Pentagoth - Revision history

Ground at 21:20, 31 July 2026

2026-07-31T21:20:27Z

← Older revision		Revision as of 21:20, 31 July 2026
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	}}		}}

	=== ~~"Gravila"~~ ===		=== Otomavila ===

	{{UserTag\|g_\|Ground\|7766ff\|This is a refinement of Mavila based on 21:24:25:26:27:31 hexad, which ~~may finally be the use case for the term "Gravila" with the Ground prefix. I need a title for this section of the article, so I guess that~~'s ~~what I'm gonna call it~~. I explain more at [[Ground's composition theory#Grave fifth chords]].}}		{{UserTag\|g_\|Ground\|7766ff\|This is a refinement of Mavila based on the otonal 21:24:25:26:27:31 hexad, which is very close to Erv Wilson's original Meta-Mavila. I explain more at [[Ground's composition theory#Grave fifth chords]].}}

	26/21 is mapped to -3 of the 31/21 fifths, tempering out 388962/387283. 21:25:31 and 21:26:31 are nearly retroversions of each other, so add 651/650. Finally, 26/21 is close to 16/13, so add 169/168. The result is the 2.21.25.13.31 16&23, which tunes 21, 13, and 31 all flat. CWE 677.518¢, CE 678.027¢.		26/21 is mapped to -3 of the 31/21 fifths, tempering out 388962/387283. 21:25:31 and 21:26:31 are nearly retroversions of each other, so add 651/650. Finally, 26/21 is close to 16/13, so add 169/168. The result is the 2.21.25.13.31 16&23, which tunes 21, 13, and 31 all flat. CWE 677.518¢, CE 678.027¢.
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	* (9/7 / (1/7) = 9) -10 - 11 = -21		* (9/7 / (1/7) = 9) -10 - 11 = -21

	This results in the GTO string: 1200 600 700 \| 2 >3 <9 25 7 13 31 \|\| 0 17 -21 10 -11 3 7, regression Generator: 676.466¢.		This results in the GTO string: 1200 600 700 \| 2 3 <9 <25 <7 <13 <31 \|\| 0 17 -21 10 -11 3 7, regression Generator: 676.466¢.

	== Composition theory ==		== Composition theory ==

Ground at 20:49, 31 July 2026

2026-07-31T20:49:29Z

← Older revision		Revision as of 20:49, 31 July 2026
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	}}		}}

	=== Gravila ===		=== "Gravila" ===

	{{UserTag\|g_\|Ground\|7766ff\|This is a refinement of Mavila based on 21:24:25:26:27:31 hexad, which may finally be the use case for the term "Gravila" with the Ground prefix. I explain more at [[Ground's composition theory#Grave fifth chords]].}}		{{UserTag\|g_\|Ground\|7766ff\|This is a refinement of Mavila based on 21:24:25:26:27:31 hexad, which may finally be the use case for the term "Gravila" with the Ground prefix. I need a title for this section of the article, so I guess that's what I'm gonna call it. I explain more at [[Ground's composition theory#Grave fifth chords]].}}

	26/21 is mapped to -3 of the 31/21 fifths, tempering out 388962/387283. 21:25:31 and 21:26:31 are nearly retroversions of each other, so add 651/650. Finally, 26/21 is close to 16/13, so add 169/168. The result is the 2.21.25.13.31 16&23, which tunes 21, 13, and 31 all flat. CWE 677.518¢, CE 678.027¢.		26/21 is mapped to -3 of the 31/21 fifths, tempering out 388962/387283. 21:25:31 and 21:26:31 are nearly retroversions of each other, so add 651/650. Finally, 26/21 is close to 16/13, so add 169/168. The result is the 2.21.25.13.31 16&23, which tunes 21, 13, and 31 all flat. CWE 677.518¢, CE 678.027¢.

