chord tones, giving the music a flowing character as well as greater variety and interest. These non-chord tones are called non-harmonic tones.
A non-harmonic tone is any tone which is not part of the immediate harmony.
Theorists have organized these non-harmonic tones into many categories depending on their function, the direction in which they move, how they are approached and left, whether they fall on accented beats or unaccented beats, and whether they remain in or move out of the diatonic pattern.
Among the classifications are (in alphabetical order) accented passing tones; accented upper neighbors; anticipations; appoggiaturas; changing tones; chromatic lower neighbors; échappées; embellishments; escape tones; inverted suspensions; neighboring tones; passing tones; pedals; retardations.
Before despair overtakes you, let it be said that some of these terms have the same meaning, and not all of them are used by any one theorist. Indeed, theorists are by no means in agreement as to terminology and usage. Moreover, these are matters which more appropriately belong in the harmony or counterpoint class than in the piano class, so, for our very practical purpose, let us reduce these confusing categories to their basic and common elements.
1. All non-harmonic tones are dissonant because they never belong to the prevailing harmony.
2. Most non-harmonic tones pass from one chord tone to another and so are called passing tones. Some passing tones occur on weak beats or weak parts of beats, as part of a free-flowing melody (unaccented passing tones), in which case the dissonance is scarcely discernible. Others occur on strong beats (accented passing tones), where their dissonant quality is more pronounced.
3. All non-harmonic tones ultimately move (resolve) to a chord tone in the same voice.
4. In general, non-harmonic tones are drawn from the degrees of the diatonic scale, but the note one half-step below a scale tone, approached from and returning to that scale tone, is used frequently. When this non-harmonic tone happens not to belong to the scale, a chromatic alteration is involved (chromatic lower neighbors).
We should bear in mind that ideas of beauty and logic in music give rise to the rules—not the reverse. Over the years our ears as well as our concepts of beauty and logic have undergone many modifications. What was once considered offensive is now not only acceptable but charming, and may even have become trite. The rules are constantly being relaxed and changed. As scholars you should be familiar with the names and habits of the non-harmonic tones, but as “functional” musicians (and especially as improvisers, required to execute a complex maneuver in a limited time-span) your purpose will be better served by listening to the finest music so that you may develop your taste, and then acquiring skill in playing what your inner ear and good taste dictate.
Before going on to the new pieces, reexamine all the songs and melodies you have harmonized up to this point and identify accented and unaccented non-harmonic tones.
Three of the pieces which follow contain accented passing tones. Those in the Menuet in F Major by W. A. Mozart are harsh because they fall on the strongest beat in the measure and are held for an appreciable time. Those in the Bach Menuet and Haydn German Dance in D Major are barely noticeable because, though falling on the strong part of the beat, they move quickly along and sometimes are unaccompanied. Listen to their effect—obvious or subtle—in the otherwise bland course of the music. They are marked * and will serve as models when you encounter similar examples in the songs which you will be harmonizing.
MINUET IN A MINOR
Henry Purcell
1658(?)-1695
GERMAN DANCE IN D MAJOR
Joseph Haydn
1732-1809
MINUET IN G MAJOR, NO. I
See section on embellishments in Chapter 10 for this and future embellishments
CIRCLE GAME
M. S. McLain
Here are two new songs to harmonize and transpose. Each has an accented non-harmonic tone where the harmony is indicated.
WINTER, GOOD-BYE
Folk Song
OLD SONG
Traditional
Since ledger lines sometimes are bothersome in sight-reading, a few hints may help you cope with them.
1. Frequently a note perched high (or low) on a ledger line is an octave away from the preceding note on the staff. This is the case in the initial part of each of the first two exercises. Do not attempt to count lines and spaces, but learn to recognize the spatial distance between the two notes, remembering always that an octave is a line-and-space, or even-numbered, interval. Regardless of staff degrees, the space between or
is not great enough for an octave. The space between
or
is too large. The space between
or
is right and you should recognize it as an octave, even though you do not know which octave. The distance between the lines of the staff and the ledger lines varies according to the kind of type used by the printer, but your eye should easily relate the octave “picture” to the dimensions, or gauge, of the staff from which you are reading.
2. When you do not have a lower note to guide you, you must establish fixed points from which to get your bearings. The best ones are the second ledger line above the treble staff and the second ledger line below the bass staff. Each is a C and from these you can make lightning calculations, knowing that every additional ledger line means an additional third, and also knowing that intervals on two lines or two spaces will be odd-numbered intervals, while those on lines and spaces will be even-numbered intervals. This may seem complicated at first, but it is infinitely quicker than laboriously counting up each line and space individually.
With practice, other ledger lines will become as familiar to you as the lines of the staff. You can hardly have a close acquaintance with and not be aware that
is A, and almost as soon know that
is the picture of E.
Play the next exercise with covered hands, and looking one measure ahead.
You may now introduce passing tones in your improvisation, but stay with the unaccented variety for the present. Develop each of the beginnings (Figures 48, 49, 50) into two four-measure phrases (eight measures in all). Each beginning could be continued in a number of different ways. How many can you invent?
- Listen carefully as someone plays a scale slowly up and down one octave. Now identify the various scale degrees when they are not played in order but at random.
- As the teacher or a student plays a familiar tune and stops at various points in its course, try to identify the scale degree on which the tune broke off.
- Play Joy to the World and Three Blind Mice by ear and tetrachord-wise in all major keys. You may also play the latter as a round.
- Play and harmonize Yankee Doodle, Silent Night, and Auld Lang Syne by ear, and in all keys.
Review and Suggested Assignments
- Define the following terms:
tetrachord; chromatic order; dominant order; subdominant order; circle of fifths; circle of fourths; enharmonic; leading tone; mediant; supertonic; submediant; subtonic; ledger line; leger line; non-harmonic tone; passing tone; accented passing tone; unaccented passing tone.
- Explain the names of the scale degrees.
- On a staff of this gauge, which pairs of notes would form octaves?
- Study the ledger-line sight-reading exercises until all the notes are familiar to you, then turn the page upside down and read the names of the notes as they now appear. (Supply an imaginary bass clef for the notes below the staff and an imaginary treble clef for the notes above the staff.)
- Place your hands over the two tetrachords of any scale and play individual scale degrees which someone else dictates, such as 6, 1, 5, 7, 2, 8, 3, 7, 4, and so on. Do this as rapidly as possible and in a number of keys.
- Again play scale degrees as rapidly as possible from dictation, but do not place your hands over tetrachords; instead, play every note with the second finger of your right hand. Do likewise with the second finger of your left hand.
- Dictate scale degrees for other students in the class to play. *
- If the Christmas Season is approaching, play three Christmas Carols of your own choice. Select them from school or community song books, or collections of carols.
*In contrast to the “rule of rows” which decrees that each pitch or number must be represented before the row can start anew, the type of dictation which is suggested here can be quite formless, returning to the same pitch repeatedly if the dictator wishes. The object is to identify scale degrees quickly. The same alertness is required of the dictator too, for he must not only call his numbers rapidly but make sure they have been played correctly.