Lemba

Lemba (the name is from Herman Miller's conlang name for the temperament) 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.

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.

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	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. L/s ratios are always precisely Φ, and MOS scales are always precisely 2Φ
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 (Prelude, Aria & Fugue)

in 8/7 eigenmonzo tuning

By Herman Miller

• Lemba Galatsia
• GPO Lemba