User talk:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning

User:Dummy index posted this on Discord overnight (Sydney time), but I decided to move it here because I would rather discuss it on a talk page:

“Hello,

> 16edo: 2 • 5 • 7 • 13 • 27 • 45

is dual-3. (and 14edo is dual-5 maybe.)

• In the first place, it is too much to have four basis elements within the 5-limit.
• 27 * 5 / 45 = 3 (76 + 37 - 88 = 25 steps) is same to nearest approximation of 3rd harmonic. Hence, it is inconsistent between 27 (76 steps) and 3 (25 steps). So we need to differentiate between 3^3 and 27 in some way, to regain regularity.
• One way, 27 = 9+ * 3- = 3++ * 3- * 3- (76 = 51 + 25 = 26 + 25 + 25). it means 2.5.7.13.27.45+ or 2.3-.3++.5.7.13 subgroup.
• Another way, 27+ = 9+ * 3 = 3+ * 3 * 3 for nearest approximation of 27th harmonic and 27 for 3^3. it means 2.5.7.13.27+.45+ or 2.3.3+.5.7.13 subgroup.”

I plan to re-read Index’s post carefully and write a reply here later today (Sydney time). It may be several hours before I can do that due to irl commitments.

--BudjarnLambeth (talk) 18:18, 7 January 2025 (UTC)