User talk:Aura: Difference between revisions

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: Did this for 159edo — the 16/11-span of 3/2 is 51. And it doesn't even work for the next EDO up or down from the last column of the tuning spectrum table of 11L 2s (135edo or 146edo, respectively) — the 3rd harmonic mapping is too unstable for EDO sizes that large in this region. Also, 51 is so many iterations of the generator that it goes well outside of the 11L 2s scale. Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 05:53, 11 April 2025 (UTC) Last Modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:52, 11 April 2025 (UTC)

:: Fascinating. I guess that means we need to look into chords with 94\159 fifths, which actually approximate 128/85 rather than 3/2. These chords clearly cannot be pure 5-limit either, but that's okay. Given that you mention 2.3.5.23 meantime, I'm now wondering if we can cobble together something for 11L 2s based on the ~128/85 archagall fifth. To start on this front, what are the modes of 11L 2s that contain ~128/85 in 159edo? --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 10:40, 11 April 2025 (UTC)

::: I need to do the modes for 11L 2s with respect to coverage of standard diatonic intervals anyway, but that's going to take a bit of time, so stay tuned. The 128/85 b fifth is going to stick out like a sore thumb in the midst of the much more accurate other intervals that 159edo has, though, so maybe it would be better to try to come up with a decent MODMOS derived from 11L&nbsp2s for 159edo? In the meantime, I figured out which equal temperaments this might apply to, since they get their fifth from 51 stacked and octave-reduced 16/11 generators, leading up to 159edo: 37b, 61, 98, 159 (have not yet tried to extend beyond 159edo to see how far you can get before the fifth mapping breaks again). Note that for 61edo, 51 stacked and octave-reduced 16/11 generators gives the same note as 10 stacked and octave-reduced 11/8 generators. I also included 37b to show the small endpoint of the series (and if you really want to go weird, include 24b); judging by the pattern, the next member of the series would be 257 (not yet sure of wart, if any). Graham Breed's temperament finder lists some more members of the series without warts, going up into the thousands, but doesn't give the temperament a name beyond [https://x31eq.com/pyscript/rt.html?ets=159_98&limit=2_3_11 159 & 98], and I don't see 61edo listed in the Nexus, Nexus clan, or Nexus family, although 159edo comes up several times (maybe one of those temperaments would be better, although they probably go from the fifth to the 11/8 rather than the other way around, and would not necessarily support 11L 2s). Its enormous complexity is presumably the reason it never got a name; I don't know the specifics for computing badness, but I am going to stick my neck out and guess that this temperament would have huge badness despite its high accuracy.

::: As for 2.3.*.23, the 23rd harmonic mapping is pretty unstable in the zone of 11L 2s — it gets better mapping stability in 17L 2s, although this region has its generators sharp enough relative to 23/16 that larger EDO sizes often flip to the next flatter approximation, so the generator for that needs to be something between 23/16 and 13/9.

::: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 15:21, 11 April 2025 (UTC)

@@ Line 395: / Line 395: @@
 : Did this for 159edo &mdash; the 16/11-span of 3/2 is 51.  And it doesn't even work for the next EDO up or down from the last column of the tuning spectrum table of 11L&nbsp;2s (135edo or 146edo, respectively) &mdash; the 3rd harmonic mapping is too unstable for EDO sizes that large in this region.  Also, 51 is so many iterations of the generator that it goes well outside of the 11L 2s scale.  Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 05:53, 11 April 2025 (UTC)  Last Modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:52, 11 April 2025 (UTC)
 :: Fascinating.  I guess that means we need to look into chords with 94\159 fifths, which actually approximate 128/85 rather than 3/2.  These chords clearly cannot be pure 5-limit either, but that's okay.  Given that you mention 2.3.5.23 meantime, I'm now wondering if we can cobble together something for 11L 2s based on the ~128/85 archagall fifth.  To start on this front, what are the modes of 11L 2s that contain ~128/85 in 159edo? --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 10:40, 11 April 2025 (UTC)
+::: I need to do the modes for 11L&nbsp;2s with respect to coverage of standard diatonic intervals anyway, but that's going to take a bit of time, so stay tuned.  The 128/85 b fifth is going to stick out like a sore thumb in the midst of the much more accurate other intervals that 159edo has, though, so maybe it would be better to try to come up with a decent MODMOS derived from 11L&nbsp2s for 159edo?  In the meantime, I figured out which equal temperaments this might apply to, since they get their fifth from 51 stacked and octave-reduced 16/11 generators, leading up to 159edo:  37b, 61, 98, 159 (have not yet tried to extend beyond 159edo to see how far you can get before the fifth mapping breaks again).  Note that for 61edo, 51 stacked and octave-reduced 16/11 generators gives the same note as 10 stacked and octave-reduced 11/8 generators.  I also included 37b to show the small endpoint of the series (and if you really want to go weird, include 24b); judging by the pattern, the next member of the series would be 257 (not yet sure of wart, if any).  Graham Breed's temperament finder lists some more members of the series without warts, going up into the thousands, but doesn't give the temperament a name beyond [https://x31eq.com/pyscript/rt.html?ets=159_98&limit=2_3_11 159 & 98], and I don't see 61edo listed in the Nexus, Nexus clan, or Nexus family, although 159edo comes up several times (maybe one of those temperaments would be better, although they probably go from the fifth to the 11/8 rather than the other way around, and would not necessarily support 11L&nbsp;2s).  Its enormous complexity is presumably the reason it never got a name; I don't know the specifics for computing badness, but I am going to stick my neck out and guess that this temperament would have huge badness despite its high accuracy.
+::: As for 2.3.*.23, the 23rd harmonic mapping is pretty unstable in the zone of 11L&nbsp;2s &mdash; it gets better mapping stability in 17L&nbsp;2s, although this region has its generators sharp enough relative to 23/16 that larger EDO sizes often flip to the next flatter approximation, so the generator for that needs to be something between 23/16 and 13/9.
+::: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 15:21, 11 April 2025 (UTC)