Talk:IFDO: Difference between revisions
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--[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:15, 9 April 2023 (UTC) | --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:15, 9 April 2023 (UTC) | ||
I have not been following your discussion with Flora, but from what I've gathered, you believe that your system using the terms EFDO and ELDO is more intuitive for musicians due to | I have not been following your discussion with Flora, but from what I've gathered, you believe that your system using the terms EFDO and ELDO is more intuitive for musicians due to naming resources frequency and length, whereas my/Flora's system using ADO and IDO is more generalizable. However, there is an easy way to have both of these advantages–use a spectrum of resources rather than means, where the "n-resource" is expressed in terms of frequency as f<sup>n</sup>. The terms EFDO, ELDO, and E(P)DO can be maintained for n = 1, n = -1, and n → 0 respectively. For any other p, the term "EnDO" (equal n-resource division of the octave) can be used, so the tuning I described on Discord can be called an E2DO, and the tuning halfway between EDO and EFDO can be called an E0.5DO. | ||
[[User:CompactStar|CompactStar]] ([[User talk:CompactStar|talk]]) 01:00, 10 April 2023 (UTC) | [[User:CompactStar|CompactStar]] ([[User talk:CompactStar|talk]]) 01:00, 10 April 2023 (UTC) | ||
: Hi CompactStar. Yes, you've gathered correctly; that is a big part of what I'm saying: that my team's system is more accessible, while yours is capable of generalizing to unexplored theoretical territory. (The other biggest part is that due to historical and contemporary usage patterns, "ADO" is unacceptable in a tuning name system.) Now, I do like how you're thinking here, attempting to synthesize the strengths of our two systems. I had not really thought of trying something like this. But unfortunately, I don't think I would recommend we go with this particular suggestion of yours. I wish I liked it... I really don't want to come across as inflexible or overly negative here. My problem is this: there's just no direct association of these resources — frequency, pitch, and length — with those numbers — 1, 0, and -1. In other words, in order to understand why a E(0.5)DO referred to a tuning between an EFDO and an EDO, one would have to understand several things: | |||
:# arithmetic/geometric/harmonic progressions in the first place, | |||
:# and then understand the blue/red/yellow "diagonal" relationship between frequency/pitch/length and arithmetic/geometric/harmonic progressions (that I showed off in the tables above), | |||
:# and then understand the relationship between such progressions and the associated means, | |||
:# 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) | |||
::: 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) | |||
:::: Dang, Fredg999, that's some good stuff! Yes, I did notice last week that some of the pages I made back in 2021 sorta take the units for granted, such as assuming frequency or length as the musical resource being sequenced or divided, just like how I'm warning against here. At the time I wrote those pages, I wasn't cognizant of all these historical usages and conceptual relationships, but now I certainly agree that this aspect needs fixing. (Also, I think my experience working with Dave on the RTT guide, where we ended up including an entire article on units analysis, has influenced me to appreciate and advocate for mindfulness of units too.) | |||
