A rasping stick is a type of stick covered with ridges, which can be rubbed on a variety of instruments to create a guiro-like effect. It has other names; rivet stick, spiral stick, probably a lot of others that I don’t know. So, if you need some clarity, here’s one that I also use in the video below.
Here’s a video of a rasping stick being used on a few different instruments:
how it works
You need the ridge to catch something; dragging the stick across a keyboard like a glissandi will just sound like any other stick. Rasping the edge of the cymbal is fine, but the bell might be much less effective. The most popular use is on the rim of the drum. It could potentially be harmful on mallet instruments or similarly fragile things, so it isn’t advisable to ask a player to use the sticks in that manner.
on notation
Historically speaking, rasping sticks aren’t particularly popular. There are plenty of examples, of course, but the following is what I would argue are the most clear. To reemphasize, this is my suggested notation and it’s not a hill that I’m about to die on. If you decide to use it yourself, the lines used are all Sibelius’s “guitar vibrato”.
Example 1

In this example, a rasping stick runs along the rim of the drum for 1 beat before another stick plays on thee head for 2 notes (then repeats).
Example 2

This is the same as example 1, with the important addition of the tremolo, indicating that the rasping stick will very quickly move up and down on the rim.
Example 3

For playing rim scrapes in time, with the addition of a stacatto mark to ensure that the rasp does not connect to the next note (though it could if you want!)
Example 4

This one is a little tricky. My intention with this example is to illustrate how to rasp on a rim while playing on the same drum simultaneously. This would work just fine in a snare solo, but if on a drum kit, this could be interpreted as playing the snare while rasping the rack tom’s rim. In this case, you’ll probably need to add some text to clarify the sound.
Example 5

Similar to above examples, but implemented on a keyboard
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A quick note on speed: Unless otherwise notated, a rasp will be executed at whatever speed is necessary to complete the duration in one movement. If you wanted the rasp to be, for example, extremely slow, that should be clarified with an expression.
dynamics
The dynamic ceiling of rasping is somewhat low, given that the stick is existing directly on top of the surface it is contacting, as opposed to sticks and mallets that are normally aided by the extra force provided by striking from higher up. That being said, the range is quite broad, with differences in force and speed allowing for very soft and abrasive tones at once. Still, cutting through dense textures may prove a bit challenging. It’s also worth notating that volume and speed an bee related; playing very loud but also very slowly isn’t possible.
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