Ground at 20:48, 31 July 2026

2026-07-31T20:48:12Z

@@ Line 215: / Line 215: @@
 * <nowiki>UNNAMED: 343/340 = 7/5 / (8/7 * 17/14) = 20/17 / (400/343)</nowiki>
 }}
 == Composition theory ==

Ground at 05:58, 31 July 2026

2026-07-31T05:58:23Z

← Older revision		Revision as of 05:58, 31 July 2026
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	== Pentagoth temperament ==		== Pentagoth temperament ==

	~~{{UserTag\|g_\|Ground\|7766ff\|The current rank~~-3 version of Pentagoth began as an extension of Sidewalk, but I realized it could be applied to other 2.5.7 temperaments I've used in the past, showing up as early as my 2020 song Wallowing in Madness in 16edo. It works very well in several edos that I have a unique affinity for, including 25, {{e\|37}}, {{e\|46}}, and 67.		<div style="float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;">

	~~The term was originally coined by UserMinusOne and me to refer to what is now called '''Vengeance''', a 2.5.17 Mavila~~-~~like temperament generated by the flat fifth 25/17. I described it in a [https~~://www.tumblr.com/groundfault/705198584894816256/the-best-mavila-probably 2022 Tumblr post]. We all independently discovered the temperament, but I agreed to let Vengeance stay even though ours came first because I had a feeling that Pentagoth was a broader category. My decision paid off. Pentagoth didn't just apply to Sidewalk, but to Mabilic (2.5.7 Mavila), Llywelyn (or Shoe or Gorgo or Laconic), and any other 2.5.7 temperament that split 7/5 in half. I've been referring to "Pentagoth triads" ~~that rely on this exact type of tempering even more often than I refer to a particular temperament.~~
	}}

	{\| class="wikitable"		{\| class="wikitable"
	\|+2.5.7.11.13.17.19.23[9 & 16 & 21] Pentagoth Lattice (49-odd-limit) generated by 13/11-up and 5/4-right		\|+2.27.5.7.11.13.17.19.23[9 & 16 & 21] Pentagoth Lattice (49-odd-limit) generated by 13/11-up and 5/4-right
	!Gens		!Gens
	!-3		!-3
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	\|		\|
	\|49/34		\|49/34
	\|		\|49/27
	\|26/23		\|26/23
	\|-		\|-
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	\|		\|
	\|49/25		\|49/25
	\|28/23 49/40		\|28/23<br>49/40
	\|26/17 35/23 49/32		\|26/17<br>35/23<br>49/32
	\|44/23		\|44/23<br>52/27
	\|-		\|-
	!3		!3
Line 79:		Line 75:
	\|		\|
	\|		\|
	\|28/17 38/23		\|28/17<br>38/23
	\|26/25 35/34		\|28/27<br>26/25<br>35/34
	\|22/17 13/10 49/38		\|22/17<br>13/10<br>35/27<br>49/38
	\|13/8		\|44/27<br>13/8
	\|-		\|-
	!2		!2
	\|		\|
	\|		\|
	\|28/25 19/17 49/44		\|28/25<br>19/17<br>49/44
	\|32/23 7/5		\|32/23<br>7/5<br>38/27
	\|40/23 7/4 44/25		\|40/23<br>7/4<br>44/25
	\|11/10 25/23 35/32		\|11/10<br>25/23<br>35/32
	\|11/8 26/19		\|11/8<br>26/19
	\|-		\|-
	!1		!1
	\|		\|
	\|38/25		\|38/25
	\|32/17 19/10 49/26		\|32/17<br>19/10<br>49/26
	\|20/17 13/11 19/16		\|32/27<br>20/17<br>13/11<br>19/16<br>27/23
	\|28/19 52/35 34/23 25/17		\|40/27<br>28/19<br>52/35<br>34/23<br>25/17
	\|13/7 35/19		\|13/7<br>50/27<br>35/19
	\|22/19		\|22/19
	\|-		\|-
	!0		!0
	\|		\|
	\|32/25 14/11		\|32/25<br>14/11
	\|8/5 35/22		\|8/5<br>27/17<br>35/22
	\|1/1		\| class="thl" \|1/1
	\|5/4 44/35		\|5/4<br>44/35<br>34/27
	\|11/7 25/16		\|11/7<br>25/16
	\|		\|
	\|-		\|-
	!-1		!-1
	\|19/11		\|19/11
	\|14/13 38/35		\|14/13<br>38/35<br>27/25
	\|34/25 19/14 23/17 35/26		\|34/25<br>19/14<br>23/17<br>27/20<br>35/26
	\|32/19 22/13 17/10		\|32/19<br>22/13<br>17/10<br>46/27<br>27/16
	\|20/19 52/49 17/16		\|20/19<br>52/49<br>17/16
	\|25/19		\|25/19
	\|		\|
	\|-		\|-
	!-2		!-2
	\|16/11 19/13		\|16/11<br>19/13
	\|64/35 20/11 46/25		\|64/35<br>20/11<br>46/25
	\|8/7 23/20 25/22		\|8/7<br>23/20<br>25/22
	\|10/7 23/16		\|10/7<br>23/16<br>27/19
	\|88/49 34/19 25/14		\|88/49<br>34/19<br>25/14
	\|		\|
	\|		\|
	\|-		\|-
	!-3		!-3
	\|16/13		\|16/13<br>27/22
	\|20/13 17/11 76/49		\|20/13<br>17/11<br>76/49<br>54/35
	\|68/35 25/13		\|68/35<br>25/13<br>27/14
	\|17/14 23/19		\|17/14<br>23/19
	\|		\|
	\|		\|
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	\|-		\|-
	!-4		!-4
	\|23/22		\|23/22<br>27/26
	\|64/49 17/13 46/35		\|64/49<br>17/13<br>46/35
	\|80/49 23/14		\|80/49<br>23/14
	\|50/49		\|50/49
	\|		\|
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	!-5		!-5
	\|23/13		\|23/13
	\|		\|54/49
	\|68/49		\|68/49
	\|		\|
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	\|		\|
	\|}		\|}
			* Vertical map: 0 -1 0 2 2 3 -1 1 -2
			* Horizontal map: 0 0 1 1 3 3 1 0 0
			</div>
			{{UserTag\|g_\|Ground\|7766ff\|The current rank-3 version of Pentagoth began as an extension of Sidewalk, but I realized it could be applied to other 2.5.7 temperaments I've used in the past, showing up as early as my 2020 song Wallowing in Madness in 16edo. It works very well in several edos that I have a unique affinity for, including 25, {{e\|37}}, {{e\|46}}, and 67.

