>There ARE modems that send a short beep now and then when they
>are originating a call to let the other end know this is a data call
>instead of fax or voice.

Even if the modem doesn't have the tone "data calling", you can simply use "ATDnumberR" to dial with reverse channel (meaning the call originating modem becomes the B-channel instead of the A-channel as normal). This means that the calling modem beeps just as the answering modem would normally have done. This enables the answering party to distinguish between modem and voice (and of course fax) calls.
ALL modems support reverse channels, but both modems of course have to be prepared that the channels will be reversed.

>        A friend has a fax machine with a very clever feature.  I think
>it automatically answers after maybe 6 rings, but it is always listening
>to the line.  If it hears the fax CNG tone, it goes off hook, sends
>answer tone and continues with receiving a fax.

I don't think it works that way, because such a fax machine would only work on special phone systems. The answering fax cannot hear the CNG tone from the calling fax without lifting the hook, and then the call is already answered! The only audio which is on the line before the call is answered, is caller-ID tones (if any).

Most "fax switches" simply answer the call at the first ring and play an audio ring tone to the caller. If it then hears a "fax calling" CNG tone, it enters fax receiving mode. Otherwise it starts the ringer in the fax telephone, so the human can answer the voice call manually.

>        You could do something like this to identify data calls.

For this to work, all phone switches need to be aware of the tone, so they can listen to it when a potential fax makes an outgoing call, and send it to the dialed fax if it's a fax calling. CNG is not implemented in the switches used in Sweden and I don't think it's implemented in any country. Neither is the tone "data calling".