Pentagoth

This portion of the page was written from the perspective of Ground. This is a new format being tested and the article is incomplete.

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: Gorgo 7 -1

16&25: Mavila 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 created surprisingly recently. Its generator is half of 7/5. Extending it is how I discovered Pentagoth, by far most important 2.5.7 extension to me, which I used as early as my 2020 song Wallowing in Madness in 16edo. The term was originally coined by UserMinusOne and me years ago to refer to what is now called Vengeance, a 2.5.17 Mavila-like temperament generated by the flat fifth 25/17. 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 Mavila, Gorgo, and any other 2.5.7 temperament that split 7/5 in half.

The base version is 2.5.7.13/11.17.19.23 25&21&16, CE generators 1200¢, 390.581¢, 288.351¢.

This can be easily extended to 2.5.7.11.13.17.19.23 25[-13]&21&16, CE generators 1200¢, 391.425¢, 288.482¢. 11 and 13 are the most complex and may not be tuned as well, such as in 25edo and thus 50edo, but this temperament is generally the same.

Here's my process. 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. 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. 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 19/16, tempering out 1805/1792, even though this makes it very flat. This makes 19/14 the octave complement of 25/17. 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.

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). I've had the thought to name Mint Sidewalk "Dandelion" and Slendric Pentagoth "Clover" after some of my favorite plants found near the sidewalk. Pentagoth Dandelion seems like it should be worse because of the very flat 3, but this allows 19/15 to be in tune.