Jtdcftul

I was playing 'In the Hall of the Mountain King' and noticed it modulates from B minor to a fifth above, F-sharp major.

I looked on wikipedia and found it says:

"After being stated, the main theme is then very slightly modified
with a few different ascending notes, but transposed up a perfect
fifth (to the key of F-sharp major, the dominant key, but with
flattened sixth) and played on different instruments."

So it got me wondering when one should say modulate versus the word transpose? I thought transposition implied all notes are shifted by the same pitch. But in Hall of Mountain King they're not the same melody, they're also different in that one is minor and one is major.

It's almost like the square and rectangle issue.

When music modulates, it just moves into a new key. Music can modulate using prior melodic material, or it can modulate with material that is completely new.

But transposition implies a copying and pasting of earlier material from one key into another key.

In other words, music that is transposed from one key to another also results in a modulation from the first key to the second. However, not every modulation is a transposition, because the music could have modulated without the use of any prior material.

(In other words, a square is a rectangle, but a rectangle isn't always a square.)

There's one added wrinkle in this distinction, but I'm saving it until the end because not all musicians agree: to me, a modulation implies a cadence in the new key. So it's possible that a transposition is not a modulation if the original music being transposed itself never reaches a cadence in the new key. As such, there's no cadence in the new key, and thus not a modulation.

Modulate means a change of key. Something needs to make clear the music is in a new key. Traditionally a cadence in a new key does the job. Importantly to the comparison of terms a modulation can be achieved with all new thematic material.

In transposition some musical material is repeated, but at a different position pitch-wise. There is both chromatic and diatonic transposing so the pitch shifting isn't necessarily by the exact same interval for each note. Either way the important thing is that some thematic material is repeated. Note the difference with modulation where repeating material isn't part of the definition.

Given that transposition can be diatonic, it won't necessarily result in a key change. (You may notice that a diatonic transposition is the same as a diatonic melodic sequence.)

If a thematic passage was chromatically transposed, it could result in a key change, but I suppose it really depends on how the passage moves, and also the length of the passage. Some might end up resulting in a sequence. This [V6/5 of IV|IV]|[V6/5 of V|V] is both a transposition of the material in brackets and a harmonic sequence. On the other hand in a simple sonata form the "second theme group" (the part of the exposition in the dominant key) can often be simply transposed down a fifth to the tonic during the recapitulation. (If that sonata stuff isn't familiar, try reading some reviews of sonata form and then look at L. Mozart's Nannerl Notebook which as many examples in small scale pieces.) Transposition in the recapitulation is a nice example where it has formal, structural meaning and a modulation, a key change, is the result.

So, there is some overlap in the meanings, but modulation doesn't really say anything about the handling of thematic material while transposition does tell specifically we are modifying thematic material.

'Modulation' is the process of getting to a different key. When we get there, we've 'modulated'.

A particular musical element, repeated literally at a different pitch, is 'transposed'.

It's not stretching the definition of 'transposed' too much to let it cover an ALMOST literal copy, and a different mode as well as a different pitch.