I purchased the FIAMM aftermarket horns models 72112 (low note) and 72102 (high note).
Both are nearly identical in dimension to the stock horns on my "05 SR5. The wiring for the stock horns is a connector containing two wires going to one of the horns, and a connector containing a single wire to the other horn. The FIAMM horns both have two terminals.
The instructions that came with the new horns were not very clear to me, so I attempted to follow them.
I cut off the connector going to the single-wire horn and attached a female blade connector.
I cut off the connector going to the dual wire horn and attached female blade connectors to both wires.
I connected the single-wire connector to one of the terminals on the high-note horn, and then a grounding wire to the other terminal and the other end of that wire to the bolt that attaches the horn to the truck's frame.
For the other horn; the low note, I simply attached each of the two wires with the blade connectors to the horn's terminals.
However, only the high-note horn sounds. I have made certain that the aires in the blade connectors are tight and secure, and the connectors are tight on the horns' terminals.
Here is a link to both horns:
https://www.amazon.com/dp/B000DINKPQ...roduct_details

https://www.amazon.com/dp/B000FKKOK6...roduct_details

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It looks to me like you have a poor ground connection. Pull the bolt out and grind away the paint around the bolt hole. The bolt also has a coating on it so take a wire wheel/brush and remove the coating.

on the factory wire, one of them is actual 12v signal, the split you cut off, verify that was not the 12v you cut out.

unplug the split one you cut off. check the 12v side. gray-black is actual 12v, connect those back together. and put a new ground wire to the horn with no ground.
304-305.pdf