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The
Origin of Musical Scales in Western Music Tradition |
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Musical scales are patterns of ascending and descending tones used as the building blocks of melody in a music composition. Many different types of scale patterns exist in various musical cultures throughout the world. In Western musical tradition, the two basic musical scales, major and minor, were developed based on scientific principles involving vibrating strings and resonating air columns of wind instruments. The Melodic Interval Structures of Major and Minor Scales The names of the notes used for the series of tones of the C major scale are C, D, E, F, G, A, B and C (in the next octave). Graph and audio example of a C Major Scale The musical intervals between the sequential tones in a major scale are as follows: 1 to 2—whole step 2 to 3—whole step 3 to 4—half step 4 to 5—whole step 5 to 6—whole step 6 to 7—whole step 7 to 8—half step (8 is also called 1 of the next octave, in this case a “C”.) The seventh tone of the major scale is often referred to as the “leading tone” and usually is followed by the tonic or key center, in this case, B to C, when played in a melody or in a series of two chords. The natural form of a minor scale is based on the major scale. Major and minor scales that have the same key signature of sharps or flats are called the relative major and minor scales. The C major scale has a relative minor scale (A minor) based on the sixth tone of the major scale. Both scales have the same key signature, no sharps or flats. In this case, A minor is the relative minor of C major and has a pattern of whole-steps and half-steps in a series that is described as the “natural form” of a minor scale. The note names for tones in the natural form of the A minor scale are A, B, C, D, E, F, G and A.
Graph and audio example of the natural form of an A
Minor Scale Melody passages in music often involve the alteration of the sixth and seventh tones of minor scales. These two scale tones often are altered to make a minor scale behave more similarly to its parallel major scale. (Note: Parallel major and minor scales have different key signatures but have the same tone or tonic note for their key centers. For example, the natural form of the C minor scale (parallel minor to C major) is shown below.) Graph and audio example of the natural form of the C
minor scale After hearing the audio examples of the above two minor scales, note that the interval patterns of the scale tones are the same: whole-step, half-step, whole-step, whole step, half-step, whole-step and whole step. In ascending melody patterns, the seventh tone of the minor scale is often raised one-half step to make the seventh tone sound more like a “leading tone”. The resulting scale with a half-step between the new seventh tone of the scale is described as the “harmonic form” of a minor scale.
Because the result is also an increase of a half-step in the interval between the sixth tone and the new seventh tone, now one and one-half steps, composers often also raise the sixth tone one-half step in the ascending melody. The final result is a new scale described as the “melodic form” of a minor scale. However, a descending melody uses the natural form of the minor scale.
Notice the difference in the intervals of tones sixth and seventh tones as they ascend and descend in the scale. A
Scientific Basis for Tones of the Major Scale: Vibrating strings The ancient Greek, Pythagoras, observed that the vibrating frequency of a tone produced on a string that is plucked or bowed is dependent on the length of the string. If the length of a vibrating string is divided in half, a different tone is produced with a vibrating frequency twice that of the full length of the string. A tone with twice the vibrating frequency of another tone is defined as an “octave”. Tones of a major scale within an octave range can be produced with string lengths longer than half the total length of the string. For example, two-thirds length of a string produces a perfect fifth (a 3:2 frequency ratio). Three-fourths of a string length produces a perfect fourth (a 4:3 frequency ratio). And, four-fifths of the length produces a major third (a 5:4 frequency ratio). In a C major scale, the perfect fifth is a G, the perfect fourth is an F and a major third is an E. Guitars have six strings of the same approximate length but different pitches due to the different thicknesses and textures of the strings. With six strings tuned to different pitches from low to high, a large number of tones can be produced on a guitar. Its ease of portability and its large range of tones makes the guitar very useful as an accompanying instrument for singers. Another advantage of the guitar involved the use of frets, metal bars placed on the fingerboard. The frets are placed at appropriate lengths of the fingerboard to facilitate the production of fixed scale tones with a series of half-step intervals. Violins have four strings of different pitch tuned a perfect fifth apart (G, D, A and E) and the fingerboard does not have frets. Violin players learn to press a finger on a string at various places on the violin fingerboard to select a certain string length to produce a tone of a specific pitch. When the four strings of a violin are tuned properly, tones produced on one of the strings with the simple frequency ratios discussed above have a more resonant tone than tones with slightly higher or lower pitch. For example, the lowest string of a violin is tuned tightening or relaxing the tension of the string to produce a tone with the note name G. If a finger is placed properly on this string to produce a D above, the interval of a perfect fifth. The second string properly tuned to a D above the G of the first string should also vibrate in sympathy when the D fingered on the G string is plucked or bowed. When the other strings resonate in “sympathy” with the fingered tone, the loudness of the tone is increased. This acoustical phenomenon is called “sympathetic vibration”. Music Keyboards Music keyboards are constructed to accommodate the major and minor scales of Western music and the practical frequency range of human hearing. The frequency range of hearing for humans is approximately 20Hz to 20,000 Hz (20KHz). The most commonly used music keyboards are the acoustic piano and various types of electronic music