We begin this routine explanation of scales and chords with a word of advice to teachers. The effectiveness of this chapter as a teaching device will be reduced if it is presented to young students in its entirety. The teacher should be thoroughly familiar with all the material to be presented, but must teach it to students a step at a time. Until they have become familiar with both the instrument and music reading, “music theory” is apt to be over their heads, and students become easily confused. In which case further effort is futile and meaningless. The student must understand and comprehend each idea presented in teaching theory before any advance can be made. Keep the final objective in mind, but approach it a step at a time. Comprehension of fundamental music theory is attained through an accumulative learning process. Each step is meaningful, and must be learned so well that it will never be forgotten. Whenever it is possible, tie up each step of the learning process and each definition with something simple and already known to the student. Stay at his level. Things that are simple to an educated musician may be mysterious and obscure to the student. Remember the days of your youth when you were befuddled and confused by some aspects of music theory.
Intervals
It is almost impossible to teach theory without learning a few simple definitions, so let us begin by defining an “interval.” In every day usage an interval is defined as the distance between two objects. We say that two objects are three feet apart—an interval of three feet. Simple, isn’t it? An “interval” in music is just as simple. It is the distance between two tones with regard to pitch. Intervals are measured by half or whole steps. An actual measurement in vibrations can be made. For example, one tone vibrates at 440 and another at 880 per second. The interval between, when measured scalewise, is found to be an octave Both notes have the same letter name, in this case “A”. “Octave” is the Italian word for eight, and when used in music means that one tone is eight scale steps distant from the other.
Scales
Now to a simple definition for the word “scale”. This is derived from the Italian word “scala”, which means a “ladder” or “terrace”. Thus, a musical scale is a “ladder” of tones. A ladder of tones progressing either up or down, made up entirely of half-tone intervals, is called a chromatic scale. Generally, although not always, ascending chromatic scales are written with sharps, and descending with flats. If the entire class reads the names of the notes in these chromatic scales aloud a few times, they will be learned and memorized, quickly and easily.
(Chromatic scales are the foundation of technical study on any instrument because they involve all the “legitimate” fingerings. Although a chromatic scale may begin on any given note in the C chromatic scale, the same notes are involved. Therefore, it may be assumed that there is only one chromatic scale, but that this may begin at different pitch levels. Time is such a precious commodity, and we are allotted so little of it during a lifetime, it strikes me that to practice one chromatic scale covering the practical playing range of the instrument, and using all the basic articulations, is making an economical use of time. Playing page after page of chromatic exercises takes time. Careful practice on one scale, using basic articulations, equips a student with all the technique needful to play any chromatic study at sight.
Scales that progress by whole rather than half steps, are called “whole tone” scales. Modern composers use whole tone scales frequently, even in easier band music. Young players may become bewildered at their first encounter with the “whole tone” scale. But a little “whole tone” scale practice will soon enable them to hear it correctly. A whole tone scale may begin on any step of the chromatic scale. When we reach the third scale step, however, we find that we are playing the same notes used in the first scale. So far as sound is concerned, there are only two whole tone scales.
The same remarks about practice on the chromatic scale are relevant to whole tone scales. Playing the first two whole tone scales illustrated, throughout the practical playing range of the instrument, using basic articulations, should equip the student to play almost any whole tone study at sight.
Now we must talk about the major and minor scales, or diatonic scales. These progress, shall we say, staff-wise. The word “diatonic” really means “through tones.” We come now to the magical word, “twelve.” There are twelve notes in the chromatic scale of an octave, and we can hear “pianistically” twelve major scales or twelve scales in any minor mode. More can be written, but these are written enharmonically, that is the tone will be the same, but different notes will be used. For example, the scales of G-flat major and F# major, although written differently, sound alike. We shall explain the structure of major scales by beginning on the note “C”. On the piano, this scale begins on C, and progresses either up or down solely on the white keys. Incidentally, “major” means in music just what it means in everyday language. We say: “The major part of the day was sunny.” Here “major” means “greater.” So it is in music. For example, when the name was coined, the “major” scales were those used mostly in writing music. By the same token, “minor” means “lesser”, both in music and when used in every day conversation. “The ‘minor’ (or ‘lesser’) part of the day was cloudy.”
Here is the C major scale. Since it is played on the white keys, no sharps or flats are used in the key signature. When we examine the intervals in this scale, we find that both whole and half tones are used.
As we examine this scale closely, we find that the four lower tones have the same intervals as the four upper, and are separated —separated by a whole tone. These two halves of the major scale are called tetrachords, the word “tetra” meaning four.
If we were to build a wooden ladder, we would endeavor to maintain the same distance between the rungs, no matter where we intended to use it, either in the basement or on the roof. So it is with music scales. These must retain the same intervals in the tetrachords no matter where the starting (tonic) tone is located.
Here is how this works in actual practice, using the upper tetrachord of the C major scale to become the lower tetrachord of the G major scale.
By the simple expedient of raising the F in the upper tetrachord one-half step, we correct the interval relationship.
So we form the scale of G major, and indicate this by using one sharp (F#) in the key signature.