:::: And Flora, I'm delighted and relieved to find that you basically agree with the main ideas I presented. I was afraid I'd been too harsh or impatient with you. As always, it's fun to confront these intellectual/musical/pedagogical challenges with you. | |||
:::: Alright, so I figure that we should give Flora and CompactStar the first pass at updating the pages they created. To be clear, this is not because I consider it their responsibility or anything, nor because I want to create work for them to do; it's because I want to give them the opportunity to make their own pages (as well as Shaahin's original pages) as clear and coherent as they can according to their own thinking. Plus this'll be a good test of the new perspective Fredg999 and I brought to the problem, to see if other people besides us can articulate it well; I bet y'all can improve further upon how we explained things, by ironing out whatever idiosyncrasies to our approaches. I expect that after they take care of this, since we seem to be largely on the same page together now, while Fredg999 and I may have some comments or tweaks here or there, probably we won't need to have too much back and forth discussion after that. | |||
:::: Meanwhile, I should do what I can to tighten up all the pages that I originally made. While I share credit for designing my system with Paul and Billy, they aren't hands-on with the wiki, so I can take care of it. That said, if Fredg999 wants to take the first pass themself — i.e. at updating the equations on the AFS, APS, and ALS pages per his comment above, like including the base frequency so as to return actual steps in frequencies rather than pure numbers — that's A-OK with me. | |||
:::: Finally, I should acknowledge that Fredg999 is 100% right about the GPS issue. In other words, I still definitely stand by the relationships I described between the arithmetic row of the tables and the frequency column of the tables, but anything else I said about the cells anywhere in the middle of those tables wasn't as carefully thought out, and I can see now that it's incorrect. Thanks for taking the time and care to lay it out so clearly, Fredg999. Well...! But now I'm just fascinated by the questions you've raised w/r/t this "geometric pitch sequence" idea. If it is to be defined, I think it has to be defined in terms of ''intervals'', not pitches (along the lines of how you showed that both <math>f</math> and <math>f_0</math> must be included in pitch-related formulae); when describing scales in cents, we often start with 0¢, but that won't work if we want to repeatedly multiply it by something, of course... so we'd need to describe a scale that, say, starts with a 100¢ interval, then goes by 90¢, then by 81¢, then 72.9¢, etc. etc. I went to my [[harmonotonic tuning]]s page because I thought I'd been pretty comprehensive about all step-monotonic tunings back when I wrote it, but as far as I can tell, scales like this are not documented yet at all. I hoped maybe this'd be equivalent to, like, some sort of a [[powharmonic series]] or [[logharmonic series]], but I really don't think so. But anyway, that stuff's really not fresh in my head, and also I really shouldn't be spending too much of my time on these sorts of experiments, but I encourage y'all to go wild. | |||
:::: First things first though... let's clean up what we've got out there already. Let me know if my plan basically makes sense to y'all too, or else I'll just keep an eye out for your changes to pages. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:32, 10 April 2023 (UTC) | |||
:::: Couple quick things. | |||
