What determines a note, and when we change to another one?
I have begun playing the tin whistle recently, and am quite curious about something in particular.
How is a note defined?
My idea is that the sharp and flat versions of a note are the "boundaries" that define the note, and any frequency in this range is considered that note.
If anyone can please help guide me, mathematical answers are fine.
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How is a note defined?
Interesting question, and perhaps not one that has an obvious answer.
Even assuming we're talking about current western music culture, I think you could answer from a few different perspectives.
If assume that we're talking about named notes in a given key - for example, in C major, the notes C-D-E-F-G-A-B - then your definition of the sharp and flat versions of a note being the "boundaries" that define the note could make sense - APART from the two places in the scale where the unaltered notes are only a semitone apart - that is, the boundary between B and C, and between E and F. In other words, C flat is B*, but we can't really clam B as being C's territory, 'cos.... it's B! In these cases, we might want to give each note a 'territory' of 50 cents on the 'narrow' side of the note (a cent is 1/100th of a semitone).
On the other hand, if by 'note' we mean one of the 12 'slots' per octave that the chromatic (12-tone) scale allows, then in each case we could give each note a territory of 50 cents either side, assuming equal temperament. However, equal temperament is not the only temperament available - different temperaments of the scale will move our boundaries up and down a bit.
And when we consider temperaments, it leads me on to the other way we could interpret this question, which is (as per Randy's answer) that a note, in a given temperament, assuming a particular concert pitch, isn't really considered to inhabit a range of frequencies at all, but instead to map to a specific frequency. Of course this might not always be the frequency that is actually hit in performance, due to inaccuracy or deliberate application of techniques such as vibrato.
*You might say that it doesn't make sense to talk about C flat in the key of C, but I'm trying to answer 'in the spirit of the question'.
Each note has an EXACT mathematical frequency.
But the human ear is not in sync with those exact frequencies - there's a range of frequencies that could be aurally accepted as the note.
So there are temperaments - 'equal', 'pythagorean', and syntonic temperament - to adjust the notes so they play more nicely together.
en.wikipedia.org/wiki/Regular_diatonic_tuning#Syntonic_temperament_and_timbre
And indeed there are tuning systems that attempt to take this into account such as Meantone. (Mean... average).
Meantone temperament is a musical temperament, that is a tuning system, obtained by slightly compromising the fifths in order to improve the thirds. Meantone temperaments are constructed the same way as Pythagorean tuning, as a stack of equal fifths, but in meantone each fifth is narrow compared to the perfect fifth of ratio 3:2."
en.wikipedia.org/wiki/Meantone_temperament
A simple example ... vibrato (violin, guitar, etc.) ... a judicious amount can strengthen the emotional impact of a melody ... too much can destroy it (as food spices).
A good explanation of the derivation of notes is here.
en.wikipedia.org/wiki/Pythagorean_tuning
If there is to be a boundary between (say) F and F#, it will be halfway between F and F#. That is where we would stop considering it an out of tune F (too sharp) and an out of tune F# (too flat).
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