Let us use the same ladder, and begin a scale on the upper tetrachord of the G major scale, starting on “D”.
By the simple expedient of raising the C one-half step to C#, we place the intervals correctly, and have the scale of D major, with F# and C# in the key signature.
By progressing, and using the upper tetrachord as the lower in a new key, we can write fourteen major scales in sharps, concluding with the scale of C double sharp, with fourteen sharps (seven double sharps) as follows. (A double sharp is indicated by an X, and raises a note a whole tone.)
The Cx major scale is enharmonic with (sounds like) the D major scale. Now, let us refer back to the C major scale in descending form, and for the purpose of convenience write it an octave higher.
In flat keys we begin with the lower tetrachord, and build down.
By the simple expedient of flatting the B (lowering it one-half step) we place the intervals correctly to make the scale of F major, with one flat (Bb) in the key signature.
Beginning with the lower tetrachord in the key of F. and flatting the E (lowering it one-half step) we place the intervals correctly to make the scale of Bb major, with two flats in the key signature.
By using the lower tetrachord to begin each new scale, and correcting the intervals by flatting the tones as indicated, we keep adding flats to the key signature until we reach the key of Cbb (double flat) as follows: (A double flat is indicated by the sign “bb” and lowers the note one whole tone.)
X: 1
L: 1/8
K: C
T: Cbb Major Scale, ascending
__C8 __D8 __E8 __F8 | __G8 __A8 __B8 __c8
w: . W W H (W) W W H
W: Enharmonic with
This scale is enharmonic with the Bb major scale. As we build all of the scales, we find that in playing the key of F# major, with six sharps in the key signature, we are using the same keys on the piano keyboard that are used to play the scale of Gb major which has six flats in the key signature. That is, although written differently, they sound alike. In the same manner, the key of B major (five sharps) is enharmonic with that of Cb major (seven flats); and C# major (seven sharps) with Db major (five flats). If the sound is the same, why use seven sharps in the key signature (C#) when five flats (Db) will get identical results? Scientists tell us that the enharmonic keys just mentioned are not the same in pitch, except in keyboard and set pitched instruments. So far as playing in the band is concerned, however, they do sound alike, and the “tempered” scale of a well-tuned piano can be used for tuning purposes.
Now to a discussion of the minor mode. “Mode” in music might be defined as meaning a family of scales. Only three minor modes are used extensively, the “pure” or “natural”, the “melodic”, and the “harmonic”. Remember that a scale is a ladder of tones, and that the intervals must remain constant in any particular family or mode of scales.
If we begin a scale on the sixth step (A) of C major, and progress up or down for an octave using only the white keys of the piano keyboard, we have the “pure” or “natural” minor scale in the key of A.
Scales built like this from the 6th step of a major scale are called “natural” minor scales. They borrow the key signature of the major scale upon which they are built.
Now we must consider an oddity in our hearing. We prefer to hear the 7th step of the scale only one-half step from the key note of the scale. It takes a little time to get accustomed to anything else. (Note that whole tone scales sound very strange at first, especially to young students.) We can change the strange feeling of the A “natural” minor scale by raising the G one-half tone to G#. We will then have started a new family or mode of scales called the “Harmonic Minor”.
If we were consistent in writing key signatures, the key of A harmonic minor would have one sharp, G# in the key signature. But it might prove confusing to use different key signatures for minor scales which begin on the same note. So tradition has it that regardless of the mode used, minor scales borrow the key signature of the major scale upon which they are built. In any case, the first five notes of minor scales are the same in each mode. In all minor scales upper and lower tetrachords do not coincide as to the intervals. Notice that in the harmonic minor scale, besides intervals of a half-tone, and whole tone, one of a whole plus a half-tone is used, called an “augmented second” . When it is played or sung, the harmonic minor scale sounds strongly minor. Perhaps this is the reason why many teachers prefer it, and use it in formal scale practice. Now. in recalling the structure of the scales, it will be remembered that the Chromatic scale is a scale of half-tones: that Whole Tone scales are made up of whole tone steps; and Major Scales are a combination of whole and half-tone intervals. The Harmonic Minor has half-tone, whole tone, and tone and a half intervals.
I would like to invent a little historical story and say that in the evolution of the minor scale system the “Natural” minors came first. But these did not feel quite right so the “Harmonic” minor scales were created. Vocalists found that singing the “augmented 2nd” in this scale was difficult, so another minor mode was invented to eliminate this interval by raising the sixth degree of the minor scale, and the result was the “Melodic” Minor Mode. (Please remember that my historical story is sheer fiction.) The Melodic minor scale descends as the “Natural” minor.
The melodic minor is not a strongly “flavored” scale. It has always seemed to me that practice on major scales enables one to play melodic minor passages at sight.
To recapitulate this discussion of scales. I repeat that a little theory tossed into the hopper during a student’s training will give him a better understanding, and will improve his ability to read and play at sight. I recommend strongly that students be lured into practice on the chromatic scale, the two whole tone scales, the twelve major scales, and by all means, the twelve Harmonic Minor scales. This practice will provide both foundation for technique and a training in listening.
