Pentagoth

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.

2.5.7: an introduction

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I've always been interested in the 2.5.7 subgroup and its extensions. Prime 3 is so central to how we tend to understand harmony that removing it is always interesting, and 5 and 7 are the next two simplest, thus the best alternatives to create a new harmonic system.

Here's a list of some of the best temperaments with their mappings of 5 and 7:

• ​6 & ​25: Didacus 2 5
• ​16 & ​21: Llywelyn 7 -1
• ​16 & ​25: Mabilic 3 -5
• ​21 & ​25: Sidewalk -7 -5
• ​21 & ​31: Miracle -7 -2
• ​15 & ​16: Rainy 5 -3
• ​15 & ​22: Porcupine -5 6

All of these are pretty well-established names, except for Sidewalk, which I came up with.

SS

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.
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).
Many apparent gaps in the temperament range are filled when 3/2 is not considered to be a target interval.

An asidewalk

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Sidewalk is likely the least well known of the basic 2.5.7 temperaments, given that I had a chance to coin the name for its comma, 823543/800000. It's one of the few temperaments with a generator in the neominor third region, half of 7/5 in this case.

• • 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.

Pentagoth temperament

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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, ​37, ​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 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.

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

Gens	-3	-2	-1	0	1	2	3
6						49/46
5					49/34		26/23
4				49/25	28/23 49/40	26/17 35/23 49/32	44/23
3				28/17 38/23	26/25 35/34	22/17 13/10 49/38	13/8
2			28/25 19/17 49/44	32/23 7/5	40/23 7/4 44/25	11/10 25/23 35/32	11/8 26/19
1		38/25	32/17 19/10 49/26	20/17 13/11 19/16	28/19 52/35 34/23 25/17	13/7 35/19	22/19
0		32/25 14/11	8/5 35/22	1/1	5/4 44/35	11/7 25/16
-1	19/11	14/13 38/35	34/25 19/14 23/17 35/26	32/19 22/13 17/10	20/19 52/49 17/16	25/19
-2	16/11 19/13	64/35 20/11 46/25	8/7 23/20 25/22	10/7 23/16	88/49 34/19 25/14
-3	16/13	20/13 17/11 76/49	68/35 25/13	17/14 23/19
-4	23/22	64/49 17/13 46/35	80/49 23/14	50/49
-5	23/13		68/49
-6		92/49

Tempering process

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All temperaments are 9 & 16 & 21 unless otherwise mentioned.

• Given a 2.5.7 temperament where 7/5 is split in half, 5/4 * sqrt(7/5) makes a flat fifth like 25/17. The supraminor third 49/40 is also close to 17/14. Equating these pairs tempers out 2023/2000, which I've decided to call the Pentagoth comma due to being the first and most obvious step.
  • 2023/2000 -> 2.5.7.17; CWE 388.049¢ 289.369¢, CE 390.556¢ 288.428¢.
• Then, 7/5 will be reasonably biased flat due to being (20/17)^2, pulling it closer to 32/23. Also, the same supraminor third is close to 28/23 as well. This tempers out the 2.5.7.23 comma 161/160.
  • 161/160 -> 2.5.7.17.23; CWE 387.534¢ 288.767¢, CE 390.950¢ 287.244¢.
• After that, things get messier. 13/11 can be easily equated to half of 7/5 by tempering out 847/845, and there are no better options than to do the same with 19/16, tempering out 1805/1792, even though this makes it very flat and the least accurate prime in the no-3 23-limit extension. It makes up for low accuracy with extremely low complexity, and makes 19/14 the octave complement of 25/17.
  • 847/845, 1805/1792 -> 2.5.7.13/11.17.19.23; CWE 386.133¢ 289.374¢, CE 390.581¢ 288.351¢.
• 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¢.
• 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. 5120/5103 (Aberschismic) tempering implies a gentle-region fifth; in fact, adding 5120/5103 to 2.5.7 Sidewalk results in 2.3.5.7[​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¢.
  • 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¢.
  • 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

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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 Vines in 2.3.5.7. Since it is also a plant, it fits perfectly into this new naming scheme.

Composition theory

TODO