Wired Doorbell Installation Question

Hello,I received a Ring Doorbell Wired model 5at375 for Christmas. I have a working electronic door chime which has a diode across the doorbell pushbutton. I have installed the Ring device following the enclosed directions (leaving off the diode and installing the supplied jumper). The electronic doorbell does not work. Is there a way to wire the Ring Doorbell so that it will sound the electronic door chime? The door chime plays a very melodious tune that I want to hear.Thank you in advance for your help!