keyboards. The piano has 88 keys used to produce a series of 88 tones, each one half-step apart in the series of tones from low to high. For example, the lowest tone of a piano music keyboard is A0, approximately 25 Hz. Playing from low to high the first eight notes of the lowest white keys of a piano keyboard would include the notes A0, B0, C1, D1, E1, F1, G1 and A1. These notes represent the interval patterns in the natural form of the A minor scale, the relative minor scale of C Major. The C major scale involves the following notes: C, D, E, F, G, A, B and C. If these notes are played on a music keyboard, they would involve the white keys starting on C and ending on the next C an octave higher. The precursor of the modern piano or pianoforte as it was originally called, was the harpsichord or cembalo. The harpsichord uses keys which operate levers that pluck a string to start the sound vibration. This method has the limitation of allowing only one loudness level for each tone. To make the sound of a tone louder, the lever could be used to pluck one of an additional set of strings tuned exactly to the first set. This is a typical Renaissance style of loudness contrast which essentially added musical instruments for increases in loudness or decreasing the number of instruments being played for softer passages. The piano uses a technology of pressing fingers on wooden keys that cause soft hammers to strike the strings. The harder or faster (key velocity) the key is struck, the louder the sound of the tones produced. Thus, a pianoforte could play both soft (“piano”) and loud (“forte”) tones. This technology was developed in the late eighteenth and nineteenth centuries and resulted in more emotional and dynamically charged music composition styles. Harpsichords and pianos also differ in the way they are tuned. Musical composition of the seventeenth and eighteenth centuries were often confined to the use of a single scale or key signature. Therefore, a harpsichord was usually tuned using the Pythagorean scale which consisted of tones tuned in pure intervals involving whole number ratios of their fundamental frequencies. The pure style of tuning a major scale sounds very good in the performance of music written in the key for that scale. If other major scales involving additional sharps or flats are played when a keyboard is tuned to a C major scale with Pathagorean intervals, the music sounds out of tune to trained musicians. However, during the development of the piano and compositions for piano, it was decided by instrument manufacturers and musicians, to tune the tones of the piano and organ to what is called an equal tempered scale. Each octave is divided into twelve equal pitch intervals. In other words, the twelve tones in an octave are “equally out of tune”. However, tones sound very sharp in the low range of a piano a very flat in the high range if tuned exactly to perfect octaves and equal tempered tuning. Piano tuners account for this psychological perception of the sharpness and flatness of extreme octaves by “stretching” the octave intervals by tuning the strings to intervals that are slightly larger than the 2:1 frequency ratio of natural octaves. Eventually, musicians have learned to accept this type of imperfect tuning because it allowed for composing more complex styles of music using a wider variety of key signature changes and their resulting scales. As mentioned before, playing a series of white and black keys in succession is called a “chromatic” scale, resulting in a series half step intervals between each of the 12 sequential tones within an octave. Chromatic scales began to be used more often after the development of the pianoforte and increasingly more by music composers of the nineteenth and twentieth century. Students interested in a more complete history of tuning systems can research the following concepts: tuning and temperament—Pythagorean, mean-tone and equal-tempered tuning. The
Natural Harmonic Series of a
Vibrating String In addition to the octave, which is produced on half the length of a vibrating string, other tones can be produced on string lengths smaller than one-half. The vibrating frequency of one third of the length of the string is three times the vibrating frequency of the full length of the string. By continuing to divide the string into simple fractions, ½, 1/3, ¼, 1/5, 1/6, etc., a series of tones can be produced with frequency ratios that are whole number multiples (2, 3, 4, 5, 6, etc.) of frequency of the full length of the string. The pattern of tones produced with increasing whole tone ratios is defined as the “natural harmonic series”. Natural
Harmonic Series of Brass Wind Instruments Tones of the natural harmonic series can also be observed by the vibrating properties of tones produced on brass and woodwind instruments. A brass instrument player can produce a tone of a specific pitch by blowing air against his or her lips in the mouthpiece of an instrument. Blowing with increased air pressure, in general, increases the vibrating frequency of the tone, resulting in a tone of higher pitch. However, the tones produced with just the mouthpiece, are not very loud. By attaching the mouthpiece to an instrument, the tone produced by the “buzzing” lips can be amplified. On brass instruments, the length of the “resonating” air column in the tube of the instrument is used to amplify the sound of a tone with a specific vibrating frequency. However, by increasing the air pressure on the vibrating lips, a series of tones that have whole number frequency ratios can be produced. Theoretically, a brass instrument player could produce the following series of tones, from low to high, notated on the Grand Staff below. (Graphic of harmonic series) The lowest theoretical tone is called a “pedal tone” but is very difficult to produce in practice. In music theory, the lowest tone is called the “fundamental”, short for fundamental harmonic. The next highest tone is called the second harmonic and has a pitch of a tone that is twice the frequency of the fundamental (an octave). Likewise, the third harmonic has a frequency that is three times that of the fundamental. The interval between the third harmonic and the fundamental is an octave plus a perfect fifth. The following table summarizes the theoretical natural harmonic series based on a tone C2 as the fundamental harmonic that can be produced on a brass instrument constructed in the key of C.
