Answer:
Dear Mr. and Mrs. __Your neighbors last name__,
Earlier today, I received your complaint letter regarding the loud commotions from my flat. It is a real shame that you had to suffer due to my oversight. I sincerely apologize for the incident and assure you that it will not happen again.
Last Wednesday I hired a firm to redecorate my place. The decorators had to relocate all my furniture to complete their work. They also used some heavy machinery to complete their endeavor. That is why you have been catching all these loud reverberations.
I understand that I owe you an apology as all these ruckuses were caused due to me and I should have informed you beforehand. Anyway, I do apologize, and I will owe you a cup of coffee for this. I hope we will still be on friendly terms after this incident.
About the sound problem, I have already instructed the personnel to tone down the noise level, so your apartment should already be relatively silent. I hope this measure is enough to appease you. By the way, the work will be completed tomorrow.
Thanks for your patience on this matter. Please come by tomorrow evening and I will show you around my freshly revamped flat.
Yours Sincerely,
__Your first and last name__
Hope it helped! :)