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	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

7-limit prime-optimized tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~8/7 = 232.9268 ¢	CWE: ~8/7 = 232.2655 ¢	POTE: ~8/7 = 232.0888 ¢

11-limit prime-optimized tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~8/7 = 231.9971 ¢	CWE: ~8/7 = 231.1781 ¢	POTE: ~8/7 = 230.9742 ¢

13-limit prime-optimized tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~8/7 = 232.0999 ¢	CWE: ~8/7 = 231.1617 ¢	POTE: ~8/7 = 230.9665 ¢

Tuning spectrum

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	5-odd-limit minimax
(2 - Φ)\2		229.179	Golden Lemba
	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

Claudi Meneghin

• Lemba Suite, for Two Organs (Prelude, Aria & Fugue) in 8/7 eigenmonzo tuning

Herman Miller

• Lemba Galatsia
• GPO Lemba