Talk:IFDO: Difference between revisions

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:# and then understand how those famous means with special names (arithmetic, geometric, harmonic) generalize as power means (where p = 1, 0, -1, respectively).

: So if we're going to discuss 0.5 tunings like this, I think we might as well use a more immediate and clear approach to it, as in the ... 2FDO, AFDO, (0.5)FDO, GFDO, (-0.5)FDO, IFDO, (-2)FDO ... continuum. If people are using this system already (i.e. your system, as revised according to my suggestions), then they should already know point #1, and possibly #3 too; it's really point #4 that's the interesting new thing. The key thing is they would never need to understand point #2. And this likens back to why I think our systems can coexist; because someone who thinks primarily in terms of frequency ratios like 7/6, 5/4, 3/1 etc. as well as in terms of these power means, well, they may potentially never have to learn anything about the "diagonal" relationship with pitch and string length (and vice versa, someone like me, for whom thinking about frequency, pitch, and length came naturally, would never have to learn about geometric and harmonic means, which indeed I had gotten away with my whole life without understanding, that is, of course, up until this whole issue came up earlier this year!) --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 03:20, 10 April 2023 (UTC)

:: I agree with most of what Cmloegcmluin said so far, especially about the importance of keeping both the type of sequence and the type of resource clearly identified. However, I'm not certain that the equivalences presented in the tables above using colour coding are all exact. This is related to Cmloegcmluin's remark that "there's just no direct association of these resources — frequency, pitch, and length — with those numbers — 1, 0, and -1"; there is an association, but it is indirect, and that may lead to misconceptions. Let's dive right in!

:: '''Frequency''': An [[AFS]]p is currently defined as <math>f_{\text{AFS}}(k) = 1 + k \cdot p</math>, which means that step <math>k</math> in the sequence has frequency <math>f(k)</math>. It is odd that this function outputs a pure number, since frequency is not a pure number, being usually expressed in Hz. Therefore this function doesn't really give the frequency, but rather the frequency ratio associated with step 0 of the scale. That is not a trivial step: it implies that the "natural" way to relate pitches to each other (i.e. intervals) is through frequency ratios, even though the purpose of this system is to avoid taking any of these things for granted! So, for the purposes of this discussion, let me redefine the function as follows: <math>f_{\text{AFS}}(k) = f_0(1 + k \cdot p)</math>, where <math>f_0</math> is the frequency of step 0, in Hz. To explain CompactStar's point, let me again rewrite the function: <math>f_{\text{AFS}}(k) = f_0(1 + k \cdot p)^1</math>. The reason for the added exponent 1 will be clearer in the next steps.

:: '''Length''': I'll skip over pitch for reasons that will be clearer later. An [[ALS]]p is currently defined as <math>L_{\text{ALS}}(k) = 1 + k \cdot p</math>. Again, to make sure that the function properly outputs a length, we'll need a length constant. Here is a first modified version of the function: <math>L_{\text{ALS}}(k) = L_0(1 + k \cdot p)</math>, where <math>L_0</math> is the length associated with step 0 of the scale. With this settled, we already see that we will need some work to compare both functions in terms of frequency, so we need to know the relationship between length and frequency. As usual in math, relationships are easier to find with some sort of constant, and what we need here is the speed of sound. (Of course, it isn't technically a constant, but we're not diving into underwater music discussions today!) The speed of sound is equal to the multiplication of wavelength by frequency. To make this more obvious, notice that length is usually expressed in m (meters), while frequency is usually expressed in Hz, which is equivalent to s−1, and finally the speed of sound is usually expressed in m/s (or m·s−1), i.e. the multiplication of the previous two units. As a formula, it looks like this: <math>c = {\lambda}f</math>, or <math>f = \frac{c}{\lambda}</math>, where <math>c</math> is the speed of sound. Now, string/tube length and wavelength are not the same property, so we have to make sure that they are proportional before we can proceed in our reasoning. We can discuss this if it becomes an issue, but for now I believe we can assume confidently that wavelength and string/tube length are proportional (e.g. twice the string length implies twice the wavelength, which is associated with half the frequency if the speed of sound is constant). Therefore, string/tube length can be derived from wavelength using a pure number constant, knowning that it inputs a length and outputs another length. That new constant largely depends on the physicality of the instrument (shape, material, etc.), and to avoid getting in the physics of strings and tubes, we'll just define <math>m = \frac{\lambda}{L}</math> (<math>m</math> for "material", and I'm carefully avoiding <math>\mu</math> which is often used in string physics equations), such that multiplying by <math>m</math> inputs a string length and outputs the corresponding wavelength. Let's see how our function looks like through the lens of frequency: taking this step by step, we first have <math>\lambda_{\text{ALS}}(k) = m \cdot L_0(1 + k \cdot p)</math>, and then <math>f_{\text{ALS}}(k) = \frac{c}{m \cdot L_0(1 + k \cdot p)}</math>. That last formula doesn't look very good with the big fraction, so let's clean it up a bit: <math>f_{\text{ALS}}(k) = \frac{c}{m \cdot L_0} (1 + k \cdot p)^{-1}</math>. The fraction now holds only the constants, which are equivalent to <math>f_0</math>, and we notice that the <math>(1 + k \cdot p)</math> bit is now written using an exponent −1. This should make it clear why I added a seemingly useless exponent 1 in the AFS function earlier, and why CompactStar is proposing to use −1 for ELDO/ALS. That said, the constants seem nontrivial to me, because they can become variables if you open up to creating new scales instead of just observing a single scale from different angles, and so this can help showcase a few different approaches to creating scales.

