__Time
Signatures, Rhythm and Tempo__

This section is dedicated to the mathematics involved in time signatures, rhythm, and tempo

- How do time signatures work?
- How are time signature rules broken?
- Why is 6/4 time and 3/2 time not the same, as it would be in mathematics?
- What is an onbeat or an offbeat?
- What are some details of note/rest divisions? (i.e whole notes, quarter notes, 64th notes, etc.)
- How does the time signatures change/affect these notes?
- What does quarter=120 mean?

**1. How do time signatures work?**

Time signatures are **very** easy to figure out.
The bottom number (equivalent to the denominator in mathematics;
think denomination), tells the length of one beat, in relation to
the whole note. For example - x/1 means each beat is worth one
whole note. x/16 means each beat is worth one sixteenth note, and
so on. Remember, we are dividing by this number, which is why as
this number increases, the length of a beat decreases.

The top number (equivalent to the numerator in mathematics, think number), tells the number of beats in each individual bar, in relation to the 1/x means there is one beat in each bar, 5/x means there are five beats in each bar, etc.

If we consider this mathematically, this is a case of very simple fractions. In math, 3/2 means the whole (1) is divided into two parts, and that there are three of these parts. In music, this would mean that the whole (the whole note) is divided into two parts or beats (half notes), and there are three of these parts in each bar.

**2. How are time signature rules broken?**

Very easily in fact. Often 'grace notes' (smaller than regular notes) are added into a bar. This is a note played very quickly as part of the next note, and does not receive any note value. Also, in the middle of a piece the composer may change the time signature, allowing for a changing of these rules. Often this time signature is for only one or two bars.

**3. Why is 6/4 time and 3/2 time not the
same?**

Technically these two time signatures are identical, if you consider it fractionally as in 1. However, if you look at it carefully, they are not the same. 6/4 time means that there are six beats in each bar, and each is worth one quarter note. 3/2 means that there are three beats, and each is worth one half note. So there are three half notes, instead of six quarter notes.

**4. What is an onbeat or an offbeat?**

An onbeat is a 'dominant' beat. An offbeat is any other beat. For example: In 4/4 time, The onbeats are 1 and 3, because those are the easiest to count and most often the strongest beat.

All notes have values in relation to the whole note (equivalent to four quarter notes) of 2 to the exponent of a zero or negative integral value.

For example:

2^{0}=1 -- 1 x (whole note) = (whole note)

2^{-1}=1/2 -- 1/2 x (whole note) = (half note)

2^{-2}=1/4 -- 1/4 x (whole note) = (quarter note)

2^{-3}=1/8 -- 1/8 x (whole note) = (eigth note)

2^{-4}=1/16 -- 1/16 x (whole note) = (16th note) [two are shown]

Similarly, the value of a with flags or bars can be determined easily using the following equation:

2^{-2-n}=Note value; where 'n' is the number of flags
or bars on a note.

For example: Note with 4 flags; substitute 4 in for n.

2^{-2-4}=2^{-6}=1/64 - 1 64th note

If a composer wishes to use notes of lengths other than these values, s/he can use a few different methods.

A triplet (most common), is worth two of the separate notes. For example, if three eighth notes are barred, and marked as a triplet (a small three over the bar), it is worth two eighth notes, or one quarter. So each individual is worth one third of a quarter note. This result in the creation of a 1/4 x 1/3 = 1/12 note. There are many combinations like this that can be put together to create notes of all sorts of fractions.

Another method of creating different lengthed-notes is the use
of the dot. A dot increases a note by one half its original
value. For example, in 4/4 time, is worth one beat. in this
time signature is worth one and a half beats. A quarter note is
worth 1/4 of a whole note. A dotted quarter note is worth 1 1/2
times that. Therefore, a dotted quarter note is the equivalent of
1/4 x 3/2=3/8 -- a three-eights-note. This method can be used to
make many notes. The end result is always a note of value 3x2^{n},
where n=Original exponent (see beginning of this section) - 1

When both these methods have been exhausted, and the
fractional note value desired still cannot be attained, a
composer may make use of a tie, which is simply the addition of
two notes:

In this example, there are two quarter notes tied together. This
means that their values are added, hence this is equivalent to a
1/4 + 1/4 = 1/2 note. This becomes more useful when tying notes
of different lengths; for example, using previous methods a
17/64th note could not be achieved. Using ties, we can tie a
quarter note with a 64th note, and adding their values: 1/4 +
1/64=17/64.

**6. How does the time signature
affect/change these notes?**

Very often the time signature is irrelevant in the length of these notes. However in some cases the time signature can confuse us. For example in 4/4 time there are four quarter notes in each bar, each beat is worth a quarter note. But in 2/2 time there are two half notes in each bar, and hence each beat is worth a half note. As a result, in 4/4 time a quarter note is worth one beat, and in 2/2 time a quarter note is worth half of a beat. This is just an example, there are a number of confusing situations like this.

represents the note divison to set the rhythm to

120 is the number of these notes per minute.

Therefore, = 120 means that there are 120 quarter notes in one minute, or two beats per second.

To determine the number of beats per second in the case of something like this, simply divide the number by sixty.