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 (supraminor and submajor), 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, odd harmonics 1–13 and their inverses are in bold.
| # | Period 0 | Period 1 | ||
|---|---|---|---|---|
| Cents* | Approximate ratios | Cents | Approximate ratios | |
| 0 | 0.0 | 1/1 | 600.0 | 7/5, 10/7 |
| 1 | 231.2 | 8/7 | 831.2 | 8/5, 13/8 |
| 2 | 462.3 | 13/10, 21/16 | 1062.3 | 11/6, 13/7, 15/8, 24/13 |
| 3 | 693.5 | 3/2 | 93.5 | 15/14 |
| 4 | 924.6 | 12/7, 22/13 | 324.6 | 6/5, 39/32 |
| 5 | 1155.8 | 39/20, 48/25, 63/32, 96/48 | 555.8 | 11/8 |
| 6 | 187.0 | 9/8, 11/10 | 787.0 | 11/7 |
| 7 | 418.1 | 9/7 | 1018.1 | 9/5 |
| 8 | 649.3 | 36/25 | 49.3 | 33/32, 36/35 |
* In 13-limit CWE tuning, octave reduced
Tunings
Prime-optimized tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~8/7 = 232.9268 ¢ | CWE: ~8/7 = 232.2655 ¢ | POTE: ~8/7 = 232.0888 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~8/7 = 231.9971 ¢ | CWE: ~8/7 = 231.1781 ¢ | POTE: ~8/7 = 230.9742 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~8/7 = 232.0999 ¢ | CWE: ~8/7 = 231.1617 ¢ | POTE: ~8/7 = 230.9665 ¢ |
Target tunings
| Target | Generator | Eigenmonzo* |
|---|---|---|
| 5-odd-limit | ~8/7 = 228.910 ¢ | 5/3 |
| 7-odd-limit | ~8/7 = 231.174 ¢ | 7/4 |
| 9-odd-limit | ~8/7 = 231.085 ¢ | 9/5 |
| 11-odd-limit | ~8/7 = 231.085 ¢ | 9/5 |
| 13-odd-limit | ~8/7 = 231.085 ¢ | 9/5 |
| 15-odd-limit | ~8/7 = 230.415 ¢ | 11/7 |
| Target | Generator | Eigenmonzo* |
|---|---|---|
| 7-odd-limit | ~8/7 = 231.298 ¢ | [0 -11 5 5⟩ |
| 9-odd-limit | ~8/7 = 232.418 ¢ | [0 17 -4 -4⟩ |
| 11-odd-limit | ~8/7 = 231.294 ¢ | [0 17 -6 -6 6⟩ |
| 13-odd-limit | ~8/7 = 231.250 ¢ | [0 63 -20 -20 22 -6⟩ |
| 15-odd-limit | ~8/7 = 231.399 ¢ | [0 66 -17 -23 25 -7⟩ |
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 5/4 | 213.686 | ||
| 15/11 | 221.016 | ||
| 11/6 | 224.681 | ||
| 3\16 | 225.000 | Lower bound of 7-odd-limit diamond monotone | |
| 13/10 | 227.107 | ||
| 11/10 | 227.501 | ||
| 13/11 | 227.698 | ||
| 8\42 | 228.571 | 42bc val | |
| 5/3 | 228.910 | 5-odd-limit minimax | |
| (2 - Φ)\2 | 229.179 | Golden Lemba[1] | |
| 21/13 | 230.253 | ||
| 11/8 | 230.264 | ||
| 11/7 | 230.415 | 15-odd-limit minimax | |
| 13/12 | 230.714 | ||
| 5\26 | 230.769 | Lower bound of 9-odd-limit diamond monotone 11- and 13-odd-limit diamond monotone (singleton) | |
| 9/5 | 231.085 | 9-, 11- and 13-odd-limit minimax | |
| 7/4 | 231.174 | 7-odd-limit minimax | |
| 17/13 | 232.213 | ||
| 12\62 | 232.258 | 62c val | |
| 13/9 | 232.676 | ||
| Φ | 233.090 | ||
| 7/6 | 233.282 | ||
| 7\36 | 233.333 | 36c val | |
| 9/7 | 233.583 | ||
| 3/2 | 233.985 | ||
| 21/17 | 234.274 | ||
| 9\46 | 234.783 | 46ce val | |
| 17/16 | 234.985 | ||
| 21/16 | 235.390 | ||
| 11\56 | 235.714 | 56ccee val | |
| 13/7 | 235.851 | ||
| 15/14 | 239.814 | ||
| 2\10 | 240.000 | Upper bound of 7- and 9-odd-limit diamond monotone | |
| 13/8 | 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