Imagine you are sat in front of a piano. After a brief explanation of letter names and location on the piano your teacher says can you play a D please. A first response I have heard many times is "Which one? " This has been more recently than when I started teaching therefore I made the decision a decade ago to use Scientific pitch notation.
The system uses the standard letter names followed by a number which explains which octave that it is found in.
For standard A440 pitch that is used today the system begins at C0 = 16.35160 Hz . This is normally called the Sub Contra Octave. The addition of the zero continues for the octave of 7 unique letter names. On the repeat of the C it becomes C1. This repeats for every octave. Every variation of the note includes the number.
"B3" and all of its possible variants (B, B♭, B, B♯, B) would properly be designated as being in octave "3".
"C4" and all of its possible variants (C, C♭, C, C♯, C) would properly be designated as being in octave "4".
In equal temperament "C♭4" is same frequency as "B3".
It provides an unambiguous way of identifying a note.
An 88-key piano, with the octaves numbered and Middle C (cyan) and A440 (yellow) highlighted.
The current international pitch standard, using A4 as exactly 440 Hz, had been informally adopted by the music industry as far back as 1926, and A440 became the official international pitch standard in 1955. SPN is routinely used to designate pitch in this system. A4 may be tuned to other frequencies under different tuning standards, and SPN octave designations still apply (ISO 16).
With changes in concert pitch and the widespread adoption of A440 as a musical standard, new scientific frequency tables were published by the Acoustical Society of America in 1939, and adopted by the International Organization for Standardization in 1955. C0, which was exactly 16 Hz under the scientific pitch standard, is now 16.352 Hz under the current international standard system