The S1 circle is a boundary that classifies pitch in the Z/12Z system as the chromatic circle, which can also be a spiral.
The string concatenary is the shape the string returns to when plucked, which is a standing wave with the boundary condition for a standing wave d2y/dt2 = 0.
The string is closed by the octave into a chromatic circle and also open to union with any pitch by adjusting the frequency of the fundamental.
The fundamental of the string has only one degree of freedom and therefore it cannot have more than one mode of vibration.
The string in classic theory is defined on the interval (0, 1) which is an unordered pair but the musical string is on an ordered pair since the frets define the lower segment.
There are 3 sets: observed pitch, fret positions value, string pitch values. The intersection of the pitch values and fret values on the string is a graph that is a restriction R x R onto R, which I think is the basis for both the S1 and d2y/dt2 = 0 boundary condition.
So I am wondering if the standing wave boundary is d2y/dt2 = (0, 1).
There must be a theorem that say the string can only have one boundary condition if it has only one degree of freedom, right?
The string is the smallest set that contains an image of every set in music so long as the string is at least 12 in size. There is the graph of pitch and position and then there are the tuning function f and the intonation function g.
The tuning function f maps the observed pitch to the string position and the intonation function g maps the string position to the observed pitch. f and g are composable functions that make an identity. This makes the string a homomorphism that is an arrow with a 1 or just a 1.
Now my question concerns how the integration of the boundary condition can be determined. It must be a simple sum if there is a smallest possible Lebesgue measure of 1 step that is countably additive so I think I can conclude the volume of the concatenary is 1. Since everything has to add up to 1, or at least a whole number. Then the boundary conditions are two ways to say the same thing. "S12 = 1".
Please help me to formalize the string finite state model. Its a 300 year old puzzle not yet solved.
I cannot see that these references go to the "corps sonore" which is the smallest possible set that contains every element, function, and relation in music. The smallest unit is an integral domain that does not depend on the frequency level at which the system in intonated.
The problem I have is that you are assuming the sound wave emitted by the string is a model of the string itself. But the frequency is merely the quantifier of the string and what we want to get at is the atomic structure. You cannot assume the string is Fourier space where waves can be added together because the string is a restriction on R.
This may seem ridiculous to you because real analysis is absolutely compelling and as an approximation in R2 cannot be proven wrong in anyway.
However, your assumption that the space of music has the real number topology is not correct in general in projective space.
The classic idea that the musical string detained between two points is open neglects the fact that the string is closed by the octave. In projective space the line is automatically closed in a circle.
There is no interest in modeling the timbre of notes, because the timbre is a secondary characteristic that is not a function of the string fundamental.
The problem is that if pitch is a real valued function then it cannot be written as a family of disjoint pairwise intervals using frequency.
I take your answer to be the string has n modes, not 1 or 2. But I doubt the string can be divided into infinitely many parts. In fact the size of the corps sonore is clearly 12, since any music in the 12-tone system can form an image in any string size 12 or more.
You see I need a system on Z2 while you are talking R2.
Thank you for the references.
- Badges
- Science topic
- Similar topics
- Solid State Physics
- Devices