You know, 8269edo is quite comparable with 8539edo, despite that the latter is more famous(?). Both are prime. Both are 27-odd-limit-consistent (and provide nothing new over 2460edo). But anyway I don't actually like using prime edos to measure the interval size.

Notice 8269 + 8539 = 16808. 16808edo is, as you've probably known if you're reading this, a 31-limit monster, using the val [16808 26640 39027 47186 58146 62197 68702 71399 76032 81653 83270>. So why don't we define the tina as 2\16808? Since it's equivalent to 1\8404 in 2.3.7.11.17.23.31 subgroup, you may as well say it's 1\8404.

Forget the eda stuff.

You may view it as a linear temperament of 8269 & 8539. 8269edo and 8539edo are flexible approximations thereof. Like what I posted before – a comparison of 224edo and 270edo – they're wonderful "edo twins".

Notation-wise, it doesn't really matter. All default ratios would remain the same using any of 1\8269, 1\8539 or 2\16808. Some would gain larger errors, compensated by others getting smaller.

1\: 10241/10240

2\: 5832/5831

3\: 4096/4095

4\: 3025/3024

5\: 2401/2400

6\: 2080/2079

7\: 1701/1700

8\: 382976/382725

9\: 131072/130977

except for the fractional-tina ratio 1515591/1515520 though, but it's easy to find one for 1\16808 (way more elegant than 1\17078).

For 8269edo, the schisma is mapped to 13 steps, not 14. In case of 16808edo, the schisma is 27 steps. That induces larger error for the schisma, yet I find it fascinating that the 5/7k is split into exactly 3 schismas, one for 5s, two for 7s (=5/7k - 5s), three for 5/7k, and four for 25/7k (225/224). It's also the case in 2460edo.

If what's already there can't be de-established, how about this? I might call 1\8269 the "major tina", 1\8539 the "minor tina" (the default), and 2\16808, the "mean tina".

Quick reference

Relative errors (prime 3 – 37)

- 8269edo: -5.49, -2.34, -1.79, -4.01, +6.40, -23.02, -11.26, -33.38, +35.52, -24.93, +3.01

- 8539edo: +0.52, +5.60, -0.37, -8.66, -5.48, +15.48, -5.30, +30.45, -29.97, +11.77, +47.77

- 16808edo: -4.97*, +3.26, -2.15, -12.66, +0.92, -7.54, -16.56, -2.94, +5.54, -13.16, -49.22

- 8269edo: 17.32, 14.55, 13.13, 11.76, 15.45, 22.27, 21.20, 27.00, 38.44, 38.20, 37.08

- 8539edo: 1.64, 10.68, 10.29, 15.62, 15.32, 19.97, 19.41, 27.62, 33.67, 32.72, 39.66

- 16808edo: 15.68*, 18.98, 16.46, 19.08, 18.27, 17.18, 18.54, 17.66, 18.37, 18.06, 28.49