Difficulty is somewhat subjective. However, the different elements that make a piece difficult are not. It's obvious that playing the C major scale at 120 BPM is harder than playing it at 20 BPM. However, it's hard to say whether it's harder to play the C major scale at the speed of 20 notes per second or to trill with 4-5 at the speed of 14 notes per second.
See. That's the
difficulty in answering the question. It does not render the question
impossible to answer. At least not from a theoretical standpoint. I'll make an analogy. Let's suppose that, instead of this question, I asked the following question:
"How many grains of sand are there on the beaches of California?"
Immediately, the overwhelming response would be a combination of two things. 1- "it is not useful to know how many grains of sand there are on the beaches of California." That's going to be true in the case of the vast, vast majority of people. The question does not lead to an answer that would be
useful to most people. However, there are certainly people who the information could be useful to, if even in a purely mathematical framework. They might be interested in knowing how to come to the correct answer. They might just be interested in thinking about the question***. Many questions are like this. In fact, some philosophical ideologies would tell you that all questions fall into this category. Using nothing but basic intuition, we can recognize that there are an infinite number of questions, but that there are a finite number of questions which apply to every person's existence
and their interest. Therefore, the average usefulness of a question approaches 0. It would be inappropriate to deem the question "useless" on such grounds, obviously, as that presents a vast array of problems, one that has just been expounded upon in the previous sentence, and several that have been expounded upon elsewhere in this thread. People can say that this question is "useless to me", but that's not what's been said here. It's all in the details.
The second issue that people would have with the California question is, "there is no way to answer it." Again, even the most basic level of intuition tells us that, of course, there is a number value of the grains of sand on the beaches of California. Again, we come to an impasse of poorly constructed statements vs. implication. There are certainly mathematical formulas that can be applied to guess,
with varying degrees of success, how many grains of sand there are on the beaches of California. At this point a multitude of issues arise, in many variations. The first issue is fairly simple; this one is not. First, if there are varying degrees of success, then an answer can be more or less incorrect (verisimilitude). Verisimilitude is defined (as simply as possible) as such:
If a question has a non-binary (meaning that the answer is not yes/no, or this/that), distinct value for an answer (not necessarily a "number" value), then incorrect answers can be more or less correct (varying degrees of verisimilitude; or, exhibiting x degree of verisimilitude). Verisimilitude is described as an a posteriori "truth-likeness" or "the degree of appearance of truth" or "proximity to the truth". For instance, I want to know a specific color. Let's go with "magenta". We must disregard the linguistic truism that people may synonymously use "pink" and "magenta"; we will assume that the terms of the question, in the way I proposed it, were exceptionally defined. One person says the color I am thinking of is "blue", while the other says the color I am thinking of is "red" (or pink, but I am trying to avoid getting into
that issue, as it's not germane to the pianistic question). There is a greater degree of verisimilitude in the answer "red" than in the answer "blue". Therefore, "red" is a truer answer than "blue". Because there is a degree of verisimilitude,
there is a correct answer. The degree of verisimilitude moves toward the correct answer in a direct relationship to its truth-likeness; as it moves towards something, this "something"
must be the correct answer. The sequence holds an infinite number of variables, but because it is linear, we can use a Dedekind cut on the equation of the line (as all variables will be accounted for). If a Dedekind cut can be used, the equation leads to a value, instead of an infinite geometric sequence, meaning that one can not simply get close to the answer, but that one can achieve its limit (the limit of degree of verisimilitude: 1, for absolute truth). Whether a person is likely to come to the answer, either by accident or incredible rigor, is not related to the statement that there is, in fact, an answer to the question. [A Dedekind cut is a mathematical term: basically, it allows us to dissect a number line at any value, including an irrational value, and its axioms prove that that in the case of an equation that does not extend infinitely (obviously, if it did, there would be an infinite number of correct answers to the sand question; in fact, it would render all answers correct), then it has a limit, as opposed to an asymptote (a number it approaches, but never reaches)]
As such, were I to propose the sand question, given the incredible complexity and technical requirements necessary to give the "correct answer", obviously I would not require "the" answer. It is possible to answer, but nobody ever would. It is too complex and tedious. Here we must redefine the incorrect statement of, "the question cannot be answered." We shall redefine it to the following (as has just been proven necessary) to, "I cannot answer the question." if we do so, statements regarding the validity and form of the question are not challenged within this wording. Because this wording does not challenge the validity of the question, it is no longer an issue. However, there is an issue with the original train of thought. It
presupposes, as in it is not implied by the question (and thus can not be used to dismerit the question; only those people's understanding) that the person who proposed the question is only interested in the absolute truth. Obviously, that is not the case. I use the word "obviously" for a number of reasons. It is simply so, patently clear. Obviously, as in, for someone to misunderstand how to answer a question so egregiously, their opinions ain't exactly worth a whole lot, if that's the extent, or at least basis, of reasoning. Again, we can look at the wording of the question. "What is your opinion. . ." "What is your guess. . ." "What do you think. . ." If these are apparent, the complaint is
formally defeated. If they are unapparent, then the complaint is
informally defeated.