:::: Previously I had requested that Flora and CompactStar's system be deprioritized relative to the one from me, Paul, and Billy, e.g. that the redirects should be altered so that AFDO redirects to ODO and on those pages the bold term ODO given first before AFDO, etc.; I no longer think this. If Flora and CompactStar are the ones exploring these tunings first, they should do it however they like. So I retract that request and apologize for making it before I understood the whole situation well enough. | |||
:::: Also, I have some more info on GPS. Here's a table: | |||
:::: {| class="wikitable" | |||
|+ | |||
!step | |||
!AFS | |||
!APS / GFS | |||
!ALS / IFS | |||
!GPS | |||
|- | |||
!0 | |||
|440. | |||
|440. | |||
|440. | |||
|440. | |||
|- | |||
!1 | |||
|550. | |||
|493.88 | |||
|469.33 | |||
|451.4 | |||
|- | |||
!2 | |||
|660. | |||
|554.37 | |||
|502.86 | |||
|466.06 | |||
|- | |||
!3 | |||
|770. | |||
|622.25 | |||
|541.54 | |||
|485.06 | |||
|- | |||
!4 | |||
|880. | |||
|698.46 | |||
|586.67 | |||
|509.9 | |||
|- | |||
!5 | |||
|990. | |||
|783.99 | |||
|640. | |||
|542.76 | |||
|- | |||
!6 | |||
|1100. | |||
|880. | |||
|704. | |||
|586.81 | |||
|- | |||
!7 | |||
|1210. | |||
|987.77 | |||
|782.22 | |||
|646.94 | |||
|- | |||
!8 | |||
|1320. | |||
|1108.73 | |||
|880. | |||
|730.85 | |||
|- | |||
!9 | |||
|1430. | |||
|1244.51 | |||
|1005.71 | |||
|851.19 | |||
|- | |||
!10 | |||
|1540. | |||
|1396.91 | |||
|1173.33 | |||
|1029.86 | |||
|- | |||
!11 | |||
|1650. | |||
|1567.98 | |||
|1408. | |||
|1306.82 | |||
|- | |||
!12 | |||
|1760. | |||
|1760. | |||
|1760. | |||
|1760. | |||
|} | |||
:::: And here's a chart of the same data: | |||
:::: [[File:Comparison of some harmonotonic tunings, in particular a potentially new geometric pitch sequence tuning.png|frameless|1000px]] | |||
:::: So it appears that GPS are on the other side yet of things from ALS/IFS. Please check my work. I couldn't find a direct formula for GPS in terms of <math>k</math> like I could for the other three. If you want to see my spreadsheet with formulas, let me know. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:54, 11 April 2023 (UTC) | |||
:::: Whoops, I meant to add a "Hz" label to the vertical axis in that previous chart. | |||
:::: Argh, I couldn't help myself from spending more time on this GPS thing. It occurred to me that while if you want to divide up some interval into <math>n</math> steps according to AF, AP/GF, or AL/IF, there is only one way to do so. However, if you want to do this for a GPS, there is a continuum of infinitely many possible ways to this, because you have two parameters to control: the initial step size (such as in cents), and the geometric factor by which it increases (or decreases) in size. The graph I shared above shows the only possible 12-EFD4/12-AFD4, 12-ED4/12-GFD4, and 12-ELD4/12-IFD4. But the 12-GPD4 I showed there is only one example of a 12-GPD4; a GPD or GPS requires more parameterization than <math>n</math> and <math>p</math> (the divided interval). It also needs one or the other of the initial step size or the geometric factor. I'm not sure which one would be more intuitive to ask people to provide. Given one, the other will be set (i.e. assuming you also already have <math>n</math> and <math>p</math>. Here's several more examples of 12-GPD4's. The one I gave in the previous graph is GPS #2 here. While in my previous post I stated GPS was on the other side yet of ALS from AFS, it looks like it may be possible to replicate any of the other harmotonic tunings. That said, I don't think it's actually possible to exactly recreate an AFS with a GPS... I can't quite tell, but it seems like maybe that nearly-straight line is actually a subtle S-curve, which I can't quite make sense of why it would be an S... anyone else got a bright idea? | |||
