Lemba
Lemba (the name is from Herman Miller's conlang name for the temperament) as a regular temperament is a natural extension of the jubilismic clan and a member of 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.
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 | 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]]
| ET generator |
eigenmonzo (unchanged interval) |
supermajor second (¢) |
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 | ||
| 234.485 | 2.3.7.17 subgroup least squares [clarification needed] | ||
| 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
By Claudi Meneghin
- Lemba Suite, for Two Organs (Prelude, Aria & Fugue)
- in 8/7 eigenmonzo tuning
- ↑ L/s ratios are always precisely Φ, and MOS scales are always precisely 2Φ