How to control a DSP-G1 programmatically/electronically?
The purpose of the MIDI output circuit is to convert the TTL signal into a 5 mA current.
The purpose of the MIDI input circuit is to convert the current back into a TTL signal.

If the Arduino and the DSP-G1 are part of the same circuit (and share the ground), you can omit these conversions, and connect the Arduino's TX directly to the LPC810's RX, without any other components needed. (The LPC810 requires a 3.3 V power supply, but its RX pin is 5 V tolerant.)

MIDI was designed to work with most microcontrollers, so it uses standard UART signals at 31250 baud. The only configuration you need is:

Serial.begin(31250);