I've heard the structure verb + mal a lot, like:
Sag mal, schau mal, warte mal
- Is it a good style? does it sound more friendly?
- What's the difference between "warte" and "warte mal"?
- Is it applicable for all verbs?
I agree with Hauser regarding the (initial) effect of using "mal": makes the imperative's roughness easy. But I think there's no connection with "einmal" in the meaning "once". "Einmal" may in fact mean "once", "zweimal" twice and so on. There are more meanings of "einmal":
So I think "einmal" in connection with the imperative means: "I need this, but not immediately" or "please do this, but only if you find time to do so".
But even using "mal", this doesn't ease the imperative in every case; in the contrary, it may aggravate it:
means it isn't a wise decision to defer that operation any longer, in other words: "do this immediately!" (furthermore, it usually doesn't denote a unique act... )
edit: to answer the initial questions:
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Probably comes from phrases like:
IMO, it denotes more a unique/singular case/situation you are unsure about what is going to happen or talking about something you want to gather information about. More meaning
An imperative wouldn't sound very friendly and/or implicate more that you are aware of what is going to happen (in case of "Warte mal"). So in both cases it eases the rough effect/meaning of an imperative. So that's where it likely comes from.
But today it's -IMO- also commonly used just to avoid the harsh imperative in daily conversations without the above original meaning.
Anti-example would be:
Here you would only use the pure imperative:
Concerning @tohu reasoning, i still think esp. because of common phrases like
that it denotes a singular case/situation. Also i dont understand why to look at the german meanings of once to understand this. verb + mal are set phrases with meaning in context, a translation and re-translation only makes this more complicated, artificial and speculative to me.
Concerning further comments and usage of mal mainly in conversations its hard to argue how mal/Mal/einmal/ein Mal is used by the speaker and understood by counterpart. You can add it to many verbs, but the exact meaning will probably be not always clear.
hardly implies dieses einzige mal nor wenn du Zeit findest. Nowadays its purpose seems mainly the easening of the imperative. I associate it mainly with Sag mal and the meaning i explained above - singular thirsting after information
maybe a hint but no proof where it comes from