The Paradox of Music
Music is a paradox. You know when you hear it, but one man's music is another man's noise. William Congreve wrote that "Music hath charm to soothe the savage breast," yet Woody Allen said, "I can't listen to that much Wagner. I start getting the urge to conquer Poland." Throughout history, mankind used music to dance, celebrate, march to battle, pray to their God, mourn their dead, win a lover's heart, and evoke social change. Why does music have so much power? How does it work? The study of music theory can answer this question.
The Science of Music
Sound is the expansion and contraction of molecules through a given medium. We hear sound when vibrating air molecules, or sound waves, reach our eardrums. There are several parameters that shape a sound wave. The three that specifically relate to music are:
- Frequency or pitch, the highness or lowness of a sound
- Amplitude, the loudness or softness of a sound
- Timbre or tone quality, the clearness or dullness of a sound
Example 1.1 is a visual representation of one sound wave. The green line above the red line represents expansion, below represents contraction. Amplitude increases when green line moves farther away from the red line, softer when it is close.
You cannot hear just one sound wave; they last for only a split second. You need a chain reaction of sound waves, as in
example 1.2. Notice that the sound gets louder as the sound waves get taller. The measurement of amplitude is called decibels or dB. In
example 1.3, the points where the green and red lines cross slowly get closer together. Notice that the pitch gets higher as the crossing points, called
nodes, narrows. Frequency is measured by the number of vibrations per second, or hertz (Hz).
In both examples, you can hear a musical tone because the shape of the waves is consistent and organized. Inconsistent and disorganized sound waves produce noise, as in example.
In example1.3 , a much softer sound wave is suddenly mixed with a sound wave. Notice that the frequency and amplitude not affected. What changes is the tone quality, or timbre. The human voice and most musical instruments naturally produce tones mixed with these very soft sound waves, called
harmonics or overtones. The amplitude of the harmonics is what gives a musical instrument its unique sound.
Harmonics do not occur randomly but rather in a set pattern called an overtone
series. The series is based upon ratios of a given pitch, called the fundamental:
- In Example 1.4, the frequency of opening sound is 100 Hz; the second is 200 Hz. Notice that the nodes to the second pitch are twice as close together, creating a
pitch ratio of 1:2. These two pitches are called octaves, or
octave notes. Even though the second pitch sounds higher than the first, acoustically they sound similar. If the lower octave note is the fundamental or first pitch in a given overtone series, then the higher octave note is the
1st harmonic (also called the 1st overtone) in this overtone series.
- In example 1.5, the frequency of opening sound is 200 Hz; the second is 300 Hz, creating a pitch ratio of 2:3. Unlike octaves, these are two differently sounding pitches. Together, they are called a
Perfect 5th. The second sound in this example is the 2nd harmonic
(2nd overtone) of this overtone series. The pitch ratio of the fundamental to the 2nd harmonic is 1:3.
- In example 1.6, the frequencies of the two pitches are 300 Hz and 400 Hz, creating a pitch ratio of 3:4. The second pitch is the 3rd harmonic of this overtone series. The fundamental, the 3rd harmonic are also octaves because:
- The fundamental and the 1st harmonic are octaves, pitch ratio of 1:2
- The pitch ratio of the 1st and 3rd harmonics is 2:4.
- 2:4 in proportion to 1:2, making the 3rd harmonic and octave note to the fundamental.
- The pitch ratio of the 3rd and 4th harmonic is 4:5. It is not an octave to the fundamental or previous harmonics, making it the second new tone of the overtone series.
- The pitch ratio of the 4th and 5th harmonic is 5:6. The 2nd and 5th harmonics are octave because 3:6 in proportion to 1:2.
In theory, the overtone series goes on to infinity. In practicality, the series stops when the overtones reach 25,000 Hz, the highest pitch heard by humans.
There is more than one overtone series. The sequence of tones in example
1.7 is an overtone series, based upon a fundamental with a frequency of 100 Hz. The overtone series in
example 1.8 is based upon a fundamental with a frequency of 125 Hz:
- The pitch proportions of each corresponding harmonic are identical.
- The frequencies of the two fundamentals are different; therefore the pitches of corresponding harmonics are different.
The Evolution of Music
Ever since the dawn of time, we as humans have had an innate ability to find order in chaos. For example,
- Have you ever seen a cloud in the form of an elephant?
- Why does Italy geographically look like a boot?
Humans first ordered music in two ways:
- By grouping together pitches that are common to an overtone series. Adjusted for
octaves, the pitches in example 1.9 are from the same overtone series:
- The third pitch is the 8th overtone
- The fourth pitch is the fundamental
- The fifth pitch is the 12th overtone
- The sixth pitch is the 2nd overtone
- The seventh pitch is the 12th overtone
- The eighth and ninth pitches are the fundamental
- The tenth pitch is the 8th overtone
- The last pitch is the 4th overtone
- By grouping articulated pitches to a repetitive unit of time, called a
beat.
When pitches and beats were grouped together, the result was a musical element called
melody.