			The term was originally coined by UserMinusOne and me to refer to what is now called '''Vengeance''', a 2.5.17 Mavila-like temperament generated by the flat fifth 25/17. I described it in a [https://www.tumblr.com/groundfault/705198584894816256/the-best-mavila-probably 2022 Tumblr post]. We all independently discovered the temperament, but I agreed to let Vengeance stay even though ours came first because I had a feeling that Pentagoth was a broader category. My decision paid off. Pentagoth didn't just apply to Sidewalk, but to Mabilic (2.5.7 Mavila), Llywelyn (or Shoe or Gorgo or Laconic), and any other 2.5.7 temperament that split 7/5 in half. I've been referring to "Pentagoth triads" that rely on this exact type of tempering even more often than I refer to a particular temperament.
			}}

	== Tempering process ==		== Tempering process ==

Ground at 05:38, 31 July 2026

2026-07-31T05:38:54Z

← Older revision		Revision as of 05:38, 31 July 2026
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	* The temperament ended up being rank-4 in the no-3 23-limit, so I looked for a good mapping for 11 and 13 with just the two important generators, and found one. I later learned that this equates 17/13 to 64/49, tempering out 833/832, which is a good choice. 11 and 13 are the most complex and may not be tuned as well, such as in 25edo and thus 50edo, but this temperament generally works.		* The temperament ended up being rank-4 in the no-3 23-limit, so I looked for a good mapping for 11 and 13 with just the two important generators, and found one. I later learned that this equates 17/13 to 64/49, tempering out 833/832, which is a good choice. 11 and 13 are the most complex and may not be tuned as well, such as in 25edo and thus 50edo, but this temperament generally works.
	** 833/832 -> 2.5.7.11.13.17.19.23; CWE 389.217¢ 289.608¢, CE 391.425¢ 288.482¢.		** 833/832 -> 2.5.7.11.13.17.19.23; CWE 389.217¢ 289.608¢, CE 391.425¢ 288.482¢.
	* ~~So what do you do~~ to ~~add~~ 3 ~~and~~ make it full 23-limit? It makes sense to either temper out 36/35 (Mint) for the low-complexity flat fifth or take advantage of the tuning range of 7 and temper out 1029/1024 (Slendric). Mint Pentagoth seems like it should be worse because of the very flat 3, but this allows 19/15 to be in tune. {{adv\|5120/5103 ([[Aberschismic]]) tempering implies a [[gentle region\|gentle-region]] fifth; in fact, adding 5120/5103 to 2.5.7 Sidewalk results in 2.3.5.7[{{e\|29}} & 46], Leapday, of whose full 23-limit Pentagoth extension 46edo is the only reasonable patent-val tuning.}} It's also possible to add an accurate alternate 9 by tempering out 126/125 (Starling).		* 27:32:40 is another triad with a grave fifth and 5/4. It can be equated to the 17:20:25 in the base version of the temperament.
			** 136/135 -> 2.27.5.7.11.13.17.19.23; CWE 389.431¢ 289.833¢, CE 391.381¢ 288.539¢.
			* The full comma list is:
			** 2023/2000
			** 161/160
			** 847/845
			** 1805/1792
			** 833/832
			** 136/135
