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E#. Wait, what?

Why are you like this?

In music, the term note has two primary meanings:

  1. A sign used in musical notation to represent the relative duration and pitch of a sound (♪, ♫);
  2. A pitched sound itself.

Those are the definitions that exist in the study of music.

A note is a musical event, it has duration and function attached to it.

A pitch is just a sound with a particular fundamental frequency.
 
Those are the definitions that exist in the study of music.

A note is a musical event, it has duration and function attached to it.

A pitch is just a sound with a particular fundamental frequency.

And when you notate a quarter note E# and a quarter note F, the player will play the exact same note.

You are arguing semantics. They both have the same pitch. They both make the same sound. The both have the same duration. They are the same note.
 
And when you notate a quarter note E# and a quarter note F, the player will play the exact same note.

You are arguing semantics. They both have the same pitch. They both make the same sound. The both have the same duration. They are the same note.

Nope. Still not the same note. You're leaving out the function of the note in the context of a key or progression.

You can't get around it. An E# will function differently than an F. It's just the way music works.

It ain't perfect, but the system of keys and note names has a logic to it that can be learned.