Chords
Now to the consideration of elementary “Harmony”. A simple definition for harmony as used in music is “the art of combining tones”. When a combination of tones sounds pleasing, it is called “concordant”. The opposite is “discordant”. When two or more tones sound simultaneously, we say they are being played “concento”; when the notes of a chord are played rapidly as a series of separate tones, we call this an “arpeggio”—that is “harp-like”.
We begin with a discussion of triads. A “triad”, similar in meaning to the usual “trio”, is a group of three notes. A triad is made of thirds. A third is an interval between two notes of either one and one-half steps (minor third) or two full steps (major third). To illustrate: In the scale of C major the note E is a third above C, counting stepwise C-D-E. Since this interval contains two whole tones, and is in the major scale, we call it a major ( or large third. In the scale of C minor, counting stepwise, we have a third which reads C-D-Eb. Since this interval occurs in a minor scale and contains only a tone and one-half, we call it a minor (or lesser) third. The upper note in a triad will be a fifth above the lowest and a third above the second note in the triad. In the examples given below, since the interval C to G is in the major scale, it is called a “perfect fifth”. Raising the G to GJ makes the interval one-half step larger, or an “augmented fifth”. Lowering it one-half step to Gb makes the interval smaller, or a “diminished fifth”.
Mathematically there are only four combinations of “true” thirds which make “true” triads.
When triads are inverted (turned over) they still retain their names. For example, the C major and E major triads are shown in all possible formations.
Before delving into “seventh chords”, I would like to present a chart of all the most used intervals in an octave, using “C” as a measuring note. These abbreviations will be used.
Perf.—Perfect Aug.—Augmented dim.—Diminished
If another third is added above a triad, the interval from the lowest tone to the new upper tone is a seventh, i.e. C-E-G-Bb C to E a 3rd; C to G a 5th; and C to Bb, a seventh. This combination of notes is called a chord.
Here is an analysis of a “Seventh Chord”.
Since the chord is used mostly in the key of F, and since it is built on the 5th or “dominant” note of the scale, it is called a “Dominant Seventh” chord in the key of F, either major or minor. If we follow the same plan used in plotting the chart for triads, it is possible to build seven “true” 7th chords having “true” thirds using any tone in the chromatic scale as a starting point. (There is an eighth 7th chord possible, but it is enharmonic with the augmented triad. So, for the sake of convenience in remembering, let us settle on seven 7th chords.
The last two examples, #6 (the dominant 7th) and #7 (the diminished 7th) are the most useful 7th chords. Many old folk tunes may be harmonized by using the tonic or first chord of the scale, and the dominant 7th, or fifth chord. For example:
Stephen Foster’s “Oh Susanna” may be harmonized with these chords, each phrase being harmonized with just one chord. Notice that these chords contain all the notes of the C major scale except the sixth step, “A”, which can be used to harmonize beautifully as the “9th” in the dominant 7th chord; and the second step, “D”, which can be omitted from the dominant 7th chord to secure a smoother progression.
It might be interesting to write out all the major and harmonic minor scales through one octave, then build triads on each note, classifying these as major, minor, augmented, or diminished. Notice the location of each kind of triad. Mark major triads with large Roman numerals, minor triads with small Roman numerals, augmented triads with large Roman numerals followed by a dash, and diminished triads with small Roman numerals and a small circle. It is hard to imagine a better way to teach students to read music rapidly. As soon as they begin writing, even if it is only copying, they will begin to read accurately and with fluency. Here are two examples:
When this is done, it will be noted that each major triad will appear in five keys. For example:
Each minor triad will appear in five different keys.
Each diminished triad will appear in three different keys.
Each augmented triad appears in only one key.
Each dominant 7th chord will be found in only two keys. A dominant 7th chord may establish the key of a piece but not its mode, for the chord is the same in both major and minor.
Each diminished triad in root position appears in only one key. It is important to remember all this, for when practicing a major triad in arpeggio form, proficiency is being acquired in five keys. I have heard harmony instructors call this “the plural significance of chords”.
N.B. There are really only four augmented triads. Each will be found in three keys, but in inverted, enharmonic form.
There are really only three diminished 7th chords. Each will be found in four keys, but in inverted enharmonic form.
To summarize this chapter, we mention the “magic twelve” again and recapitulate briefly. There are only:
- 12 basic note patterns
- 12 basic articulations
- 12 notes in the chromatic scale
- 12 chromatic scales (resolving to one)
- 12 whole tone scales (resolving to one)
- 12 major scales (when enharmonic scales are excluded)
- 12 minor scales in any mode
- 12 major triads
- 12 minor triads
- 12 diminished triads
- 12 augmented triads (resolving to four)
- 12 each of the seven different 7th chords
- and finally 12 items in the list of “twelves” just enumerated!
This taste of elementary theory has been placed here to guide the director in teaching this subject a bit at a time as the young student progresses. Eventually, they gain an understanding of music, and, even more important, lose all fear of music fundamentals. Perhaps this small taste will whet their appetites for a more substantial portion of the same.
Can anyone question seriously the value of a knowledge of the structural formation of scales and chords to the music student who seeks to improve his “sight playing” ability?
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