			* Alternatively, prime 3 itself can be added to make Pentagoth full 23-limit, although it was made for a no-3 context. It makes sense to either temper out 36/35 (Mint) for the low-complexity flat fifth or take advantage of the tuning range of 7 and temper out 1029/1024 (Slendric). Mint Pentagoth seems like it should be worse because of the very flat 3, but this allows 19/15 to be in tune. {{adv\|5120/5103 ([[Aberschismic]]) tempering implies a [[gentle region\|gentle-region]] fifth; in fact, adding 5120/5103 to 2.5.7 Sidewalk results in 2.3.5.7[{{e\|29}} & 46], Leapday, of whose full 23-limit Pentagoth extension 46edo is the only reasonable patent-val tuning.}} It's also possible to add an accurate alternate 9 by tempering out 126/125 (Starling).
	** 36/35 -> 23-limit; CWE 682.871¢ 390.965¢, CE 681.014¢ 392.071¢.		** 36/35 -> 23-limit; CWE 682.871¢ 390.965¢, CE 681.014¢ 392.071¢.
	** 1029/1024 -> 23-limit[16 & 21 & 30]; CWE 677.955¢ 233.348¢, CE 679.315¢ 233.108¢.		** 1029/1024 -> 23-limit[16 & 21 & 30]; CWE 677.955¢ 233.348¢, CE 679.315¢ 233.108¢.

Inthar: /* Composition theory */

2026-07-07T03:04:06Z

Composition theory

← Older revision		Revision as of 03:04, 7 July 2026
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	{{Navbox regtemp}}		{{Navbox regtemp}}
			{{Cat\|Temperaments}}

Ground at 19:26, 5 July 2026

2026-07-05T19:26:44Z

← Older revision		Revision as of 19:26, 5 July 2026
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	* {{e\|15}} & {{e\|16}}: Rainy 5 -3		* {{e\|15}} & {{e\|16}}: Rainy 5 -3
	* {{e\|15}} & {{e\|22}}: [[Porcupine]] -5 6		* {{e\|15}} & {{e\|22}}: [[Porcupine]] -5 6
			* {{e\|16}} & {{e\|27}}: Kumonga restriction -9 -1
	All of these are pretty well-established names, except for Sidewalk, which I came up with.		All of these are pretty well-established names, except for Sidewalk, which I came up with.
	}}		}}
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	{{UserTag\|g_\|Ground\|7766ff\|Vengeance is a very efficient temperament. 34edo's tuning is very good for optimising 17:20:25, but adding 7 becomes difficult because 34edo almost perfectly straddles it. On one side is 25edo, mapping 7 to -5 generators. While the original discussions of this temperament were going on, I gravitated towards 25edo even though it wasn't optimal, which I now know is because of Pentagoth. On the other side is 43edo, mapping 7 to -14 generators. It is known for Amavil and Cohemimabila (the temperament splitting its grave fifth in half), both of which I independently discovered while researching 43edo and this subgroup a while before learning of their names.		{{UserTag\|g_\|Ground\|7766ff\|Vengeance is a very efficient temperament. 34edo's tuning is very good for optimising 17:20:25, but adding 7 becomes difficult because 34edo almost perfectly straddles it. On one side is 25edo, mapping 7 to -5 generators. While the original discussions of this temperament were going on, I gravitated towards 25edo even though it wasn't optimal, which I now know is because of Pentagoth. On the other side is 43edo, mapping 7 to -14 generators. It is known for Amavil and Cohemimabila (the temperament splitting its grave fifth in half), both of which I independently discovered while researching 43edo and this subgroup a while before learning of their names.

