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:::: 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)
:::: 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)
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