:::: [[File:GPSs can replicate any of the other harmonotonic tunings .png|frameless|1000px]] | |||
:::: Maybe it was dumb not to show my work in the previous post. Here's the raw data for that chart: | |||
:::: {| class="wikitable" | |||
|+ | |||
! | |||
! colspan="4" rowspan="1" |GPS #1: initial step is 1 ¢, geometric factor is 1.90 | |||
! colspan="4" rowspan="1" |GPS #2: initial step is 44.27 ¢, geometric factor is 1.25 | |||
! colspan="4" rowspan="1" |GPS #3: initial step is 97.97 ¢, geometric factor is 1.12 | |||
! colspan="4" rowspan="1" |GPS #4: initial step is 189.24 ¢, geometric factor is 1.01 | |||
! colspan="4" rowspan="1" |GPS #5: initial step is 350 ¢, geometric factor is 0.89 | |||
! colspan="4" rowspan="1" |GPS #6: initial step is 666 ¢, geometric factor is 0.73 | |||
|- | |||
!step | |||
!cents | |||
!interval cents | |||
!frequency ratio | |||
!Hz | |||
!cents | |||
!interval cents | |||
!frequency ratio | |||
!Hz | |||
!cents | |||
!interval cents | |||
!frequency ratio | |||
!Hz | |||
!cents | |||
!interval cents | |||
!frequency ratio | |||
!Hz | |||
!cents | |||
!interval cents | |||
!frequency ratio | |||
!Hz | |||
!cents | |||
!interval cents | |||
!frequency ratio | |||
!Hz | |||
|- | |||
!0 | |||
|0 | |||
| | |||
| | |||
|440. | |||
|0 | |||
| | |||
| | |||
|440. | |||
|0 | |||
| | |||
| | |||
|440. | |||
|0 | |||
| | |||
| | |||
|440. | |||
|. | |||
| | |||
| | |||
|440. | |||
|0 | |||
| | |||
| | |||
|440. | |||
|- | |||
! | |||
| | |||
|1. | |||
|1. | |||
| | |||
| | |||
|44.27 | |||
|1.03 | |||
| | |||
| | |||
|97.97 | |||
|1.06 | |||
| | |||
| | |||
|189.24 | |||
|1.12 | |||
| | |||
| | |||
|350. | |||
|1.22 | |||
| | |||
| | |||
|666. | |||
|1.47 | |||
| | |||
|- | |||
!1 | |||
|1. | |||
| | |||
| | |||
|440.25 | |||
|44.27 | |||
| | |||
| | |||
|451.4 | |||
|97.97 | |||
| | |||
| | |||
|465.62 | |||
|189.24 | |||
| | |||
| | |||
|490.82 | |||
|350. | |||
| | |||
| | |||
|538.58 | |||
|666. | |||
| | |||
| | |||
|646.43 | |||
|- | |||
! | |||
| | |||
|1.9 | |||
|1. | |||
| | |||
| | |||
|55.34 | |||
|1.03 | |||
| | |||
| | |||
|109.97 | |||
|1.07 | |||
| | |||
| | |||
|191.13 | |||
|1.12 | |||
| | |||
| | |||
|311.63 | |||
|1.2 | |||
| | |||
| | |||
|485.33 | |||
|1.32 | |||
| | |||
|- | |||
!2 | |||
|2.9 | |||
| | |||
| | |||
|440.74 | |||
|99.62 | |||
| | |||
| | |||
|466.06 | |||
|207.94 | |||
| | |||
| | |||
|496.15 | |||
|380.37 | |||
| | |||
| | |||
|548.11 | |||
|661.63 | |||
| | |||
| | |||
|644.8 | |||
|1151.33 | |||
| | |||
| | |||
|855.6 | |||
|- | |||
! | |||
| | |||
|3.59 | |||
|1. | |||
| | |||
| | |||
|69.18 | |||
|1.04 | |||
| | |||
| | |||
|123.43 | |||
|1.07 | |||
| | |||
| | |||
|193.04 | |||
|1.12 | |||
| | |||
| | |||
|277.46 | |||
|1.17 | |||
| | |||
| | |||
|353.67 | |||
|1.23 | |||
| | |||
|- | |||
!3 | |||
|6.49 | |||
| | |||
| | |||
|441.65 | |||
|168.8 | |||
| | |||
| | |||
|485.06 | |||
|331.37 | |||
| | |||
| | |||
|532.82 | |||
|573.41 | |||
| | |||
| | |||
|612.77 | |||
|939.09 | |||
| | |||
| | |||
|756.89 | |||
|1505. | |||
| | |||
| | |||
|1049.53 | |||
|- | |||
! | |||
| | |||
|6.81 | |||
|1. | |||
| | |||
| | |||
|86.47 | |||
|1.05 | |||
| | |||
| | |||
|138.55 | |||
|1.08 | |||
| | |||
| | |||
|194.97 | |||
|1.12 | |||
| | |||
| | |||
|247.04 | |||
|1.15 | |||
| | |||
| | |||
|257.73 | |||
|1.16 | |||
| | |||
|- | |||
!4 | |||
|13.3 | |||
| | |||
| | |||
|443.39 | |||
|255.27 | |||
| | |||
| | |||
|509.9 | |||
|469.92 | |||
| | |||
| | |||
|577.21 | |||
|768.38 | |||
| | |||
| | |||
|685.81 | |||
|1186.13 | |||
| | |||
| | |||
|872.98 | |||
|1762.73 | |||
| | |||
| | |||
|1218. | |||
|- | |||
! | |||
