Talk:IFDO: Difference between revisions

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:::: --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:39, 11 April 2023 (UTC)

::::: I'm not sure if this talk page is the best place to continue this discussion about GPS, but here goes anyway. A few thoughts on the above:

::::: 1. The domain and range of AFS, APS, ALS and GPS have different peculiarities. AFS have only one half of their domain which is musically usable, because the other half results in negative frequencies. APS can be used on its whole domain, since it only outputs positive frequencies, and notably they converge to 0 Hz (−∞{{cent}}) at one end of the domain. ALS have an issue similar to AFS, because the half of the domain which is associated with negative lengths results in negative frequencies too. GPS behaves a bit like APS, since APS is equivalent to GFS, but on the pitch scale, so it converges to 0{{cent}} at one end of the domain, and therefore GPS have either a minimum or a maximum frequency, which is not the case for any of AFS, APS and ALS. In the tables above, the 0{{cent}} mark was attributed to step 0 of the scales, so the minimum or maximum pitch would be represented with a different value in cents, but it would still be a finite quantity. For reference, moving the 0{{cent}} mark to the value of convergence of each GPS would result in transposing the scales (as opposed to shifting or stretching), if you keep the same frequency for 0{{cent}}.

::::: 2. In an APS, the pitch of each degree increases linearly and the size of the steps is constant. In terms of functions, for <math>p(x) = ax + b</math>, the derivative is <math>p'(x) = a</math> (constant). In a GPS, the pitch of each degree increases (or decreases) exponentially, and the size of the steps also increases (or decreases) exponentially. In terms of functions, for <math>p(x) = a b^x + k</math>, the derivative is <math>p'(x) = a \ln(b) b^x</math> (proportional to <math>p(x)</math>).

::::: 3. This indicates that there are two ways to think of a GPS: defining its exponentially varying steps (intervals) or defining its exponentially varying degrees (pitches). As we saw, the same "geometric factor" applies to both steps and degrees. In your examples, you chose to define the steps, so if we defined a "steps" function, we would have to take the indefinite integral to find the "degrees" function, which would feature an arbitary constant that roughly corresponds to the reference pitch.

::::: 4. With these function models in mind, it becomes clearer that only one line passes through two given points on a pitch graph, since there is only the parameter <math>a</math> to work with, but infinitely many exponential graphs pass through these two points, since there are two parameters <math>a</math> and <math>b</math> to work with.

::::: 5. GPS with a "geometric factor" of 1 are degenerate, because in that case they are constant functions. This is a bit weird though, because constant steps correspond to APS, while constant degrees would be just a single pitch.

::::: I'm pretty sure I noticed more things, but it's getting late and this is probably enough food for thought for now. I think that framing this in functions with various parameters, experimenting with negative values and with absolute values larger or smaller than 1, and making a difference between the step function and the degree function will help clear out a lot of this. I feel like GPS are a sort of rank-2 family of scales, in the sense that there are two variables to play with, even though rank-2 is probably not the best way to describe how this is behaving. Anyway, that's all for today. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 05:39, 12 April 2023 (UTC)

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 :::: --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:39, 11 April 2023 (UTC)
+::::: I'm not sure if this talk page is the best place to continue this discussion about GPS, but here goes anyway. A few thoughts on the above:
+::::: 1. The domain and range of AFS, APS, ALS and GPS have different peculiarities. AFS have only one half of their domain which is musically usable, because the other half results in negative frequencies. APS can be used on its whole domain, since it only outputs positive frequencies, and notably they converge to 0 Hz (−∞{{cent}}) at one end of the domain. ALS have an issue similar to AFS, because the half of the domain which is associated with negative lengths results in negative frequencies too. GPS behaves a bit like APS, since APS is equivalent to GFS, but on the pitch scale, so it converges to 0{{cent}} at one end of the domain, and therefore GPS have either a minimum or a maximum frequency, which is not the case for any of AFS, APS and ALS. In the tables above, the 0{{cent}} mark was attributed to step 0 of the scales, so the minimum or maximum pitch would be represented with a different value in cents, but it would still be a finite quantity. For reference, moving the 0{{cent}} mark to the value of convergence of each GPS would result in transposing the scales (as opposed to shifting or stretching), if you keep the same frequency for 0{{cent}}.
+::::: 2. In an APS, the pitch of each degree increases linearly and the size of the steps is constant. In terms of functions, for <math>p(x) = ax + b</math>, the derivative is <math>p'(x) = a</math> (constant). In a GPS, the pitch of each degree increases (or decreases) exponentially, and the size of the steps also increases (or decreases) exponentially. In terms of functions, for <math>p(x) = a b^x + k</math>, the derivative is <math>p'(x) = a \ln(b) b^x</math> (proportional to <math>p(x)</math>).
+::::: 3. This indicates that there are two ways to think of a GPS: defining its exponentially varying steps (intervals) or defining its exponentially varying degrees (pitches). As we saw, the same "geometric factor" applies to both steps and degrees. In your examples, you chose to define the steps, so if we defined a "steps" function, we would have to take the indefinite integral to find the "degrees" function, which would feature an arbitary constant that roughly corresponds to the reference pitch.
+::::: 4. With these function models in mind, it becomes clearer that only one line passes through two given points on a pitch graph, since there is only the parameter <math>a</math> to work with, but infinitely many exponential graphs pass through these two points, since there are two parameters <math>a</math> and <math>b</math> to work with.
+::::: 5. GPS with a "geometric factor" of 1 are degenerate, because in that case they are constant functions. This is a bit weird though, because constant steps correspond to APS, while constant degrees would be just a single pitch.
+::::: I'm pretty sure I noticed more things, but it's getting late and this is probably enough food for thought for now. I think that framing this in functions with various parameters, experimenting with negative values and with absolute values larger or smaller than 1, and making a difference between the step function and the degree function will help clear out a lot of this. I feel like GPS are a sort of rank-2 family of scales, in the sense that there are two variables to play with, even though rank-2 is probably not the best way to describe how this is behaving. Anyway, that's all for today. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 05:39, 12 April 2023 (UTC)