Frequency and amplitude are generally considered to be scalar values and not vectors, implying they can be defined by a point on a real number line scale.
But there is another mathematical view in which the frequency spectrum does not have the topology of the real number line. Real analysis of frequency is compelling and successful theory but does not apply when objects in the frequency domain are defined by a discrete topology, because the discrete values like pitch values in music are vectors and not scalar values. In harmonic systems all defined values have a common point of origin which is the fundamental. Any two vectors are perpendicular, if they are not the same.
The implications of the real and discrete topology are quite different, but it does not seem to be well-known.
Using the language of category theory, if every defined value in the harmonic system is a simple multiple of the fundamental frequency, and if the fundamental is thought of as an arrow with a 1, or just a 1, then the fundamental F is itself a vector in a field and every frequency defined by a simple multiple of the fundamental must also be a vector. (please see attached diagram) My question is why isn't that true?
Similarly, the amplitude of a vibrating string is always defined as 1 since the string has a fixed boundary condition along its entire length and not point on a real line. The string amplitude is subject to the boundary condition which is 0 1 0; it is not like a vibrating column of air which has no fixed points it its boundary and assume the shape of its container.
It seems to me that frequency and amplitude are properly scalar values only in the topology of real numbers in which the frequency of the fundamental is considered a point defined in the frequency domain. The problem with this is the spectral resolution theorem states that the frequency domain cannot be resolved to a point without making some kind of assumption about the mathematical topology. However, the axiom of choice suggests that it is possible to choose a frequency and define the fundamental as a vector between the fundamental and that frequency which defines the fundamental. This view, which is not the topology of real number, makes the fundamental F a vector that is orthogonal to the frequency domain.
For instance, if middle C is defined as 256 Hz it seems as if the pitch value C4 is a scalar value when it fact it is a vector. But in fact C4 is defined by the 4th multiple of C1 and not by 256 Hz which only applies at concert pitch. So C4 could also be 257 Hz and is not defined by the frequency but as a multiple of F
This problem is in part significant because it relates to Euler’s tone net which is a vector field in music based on his observation that for Bach’s tempered key board every musical key can form a union with every other musical key by a change in intonation, that is to say every key sounds in tune with every other key. This lead Euler to think the vector field of pitch values in music is defined in effect by two scalars, key and pitch but this is clearly wrong if both the key and pitch are multiple of the one and only fundamental, which creates a different topology based on a sphere and not a torus. Note the sphere reduces to a point but the tone net torus cannot.
The question I am asking seems to me to be like the continuum hypothesis problem because in effect Euler postulated a vector field where there are two real number lines when there can only be one frequency domain.
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