Are C♯ and D♭ same? Interval & Octave explained: Music & law of frequencies

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Octave is derived from a word that means eight. Why are there twelve notes? Why are there no black keys between E and F, and between B and C? Why is there only a single key for both sharp and flat? Why does the concept exist?
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bisqwit
joel yliluoma
music theory
intervals
octave
sharps and flats
c-sharp
pure intervals
major chord
equal temperament
zhu zayin
pythagoras
perfect fourth
perfect fifth
major second
minor second
perfect sixth
perfect third
major third
augmented interval
augmented fourth
diminished fifth
five-limit tuning
prime numbers in music
cello
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bowing or plucking