Yes, it can. And even more than that, it will inevitably be both.
Most of the great composers had a very precise idea of their piece, and most were very skilled in writing down a text which would capture the essence of this idea in outstanding detail and accuracy. So the notion of objectivity comes into interpretation surprisingly easily: Whatever goes against the text of a composition is objectively wrong — as long as the interpreter’s goal is to present the composer’s music, not his own. (Not every pianist feels obliged to that goal, but I would hesitate to call any more liberal approach to piano playing “interpretation” in its strictest sense.)
Things become more sophisticated when we start to ask what is right instead of what is wrong. Obviously, there are many different ways to play a piece while producing perfectly convincing and beautiful music, and doing nothing whatsoever against the text. All of these interpretations will be objectively right, but still different. And this is where personality comes in: Each musician’s personality is unique. There will be overlaps between a composer’s and his interpreter’s personality — then the interpreter will probably play very much what the composer had in mind. There will be differences in their personalities — then the interpreter will perhaps not come close to the composer’s idea, but there is still a chance that the result will make sense and even be of a kind to find the composer’s surprised approval (and there is a chance that it will not; if a composer’s personality is too far from my own, I had, in some cases, better leave him alone).
To judge the quality of an interpretation concerning objective truthfulness, there is very simple criterion — according to which, however, the vast majority of piano interpretations fail sadly: After hearing an objectively right, i. e., text-adherent interpretation, a sufficiently skilled listener should be able to write down exactly the text the composer actually wrote. A striking example: Isn’t Horowitz one of those pianists most famous for an excessively personal, free, romantic, rubato, etc., way of interpreting his pieces? And yet: There is a number of major works of piano literature which I got to know first in Horowitz’ recording, while seeing the score only later. Many a time I was more than taken aback by how exactly the text looked as I had anticipated it from listening to Horowitz, up to the tiniest detail. This could be demonstrated in thousands of instances; one shall suffice: In the last twelve bars before the final return of the main subject of the A flat major Polonaise op. 53, when the last interlude comes to an end in F minor, Chopin’s whimsy took him to put an accent on each and every C’’, no matter where it occured in the uninterrupted series of semiquavers, and no matter whether the accentuation made any sense locally:
Take any of Horowitz’ recordings of the piece and verify that no music-literate listener would fail to note these accents. Now put any other pianist’s recording to the test whether these accents are audible at all. Was Horowitz really the only one to notice this highly interesting, curious, and original idea of Chopin’s, make sense of it, and convey his understanding to the audience? It is really sad what a vast collection of examples of unique and curious features of compositions, being neglected by almost every interpreter, could be put together.
It may be added that Horowitz (or Rachmaninov, or Rubinstein, or any of the truly great), with all his rubato, never obscures the rhythm; our able listener will hardly ever have difficulties in determining the intended note values, and, upon writing them down, find out that his text will be virtually identical with the composer’s. Do I have to name any of those innumerable examples of pianists who seem to believe that rubato means the licence to distort note values at will?
Of course, it is more convenient to refer to the interpreter’s licence rather than taking pains to study a text thoroughly. I have heard pianists saying: “I have worked with a number of composers, and you know what: They never really cared what I played; they told me to play as I felt was right. Their text just need not be taken that strictly.” Now, I cannot speak for anyone else but me, but I know that some musicians who studied my compositions with me will probably say something similar about me: It is not true, however. I have very precise ideas of how my pieces should be played, and these ideas are written down in the text (which took me lots and lots of nerve-consuming pondering over the details); but when I meet somebody who is so kind as to put practising effort into my pieces, I will certainly try not to annoy him any more than necessary; and when I see that his personality is different from mine in some aspect, I would not make him play in a way which is not his, but I will say something to the effect “Play as feels right to you”. Nonetheless, I might silently hope that some other personality, more similar to mine, come some day and study my piece anew … For many composers, by the way, the above-quoted statement seems highly improbable to me, given what a consistent, logical, and coherent picture emerges from an in-depth analysis of their text up to the smallest details. I always end up asking myself: Why on earth should Chopin have cared to write these accents (certainly bewildering to some) if they were not important to him? Or if he would have expected them to be inaudible? Or not clearly distinguishable from unaccented notes? And I cannot spare most pianists here from the accusation of either not having studied the text carefully, or having failed to make sense of it, or having failed to make their understanding actually reach the audience.
Each major work of art is infinitely rich in content, which means that each interpreter can, and will, discover features that are truly and a-priori in it but have not been discovered before. Bringing out such features results in an interpretation that is both objective and personal.
A memorable experience, for which I am grateful until today, was seeing Wagner’s complete “Ring” at Bonn Opera House in 2001. German opera stage directors tend to be as arbitrary in their “interpretations” as could possibly be conceived, but Siegfried Schoenbohm’s staging of Wagner was an admirable exception. It was full of highly original, entertaining, and creative ideas; none of these, though, were against Wagner’s work. In fact, they all, surprising as they were at first, just opened a view on what was really in the composition. When listening to audio recordings of Siegfried’s death in “Götterdämmerung”, I had always asked myself why, in the music, Siegfried appeared to die twice. Those heavy chords from the orchestra, indicating the event, occur not once but two times, each followed by the scary lines “Hagen, was tust du?” (“What are you doing?”) and “Hagen, was tatest du?” (“What have you done?”) respectively. In Schoenbohm’s staging, Siegfried rose in terror with the first chords, struck by Hagen’s spear, and broke down to the floor with the second. Now all of a sudden the double blow in the music not only made sense, but even more, the music appeared as if intelligently and aptly written for the action as seen on stage. Many more examples could be given to show how Schoenbohm’s interpretation, unusual and personal as it was, did not do anything but just bring out what had always been in Wagner’s work. To my best knowledge (and regret), no performance of this “Ring” has been recorded on video.
Being personal is, by the way, unavoidable. It is therefore unnecessary to spend any time trying to be original and innovative (let alone, disregard textual details deliberately); you will inevitably be original, just because you are a unique personality. Instead, time should be spent on reading and understanding the text as deeply as possible; this task is nearly infinite.