In your first post, you stated that the horn relay clicked. That would be a sign that the relay is good. The relay doesn't "step down" the voltage. In fact, it should allow full battery or charging voltage, less tenths of a volt, pass through the relay from the attached battery cable on the terminal at the bottom of the relay to the green wire terminal. The black wire terminal as you are referring to, is a ground that activates the relay. That black wire goes up to the horn bar. When the horn bar is pushed, the relay is grounded and activated.
As been previously stated in another reply, simply check for power at the wire connection at the horn. I like to use a test light rather than a volt/ohm meter when tracking down a problem. If you have power at the horn connection when you press the horn bar, the circuit is fine. Now you have either bad ground at the horn or somewhere in the sheet metal. Or the horns themselves are bad. And as previously stated, the horns can be tested by adding 12 volts to the terminal and grounding the body of the horn. It should sound. If it doesn't sound, the horn is bad. The horn is simply a vibrating electro-magnet. The internals can corrode, and then it can't vibrate.
Best of luck with finding your solution,
Loren@65GS.com