	In adding 7 to Vengeance to get 17/14 and such, the best temperament might be Cohemimabila, which is represented well by edos 43, 68, and 25, but not 34. 2.5.7.17 Cohemimabila is Didacus plus 4913/4900, which is like an analogue to the Pentagoth comma. Combining 4913/4900 with Sidewalk results in Pentagoth with 343/340 or Pentagoth Sidewalk.		In adding 7 to Vengeance to get 17/14 and such, the best temperament might be Cohemimabila, which is represented well by edos 43, 68, and 25, but not 34. 2.5.7.17 Cohemimabila is Didacus plus 4913/4900, which is like an analogue to the Pentagoth comma. Combining 4913/4900 with Sidewalk results in Pentagoth with 343/340 or Pentagoth Sidewalk. Pentagoth Didacus splits the whole tone generator in half.

	* <nowiki>Vengeance comma: 78608/78125 = 16/5 / (25/17)^3</nowiki>		* <nowiki>Vengeance comma: 78608/78125 = 16/5 / (25/17)^3</nowiki>

Ground at 19:12, 5 July 2026

2026-07-05T19:12:49Z

← Older revision		Revision as of 19:12, 5 July 2026
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	* {{e\|15}} & {{e\|22}}: [[Porcupine]] -5 6		* {{e\|15}} & {{e\|22}}: [[Porcupine]] -5 6
	All of these are pretty well-established names, except for Sidewalk, which I came up with.		All of these are pretty well-established names, except for Sidewalk, which I came up with.
			}}

			{{UserTag\|SS\|Vector\|cebaff\|2.5.7 is useful to explore in the context of temperaments, due to the fact that often times, 3/2 is sort of shoehorned into temperaments that don't tune it accurately. The most egregious example is Mabilic, which at its best tunes 5/4 and 7/4 to within 5 cents of just, but when extended to Mavila in the full 7-limit uses a much less accurate 3/2 with 29 cents of error. <br> 2.5.7 also generally takes the 6-form, with didacus serving a similar role for it as meantone does for 2.3.5 (in fact, didacus can be seen as a much more accurate restriction of septimal meantone, via the logic above). <br> Many apparent gaps in the temperament range are filled when 3/2 is not considered to be a target interval.
	}}		}}

	{{UserTag\|SS\|Vector\|cebaff\|2.5.7 is useful to explore in the context of temperaments, due to the fact that often times, 3/2 is sort of shoehorned into temperaments that don't tune it accurately. The most egregious example is Mabilic, which at its best tunes 5/4 and 7/4 to within 5 cents of just, but when extended to Mavila in the full 7-limit uses a much less accurate 3/2 with 29 cents of error. <br> 2.5.7 also generally takes the 6-form, with didacus serving a similar role for it as meantone does for 2.3.5 (in fact, didacus can be seen as a much more accurate restriction of septimal meantone, via the logic above). <br> Many apparent gaps in the temperament range are filled when 3/2 is not considered to be a target interval.}}

	== An asidewalk ==		== An asidewalk ==
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	** 823543/800000 -> 2.5.7[21 & 25]; CWE 287.441¢, CE 287.185¢.		** 823543/800000 -> 2.5.7[21 & 25]; CWE 287.441¢, CE 287.185¢.

