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coffeem
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Homework Statement
musical notes are defined by the frequency of oscillation. however notes are usually not qualifies in terms of absolute frequency, but the ratio (called interval) of their frequency to that of a reference notes called root. Pythagorus discovered that intervals that please the ear are characterised by simple frequency ratios. Th most important intervals are the octace, the perfect fifth (P5), and the perfect fourth (P4), defined by simple ratios: 2 (octave), 3/2 (P5) and 4/3 (P4).
a) Show that P5, followed by P4, equal an octave.
b) In western music, the octave is divided into twelve equal intervals, called semitones. Give the frequency ratio, h, that characterises the semitone.
c) How many integer semitones above the root do you get the best possible approximation of P4 and P5? Show you working you cannot just quote from musical knowledge.
d) Assume the octave were equally divided into 15 alternative semitones, rather than 12. Calculate the alternative semitone frequency ratio h', fine the best integer alternative semitone approximation for P4 and P5. How close is that approximation, compared to the conventional 12 semitone octave? Why is the conventional division of the octave into 12 semitones not as arbitrary as it may first appear?
The Attempt at a Solution
I am struggling with this. I know that if I multiply P4 and P5 together I get the correct answer, 2. However I am unsure about the theory behind this. I have looked in my notes and have also tried searching on the internet for what I need to do, but have found nothing.