| | |||
|12.91 | |||
|1.01 | |||
| | |||
| | |||
|108.09 | |||
|1.06 | |||
| | |||
| | |||
|155.52 | |||
|1.09 | |||
| | |||
| | |||
|196.92 | |||
|1.12 | |||
| | |||
| | |||
|219.96 | |||
|1.14 | |||
| | |||
| | |||
|187.81 | |||
|1.11 | |||
| | |||
|- | |||
!5 | |||
|26.2 | |||
| | |||
| | |||
|446.71 | |||
|363.36 | |||
| | |||
| | |||
|542.76 | |||
|625.44 | |||
| | |||
| | |||
|631.46 | |||
|965.3 | |||
| | |||
| | |||
|768.43 | |||
|1406.08 | |||
| | |||
| | |||
|991.24 | |||
|1950.54 | |||
| | |||
| | |||
|1357.57 | |||
|- | |||
! | |||
| | |||
|24.46 | |||
|1.01 | |||
| | |||
| | |||
|135.11 | |||
|1.08 | |||
| | |||
| | |||
|174.56 | |||
|1.11 | |||
| | |||
| | |||
|198.89 | |||
|1.12 | |||
| | |||
| | |||
|195.84 | |||
|1.12 | |||
| | |||
| | |||
|136.86 | |||
|1.08 | |||
| | |||
|- | |||
!6 | |||
|50.67 | |||
| | |||
| | |||
|453.07 | |||
|498.47 | |||
| | |||
| | |||
|586.81 | |||
|800. | |||
| | |||
| | |||
|698.46 | |||
|1164.19 | |||
| | |||
| | |||
|861.98 | |||
|1601.92 | |||
| | |||
| | |||
|1109.96 | |||
|2087.4 | |||
| | |||
| | |||
|1469.25 | |||
|- | |||
! | |||
| | |||
|46.37 | |||
|1.03 | |||
| | |||
| | |||
|168.89 | |||
|1.1 | |||
| | |||
| | |||
|195.94 | |||
|1.12 | |||
| | |||
| | |||
|200.88 | |||
|1.12 | |||
| | |||
| | |||
|174.37 | |||
|1.11 | |||
| | |||
| | |||
|99.74 | |||
|1.06 | |||
| | |||
|- | |||
!7 | |||
|97.03 | |||
| | |||
| | |||
|465.37 | |||
|667.37 | |||
| | |||
| | |||
|646.94 | |||
|995.94 | |||
| | |||
| | |||
|782.15 | |||
|1365.07 | |||
| | |||
| | |||
|968.04 | |||
|1776.29 | |||
| | |||
| | |||
|1227.58 | |||
|2187.14 | |||
| | |||
| | |||
|1556.38 | |||
|- | |||
! | |||
| | |||
|87.89 | |||
|1.05 | |||
| | |||
| | |||
|211.12 | |||
|1.13 | |||
| | |||
| | |||
|219.93 | |||
|1.14 | |||
| | |||
| | |||
|202.89 | |||
|1.12 | |||
| | |||
| | |||
|155.25 | |||
|1.09 | |||
| | |||
| | |||
|72.68 | |||
|1.04 | |||
| | |||
|- | |||
!8 | |||
|184.92 | |||
| | |||
| | |||
|489.6 | |||
|878.48 | |||
| | |||
| | |||
|730.85 | |||
|1215.87 | |||
| | |||
| | |||
|888.11 | |||
|1567.96 | |||
| | |||
| | |||
|1088.4 | |||
|1931.55 | |||
| | |||
| | |||
|1342.76 | |||
|2259.82 | |||
| | |||
| | |||
|1623.11 | |||
|- | |||
! | |||
| | |||
|166.58 | |||
|1.1 | |||
| | |||
| | |||
|263.89 | |||
|1.16 | |||
| | |||
| | |||
|246.87 | |||
|1.15 | |||
| | |||
| | |||
|204.92 | |||
|1.13 | |||
| | |||
| | |||
|138.23 | |||
|1.08 | |||
| | |||
| | |||
|52.96 | |||
|1.03 | |||
| | |||
|- | |||
!9 | |||
|351.5 | |||
| | |||
| | |||
|539.05 | |||
|1142.38 | |||
| | |||
| | |||
|851.19 | |||
|1462.74 | |||
| | |||
| | |||
|1024.22 | |||
|1772.87 | |||
| | |||
| | |||
|1225.16 | |||
|2069.78 | |||
| | |||
| | |||
|1454.36 | |||
|2312.78 | |||
| | |||
| | |||
|1673.53 | |||
|- | |||
! | |||
| | |||
|315.73 | |||
|1.2 | |||
| | |||
| | |||
|329.87 | |||
|1.21 | |||
| | |||
| | |||
|277.1 | |||
|1.17 | |||
| | |||
| | |||
|206.97 | |||
|1.13 | |||
| | |||
| | |||
|123.07 | |||
|1.07 | |||
| | |||
| | |||
|38.6 | |||
|1.02 | |||
| | |||
|- | |||
!10 | |||
|667.23 | |||
| | |||
| | |||
|646.9 | |||
|1472.25 | |||
| | |||
| | |||
|1029.86 | |||
|1739.84 | |||
| | |||
| | |||
|1202.01 | |||
|1979.84 | |||
| | |||
| | |||
|1380.74 | |||
|2192.85 | |||
| | |||
| | |||
|1561.52 | |||
|2351.38 | |||
| | |||
| | |||
|1711.26 | |||
|- | |||
! | |||
| | |||
|598.45 | |||
|1.41 | |||
| | |||
| | |||
|412.34 | |||
|1.27 | |||
| | |||
| | |||
|311.03 | |||
|1.2 | |||
| | |||
| | |||
|209.04 | |||
|1.13 | |||
| | |||
| | |||
|109.58 | |||
|1.07 | |||
| | |||
| | |||
|28.13 | |||
|1.02 | |||
| | |||
|- | |||
!11 | |||
|1265.68 | |||
| | |||
| | |||
|914.03 | |||
|1884.58 | |||
| | |||
| | |||
