Chapter R: Prealgebra Review
Numbers in the Real World

Note Taking


Everyone loves to hear music of some kind.  We know that music is made up of notes, and you may even be able to tell when a "wrong" note is played, but what is a note?  Is a note the same in all cultures?  And how are notes and music related to math?

All music is produced in frequencies.  Some musical instruments produce frequencies higher than the human ear can hear.  The range of frequencies that human hearing can discern is split into subranges, called octaves.  An octave is defined as a frequency range such that the ending frequency is twice the beginning frequency.  Octaves are defined in this way due to the logarithmic characteristics of human hearing.  (You will study logarithms in College Algebra).  Frequencies are measured in hertz (Hz).

Piano keyboards are tuned based on a certain note, middle C, as shown in the figure below.  Middle C has a frequency of 240 Hz.  If Middle C is the beginning of an octave, what is the ending frequency of that octave?



In Western culture, each octave is split into 12 tones, which we usually call notes.  The first note in an octave is C; the last note in an octave is B.  Consider the octave shown in gray above.  Count each key, white and black in the octave.  You should have counted 12. Draw an octave on paper.  Label the keys in the first octave from 1 to 12.  A black key is identified as a sharp or flat, depending upon its relationship to the white key before or after it.  If a black key is considered in relationship to the white key before it, it is a sharp of that note.  Likewise if a black key is considered in relationship to the white key after it, it is a flat of that note.

What is the seventh key in the above octave called?   

Each note has a frequency that is the twelfth root of two times the frequency of the previous note.  To compute this value on your calculator, enter 2 to the 1/12 power.  You should get an approximate value of 1.05946.

So the C sharp key (first black key above middle C) has a frequency of 240(1.05946) = 254 Hz.   What is the frequency of the fourth key above middle C?

What is the frequency of the C in the octave below middle C?

What is the frequency of G sharp in that octave?

Since there are black keys between most of the white keys, the black key represents a half tone or half step above or below the white key before or after it. In other cultures, such as India, there are quarter and three-quarter tones, called microtones, that represent smaller changes in pitch.  In the Indian culture, there are actually 22 tones per octave.  Their octaves are still the same length, however, since the octaves are based on human hearing.  To study more about Indian music, check out this Web page.

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