User:Overthink/EDO impressions

This is a page about my thoughts on various EDOs. An asterisk means my opinion has changed, and I may potentially rewrite this.

Ideas

In my opinion, an interval is melodically represented well if it is within ~15 cents, but harmonically represented decently only if within ~5-6 cents, the more accurate the better. It is absolutely essential for any system to approximate the perfect fifth within this 5-6 cent bound*. Octave stretch may help improve the perfect fifth, though the octave itself is even more important, so it shouldn't be stretched too much.

EDO Impressions

Note: They are by no means final!

EDOs I should try: 27, 29, 31, 34, 43, 46, 50, 55?, 56?, 58, 62?, 72, 84, 87?, 94, 111

12: We've all heard this system thousands of times before. It has a good perfect fifth, which is essential. In my opinion, this system sounds much more like pythagorean tuning than 5-limit, though the slight tempering of the fifth reduces the dissonance of the thirds by pushing them towards the 5-limit ones, but they still don't really show 5-limit flavor. If you want true 5-limit use 53edo (or possibly a more tempered meantone system) instead. Still, the expressive possibilities are vast, though that doesn't mean you shouldn't use other systems.

17: The diatonic scale has a septimal flavor to it, with major thirds sounding nothing like 12edo ones. The minor second is the perfect size melodically, though it may take a while to get used to it. Haven't really experimented with harmony in this system.

19: Due to the flat tendency of the system, melody seems to have a melancholy feeling that you can't replicate with 12edo. Unfortunately, this includes the perfect fifth, which I feel is too off, though it may be mitigated by octave stretch. Another issue is that the minor seconds are too wide melodically.

22: The diatonic scale sounds even brighter than in 17, with thirds and sixths having a more obvious septimal flavor. The diatonic semitone is a little smaller than ideal, but still usable. I haven't looked too much into it harmonically. A possibly interesting property is that it supports pajara. The fifth is not very good though, though a compressed octave may help. For example, 57ed6 has the fifth less sharp at 707.5 cents compared to 22edo's 709.1, and sounds noticeably more consonant. This also improves intervals like 6/5 and 7/4, though 6/5 still feels too off.

31: A meantone tuning with near-pure 5/4 and 7/4. I don't like the fact that, in the major triad, the major third stands out more than the fifth. There are two ways to solve this; one is to sharpen the fifth to around 55edo, while another is to flatten the fifth close to 695.63 ¢ to get DR 4:5:6. Personally, the second option sounds much more in tune, while the first sounds 12edo-ish but somewhat less out of tune. However, these tunings don't approximate septimal meantone as well, so if that's your goal, use a stretched-octave version around 111ed12, which mitigates this as much as possible.

41: While this EDO has be cited as good for approximating JI, it has quite a few problems. For example, 5/4 is quite flat, making 5-limit chords sound less concordant, which isn't good for an EDO this size. 7/4 is also a bit flat, making septimal intervals sound more ambiguous and less concordant. In general pental and septimal intervals are too far from their nearest pythagorean intervals, as the comma step is over 29 cents. I prefer 53edo instead.

43: A meantone tuning around 1/5-comma. The thirds are slightly more off than in 31, and the fifth is closer, but the error still feels problematic*. Septimal meantone tunes 7/4 too sharp here.

53: Excellent 5- and 7-limit harmonies. 5/4 is now only slightly flat, compared to nearly 6 cents in 41edo. The 7/4 is a bit sharp, but this makes septimal harmonies more concordant compared to 41. The 49/48 interval is also more noticeable at 45 cents, compared to 29 cents in 41edo. Chords are significantly more consonant than in 41. Definitely would recommend.

55: Around the best meantone tuning for most music*. The fifth is slightly out of tune, but would probably sound normal to most people. Any sharper and the error of the thirds becomes problematic.

72: Haven't really tried. This edo seems to approximate the 11-limit well. The 7th harmonic is a little flat like in 41edo, though due to the slightly flat fifth 7/6 and 9/7 may be a bit more concordant than in 41.