Lemba: Difference between revisions

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 '''Lemba''' (the name is from [[Herman Miller]]'s conlang name for the temperament) as a regular temperament is the intersection of the [[Jubilismic clan #Lemba|Jubilismic clan]] and the [[Gamelismic clan #Lemba|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.
-==Spectrum of Lemba Tunings by Eigenmonzos==
+== Spectrum of Lemba Tunings by Eigenmonzos ==
 Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77]
@@ Line 7: / Line 7: @@
 Gencom map: [&lt;2 2 5 6 5 7|,[&lt;0 3 -1 -1 5 1|]
-{| class="wikitable"
+{| class="wikitable center-1 right-2"
 |-
 ! 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\42b
 | 228.571
+|
 |-
 | 6/5
 | 228.910
+|
 |-
 | (2-Φ)x600
-| 229.179 (Golden Lemba. L/s ratios are always precisely Φ, and MOS scales are always precisely 2xΦ)
+| 229.179
+| Golden Lemba. L/s ratios are always precisely Φ, and MOS scales are always precisely 2xΦ
 |-
 | 21/13
 | 230.253
+|
 |-
 | 11/8
 | 230.264
+|
 |-
 | 14/11
-| 230.415 (15 limit minimax)
+| 230.415
+| 15 limit minimax
 |-
 | 13/12
 | 230.714
+|
 |-
 | 5\26
 | 230.769
+|
 |-
 | 10/9
-| 231.085 (9, 11 and 13 limit minimax)
+| 231.085
+| 9, 11 and 13 limit minimax
 |-
 | 8/7
-| 231.174 (7 limit minimax)
+| 231.174
+| 7 limit minimax
 |-
 | [0 63 -20 -20 22 -6&gt;
-| 231.250 (13 limit least squares)
+| 231.250
+| 13 limit least squares
 |-
 | [0 17 -6 -6 6&gt;
-| 231.294 (11 limit least squares)
+| 231.294
+| 11 limit least squares
 |-
 | 52521875/177147
-| 231.298 (7 limit least squares)
+| 231.298
+| 7 limit least squares
 |-
 | [0 66 -17 -23 25 -7&gt;
-| 231.399 (15 limit least squares)
+| 231.399
+| 15 limit least squares
 |-
 | 17/13
 | 232.213
+|
 |-
 | 12\62
 | 232.258
+|
 |-
 | 129140163/1500625
-| 232.418 (9 limit least squares)
+| 232.418
+| 9 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)
+| 234.485
+| 2.3.7.17 subgroup least squares
 |-
 | 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
+|
 |}