My best shot: Natural Pan Temperament.

Before I joined this group I was a strict Just Intonation man and my best shot at a scale was:

1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 2/1.

I call this scale NPJ (Natural Pan Just).The notes going up from 1/1 are the mirror image of the notes going down from 3/2 so you have two tonics of equal strength, 1/1 and 3/2. 1/1 is better for music that ascends from and then descends to 1/1 and 3/2 is better for music that descends from and then ascends to 3/2. A slightly better system has 16/15 instead of 15/14 but this way you have only one tonic (1/1), not two, so it's not symmetric. Also in this case 1/1 is stronger than 2/1. I found that it is impossible to have a symmetric 12 key scale where 1/1 and 2/1 have equal strength unless you use the sqrt(2) as the tritone or omit the tritone altogether. I found symmetry in another way where the notes going up from 1/1 are the mirror image of the notes going down from 3/2.

Why not have more than 12 keys? I play guitar and I modified a cheap guitar to produce NPJ notes. When playing I looked at the frets and wondered how tricky a regular chord would be to play if I placed another fret in between the existing frets and decided that 12 was the golden mean between simplicity and complexity. Also with a 12 key system the transition from 12TET to my microtonal system should be relatively easy.

The problem with my NPJ system (and I suspect with all Just 12 key systems) is that (if the tonic, 1/1, is E) there are woefully few chords in the keys of F, F#, A#, D and D#.

Michael turned me on to the idea of tempering so I decided to temper my NPJ scale and hey presto, a lot more chords in the keys of F, F#, A#, D and D# occur.

With my just NPJ system, of all intervals (an octave wide or less, including the octave and excluding the unison) that occur over a 12 key range (144 intervals), exactly 72 of these (50%) are good and are all perfectly in tune. There's also a 13/7 lurking in there somewhere, not perfectly in tune but within 6.775 cents accuracy. What are the good intervals. According to my Interval Calculator program (in the JohnOSullivan folder in the Files section) the following intervals are good:
9/8, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5, 10/7, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 11/6, 13/7. I have tested all of these and to me they sound acceptable. Some would include intervals narrower than 9/8 (e.g.10/9, 11/10) but to my ear they are dissonant.

When I tempered the just NPJ system I preserved symmetry as follows: I left the 1/1 and 3/2 alone. If I raised the first note going up from 1/1 by `a' cents then I lowered the first note going down from 3/2 by `a' cents. Or vice versa. If I raised the second note going up from 1/1 by `b' cents then I lowered the second note going down from 3/2 by `b' cents. Or vice versa. And so on. Here's what I got...

0.0c
121.56c
200.73c
313.52c
388.43c
501.22c
580.39c
701.96c
816.86c
889.44c
1012.51c
1085.09c
1200.0c

1.0, 1.07274, 1.122938, 1.198533, 1.25153, 1.335782, 1.398289,
1.5, 1.602938, 1.671566, 1.794724, 1.871563, 2.0.

I call it "Natural Pan Temperament" or NPT for short.

Of the 144 possible intervals outlined above, 97 (67%) of these are within 6.775877 cents of any of the good intervals listed above. 6.775877 cents expressed as an interval is 256/255. To me, 2, 4, 16 and 256 are "Magic numbers". I tested a slightly wider threshold (8 cents) with the 3/2 interval but I could detect a slight dissonance so I settled for a threshold of 6.775877 cents (256/255) which seems reasonable when tested. If I'm erring, then I'm erring on the side of caution.

Here again are the good intervals and how often they occur in Natural Pan Temperament (within an accuracy of 6.775877 cents)

9/8.... 2 times
8/7.... 2 times
7/6.... 3 times
6/5.... 8
5/4.... 8
9/7.... 4
4/3.... 9
11/8.... 0
7/5.... 6
10/7.... 6
3/2.... 9
11/7.... 0
8/5.... 8
5/3.... 8
12/7.... 3
7/4.... 2
9/5.... 4
11/6.... 0
13/7.... 3
2/1.... 12

The three 11 limit intervals don't occur.

If 1/1 is E then the best keys to play in are E and B but the keys of G, G#, C and D# can also be used because these keys have a Perfect Fourth and a Perfect Fifth available (within 6.775 cents accuracy).

I can't test my NPT system because I can't play piano/keyboards so if someone out there would like to try it let me know what you think of it. A lot of the regular chords from 12TET should work with my system and are much more in tune.

I plan to modify a guitar at some stage to conform to NPT but this will take some time.

Finally, I suspect that if I use 16/15 instead of 15/14 in my just scale, and then temper it, I might be able to squeeze a few more good intervals out of it but this scale would not be symmetric.

John.