Note that harmonics 8, 9, 10, 11, 12 13, 15 and 16 approximate the tones of a C major scale. However, the harmonic 11 is a tone that sounds like a sharpened F or flatted F-sharp, a tone with a pitch between F and F-sharp. In music practice, harmonics 3 through 16, can easily be played on a natural horn which is a brass instrument with a very long and thin resonance tube and no valves, the precursor of the modern French horn which does have valves. This type of horn was very useful to composers in eighteenth century classical music because it produced tones similar to a major scale. On the other hand, certain tones like harmonics 11 and 15 were out of tune and had to be raised or lowered by placing a hand at different positions inside the bell of the instrument. The natural horn could also play in major scales with different key signatures by changing the overall length of the resonating tube. The length of tubing for a natural horn can be changed by adding short lengths of tubing called “crooks” between the mouthpiece and the main tube of the instrument to play tones in the natural harmonic series of other major scales. The development of brass instrument technology also involves more modern methods of adding varying lengths of tubing to change the pitch of a tone. The slide trombone is a obvious example. The slide is a double tube, a length of tubing placed inside a part of the main resonating tube of the trombone. When the outside of the slide is gradually pulled out, the overall length of the resonating tube is increased and the pitch of the tone is lowered. The trombone player must learn the specific “slide positions” of the slide that correspond to tones of musical scales. The invention of valves for brass instruments was a major technological development and allowed the development of more complex styles of music composition for brass instruments. A valve switches the airflow of a brass instrument into a small length of additional tubing. Increasing the length of the resonating air column lowers the “resonance frequency” of the instrument and results the production of a tone of lower pitch. Most valved brass instruments have three valves. First valve played with the index finger lowers the pitch of a tone one whole-step (a major second—C to B-flat). The second valve played with the middle finger lowers the pitch one half-step (a minor second—C to B). The third valve played with the ring finger lowers the pitch one and one-half steps, the equivalent of valves 1 and 2 combined (a minor third—C to A). The combination of valves 2 and 3 lowers the pitch two whole-steps (a major third—C to A-flat). A combination of valves 1 and 3 lowers the pitch two and one-half steps (a perfect fourth—C to G). A combination of all three valves lowers the pitch three whole steps (an augmented fourth or a diminished fifth—C to F-sharp or G-flat). The above examples use C as the harmonic partial that is changed by using valves. If the valves are added when a G is the starting harmonic partial, the resulting tones with these six different valve combinations would be F-sharp, F , E, E-flat, D and D-flat or C-sharp. Valved instrument brass players must memorize the appropriate valve fingering combinations for various scale tones used in a musical composition. Some brass instruments have an additional fourth valve. The modern French horn is called a double horn, one horn pitched in the key of F and the other in the key of B-flat, using a thumb valve to switch from one horn to the other. The reason for the double French horn is to improve intonation by choosing the horn and valve combination that has the best pitch for achieving good intonation in an ensemble such as a band or orchestra. Some tubas, euphoniums and trombones have similar additional fourth valves. Resonating
Properties of Woodwind Instruments In general, the vibrating frequency of a tone produced on a woodwind instrument such as a flute or oboe is dependent on the length of the vibrating column of air within the hollow tube of the instrument. For example, if the length of a flute is cut in half, the resulting tone is an octave higher than the tone produced on the full length of the original flute. A pan flute is a series of flutes tied together in a series of gradually decreasing lengths that can produce tones of various pitches in a musical scale. Tones of different pitch also can be produced on a vertical flute (called a recorder) or on a traditional metal flute (transverse flute) by drilling a series of holes in the tube of the instrument. The exact location of each hole determines the frequency of vibration of a tone. By covering all of the open holes of the instrument, the lowest possible frequency tone is produced. Covering the holes that allow a tone to be produced on half the length of the tube produces an octave. The technology of drilling holes and adding metal keys to cover the holes is very complex and required several centuries of development. Woodwind instruments also can produce tones of a harmonic series but with a more limited range of maybe three or four harmonics. For example, a tone of octave interval can be produced on a flute, oboe, bassoon or saxopone, by blowing with more air pressure, causing the second harmonic tone to sound. This is called “overblowing the octave”. Overblowing can also produce the octave plus a perfect fifth. Overblowing the next octave is much more difficult, but possible. Thus, the first, second, third and fourth harmonics can be produced, for example, C, the second octave C, the second octave G and third octave C using the same basic note fingering. The clarinet can only produce the first, third and fifth harmonic, for example, C, the second octave G and the third octave E using the same basic fingering. |
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