	The name Sidewalk comes from its edo join of 21 & 25, the ages to drink alcohol and rent a car in the USA. Instead of drinking and driving, you should use the sidewalk. I originally called it Gridacus for "Ground Didacus" as a half-joke, which became Gridwalker after a character, which became Sidewalk again. Sidewalk is also part of the ground, which is me. It reminds me of urbanism and the excessive number of walks I go on. I initially found it while looking through possible generators of 2.<5.7 or 2.>5.7 (see [[straddle primes]]) in [[67edo]] and being surprised by its low complexity, then again later while creating a 4L1M2s scale in the same tuning as a replacement for Gorgo.}}		The name Sidewalk comes from its edo join of 21 & 25, the ages to drink alcohol and rent a car in the USA. Instead of drinking and driving, you should use the sidewalk. I originally called it Gridacus for "Ground Didacus" as a half-joke, which became Gridwalker after a character, which became Sidewalk again. Sidewalk is also part of the ground, which is me. It reminds me of urbanism and the excessive number of walks I go on. I initially found it while looking through possible generators of 2.<5.7 or 2.>5.7 (see [[straddle primes]]) in [[67edo]] and being surprised by its low complexity, then again later while creating a 4L1M2s scale in the same tuning as a replacement for Gorgo.
			}}

	== Pentagoth temperament ==		== Pentagoth temperament ==
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	{{UserTag\|g_\|Ground\|7766ff\|The current rank-3 version of Pentagoth began as an extension of Sidewalk, but I realized it could be applied to other 2.5.7 temperaments I've used in the past, showing up as early as my 2020 song Wallowing in Madness in 16edo. It works very well in several edos that I have a unique affinity for, including 25, {{e\|37}}, {{e\|46}}, and 67.		{{UserTag\|g_\|Ground\|7766ff\|The current rank-3 version of Pentagoth began as an extension of Sidewalk, but I realized it could be applied to other 2.5.7 temperaments I've used in the past, showing up as early as my 2020 song Wallowing in Madness in 16edo. It works very well in several edos that I have a unique affinity for, including 25, {{e\|37}}, {{e\|46}}, and 67.

	The term was originally coined by UserMinusOne and me to refer to what is now called '''Vengeance''', a 2.5.17 Mavila-like temperament generated by the flat fifth 25/17. I described it in a [https://www.tumblr.com/groundfault/705198584894816256/the-best-mavila-probably 2022 Tumblr post]. We all independently discovered the temperament, but I agreed to let Vengeance stay even though ours came first because I had a feeling that Pentagoth was a broader category. My decision paid off. Pentagoth didn't just apply to Sidewalk, but to Mabilic (2.5.7 Mavila), Llywelyn (or Shoe or Gorgo or Laconic), and any other 2.5.7 temperament that split 7/5 in half.}}		The term was originally coined by UserMinusOne and me to refer to what is now called '''Vengeance''', a 2.5.17 Mavila-like temperament generated by the flat fifth 25/17. I described it in a [https://www.tumblr.com/groundfault/705198584894816256/the-best-mavila-probably 2022 Tumblr post]. We all independently discovered the temperament, but I agreed to let Vengeance stay even though ours came first because I had a feeling that Pentagoth was a broader category. My decision paid off. Pentagoth didn't just apply to Sidewalk, but to Mabilic (2.5.7 Mavila), Llywelyn (or Shoe or Gorgo or Laconic), and any other 2.5.7 temperament that split 7/5 in half. I've been referring to "Pentagoth triads" that rely on this exact type of tempering even more often than I refer to a particular temperament.
			}}