:: '''Pitch''': We're getting to the good part now. An [[APS]]p is currently defined as <math>P_{\text{APS}}(k) = k \cdot p</math>. The absence of the <math>1 +</math> in this formula should already alert us that something is different about this sequence compared to the previous two. As we know, pitch perception is mostly logarithmic, and while there are complicated functions that try to show precisely how the human ear perceives pitch, most xenharmonists are used to simply take a logarithmic ratio of frequencies, commonly expressed in cents, a dimensionless unit, making pitch a pure number: <math>P = \log_b \left(\frac{f}{f_0}\right)</math>. It is important to note that you ''cannot'' take the logarithm of a frequency, because the logarithm of the unit Hertz is not defined in any way that would meaningful to us. So whereas there is a relationship between frequency and length, there is no direct relationship between frequency and pitch, but only one between frequency ''ratio'' and pitch. So we'll need to carry <math>f_0</math> with us now. We need the reciprocal of the previous formula to be able to express frequency as a function of pitch: <math>f = f_0 \cdot b^P</math>. We're ready to write our APS function in terms of frequency: <math>f_{\text{APS}}(k) = f_0 \cdot b^{k \cdot p}</math>. We can see very clearly that this function cannot be written using the same structure as the previous two functions but using an exponent 0 instead of 1 or −1, otherwise we would end up with a constant function.

:: '''Let's check the colour coding'''. There is only one blue cell, so there's nothing to do here. The two red cells are APS and GFS. An APS is equivalent to any familiar [[equal-step tuning]], and we know that each step of such a tuning has a constant frequency ratio, therefore the GFS checks out. Now, there are three yellow cells, so it might be a bit trickier. ALS and IFS are easy to check, because they're basically equivalent by definition: length and frequency are inversely proportional, and inverse-arithmetic is called that way for a reason. The actual tricky part is with GPS. Let's take ALS and GPS. I ran a quick example in a spreadsheet, only to find that the ratios between the pitches of each step of the ALS were not constant as one would expect from a geometric progression, but rather took the form of a decreasing sequence that converged to 1. We can see why that happened by working through our formulas. Let's recall our frequency-based ALS formula: <math>f_{\text{ALS}}(k) = f_0 (1 + k \cdot p)^{-1}</math>. Now, what does a GPS look like? Well, already in terms of pitch it should be something like this: <math>P_{\text{GPS}}(k) = P_0 \cdot p^k</math>, for some constant <math>p</math> and starting pitch <math>P_0</math>... Hang on, wouldn't <math>P_0</math> just be 0? And multiplying anything by 0 results in 0? Clearly, something weird is happening here. To avoid this problem, let's rewrite the ALS in terms of pitch and check if it looks like a geometric progression: <math>P_{\text{ALS}}(k) = \log_b((1 + k \cdot p)^{-1})</math>. This can be rewritten to get rid of the negative exponent using logarithm laws: <math>P_{\text{ALS}}(k) = -\log_b(1 + k \cdot p)</math>. This function shows that pitch decreases monotonically, diverging to negative infinity, but doing so more and more slowly, which explains why the ratios between consecutive pitches converged to 1. In a geometric pitch sequence with p < 1, you can get a decreasing sequence of pitches, but that sequence will converge to 0, which is different from the behaviour of an ALS. Clearly, GPS is doing something completely different from ALS/IFS, and that is mostly because of how logarithms work.

