Lemba
Lemba as a regular temperament is the intersection of the Jubilismic clan and the Gamelismic clan. This means that the perfect fifth is split into three equal parts, each approximately an 8/7. It also means the period is half an octave, and repeats precisely a tritone apart, tempering out 50/49. A generator plus a period comes very close to the golden ratio phi, which means ratios in the sequence 8:13:21:34:55 etc are also well approximated, and any one of these can be made just by choosing the right eigenmonzo. The combination of these factors means many composite ratios in the 2.3.5.7.13.17 subgroup are both well approximated and accessible with a relatively small gamut, giving you a strong selection of chords to choose from. It's main weaknesses are that ratios of 5 and 13 are conflated by the tempering out of 65/64, favoring 13 in the better tunings, so traditional major and minor chords are strongly neutral flavoured, and ratios involving 11 are not approximated at all until you have a large gamut. However, ignoring the 5 and 13, and focusing purely on the 2.3.7.17 subgroup, it can be highly accurate, with a total error of less than 7 cents in the tonality diamond in the least squares tuning. It forms mode of symmetry scales that are always double a fibonacci sequence number, at 4, 6, 10, 16, 26, etc, which means L/s ratios remain well mixed and clearly distinct many iterations down.
Spectrum of Lemba Tunings by Eigenmonzos
Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77]
Gencom map: [<2 2 5 6 5 7|,[<0 3 -1 -1 5 1|]
| Eigenmonzo | Supermajor Second |
|---|---|
| 5/4 | 213.686 |
| 15/11 | 221.016 |
| 12/11 | 224.681 |
| 3\16 | 225.000 |
| 13/10 | 227.107 |
| 11/10 | 227.501 |
| 13/11 | 227.698 |
| 8\42b | 228.571 |
| 6/5 | 228.910 |
| 21/13 | 230.253 |
| 11/8 | 230.264 |
| 14/11 | 230.415 (15 limit minimax) |
| 13/12 | 230.714 |
| 5\26 | 230.769 |
| 10/9 | 231.085 (9, 11 and 13 limit minimax) |
| 8/7 | 231.174 (7 limit minimax) |
| [0 63 -20 -20 22 -6> | 231.250 (13 limit least squares) |
| [0 17 -6 -6 6> | 231.294 (11 limit least squares) |
| 52521875/177147 | 231.298 (7 limit least squares) |
| [0 66 -17 -23 25 -7> | 231.399 (15 limit least squares) |
| 17/13 | 232.213 |
| 12\62 | 232.258 |
| 129140163/1500625 | 232.418 (9 limit least squares) |
| 18/13 | 232.676 |
| Φ | 233.090 |
| 7/6 | 233.282 |
| 7\36 | 233.333 |
| 9/7 | 233.583 |
| 4/3 | 233.985 |
| 21/17 | 234.274 |
| 234.485 (2.3.7.17 subgroup least squares) | |
| 9\46 | 234.783 |
| 17/16 | 234.985 |
| 21/16 | 235.390 |
| 11\56 | 235.714 |
| 14/13 | 235.851 |
| 11/9 | 236.851 |
| 16/15 | 237.243 |
| 15/14 | 239.814 |
| 16/13 | 240.528 |
| 15/13 | 247.741 |
Music
Lemba Suite (Prelude, Aria & Fugue) by Claudi Meneghin
In 8/7 eigenmonzo tuning