There have been a number of piano and music firms named Harper, including...
HARPER & CASTLE
HARPER & Co., Newmarket
HARPER & DAVIES, LIVERPOOL
HARPER & SMITH
HARPER (Geoffrey) Wimbledon
HARPER (Thomas W.) Hornsey
HARPER (William) London
HARPER ALLEN, Banbury
HARPER Brothers, Brighton, Sussex
HARPER Electric Piano Co. Ltd., Holloway
HARPER Iron Foundry
HARPER Piano Co. Ltd., Holloway, London
HARPER, Bury St.Edmunds, Suffolk
The following piano or music firms were all in business in 1928:
HARPER (A.C.) & Co., SEREMHAM
HARPER (C.E.) SIDNEY
HARPER (CHARLES) & Co., PENRITH & BERTH
HARPER (D.) & Co., LONDON
HARPER (F.W.) DELANO
HARPER (G.) LIVERPOOL
HARPER (G.H.) ORANGE & UTTOXETER
HARPER (J.E.) HOT SPRINGS
HARPER (J.G.) & Co., LETHBRIDGE
HARPER MUSIC SHOP, DETROIT
HARPER (N.) JUNAH
HARPER PIANO Co., LONDON
HARPER'S MUSIC SALONS, BURY St. EDMUNDS, ELY & NEWMARKET
HARPER (THOMAS) LONDON
HARPER (W.J.) CALAIS
HARPER & BROTHERS, NEW YORK
Circa 1915 #3700
Circa 1920 #5000 has the "London N" address with no suffix number, which suggests that the transfer was made before 1918, but the piano may be later...
1928 #5266 The action was made by Malcolm, and is dated by the Malcolm number on the rear of the action.
Some were made by the wholesalers Brasted Brothers, but don't have the same serial number sequence.
but although the facts are accurate, don't assume that this will make your life easier!
#71013 could be around 1929... or not!