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In many sciences there a are a number open questions that are well known in the field, and for which a solid answer would gain the answerer much Prestige. For example, maths has many open questions, including a few very important ones, for which there are large sums of money on offer.
Are there any important open questions in music theory, or music-related sciences (e.g. musical acoustics; or music-related cognitive or physiological science)? Or was everything solved in the Middle Ages?
Dissonance is a keenly debated subject within music theory. Because of the lack of a proper definition, and due to the general subjectivity of the general understanding of the term, it's really hard to define any chord as consonant/dissonant.
Dissonance has been explained many ways and as different phenomenons (e.g. the coincidence of partials), but none of these definitions explain the term thoroughly and generally.
In addition to the mere definition of the term, there's also the perception of dissonance — what makes one person find the microtonal intervals pleasant, and another person find them unpleasant?
This may not be what you're seeking, but since music theory doesn't operate with the same rules as mathematics (such as axioms), any term can be defined in its own right and does not need to be defined in terms of another term. Thus everything can be defined as the mere "empirical" perception of the phenomenon.
This answer might spawn a whole new question itself, but I believe the notion of Perfect Pitch vs. Relative Pitch and how one obtains Perfect Pitch is a very debatable topic.
Until recent, it had been thought that one is either born with Pitch or was not and that it can not be learned, and that any attempt of learning it would only result in a person having Relative Pitch. However a study has been released that refutes this.
Quotes of the article are as follows:
In a study published in the Journal of the Acoustical Society of America and being presented at the ASA meeting in Portland on May 21, Deutsch and her coauthors find that musicians who speak an East Asian tone language fluently are much more likely to have perfect pitch.
“Perfect pitch for years seemed like a beautiful gift – given only to a few genetically endowed people. But our research suggests that it might be available to virtually everybody,” Deutsch said.
Unlike English, many East Asian languages, such as Mandarin, Cantonese and Vietnamese, are "tonal," so that a word’s meaning often depends on the tone in which it is said (not to be confused with intonation such as sarcasm). Deutsch surmises that learning perfect pitch is, for fluent speakers of a tone language, akin to learning a second tone language.
The study follows up on one Deutsch did in 1999, which found that native speakers of Vietnamese and Mandarin exhibited a form of perfect pitch in enunciating words, which led her to hypothesize that pitch was an extra-musical ability. Deutsch then set out to investigate perfect pitch in music. In 2004, she found that students at the Central Conservatory of Music in Beijing, China, all of whom spoke Mandarin, were almost nine times more likely to have perfect pitch than students at the Eastman School of Music in New York. That last study, however, left open the question of whether perfect pitch might be a genetic trait – since all the Mandarin speakers were East Asian.
Overall I believe this is still a highly debated topic that I find extremely interesting and takes a role in the comparison between those with musical talent and those who who are without musical talent and if it is not genetic at all.
I guess these are "open questions":
Further exploring and utilizing advanced or uncommon temperaments such as non-twelve-tone octaves (equal or non-equal tempered) or temperaments that don't use a 2/1 octave as a fundamental basis.
A specific challenge in this is to find what sounds musically "natural" and appealing to an accustomed listener rather than just interesting but non-musical. Creating new or augmenting existing musical theory comes with this. Add to this the challenge of fascilitating the creation of such music with musical instruments (e.g. what would an appropriate keyboard look like?) as well as the challenge of reaching out to and getting acceptance from a wide audience.
Creating better means of musical notation that (1) can handle musical styles that don't turn out well in modern western musical notation, (2) is cognitively superior to modern western musical notation in terms of ease of learning and reading, (3) can catch things that cannot (easily and/or unambiguously) be notated in current modern western notation, (4) fascilitates a clear notation of advanced scale temperaments according to the above and including non-western music.
However none of these can be solved in the style of a "final proof" such as a mathematical proof of e.g. Fermat's last theorem, but would rather constitute a development of musical practices.
Computer programs are quite worse at transcribing music from audio recordings than humans are. So psychoacoustic modelling seems to have problems leaving its infancy.
