Talk:IFDO: Difference between revisions

Line 449:

:: '''Why 0 then?''' A good reason one might want to assign a value of 0 to pitch, placing it in between the 1 of frequency and the −1 of length, is by considering the derivatives of the frequency functions. Derivatives express how a function varies over its domain, which describes musically how the steps increase or decrease over its range, and typically taking a derivative of a polynomial brings the exponents down by 1. A special case is <math>\log_b(x)</math>, whose derivative is <math>\frac{1}{x}</math>, or <math>x^{-1}</math>. So even though pitch isn't <math>x^0</math>, if you consider its derivative to be <math>x^{-1}</math> and shift everything up by 1, then you get the index number 0. Similarly, the derivative of <math>x^{-1}</math> is <math>-x^{-2}</math>, and disregarding the sign, you can see that the exponent went down by 1 again, so you can move it up by 1 and get the index number −1. The frequency case is basically the same, but without a sign change, and you find the index number 1.

:: '''What's the takeaway?''' First, as I often say, it's always a good idea to carry the units around, and that could be improved on the individual pages for ALS and such. Second, I think we can learn some insight from the derivatives of the frequency functions, but we should check properly which scale structures are truly equivalent, especially if we're going to generalize this system to all power means and write on the Xen Wiki about it. The "diagonals" in the table aren't as simple as one would first expect, mostly because log spaces and linear spaces don't behave the same way, namely in terms of how they handle 0 and negative numbers, and making a continuous transition between both worlds isn't as simple as putting numbers in the middle and hoping it works out of the box. I don't think GPS is a well-defined structure at all, for that matter. It will be interesting to see what kinds of structures work and which ones don't, in terms of combining the horizontal "resource" axis with the vertical "power mean" axis. There might be other things to keep in mind, but I think it's important to at least make sure that linear/dimension-1 resources (those that have units with exponent 1, so length and frequency) remain strictly positive, while log resources (dimensionless, so pitch), can take any real value. It's also good to ponder what the units will look like: are the elements of an E(0.5)DO expressed in Hz<sup>0.5</sup>? Does that have any physical or psychoacoustic significance or is it purely recreational mathematics at this point? --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 08:58, 10 April 2023 (UTC)

::: Woa I'm really impressed by all the researches you guys have done here. Those tables are super helpful. You know, I vaguely felt the underlying structure but never tried to clear my mind or to even convey it. Anyway, to keep the convo short: I'm fully convinced that my terms need to be fixed, that ''ID'' must be ''IFD'', that there's the equivalence of AP(D/S) and GF(D/S), and of AL(D/S) and IF(D/S). If I interpreted it correctly, Fred pointed out that GP(D/S) is different from AL(D/S)~IF(D/S), which makes sense to me.

::: Let's edit the pages accordingly. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:01, 10 April 2023 (UTC)

@@ Line 449: / Line 449: @@
 :: '''Why 0 then?''' A good reason one might want to assign a value of 0 to pitch, placing it in between the 1 of frequency and the −1 of length, is by considering the derivatives of the frequency functions. Derivatives express how a function varies over its domain, which describes musically how the steps increase or decrease over its range, and typically taking a derivative of a polynomial brings the exponents down by 1. A special case is <math>\log_b(x)</math>, whose derivative is <math>\frac{1}{x}</math>, or <math>x^{-1}</math>. So even though pitch isn't <math>x^0</math>, if you consider its derivative to be <math>x^{-1}</math> and shift everything up by 1, then you get the index number 0. Similarly, the derivative of <math>x^{-1}</math> is <math>-x^{-2}</math>, and disregarding the sign, you can see that the exponent went down by 1 again, so you can move it up by 1 and get the index number −1. The frequency case is basically the same, but without a sign change, and you find the index number 1.
 :: '''What's the takeaway?''' First, as I often say, it's always a good idea to carry the units around, and that could be improved on the individual pages for ALS and such. Second, I think we can learn some insight from the derivatives of the frequency functions, but we should check properly which scale structures are truly equivalent, especially if we're going to generalize this system to all power means and write on the Xen Wiki about it. The "diagonals" in the table aren't as simple as one would first expect, mostly because log spaces and linear spaces don't behave the same way, namely in terms of how they handle 0 and negative numbers, and making a continuous transition between both worlds isn't as simple as putting numbers in the middle and hoping it works out of the box. I don't think GPS is a well-defined structure at all, for that matter. It will be interesting to see what kinds of structures work and which ones don't, in terms of combining the horizontal "resource" axis with the vertical "power mean" axis. There might be other things to keep in mind, but I think it's important to at least make sure that linear/dimension-1 resources (those that have units with exponent 1, so length and frequency) remain strictly positive, while log resources (dimensionless, so pitch), can take any real value. It's also good to ponder what the units will look like: are the elements of an E(0.5)DO expressed in Hz<sup>0.5</sup>? Does that have any physical or psychoacoustic significance or is it purely recreational mathematics at this point? --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 08:58, 10 April 2023 (UTC)
+::: Woa I'm really impressed by all the researches you guys have done here. Those tables are super helpful. You know, I vaguely felt the underlying structure but never tried to clear my mind or to even convey it. Anyway, to keep the convo short: I'm fully convinced that my terms need to be fixed, that ''ID'' must be ''IFD'', that there's the equivalence of AP(D/S) and GF(D/S), and of AL(D/S) and IF(D/S). If I interpreted it correctly, Fred pointed out that GP(D/S) is different from AL(D/S)~IF(D/S), which makes sense to me.
+::: Let's edit the pages accordingly. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:01, 10 April 2023 (UTC)