Lemba
Lemba is a regular temperament which is a natural extension of the jubilismic clan and a member of the gamelismic clan. This means that the perfect fifth of ~3/2 is split into three equal parts, each approximating 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 a suitable eigenmonzo (unchanged-interval). 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. Its 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. It forms mos 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. Related temperaments include baladic, which discards the 5 and 11 to improve the accuracy of the 2.3.7.13.17 subgroup, and is very accurate in all of these except the 13. Or Echidnic, which has a slightly sharper generator that sacrifices the precision of 7 and logarithmic phi to map most other intervals in a more accurate but complex way.
The name is from Herman Miller's conlang name for the temperament.
See Jubilismic clan #Lemba for more technical data.
Interval chain
In the following table, prime harmonics are in bold.
| # gens | Period 0 | Period 1 | ||
|---|---|---|---|---|
| Cents* | Approximate Ratios | Cents | Approximate Ratios | |
| 0 | 0.000 | 1/1 | 600.000 | 7/5, 10/7 |
| 1 | 230.966 | 8/7 | 830.966 | 8/5, 13/8 |
| 2 | 461.932 | 21/16, 13/10 | 1061.932 | 11/6, 15/8, 24/13 |
| 3 | 692.898 | 3/2 | 92.898 | 15/14 |
| 4 | 923.864 | 12/7, 22/13 | 323.864 | 6/5, 39/32 |
| 5 | 1154.830 | 48/25, 63/32, 96/48 | 554.830 | 11/8 |
| 6 | 185.796 | 9/8 | 785.796 | 11/7 |
| 7 | 416.762 | 9/7 | 1016.762 | 9/5 |
| 8 | 647.728 | 36/25 | 47.728 | 33/32, 36/35 |
* In 13-limit POTE tuning. This tuning is very close to 26edo.
Tuning spectrum
Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77]
Gencom mapping: [⟨2 2 5 6 5 7], ⟨0 3 -1 -1 5 1]]
| EDO generator |
eigenmonzo (unchanged-interval) |
generator (¢) | comments |
|---|---|---|---|
| 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\42 | 228.571 | ||
| 6/5 | 228.910 | ||
| (2 - Φ)\2 | 229.179 | Golden Lemba[1] | |
| 21/13 | 230.253 | ||
| 11/8 | 230.264 | ||
| 14/11 | 230.415 | 15-odd-limit minimax | |
| 13/12 | 230.714 | ||
| 5\26 | 230.769 | ||
| 10/9 | 231.085 | 9-, 11- and 13-odd-limit minimax | |
| 8/7 | 231.174 | 7-odd-limit minimax | |
| [0 63 -20 -20 22 -6⟩ | 231.250 | 13-odd-limit least squares | |
| [0 17 -6 -6 6⟩ | 231.294 | 11-odd-limit least squares | |
| 52521875/177147 | 231.298 | 7-odd-limit least squares | |
| [0 66 -17 -23 25 -7⟩ | 231.399 | 15-odd-limit least squares | |
| 17/13 | 232.213 | ||
| 12\62 | 232.258 | ||
| 129140163/1500625 | 232.418 | 9-odd-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 | ||
| 9\46 | 234.783 | ||
| 17/16 | 234.985 | ||
| 21/16 | 235.390 | ||
| 11\56 | 235.714 | ||
| 14/13 | 235.851 | ||
| 15/14 | 239.814 | ||
| 16/13 | 240.528 | ||
| 15/13 | 247.741 | ||
| 11/9 | 252.592 |
Music
- Lemba Suite, for Two Organs (Prelude, Aria & Fugue) in 8/7 eigenmonzo tuning
Notes
- ↑ L/s ratios are always precisely Φ, and mos scales are always precisely 2Φ