Whitewood family
The whitewood family of temperaments tempers out the apotome, 2187/2048. Consequently the fifths are always 4/7 of an octave, a distinctly flat 685.714 cents. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.
The 5-limit version of this temperament is called whitewood, to serve in contrast with the blackwood temperament which tempers out 256/243, the pythagorean limma. Whereas blackwood temperament can be thought of as a closed chain of 5 fifths and a major third generator, whitewood is a closed chain of 7 fifths and a major third generator. This means that blackwood is generally supported by 5n-edos, and whitewood is supported by 7n-edos, and the mos of both scales follow a similar pattern.
The 14-note mos of whitewood, like the 10-note mos of blackwood, shares a number of interesting properties which derive from the relatively small circle of fifths common to both. From any major or minor triad in the scale, one can always move away by ~3/2 or ~4/3 to reach another triad of the same type. This contrasts with the diatonic scale, in which one will eventually "hit a wall" if one moves by perfect fifth for long enough; the chain of fifths will eventually "stop" and make the next fifth a diminished fifth. This means that this scale is, in a sense, "pantonal", since resolutions that work in one key will work in all other keys in the scale, at least keys that share the same chord quality.
Another interesting property is that it becomes possible to construct "super-linked" 5-limit chords. In whitewood[14], or blackwood[10], if one stacks alternating major and minor thirds on top of one another, one will eventually come back to the root without ever hitting a wall, and hence the pattern can continue forever. Since all of the diatonic modes can be thought of as a stacked chain of 7 alternating thirds, placed in inversion, this means that whitewood[14] and blackwood[10] also make for excellent "panmodal" scales, in which you can construct "modal" sounding sonorities in one key that will work in all keys.
Lastly, while blackwood fifths are sharp and thus necessitate the tuning as a whole to be sharp-leaning, whitewood fifths are flat and thus this tuning is generally flat-leaning.
Whitewood
Subgroup: 2.3.5
Comma list: 2187/2048
Mapping: [⟨7 11 16], ⟨0 0 1]]
- mapping generators: ~9/8, ~5
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 374.469
Optimal ET sequence: 7, 21, 28, 35, 77bb
Badness: 0.154651
Septimal whitewood
Subgroup: 2.3.5.7
Comma list: 36/35, 2187/2048
Mapping: [⟨7 11 16 20], ⟨0 0 1 -1]]
Wedgie: ⟨⟨0 7 -7 11 -11 -36]]
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 392.700
Optimal ET sequence: 7, 14, 21, 28, 49b
Badness: 0.113987
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 45/44, 2079/2048
Mapping: [⟨7 11 16 20 24], ⟨0 0 1 -1 1]]
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 389.968
Optimal ET sequence: 7, 14e, 21, 28, 49b
Badness: 0.060908
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 36/35, 45/44, 512/507
Mapping: [⟨7 11 16 20 24 26], ⟨0 0 1 -1 1 0]]
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 390.735
Optimal ET sequence: 7, 14e, 21, 28, 49bf
Badness: 0.039956
Redwood
Subgroup: 2.3.5.7
Comma list: 525/512, 729/700
Mapping: [⟨7 11 16 20], ⟨0 0 1 -2]]
Wedgie: ⟨⟨0 7 -14 11 -22 -52]]
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 378.152
Optimal ET sequence: 7, 21d, 28d, 35
Badness: 0.165257
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 385/384, 729/700
Mapping: [⟨7 11 16 20 24], ⟨0 0 1 -2 1]]
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 376.711
Optimal ET sequence: 7, 21d, 28d, 35
Badness: 0.078193
Mujannab
Subgroup: 2.3.5.7
Comma list: 54/49, 64/63
Mapping: [⟨7 11 16 20], ⟨0 0 1 0]]
Wedgie: ⟨⟨0 7 0 11 0 -20]]
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 395.187
Optimal ET sequence: 7, 14d, 21dd
Badness: 0.105820
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 54/49, 64/63
Mapping: [⟨7 11 16 20 24], ⟨0 0 1 0 1]]
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 394.661
Optimal ET sequence: 7, 14de, 21dd
Badness: 0.060985
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 45/44, 52/49, 64/63
Mapping: [⟨7 11 16 20 24 26], ⟨0 0 1 0 1 0]]
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 395.071
Optimal ET sequence: 7, 14de, 21dd
Badness: 0.042830
Greenwood
Subgroup: 2.3.5.7
Comma list: 405/392, 1323/1280
Mapping: [⟨7 11 1 12], ⟨0 0 2 1]]
- mapping generators: ~9/8, ~15/7
Wedgie: ⟨⟨0 14 7 22 11 -23]]
Optimal tuning (CTE): ~9/8 = 1\7, ~15/14 = 108.062
Optimal ET sequence: 14c, 21, 35
Badness: 0.121752
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 99/98, 1323/1280
Mapping: [⟨7 11 1 12 9], ⟨0 0 2 1 2]]
Optimal tuning (CTE): ~9/8 = 1\7, ~15/14 = 106.997
Optimal ET sequence: 14c, 21, 35, 49bcde, 84bbccde
Badness: 0.057471
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 45/44, 99/98, 640/637
Mapping: [⟨7 11 1 12 9 26], ⟨0 0 2 1 2 0]]
Optimal tuning (CTE): ~9/8 = 1\7, ~15/14 = 106.997
Optimal ET sequence: 14c, 21, 35
Badness: 0.054009