It is the latter that is the issue. A formal error can be proven incorrect using basic analysis of the propositions. An informal error is much more complex. Many fallacies that are, or were, categorized as "informal" are actually formal; newer set theories and logical axioms (fuzzy logic, for instance) can disprove them. Basically, an informal error presents a non-binary variable, and if one attempts to define the variable so that it can be replicated in a formal construction, one is only addressing one of the possible variables, or a set of the possible variables. 0 and infinity don't play nice in logic, which necessarily arise in an informal fallacy. Even worse, they often operate outside the actual constructs of the argument. You'll often get a -> p, where "a" represents the error and p represents the proposition, a -> p meaning "a" proves p (or, p from a; if a, then p). You can't just shove "a"
into p. Informal fallacies, in a more technical way, deal with indefinite terms and/or the lingual (rhetorical) aspects of proposing an argument, as opposed to the propositions themselves. Loki's Wager is a favorite example, because it is extremely simple:
The Norse god Loki promises to give his head to a group of dwarves. There's more story than that, but let's skip straight to the important part. He shows up, they sharpen their axes, he bends over the block. But before they swing, he asks where they are going to cut. They say neck, and he says that he only promised them his "head", not part of his neck. They say they'll start at the chin, and he says that's still part of his neck. This goes on ad infinitum, until they end up getting nothing, and Loki walks away unharmed.
That is informal. One cannot retroactively define a term. There isn't an actual error in argument, which is why it is not a formal fallacy. So, back to California. "I cannot answer the question." The specific names of the informal fallacies that can be used to a priori deride the question, on the basis that it's likely that nobody can answer the question, are the fallacy of distribution, and the suppression of the correlative. "Because I can't answer it, it can't be answered." - and - "Because I can't answer it, other people can't answer it." Those look very similar, but they're slightly different. The first states that the question can't be answered at all. The second states that other people are also incapable of answering it; however, hermeneutic analysis shows that the question may still have an answer, regardless. An example: is M theory correct? Nobody can answer that right now, but it's possible that it's correct. The second is more problematic, as it will necessary attempt to apply itself to the gauging of the verisimilitude of an answer. As it does so, it necessarily implicates verisimilitude, thus circularly proving itself incorrect.
We can further place this into the framework of a Sorites Paradox. As we verify the validity of terms working toward an objectively derived answer (so-as to conform to the rules of empirical epistemology), one can only keep saying that we're no closer to the truth so many times until their statements are meaningless. Think of it as counting the grains of sand, one by one (let's disregard the fact that, in reality, the number would change by the time we finished; anachronistic issues don't apply to the real question in this thread). We keep counting them and asking if we're closer to the answer. Each grain of sand only brings us infinitesimally closer, but eventually, when there's only one grain left, if you're still saying no, then you're just plain dumb, sorry to say (not saying
you are).
So, back to piano. As we locate and define difficulties, we move closer to the truth. Answers can be closer or further from the truth. Therefore, there is a truth. I would think this would be patently obvious. Your question, specifically, refers to valuing one criterion to the other. Here's my answer: I don't know. Which is harder: octaves at these intervals, or scales at this tempo? I don't personally know the answer to every single one of such examples you could bring up. However, there are answers to those questions. The proof is as follows: please remember the constraint of the question. You again fall into the convenient trap of talking about someone who is "left-handed" vs. "right-handed". Immediately, you must see that you are no longer answering the question proposed. The question to which you refer is not mine, as I made clear that I understand the reality of the situation: one piece might be harder for one pianist, and vice-versa. However, "well-rounded pianist" is a term I used, as well as mentioning this issue in the comment about Mozart. Perhaps the term is not completely defined, I will admit that. It's simply that I did not think excessive explanation was necessary; I understood the way I used the term to be rather self-evident, although I suppose it's not (unless you're just playing devil's advocate, which is actually the feeling I get). Let's change it to "the average pianist". Amplified as, "a single pianist, whose technique is composed of the average abilities of all pianists who possess the capability to perform these works to an adequate degree." The phrase "perform to an adequate degree"
has been well-defined. Therefore, for that "pianist", there will be an answer to your examples. And that is the question.
Poll
Topic: Most Difficult Piece in Standard Repertoire