Later, as civilization evolved and became more complex, so did music. Over time, more musical elements were created:
- Rhythm: The subdivision of the beat
- Harmony: More than one pitch heard at the same time, often called chords
- Texture: The of number melodies heard at the same time
- Form: The organization of identical, different, and similar melodies
- Timbre: The clearness or dullness of a sound
The evolution of these musical elements was unique from culture to culture. For example:
- In West Africa, rhythm is the most complex element but harmony and melody are less developed.
- In both Arabic and Indian music, melody and rhythm are extremely intricate but harmony is almost non-existent.
- In China, the subtle changes of timbre are the most complex.
- In Western Europe, melody and rhythm are very simple, but harmony is the most complex of all cultures. It is the
defining element of Western (European) music.
The seeds of Western harmony started in Ancient Greece with the mathematician, Pythagoras. By striking strips of metal of various proportions, he discovered the ratios of the overtones series and came to these conclusions:
- Musical tones come from the harmonics of the overtone series.
- The fundamental was acoustically the most important pitch. The other tones center around it.
- Specific sequences of pitches, from a given tone to its octave note (what we know today as
scales) created specific emotional responses. Pythagoras categorized each these sequences and called them
modes.
- When struck together, some overtone combinations created a pleasant, harmonious sound. Others create an unpleasant, discordant sound.
- Pitch ratio combinations from the 3rd, 4th, and 5th harmonics were the most pleasant:
- Harmonics 3 and 5 create the pitch ratio 2:3 (what we know today as Perfect 5th)
- Harmonics 3 and 4 create the pitch ratio 4:5 (what we know today as
Major 3rd)
- Harmonics 4 and 5 create the pitch ratio 5:6 (what we know today as
Major 3rd)
- Pythagoras noticed that harmonics 3, 4, and 5 were also the numbers he used in the Pythagorean Theorem.
- The universe, therefore, is a held together by parallel musical and mathematical relationships, which he called the “music of the spheres.”
The birth of Western harmony happened around 1000. Catholic monks created a crude form of harmony by simultaneously singing two melodic lines (example):
- The higher sounding melody was a Gregorian chant
- The lower sounding melody was the Gregorian chant’s fundamental, sung long and augmented (what we know today as a
drone).
Gradually, the drone started to copy the melodic contour of the Gregorian chant, using the pitch ratio of 2:3 as a guide (example). Monks chose this ratio because:
- God created nature; the overtone series is natural occurrence, hence God created the overtone series.
- The fundamental represents God, or perfection.
- 1:2, 2:3, and 3:4 are the closest pitch ratios to the fundamental. They called these ratio perfectus, or perfect.
- All other pitch ratios (including 4:5 and 5:6) were discordant called imperfectus, or imperfect.
As Greek science, mathematics, and philosophy began to influence the Italian Renaissance, 1400 – 1600, Pythagorean harmony influenced music:
- Ratios 4:5 and 5:6 became harmonious, just as Pythagoras prescribed.
- The Ancient Greek modes were simplified into two, each with its own emotional response:
- A “happy” sounding mode was called major
- A “sad” sounding mode called minor
The years 1600 – 1900 are called the Common Practice Period, when Western harmony reaches its maturity. Bach, Mozart, Beethoven, and other famous European composers lived during the Common Practice Period. They all used the same “musical pallet” to create their great masterworks:
- Music is based upon tonality, which is a hierarchy of pitches that gravitate to a
tonal center, or tonic (the fundamental to an overtone series).
- The best method to establish tonality to write harmony that continual moves away, then back towards the tonic
- Chromaticism, or notes outside a diatonic scale, expands the composers “emotional pallet.”
As chromaticism increased, the importance of tonality decreased. The Common Practice Period ending in the early 20th century, when some composers began to write non-tonal music:
- Atonality, music without a tonal center
- Polytonality, music multiple tonal centers
Other composers rejected non-tonal music and continued to write tonal music. Still others composed music with fused with both tonal and non-tonal elements, called modern tonal music. The majority of new music today is in a modern tonal style.
The Arrogance of Western Music
Music theorists of the Common Practice Period believed that their system of music was superior to all other cultures. Even today, other world cultures base their harmony on Western Europe. Common Practice Period theorists justified this notion by linking harmony to nature, then to God, through the overtone series. There are, however, flaws in this way of thinking:
- The fourth note of the major scale is not found in the overtone series
- The emphasis on the 7th note of a scale, called the leading
tone, is difficult to justify since its pitch ratio, 14:15, is not close to the fundamental.
- Minor scales were created by mutating a Greek diatonic mode. Mutations are not natural; therefore not of God.
- During the Common Practice Period, Pythagorean tuning, tuning based upon the overtone series, was replaced with
Equal temperament tuning, a tuning method that slightly mutates that frequency of each tone.
Western harmony is one of many musical systems. It is primarily based upon nature (the overtone series), but also human imperfection (tradition). What theorists did not realize is that perfectly played music does not sound human, but rather robotic. It is this tradition, along with nature, that creates the emotions of music.
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