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Double Whole Notes and Double Dots
(Click here to read this lesson with UK terminology - "Breves and Double Dots".)
Double Whole Notes
Up to now, the longest note we have come across is the whole note - written as an open note-head without a stem.
We'll now learn about a note which is twice as long as a whole note - the DOUBLE WHOLE NOTE.
The double whole note is written just like a whole note, plus 2 short, vertical lines on each side of the note head, like this:

A double whole note is worth 2 whole notes, or 8 quarter notes.
We don't see double whole notes very often - mainly because they last longer than a complete measure in the majority of time signatures. They are just too big to use in 4/4 for example! Double Whole Notes can be found in 4/2 (four half notes per measure = 1 double whole note), for instance.
The double whole rest is a solid block which fills in the gap of the C space (treble clef):

In the grade 4 exam you might be asked to write a note as a double whole note. Quite often the question will ask you to write the enharmonic equivalent of a note from a score, as a double whole note. Make sure you have practiced drawing them before your exam - they are not difficult, but it's easy to forget how to do them because we hardly ever see them!
Why is a double whole note called a "breve" in the UK? Many students wonder this, since the word sounds a lot like "brief", which means short!
"Breve" and "brief" are indeed connected - in the 13th century, the note we call a breve (double whole note) today was the shortest note available to composers. There were notes that were longer than the breve, which were called "longa" and "maxima". You can read more about how musical notation has changed over the centuries here.
Double Dots
A single dot to the right of a note head increases the length of the note by 50% (the note plus half of itself).
So, a dotted half note = 1.5 half notes = 3 quarter notes:

Double dots increase the value by 75% (the note plus three quarters of itself).
So a double dotted half note = 1.75 half notes = 3 quarter notes + 1 eighth note:
Rests can also have dots and double dots added to them, of course.
In the grade 4 exam you might be asked about equivalents, for example:
How many eighth notes is a double dotted whole note worth? 
First calculate how many eighth notes there are in a normal whole note (8)
Then add on the number of eighth notes in half note (4)
Then add on the number of eighth notes in a quarter note (2).
8+4+2=14
The quick way to do this is simply to remember (number)+(number1 divided by 2)+(number2 divided by 2)
Here's a crazy example to show you as an example - how many sixteenth notes are there in a double dotted double whole note?
1 double whole note = 32 sixteenth notes
32+16+8=56.
There are 56 sixteenth notes in a double dotted double whole note!
Another common question in the grade 4 paper is to explain the meaning of dots and double dots. All you need to write is:
A dotted note lasts 1.5 times longer than an undotted one.
A double dotted note lasts 1.75 times longer than an undotted one.
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