	{\| class="wikitable"		{\| class="wikitable"
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	** 1029/1024 -> 23-limit[16 & 21 & 30]; CWE 677.955¢ 233.348¢, CE 679.315¢ 233.108¢.		** 1029/1024 -> 23-limit[16 & 21 & 30]; CWE 677.955¢ 233.348¢, CE 679.315¢ 233.108¢.
	** 5120/5103 -> 23-limit[46 & 53[-17, -23] & 58]; CWE 703.389¢ 389.431¢, CE 703.820¢ 391.381¢.		** 5120/5103 -> 23-limit[46 & 53[-17, -23] & 58]; CWE 703.389¢ 389.431¢, CE 703.820¢ 391.381¢.
	** 126/125 -> 2.9.5.7.11.13.17.19.23[9 & 21 & 37]; CWE 389.680¢ 100.487¢, CE 391.298¢ 102.695¢.}}		** 126/125 -> 2.9.5.7.11.13.17.19.23[9 & 21 & 37]; CWE 389.680¢ 100.487¢, CE 391.298¢ 102.695¢.
			}}

	== Sidewalk again ==		== Sidewalk again ==
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	{{UserTag\|g_\|Ground\|7766ff\|As much as superfluous temperament names bother me, I'll propose these extensions to Sidewalk just to avoid resulting in 3-word names when combined with Pentagoth: Mint Sidewalk "Dandelion" and Slendric Pentagoth "Clover", after some of my favorite plants found near the sidewalk. The naming occurred immediately after a period in which I found dozens of clovers with at least 4 leaves. Coincidence? Yeah.		{{UserTag\|g_\|Ground\|7766ff\|As much as superfluous temperament names bother me, I'll propose these extensions to Sidewalk just to avoid resulting in 3-word names when combined with Pentagoth: Mint Sidewalk "Dandelion" and Slendric Pentagoth "Clover", after some of my favorite plants found near the sidewalk. The naming occurred immediately after a period in which I found dozens of clovers with at least 4 leaves. Coincidence? Yeah.

	Starling Sidewalk is unique in the 2.9.5.7 subgroup and is a weak restriction of the half-octave temperament [https://en.xen.wiki/w/Starling_temperaments#Vines Vines] in 2.3.5.7. Since it is also a plant, it fits perfectly into this new naming scheme.}}		Starling Sidewalk is unique in the 2.9.5.7 subgroup and is a weak restriction of the half-octave temperament [https://en.xen.wiki/w/Starling_temperaments#Vines Vines] in 2.3.5.7. Since it is also a plant, it fits perfectly into this new naming scheme.
			}}

			== More commas in 2.5.7.17 ==

			{{UserTag\|g_\|Ground\|7766ff\|Vengeance is a very efficient temperament. 34edo's tuning is very good for optimising 17:20:25, but adding 7 becomes difficult because 34edo almost perfectly straddles it. On one side is 25edo, mapping 7 to -5 generators. While the original discussions of this temperament were going on, I gravitated towards 25edo even though it wasn't optimal, which I now know is because of Pentagoth. On the other side is 43edo, mapping 7 to -14 generators. It is known for Amavil and Cohemimabila (the temperament splitting its grave fifth in half), both of which I independently discovered while researching 43edo and this subgroup a while before learning of their names.

			In adding 7 to Vengeance to get 17/14 and such, the best temperament might be Cohemimabila, which is represented well by edos 43, 68, and 25, but not 34. 2.5.7.17 Cohemimabila is Didacus plus 4913/4900, which is like an analogue to the Pentagoth comma. Combining 4913/4900 with Sidewalk results in Pentagoth with 343/340 or Pentagoth Sidewalk.

			* <nowiki>Vengeance comma: 78608/78125 = 16/5 / (25/17)^3</nowiki>
			* <nowiki>Sidewalk comma: 823543/800000 = 7/5 / (400/343)^2</nowiki>
			* <nowiki>Pentagoth comma: 2023/2000 = 7/5 / (20/17)^2</nowiki>
			* <nowiki>UNNAMED: 4913/4900 = (17/10)^2 * 17/14 / (7/2)</nowiki>
			* <nowiki>UNNAMED: 343/340 = 7/5 / (8/7 * 17/14) = 20/17 / (400/343)</nowiki>
			}}

	== Composition theory ==		== Composition theory ==

	~~TODO~~		{{UserTag\|g_\|Ground\|7766ff\|Pentagoth temperament represents a generalization of the '''pentagoth triad''' which includes a pental major third and neominor third. This triad is the center of Pentagoth composition.

