(MAF) Music, Math, and Science Lesson
Music, Math, and Science
Music theorists sometimes use mathematics to understand music, and although music has no foundation in modern mathematics, mathematics is "the basis of sound" and therefore music. Music exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical".
The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory. Some composers have incorporated the golden ratio and Fibonacci numbers into their work.
History
Though ancient Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound, the Pythagoreans of ancient Greece were the first researchers known to have investigated the expression of musical scales in terms of numerical ratios, particularly the ratios of small integers. Their central doctrine was that "all nature consists of harmony arising out of numbers".
From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics. Early Indian and Chinese theorists show similar approaches: all sought to show that the mathematical laws of harmonics and rhythms were fundamental not only to our understanding of the world but to human well-being. Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.
Time, Rhythm, and Meter
Without the boundaries of rhythmic structure - a fundamental equal and regular arrangement of pulse, accent, phrase and duration - music would not be possible. In Old English the word "rhyme", derived to "rhythm", became associated and confused with rim - "number"- and modern musical use of terms like meter and measure also reflects the historical importance of music, along with astronomy (science), in the development of counting, arithmetic and the exact measurement of time.
Musical Form
Musical form is the plan of a piece of music. The term "plan" is also used in architecture, to which musical form is often compared. Like the architect, the composer must take into account the function for which the work is intended and the means available, making use of repetition and order. The common types of form known as binary and ternary ("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.
A musical scale is a set of pitches used in making or describing music. The most important scale in the Western tradition is the diatonic scale, but many others have been used and proposed in various historical eras and parts of the world. Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.). A scale has an interval of repetition, normally the octave. The octave of any pitch refers to a frequency exactly twice that of the given pitch.
Succeeding superoctaves are pitches found at frequencies four, eight, sixteen times and so on , (higher), of the fundamental frequency. Pitches at frequencies of half, a quarter, an eighth and so on (lower) of the fundamental are called suboctaves. Therefore, any note and its octaves will have the same name.
When expressed as a frequency bandwidth an octave A2 -A3 spans from 110 Hz to 220 Hz. The next octave will span from 220 Hz to 440 Hz. The third octave spans from 440 Hz to 880 Hz and so on. Each successive octave spans twice the frequency range of the previous octave.
Because we are often interested in the relations or ratios between the pitches (known as intervals), it is usual to refer to all the scale pitches in terms of their ratio from a particular pitch, which is given the value of one (often written 1/1), generally a note which functions as the tonic or main note of the scale.
The chart below shows the intervals created within a major scale, with frequencies and ratios.
Semitone |
Ratio |
Interval |
Natural |
Half Step |
---|---|---|---|---|
0 |
1:1 |
unison |
480 |
0 |
1 |
16:15 |
minor second |
512 |
16:15 |
2 |
9:8 |
major second |
540 |
135:128 |
3 |
6:5 |
minor third |
576 |
16:15 |
4 |
5:4 |
major third |
600 |
25:24 |
5 |
4:3 |
perfect fourth |
640 |
16:15 |
6 |
45:32 |
tritone |
675 |
135:128 |
7 |
3:2 |
perfect fifth |
720 |
16:15 |
8 |
8:5 |
minor sixth |
768 |
16:15 |
9 |
5:3 |
major sixth |
800 |
25:24 |
10 |
9:5 |
minor seventh |
864 |
27:25 |
11 |
15:8 |
major seventh |
900 |
25:24 |
12 |
2:1 |
octave |
960 |
16:15 |
Pythagorean tuning is tuning based only on the perfect consonances, the (perfect) octave, perfect fifth, and perfect fourth. Western common practice music requires a systematically tempered scale. In equal temperament, the octave is divided into twelve equal parts called semitones or half-steps. Each of these twelve equal half steps adds up to exactly an octave. Because of this historical force, twelve-tone equal temperament is now the dominant intonation system in the Western, and much of the non-Western, world.
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