Understanding Rhythms

This is a rhythm tree. It shows how different notes relate to each other. The whole note at the top of the tree is equally divided into two half notes, or four quarter notes. The names of the notes are the same as common fractions. Therefore, a single sixteenth note at the bottom of the tree is the same as 1/16 of a whole note. 

beat division

 

This is the same rhythm tree, but instead of notes (which make sound) we have rests which are silent. The whole rest, like the whole note, is at the top of the tree followed by the half rest, the quarter rest, the eighth rest, and the sixteenth rest.

Rest division

It is possible to continue this pattern down to thirty-second and sixty-fourth notes/rests but those are less common. While theoretically possible, a 128th note is generally not used in common standardized notation.

 

Subdivision - To break up a larger metrical pattern into smaller parts so that it may be more easily understood. The division of the beat in simple meter (div. by two) into four equal parts or in compound meter (div. by three) into six equal parts. The division of the beat in simple meter (div. by two) into four equal parts or in compound meter (div. by three) into six equal parts.

 

Example: You could count 4 quarter notes: 1-2-3-4. Or you could subdivide and count the quarter notes as if they were eighth notes 1 & 2 & 3 & 4 &. This is useful if you want to prepare for upcoming rhythms, or to avoid rushing your part.

 

Time Signatures helps us know how many beats are in a measure (the top number) and what note gets the beat (the bottom number). Here are some examples of time signatures. 
Picture
C stands for common time, or 4/4. It is the most commonly used time signature. This means there are four beats in each measure, and the quarter note gets one beat. 
common time
 
If you get going really fast in common time, it might be easier to count the half note as one beat. This is called cut time.
cut time
There are still four quarter notes in a measure, but you would now count them as 1 & 2 & since the half note is now the pulse.