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I've seen in a few places that the subject of a fugue begins on either the tonic (first degree) or dominant (fifth degree).

From what I've read, the stated reason is that the answer (a transposed subject) begins on either the dominant (if the subject began on the tonic, called a "real answer") or the tonic (if the subject begins on the dominant, called a "tonal answer"). The answer then continues as though transposed a fifth higher.

It's stated that there are harmonic reasons that these two first notes are used, but I don't see any statement in my research for why you couldn't begin on the mediant (third degree) and then have the answer begin on the submediant (sixth degree) or leading tone (seventh degree) with a similar transposition to the real or tonal answer. Am I missing something?

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    For questions like this, it really helps if you try it out first. It’s not difficult to see that starting on the mediant precludes ending the theme on the tonic (so that seems like a good reason to avoid the mediant already) and even if you also end on the mediant (melodically, that seems suspicious already) you have to have an open fifth on the third degree at the start of the comes (harmonically suspicious). Ending the theme on other notes of the scale would be highly unusual for melodic reasons. – 11684 Nov 24 at 7:19
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Well, when one wants an exhaustive list of obscure exceptions to every conceivable musical principle, often the person to turn to is the great Ebenezer Prout, whose books are full of just that sort of stuff. Prout doesn't disappoint in his book on Fugue, where he notes:

Though there are limitations as to the note of the scale on which a fugue subject should end, there are none as to that on which it should begin. In an enormous majority of cases, the subject begins on either the tonic or dominant; but numerous examples are to be met with of the employment of the other degrees of the scale for the initial note. We give a few instances of each...

Here Prout notes that subjects beginning on the supertonic note are "rather rare" while those beginning on the mediant "are also not very often met with." The two examples he provides for the mediant are from Beethoven's Missa Solemnis and from Cherubini's 4th Mass:

Beethoven. Mass in D.

Cherubini. 4th Mass.

I don't think fugue subjects beginning on a mediant are a large enough group to generalize about their treatment. But yes, there is no reason you can't begin on the mediant, just as composers began fugue subjects on every other note of the scale (see Prout link above; also, another answer mentions a WTCII fugue that begins on the leading tone, while another fugue in WTCII begins on the supertonic).

The standard notes are tonic and dominant for the obvious reasons that they work best within the normal world of fugue subjects and answers that generally have strong tonic/dominant relationships. Also, most (though certainly not all) fugue subjects begin by strongly identifying their key, and tonic/dominant notes tend to do that the best. Beginning elsewhere may introduce a few complexities in counterpoint and harmonization, but nothing that a skilled fugue composer wouldn't be able to deal with.

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There are always exceptions. In WTC Book 2 Fugue 13, Bach starts the subject with a long trill on the leading note, and passes through the subdominant and back to the tonic before the end of the subject. The "real" answer starts on the leading note of the dominant (and at an augmented fourth above to tonic in two part counterpoint - so much for textbook theory!)

Starting on the mediant is likely to be tonally ambiguous, since until there is an unambiguous leading-note, there is not much to differentiate starting on the mediant of a major key or the dominant of the relative (melodic) minor.

Strongly unambiguous tonality not really a good idea in a fugue subject, because it limits the possibilities for counterpoint. You need to find a path between too much ambiguity and too little, but being unsure if the fugue starts in a major or minor key is usually "too little."

The same problem can apply to subjects that start on the dominant of a minor key, unless the "sharp" 6th and 7th degrees of the scale appear.

This example (by Bach) makes the point. On paper there is no reason why it shouldn't be in C major, but it sounds as if it is in A minor, and that interpretation turns out to be correct. But altering just one note, changing the A to C, pushes it into the major key instead.

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I don't have a complete answer yet, but I'm excited to see what others say; this is a wonderful question, and one I'm sorry to say I've somehow never been confronted with.

I'll simply share two quotes from two famous modern textbooks on fugue writing:

[The subject] almost invariably opens with either scale degree 1 or 5 (rarely 3) and will normally close or cadence on either 1, 3, or a tonicized 5 (implying a modulation to the dominant).

The above is from Bob Gauldin's A Practical Approach to Eighteenth-Century Counterpoint, pg. 211. Although he doesn't cite an example, he does not seem to forbid beginning on scale-degree 3.

[The subject] may begin on the root (strongest), or 3rd of the tonic chord, or it may begin on the 5th scale step, either as root of the dominant chord or as 5th of the tonic.

This quote is from Neale Mason's Essentials of Eighteenth-Century Counterpoint, pg. 120. Like Gauldin, he does not cite an example.

I wish I had some examples in my head to discuss what needs to be done differently with a subject that begins on 3, but none are coming to mind. Hopefully others here can help!

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Percy Goetschius also suggests that a fugue subject usually starts on steps 1, 3, or 5 (and suggests that step 1 should occur early in the subject.)

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