The symbols for the half rest and whole rest are pretty much the same and in the literature my students play we rarely encounter them.
Is there a good, memorable way to describe which is which to students?
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There was a trick for these that I used all the time based on what the rests look like.
The whole rest looks like a hole. The words sound the same so it's a good way to equate them.
The half rest looks like a hat and since hat and half both start with the letter 'h' they go together.
I like this trick a lot because it associates the rests more with objects that are very distinguishable a it's little more fun to ask if the rest is a hat or a hole then if it's a half or whole.
In elementary school, I was taught to think of the rest like a raft in water. Since a half rest gets two beats, it's like a raft carrying two people - light enough to float on top of the water:
The whole rest, on the other hand, gets four beats (in common time, anyway) and so it's like a raft carrying four people - enough weight such that it sinks down below the waterline:
The mnemonic device I remember seeing in elementary school music class was that the half rest is "weaker" so it has to lay on the ground (the staff line), while the whole rest is "stronger" so it can cling to the ceiling. This was illustrated with some cartoon block dudes sleeping, though I can't remember what mechanism was holding Mr. Whole Rest to the top.
I heard from a music teacher that he tells his students that the whole rest is somehow "bigger" than half rest, so it is "heavier"(!) and can not stay above the line (4th line of stave) and it has dropped down, while half rest is not that heavy and can stay above the line!
I'm not sure if it would be a good way for you too, since his students were all little children.
My music teacher had a good way to remember them based on the fact it looks like a hat.
A man came over to visit. He was only planning on visiting for a short time, so he put his hat on top of the chair (half rest). After a while, he realized he wanted to stay later and needed a place to sit, so he had to move his hat under the chair (whole rest) so he could sit down.
This is fun! Here's the goofy mnemonic device I was taught in elementary school:
Imagine the rest as a ham. Yes, a ham. As long as it's whole and uncut, it hangs (on a hook under the ceiling of the larder). When you take it down and cut it in half, it rests on top (of the kitchen table) - just like the whole rest hangs below its line and the half sits on top of it.
If all you want is a quick way to tell which is which while sight reading, then if there's any notes in the measure along with the rest, it's a half rest. Reason being, a whole rest signifies that you rest for the whole measure and therefore by definition it will be by itself.
It doesn't seem to make good sense. But in fact it is common now to use a whole rest to mean "one measure of rest" regardless of the meter.
But you are asking, How common is this really?
It's hard to come up with a metric for that, but let's rely on the testimony of Gardner Read, whose "Music Notation" is something of a Bible for music notation. He says of the whole rest,
"...it now commonly serves as the symbol for any completely silent measure, regardless of the meter or time signature." (2nd Edition, p. 98). He adds that given this current custom, the whole rest must not be used to represent a partial measure, except perhaps in a meter where the whole note is the denominator of the meter signature, like 2/1 or 3/1.
And I've seen it a lot in orchestra parts. I'd answer your question: "pretty common." But I wouldn't write it that way in a piece where there were frequent changes of meter. Clarity is the goal, after all.
A full gentleman takes his hat off to greet a lady (full rest looks like an upside down hat), but a half gentleman leaves his hat on his head (half rest looks like a riteside up hat). Credit for this goes to my second grade music teacher who I can remember only as Large, with a blond ponytail to her waist and drawing music notation on the chalkboard.
The book Usborne First Book of the Keyboard is full of memorable cartoons for remembering this kind of stuff.
The cartoon for remembering rests has two figures:
The first figure is lazy and sits on top of the line, so he's a half rest.
The second figure is strong and can hang below the line, so he needs a full rest.
I used to think of it this way:
If I lie in a bed, I can get rested quickly. If I try to sleep while hanging from a rope, it will take a lot longer to "rest".
I actually like some of the other answers given better (DOWN=4, UP=2 - but since I didn't grow up with English as my first language, that was not available to 5 year old me) - but you never now what mnemonic works for whom, so I offer this alternative.