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:::: 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)
:::: 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)
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