Is it absolutely necessary for me to study harmony in connection with my piano? My teacher wants me to do it, but I don't see the use! Of what benefit is harmony?
Of what benefit is the general school-work a child has to go through? To play the piano well a good hand and so many hours of practice are not sufficient; it requires a general musical education. This means, first and foremost, a knowledge of harmony, to which you may later add the study of counterpoint and forms. Your teacher is absolutely right.
Text-Books on Harmony
Would you care to recommend two or three of the best books on the study of harmony?
The doctrine of harmony is ever the same, but the modes of teaching it are constantly changing and, I trust, improving. For this reason I feel a certain hesitation in recommending at this time the text-books which I studied many years ago, especially as I am not certain that they have been translated into English. I advise you, therefore, to inquire of some good teacher of harmony or, at least, of a reliable music publisher or dealer. E. F. Richter and Bussler wrote works of recognized merit, which, though no longer modern, may be safely studied.
Learning to Modulate
Is it possible to learn modulating from a book without the aid of a teacher, so as to connect two pieces of different tonality?
Possible, yes, but not probable; for since in your written exercises you are likely to err at times, you will need some one to point out your errors and so show you the way to correct them. Generally speaking, I do not think much of studying the rudiments of anything without the aid of an experienced adviser.
Studying Counterpoint by One's Self
Is it possible to study counterpoint without a teacher, and, if so, what book can you recommend for its study?