Musical pieces are read much like you would read math symbols.  The symbols represent some bit of information about the piece.  Musical pieces are divided into sections called measures or bars.  Each measure embodies an equal amount of time.  Furthermore, each measure is divided into equal portions called beats.  These are all mathematical divisions of time.

Fractions are used in music to indicate lengths of notes.  In a musical piece, the time signature tells the musician information about the rhythm of the piece.  A time signature is generally written as two integers, one above the other.  The number on the bottom tells the musician which note in the piece gets a single beat (count).  The top number tells the musician how many of this note is in each measure.  Numbers can tell us a lot about musical pieces.

Each note has a different shape to indicate its beat length or time.  Notes are classified in terms of numbers as well.  There are whole notes (one note per measure), half notes (two notes per measure), quarter notes (four notes per measure), eighth notes (eight notes per measure), and sixteenth notes (sixteen notes per measure).  These numbers signify how long the notes last.  That is, a whole note would last through the entire measure whereas a quarter note would only last one quarter of the measure and thus there is enough time for four quarter notes in one measure.  This can be expressed mathematically since 4 x 1/4 = 1.  A note with a dot after it lengthens the note by half.  For example, a quarter note with a dot after it would be held for 3/8 of a measure, since: 1/4 + 1/2(1/4) = 3/8.

Three eigths of a measure is midway between a quarter note and a half note.  It is important for musicians to understand the relationships and values of fractions in order to correctly hold a note.

Fibonacci: The Fibonacci sequence is a famous and well-known sequence that follows as: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,   and so on, adding each term to the one before it to create the next term.  That is, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34, and continuing infinitely.  In music, the Fibonacci sequence can be seen in piano scales.  For example, the C scale on the piano consists of 13 keys from C to C; eight white keys and five black keys, with black keys arranged in groups of three and two.  In the Fibonacci sequence, the ratio between each term is very close to 0.618, which is known as the golden ratio.

Pythagoras and Frequency:  It was Pythagoras who realized that different sounds can be made with different weights and vibrations.  This led to his discovery that the pitch of a vibrating string is proportional to and can be controlled by its length.  Strings that are halved in length are one octave higher than the original.  In essence, the shorter the string, the higher the pitch.  He also realized that notes of certain frequencies sound best with multiple frequencies of that note.  For example, a note of 220Hz sounds best with notes of 440Hz, 660Hz, and so on.

The closest tie between music and math is patterns.  Musical pieces often have repeating choruses or bars, similar to patterns.  In mathematics, we look for patterns to explain and predict the unknown.  Music uses similar strategies.  When looking at a musical piece, musicians look for notes they recognize to find notes that are rare (high or low) and less familiar.  In this way, notes relate to each other.  Relationships are fundamental to mathematics and create an interesting link between music and math.  See http://www.italmeds.com/index.php?option=com_content&view=article&id=110:music-and-mathematics&catid=87:music-philosophy

Octaves

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems".  It may be derived from the harmonic series as the interval between the first and second harmonics.

Three commonly cited examples of melodies featuring the perfect octave as their opening interval are "Singin' in the Rain", "Somewhere Over the Rainbow", and "Stranger on the Shore".

The frequency of middle C on a piano is often set at 261.6 Hz.  There are twelve semitones in an octave.   A piano keyboard has 7 white keys and 5 black keys to play notes within any octave.  A trumpet has 3 valves that can be all open, two closed, etc. with 7 combinations of fingering, fingering, so a trumpet can only play 7 tones within an octave.

Also the trumpet reference is mathematically correct about the valve openings, but trumpet players use different techniques and can play a full range of notes within an octave, and also over several octaves."

The treble clef and bass clef are most commonly used in sheet music.  The numbers below indicate which octave the note is in. There are 12 semitones in an octave. There are five lines and four spaces in a clef.. In the treble clef the spaces contain F, A, C, E and the lines contain  E, G, B, D, and F.  In an octave, not counting E and F twice, the lines and spaces handle 7 of the semitones, and special sharp or flat symbols are used to designate the other 5 semitones in the octave.

Don Francis writes: "The ratio of the interval ending on F sharp is sometimes called the devil's interval, probably because there is no good ratio to define it.  It was rarely used in pre 20th century music because of perceived dissonance.  It came to typify some types of jazz (the famous flatted fifth of bebop).  A good reference for remembering the interval is the first two notes of Maria."

A fourth is a musical interval encompassing four staff positions in sheet music. For example, the ascending interval from C to F is a perfect fourth, as there are four staff positions including C to F (counting C, D, E, and F). F lies five semitones above C (counting C sharp, D, D sharp, E, and F).

A fifth is a musical interval encompassing five staff positions in sheet music. For example, the ascending  interval from C to G is a perfect fifth, as there are five staff positions including C to G (counting C, D, E, F, and G).  G lies seven semitones above C (counting C sharp, D, D sharp, E, F, F sharp, and G).

Heinrich Rudolf Hertz (1857-1894):   A Hertz, or Hz, is named after this German physicist.  It is a measure of the frequency, the number of vibrations of a string per second.  People in different musical traditions have different ideas about which notes they think sound good together.  If you double the frequency, the human ear tends to hear both notes  as the same. This is called "octave equivalency". The doubled frequency is called a higher octave. This "octave"  is two times higher, not eight times higher. In the "diatonic scale", there are 8 notes counting both ends of the octave hence the term "octave".  In the "chromatic scale" there are 13 notes counting both ends, and the "Arab classical scale" has 17, 19, or even 24 notes in its "octave.

Scales can be classified as "Just" (See  http://en.wikipedia.org/wiki/Just_intonation (if the ratios between the frequencies are ratios of integers), "tempered" (if the just scale is tempered), and "practice-based" (if it reflects musical practice)   In the Even Tempered Scale going from one semitone to  the next is the 12th root of 2, or 1.05946...  (Pythagoras discovered that frequencies whose ratio is equal to  the ratio of two simple whole numbers yield "harmonious"  and pleasing sounds.  The ratio of 3:2 is 1.5.  A "perfect fifth" corresponds to a separation of seven semitones, as the seventh power of 1.05946 is close to 1.5.  A "perfect fourth" corresponds to a frequency ratio of 4/3 and five semitones.)