If a string vibrates at 256 cycles per seconds then counting 256 cycles is the measure of 1 second. The number is real because it measures time and the number is arbitrary because it does not have to be 1 second that is used.
This establishes that the pitch is a point with the real number topology, right?
Terence,
I'm not sure what you mean by "real number topology".
A sound with a frequency of 256 Hz is no more special than one with a frequency of 3.14159... Hz.
Neither are especially meaningful as (you rightly point out) the second is an arbitrary construct.
Both frequencies are 'real' numbers in that they have no imaginary part (ie: complex numbers being X = a+ib where i^2 is -1) - nor can they have as the second is a 'real' number too.
Thank you for your answer, Mr Garry.
There is a question about whether 256 is a real number or a counting number. Is 256 hz part of a continuous function? Because we don't know what 256.5 cycles means, unless we assume some underlying real-value function such as a sine wave.
The real number topology is defined by open and closed sets, compactness, etc.
The frequency of string vibration is special because it happens every time the string is plucked. The frequency of vibration is a property of the equation of motion.
Of course 256 Hz is also a special number because it defines middle C (relative to concert pitch at 440 hz). Middle C or C4 is special because it is the 4th octave of C1.
My question here is whether pitch is a time-like variable.
I am looking for a simple statement about musical topology that is general accepted.
Some mathematicians say that musical pitch cannot be defined mathematically.
Is there something we can say about pitch that can be used to make logical inference about the mathematical structure tablature? Tablature is just fret numbers written on lines. The fret numbers are given pitch values by the tuning. So we have in tablature a mathematic object which is only strings, fret numbers, and pitch values. So tabs are purely matheatmical but what sort or mathematical object is tablature?
If pitch has the real number topology then we can understand tabs as open and closed sets on the real line.
To Mr Gerck
But pitch is continuous because between any two pitch values is an interval and between two intervals is a pitch value, so we have the real line minus finitely many points.
I am confused by the statement that real numbers are non-computable. real numbers are analytic.
My idea here is that music theory must be a theory of real numbers rather than frequencies. I think you are saying there is no theory of Q.
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