			This section is incomplete.
			}}

	{{Navbox regtemp}}		{{Navbox regtemp}}

Inthar at 17:52, 16 June 2026

2026-06-16T17:52:52Z

← Older revision		Revision as of 17:52, 16 June 2026
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	'''Pentagoth''' is the rank-3 2.5.7.17(.11.13.19.23)[9 & 16 & 21] temperament and its variants, which can be used to extend existing 7-limit temperaments. It tempers out 2023/2000, the '''Pentagoth comma'''. The rank-3 temperament is generated by a sharp 5/4 (~390¢) and half of 7/5 (~287¢ * 2 = ~574¢), interpreted as a sharp 20/17, a roughly in-tune 13/11, and a flat 19/16. 7/4 is found at two 20/17s stacked with a 5/4, and 17/16 itself is the ~103¢ semitone between the two generators. Stacking the two generators results in a flat fifth of ~677¢, used in the 17:20:25 triad.		'''Pentagoth''' is the rank-3 2.5.7.17(.11.13.19.23)[9 & 16 & 21] temperament and its variants, which can be used to extend existing 7-limit temperaments. It tempers out 2023/2000, the '''Pentagoth comma'''. The rank-3 temperament is generated by a sharp 5/4 (~390¢) and half of 7/5 (~287¢ * 2 = ~574¢), interpreted as a sharp 20/17, a roughly in-tune 13/11, and a flat 19/16. 7/4 is found at two 20/17s stacked with a 5/4, and 17/16 itself is the ~103¢ semitone between the two generators. Stacking the two generators results in a flat fifth of ~677¢, used in the 17:20:25 triad.

	'''Sidewalk''' is a rank-2 2.5.7.17... temperament that supports Pentagoth; it is generated by a neominor third, two of which stack to a flat 7/5.		'''Sidewalk''' is a rank-2 2.5.7.17...[21 & 25] temperament that supports Pentagoth; it is generated by a neominor third, two of which stack to a flat 7/5.

	== 2.5.7: an introduction ==		== 2.5.7: an introduction ==

Inthar at 17:52, 16 June 2026

2026-06-16T17:52:15Z

← Older revision		Revision as of 17:52, 16 June 2026
Line 1:		Line 1:
	'''Pentagoth''' is the rank-3 2.5.7.17(.11.13.19.23)[9 & 16 & 21] temperament and its variants, which can be used to extend existing 7-limit temperaments. It tempers out 2023/2000, the '''Pentagoth comma'''. The rank-3 temperament is generated by a sharp 5/4 (~390¢) and half of 7/5 (~287¢ * 2 = ~574¢), interpreted as a sharp 20/17, a roughly in-tune 13/11, and a flat 19/16. 7/4 is found at two 20/17s stacked with a 5/4, and 17/16 itself is the ~103¢ semitone between the two generators. Stacking the two generators results in a flat fifth of ~677¢, used in the 17:20:25 triad.		'''Pentagoth''' is the rank-3 2.5.7.17(.11.13.19.23)[9 & 16 & 21] temperament and its variants, which can be used to extend existing 7-limit temperaments. It tempers out 2023/2000, the '''Pentagoth comma'''. The rank-3 temperament is generated by a sharp 5/4 (~390¢) and half of 7/5 (~287¢ * 2 = ~574¢), interpreted as a sharp 20/17, a roughly in-tune 13/11, and a flat 19/16. 7/4 is found at two 20/17s stacked with a 5/4, and 17/16 itself is the ~103¢ semitone between the two generators. Stacking the two generators results in a flat fifth of ~677¢, used in the 17:20:25 triad.

	'''Sidewalk''' is a rank-2 2.5.7.17... temperament that supports Pentagoth; it is generated by a neominor third, two of which ~~stacks~~ to a flat 7/5.		'''Sidewalk''' is a rank-2 2.5.7.17... temperament that supports Pentagoth; it is generated by a neominor third, two of which stack to a flat 7/5.

	== 2.5.7: an introduction ==		== 2.5.7: an introduction ==