|1306.82 | |||
|2050.88 | |||
| | |||
| | |||
|1438.57 | |||
|2188.87 | |||
| | |||
| | |||
|1557.94 | |||
|2302.43 | |||
| | |||
| | |||
|1663.55 | |||
|2379.5 | |||
| | |||
| | |||
|1739.29 | |||
|- | |||
! | |||
| | |||
|1134.3 | |||
|1.93 | |||
| | |||
| | |||
|515.42 | |||
|1.35 | |||
| | |||
| | |||
|349.12 | |||
|1.22 | |||
| | |||
| | |||
|211.13 | |||
|1.13 | |||
| | |||
| | |||
|97.57 | |||
|1.06 | |||
| | |||
| | |||
|20.5 | |||
|1.01 | |||
| | |||
|- | |||
!12 | |||
|2400. | |||
| | |||
| | |||
|1760. | |||
|2400. | |||
| | |||
| | |||
|1760. | |||
|2400. | |||
| | |||
| | |||
|1760. | |||
|2400. | |||
| | |||
| | |||
|1760. | |||
|2400. | |||
| | |||
| | |||
|1760. | |||
|2400. | |||
| | |||
| | |||
|1760. | |||
|} | |||
:::: --[[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) | |||
:::::: Thanks for the detailed report! I understand and agree with all above. Yeah, I definitely steered this thread away from its original purpose... Perhaps we should continue this part of the conversation on Discord. I know you started some talk in the #wiki channel there, but perhaps we should start a thread? We could include the relevant materials from here to kick it off. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:21, 12 April 2023 (UTC) | |||
:::::: Good work so far on the edits. I see that all the ADO and IDO pages have been moved to AFDO and IFDO. I had expected that the ADO page would be preserved for Shaahin's original concept, and that you'd start anew with AFDO, but I think you handled things just fine in how you wrote the History section of the page. I just posted some follow-up messages on discussion pages for "Arithmetic MOS" and "Arithmetic interval chain", both of which still need some attention. | |||
:::::: Per Fredg999's suggestion, I have started a discussion for GPS on the Discord server: https://discord.com/channels/332357996569034752/1096541142167846962 --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 21:45, 14 April 2023 (UTC) | |||
:::::: Okay, I have a few updates from Dave, who asked for a link to this discussion and then emailed me about it: | |||
:::::: "The equivalences in this table are strictly by reflection across the main diagonal, not by merely being on the same anti-diagonal." I think that's an excellent generalization, and it explains why I got the color wrong for the GPS cell. | |||
:::::: He also noted that in my tables, I should have had "p → 0", not "p = 0" for geometric sequences. | |||
:::::: He also notes that "quadratic" would be a preferable way to refer to the case of p = 2, rather than RMS, as in "quadratic mean" and "quadratic progression", for better parallelism. | |||
:::::: He also notes that "reciprocal-arithmetic mean" would have been better than "inverse-arithmetic mean". In other words, he thinks RFDO would be clearer than IFDO. For example, if we introduce CompactStar's p=2 scale as a QFDO (quadratic frequency division of the octave), then the "inverse-quadratic mean" would be square-root (p = ½) while "reciprocal-quadratic" would be λx.1/x² (p = -2). But I hesitate even to bring this up, since there's been so much critique of this new naming system already. | |||
:::::: --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:13, 19 April 2023 (UTC) | |||
:::::: Dave sent me another email. He points out another issue with the tables I presented above. In the bottom right I have "(of low interest)", but in fact, an ILD is the same thing as an AFD (or as Dave would prefer, because of the "reciprocal" preference over "inverse" issue, an RLD = AFD). --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 17:37, 20 April 2023 (UTC) | |||