:: '''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 Hz0.5? 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)

@@ Line 442: / Line 442: @@
 :# and then understand how those famous means with special names (arithmetic, geometric, harmonic) generalize as power means (where p = 1, 0, -1, respectively).
 : So if we're going to discuss 0.5 tunings like this, I think we might as well use a more immediate and clear approach to it, as in the ... 2FDO, AFDO, (0.5)FDO, GFDO, (-0.5)FDO, IFDO, (-2)FDO ... continuum. If people are using this system already (i.e. your system, as revised according to my suggestions), then they should already know point #1, and possibly #3 too; it's really point #4 that's the interesting new thing. The key thing is they would never need to understand point #2. And this likens back to why I think our systems can coexist; because someone who thinks primarily in terms of frequency ratios like 7/6, 5/4, 3/1 etc. as well as in terms of these power means, well, they may potentially never have to learn anything about the "diagonal" relationship with pitch and string length (and vice versa, someone like me, for whom thinking about frequency, pitch, and length came naturally, would never have to learn about geometric and harmonic means, which indeed I had gotten away with my whole life without understanding, that is, of course, up until this whole issue came up earlier this year!) --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 03:20, 10 April 2023 (UTC)
+:: I agree with most of what Cmloegcmluin said so far, especially about the importance of keeping both the type of sequence and the type of resource clearly identified. However, I'm not certain that the equivalences presented in the tables above using colour coding are all exact. This is related to Cmloegcmluin's remark that "there's just no direct association of these resources — frequency, pitch, and length — with those numbers — 1, 0, and -1"; there is an association, but it is indirect, and that may lead to misconceptions. Let's dive right in!
+:: '''Frequency''': An [[AFS]]p is currently defined as <math>f_{\text{AFS}}(k) = 1 + k \cdot p</math>, which means that step <math>k</math> in the sequence has frequency <math>f(k)</math>. It is odd that this function outputs a pure number, since frequency is not a pure number, being usually expressed in Hz. Therefore this function doesn't really give the frequency, but rather the frequency ratio associated with step 0 of the scale. That is not a trivial step: it implies that the "natural" way to relate pitches to each other (i.e. intervals) is through frequency ratios, even though the purpose of this system is to avoid taking any of these things for granted! So, for the purposes of this discussion, let me redefine the function as follows: <math>f_{\text{AFS}}(k) = f_0(1 + k \cdot p)</math>, where <math>f_0</math> is the frequency of step 0, in Hz. To explain CompactStar's point, let me again rewrite the function: <math>f_{\text{AFS}}(k) = f_0(1 + k \cdot p)^1</math>. The reason for the added exponent 1 will be clearer in the next steps.
+:: '''Length''': I'll skip over pitch for reasons that will be clearer later. An [[ALS]]p is currently defined as <math>L_{\text{ALS}}(k) = 1 + k \cdot p</math>. Again, to make sure that the function properly outputs a length, we'll need a length constant. Here is a first modified version of the function: <math>L_{\text{ALS}}(k) = L_0(1 + k \cdot p)</math>, where <math>L_0</math> is the length associated with step 0 of the scale. With this settled, we already see that we will need some work to compare both functions in terms of frequency, so we need to know the relationship between length and frequency. As usual in math, relationships are easier to find with some sort of constant, and what we need here is the speed of sound. (Of course, it isn't technically a constant, but we're not diving into underwater music discussions today!) The speed of sound is equal to the multiplication of wavelength by frequency. To make this more obvious, notice that length is usually expressed in m (meters), while frequency is usually expressed in Hz, which is equivalent to s<sup>−1</sup>, and finally the speed of sound is usually expressed in m/s (or m·s<sup>−1</sup>), i.e. the multiplication of the previous two units. As a formula, it looks like this: <math>c = {\lambda}f</math>, or <math>f = \frac{c}{\lambda}</math>, where <math>c</math> is the speed of sound. Now, string/tube length and wavelength are not the same property, so we have to make sure that they are proportional before we can proceed in our reasoning. We can discuss this if it becomes an issue, but for now I believe we can assume confidently that wavelength and string/tube length are proportional (e.g. twice the string length implies twice the wavelength, which is associated with half the frequency if the speed of sound is constant). Therefore, string/tube length can be derived from wavelength using a pure number constant, knowning that it inputs a length and outputs another length. That new constant largely depends on the physicality of the instrument (shape, material, etc.), and to avoid getting in the physics of strings and tubes, we'll just define <math>m = \frac{\lambda}{L}</math> (<math>m</math> for "material", and I'm carefully avoiding <math>\mu</math> which is often used in string physics equations), such that multiplying by <math>m</math> inputs a string length and outputs the corresponding wavelength. Let's see how our function looks like through the lens of frequency: taking this step by step, we first have <math>\lambda_{\text{ALS}}(k) = m \cdot L_0(1 + k \cdot p)</math>, and then <math>f_{\text{ALS}}(k) = \frac{c}{m \cdot L_0(1 + k \cdot p)}</math>. That last formula doesn't look very good with the big fraction, so let's clean it up a bit: <math>f_{\text{ALS}}(k) = \frac{c}{m \cdot L_0} (1 + k \cdot p)^{-1}</math>. The fraction now holds only the constants, which are equivalent to <math>f_0</math>, and we notice that the <math>(1 + k \cdot p)</math> bit is now written using an exponent −1. This should make it clear why I added a seemingly useless exponent 1 in the AFS function earlier, and why CompactStar is proposing to use −1 for ELDO/ALS. That said, the constants seem nontrivial to me, because they can become variables if you open up to creating new scales instead of just observing a single scale from different angles, and so this can help showcase a few different approaches to creating scales.
+:: '''Pitch''': We're getting to the good part now. An [[APS]]p is currently defined as <math>P_{\text{APS}}(k) = k \cdot p</math>. The absence of the <math>1 +</math> in this formula should already alert us that something is different about this sequence compared to the previous two. As we know, pitch perception is mostly logarithmic, and while there are complicated functions that try to show precisely how the human ear perceives pitch, most xenharmonists are used to simply take a logarithmic ratio of frequencies, commonly expressed in cents, a dimensionless unit, making pitch a pure number: <math>P = \log_b \left(\frac{f}{f_0}\right)</math>. It is important to note that you ''cannot'' take the logarithm of a frequency, because the logarithm of the unit Hertz is not defined in any way that would meaningful to us. So whereas there is a relationship between frequency and length, there is no direct relationship between frequency and pitch, but only one between frequency ''ratio'' and pitch. So we'll need to carry <math>f_0</math> with us now. We need the reciprocal of the previous formula to be able to express frequency as a function of pitch: <math>f = f_0 \cdot b^P</math>. We're ready to write our APS function in terms of frequency: <math>f_{\text{APS}}(k) = f_0 \cdot b^{k \cdot p}</math>. We can see very clearly that this function cannot be written using the same structure as the previous two functions but using an exponent 0 instead of 1 or −1, otherwise we would end up with a constant function.
+:: '''Let's check the colour coding'''. There is only one blue cell, so there's nothing to do here. The two red cells are APS and GFS. An APS is equivalent to any familiar [[equal-step tuning]], and we know that each step of such a tuning has a constant frequency ratio, therefore the GFS checks out. Now, there are three yellow cells, so it might be a bit trickier. ALS and IFS are easy to check, because they're basically equivalent by definition: length and frequency are inversely proportional, and inverse-arithmetic is called that way for a reason. The actual tricky part is with GPS. Let's take ALS and GPS. I ran a quick example in a spreadsheet, only to find that the ratios between the pitches of each step of the ALS were not constant as one would expect from a geometric progression, but rather took the form of a decreasing sequence that converged to 1. We can see why that happened by working through our formulas. Let's recall our frequency-based ALS formula: <math>f_{\text{ALS}}(k) = f_0 (1 + k \cdot p)^{-1}</math>. Now, what does a GPS look like? Well, already in terms of pitch it should be something like this: <math>P_{\text{GPS}}(k) = P_0 \cdot p^k</math>, for some constant <math>p</math> and starting pitch <math>P_0</math>... Hang on, wouldn't <math>P_0</math> just be 0? And multiplying anything by 0 results in 0? Clearly, something weird is happening here. To avoid this problem, let's rewrite the ALS in terms of pitch and check if it looks like a geometric progression: <math>P_{\text{ALS}}(k) = \log_b((1 + k \cdot p)^{-1})</math>. This can be rewritten to get rid of the negative exponent using logarithm laws: <math>P_{\text{ALS}}(k) = -\log_b(1 + k \cdot p)</math>. This function shows that pitch decreases monotonically, diverging to negative infinity, but doing so more and more slowly, which explains why the ratios between consecutive pitches converged to 1. In a geometric pitch sequence with p < 1, you can get a decreasing sequence of pitches, but that sequence will converge to 0, which is different from the behaviour of an ALS. Clearly, GPS is doing something completely different from ALS/IFS, and that is mostly because of how logarithms work.
+:: '''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)