198edo
Posted: Sun Nov 08, 2020 4:00 am
Hello. I'm new here but you may have known about me. Straight to the topic, 198edo has become my latest mega-fav-edo and I'd like to work out a solution in sagittal notation.
Here's a brief intro to 198edo.
I'll call a step of 198edo, ~6 cents, a quarter-comma, for the size of 4 steps is extremely close to the mean of 81/80 and 64/63. Note also 198b val supports meantone and is close to the quarter-comma tuning (slightly sharp than 31edo). 198bbdddd val on the other hand supports superpyth and is also close to the quarter-comma tuning (literally identical to 22edo for 198 = 22*9). It's a sharp-20 system; here are the steps:
1\198: 176/175, 385/384, 540/539, 896/891, 196/195, 325/324, 351/350, 364/363
2\198: schisma (way too large), 121/120, 243/242, 144/143, 169/168
3\198: 99/98, 100/99
4\198: 81/80 , 64/63, 66/65, 78/77
5\198: 55/54 , 56/55, 65/64
6\198: Pyth comma (too large), 49/48, 50/49
7\198: 45/44 , 40/39
8\198: (81/80)^2 , 36/35
9\198: 33/32 , 1053/1024
10\198: half-apotome, 28/27
11\198: 729/704 , 26/25, 27/26
12\198: 25/24
etc.
Several problems can already be seen through this.
Here's a brief intro to 198edo.
- It's a nice 13-limit temperament yet to receive much attention.
- It's distinctly consistent in the 15-odd-limit.
- Its 13-limit TE error is lower than any previous edos.
- Its 13-limit TE simple badness is only outperformed by two previous edos (72 and 130).
- It's contorted in the 7-limit with the same mapping as 99edo. An interesting property therefore follows that all odd-ordered undecimal and tridecimal intervals are odd steps, whereas all 7-limit intervals are even steps.
I'll call a step of 198edo, ~6 cents, a quarter-comma, for the size of 4 steps is extremely close to the mean of 81/80 and 64/63. Note also 198b val supports meantone and is close to the quarter-comma tuning (slightly sharp than 31edo). 198bbdddd val on the other hand supports superpyth and is also close to the quarter-comma tuning (literally identical to 22edo for 198 = 22*9). It's a sharp-20 system; here are the steps:
1\198: 176/175, 385/384, 540/539, 896/891, 196/195, 325/324, 351/350, 364/363
2\198: schisma (way too large), 121/120, 243/242, 144/143, 169/168
3\198: 99/98, 100/99
4\198: 81/80 , 64/63, 66/65, 78/77
5\198: 55/54 , 56/55, 65/64
6\198: Pyth comma (too large), 49/48, 50/49
7\198: 45/44 , 40/39
8\198: (81/80)^2 , 36/35
9\198: 33/32 , 1053/1024
10\198: half-apotome, 28/27
11\198: 729/704 , 26/25, 27/26
12\198: 25/24
etc.
Several problems can already be seen through this.
- As mentioned above, both 7/5k and 13/11k are tempered out. Very unusual for edos of this size. Consequently 81/80 and 64/63 share a single degree , same for 33/32 and 1053/1024 .
- 4096/4095 is -1 step, so we see 36/35 is 8 steps whereas 1053/1024 is 9 steps. A possible solution is to assign to 8\ and to 9\ – Unfortunately, (81/80)^2 is closer to 7\. We can safely assign to 5\ and to 7\.
- We must fill in the blanks of 1\, 2\, 3\, 6\ and \10. Unfortunately, there're two possible mappings for 17. By the fact that is tempered out, 4131/4096 and 2187/2176 should be mapped to the same degree by what I learned is known as flag arithmetics. Since and make a pyth comma, both should be 3 steps. That leads to the 198 patent val in the 17-limit. Similarly, there're two possible mappings for 19. Adopting 896/891 as 1\ indicates 513/512 is also 1\. That leads to 198 patent val in the 19-limit.
- For 2\, we got 144/143 , yet it contradicts the flag arithmetics of and established above.
- I'm completely clueless about 6\ and 10\. I confess I'm not familiar with all the symbols.