Lemba: Difference between revisions

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→Spectrum of lemba tunings by eigenmonzos: improve and standardize tuning spectra tables
Line 72: Line 72:
<nowiki>*</nowiki> In 13-limit POTE tuning. This tuning is very close to [[26edo]].
<nowiki>*</nowiki> In 13-limit POTE tuning. This tuning is very close to [[26edo]].


== Spectrum of lemba tunings by eigenmonzos ==
== Tuning spectrum ==


Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77]
Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77]
Line 78: Line 78:
Gencom map: [{{val| 2 2 5 6 5 7 }},[{{val| 0 3 -1 -1 5 1 }}]
Gencom map: [{{val| 2 2 5 6 5 7 }},[{{val| 0 3 -1 -1 5 1 }}]


{| class="wikitable center-1 right-2"
{| class="wikitable center-all"
|-
|-
! Eigenmonzo
! ET<br>generator
! Supermajor<br>Second
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! Comments
! supermajor<br>second (¢)
! comments
|-
|-
|
| 5/4
| 5/4
| 213.686
| 213.686
|  
|  
|-
|-
|
| 15/11
| 15/11
| 221.016
| 221.016
|  
|  
|-
|-
|
| 12/11
| 12/11
| 224.681
| 224.681
|  
|  
|-
|-
| (3\16)
| 3\16
|
| 225.000
| 225.000
|  
|  
|-
|-
|
| 13/10
| 13/10
| 227.107
| 227.107
|  
|  
|-
|-
|
| 11/10
| 11/10
| 227.501
| 227.501
|  
|  
|-
|-
|
| 13/11
| 13/11
| 227.698
| 227.698
|  
|  
|-
|-
| (8\42)
| 8\42
|
| 228.571
| 228.571
|  
|  
|-
|-
|
| 6/5
| 6/5
| 228.910
| 228.910
Line 121: Line 131:
|-
|-
| (2 - Φ)\2
| (2 - Φ)\2
|
| 229.179
| 229.179
| Golden Lemba. L/s ratios are always precisely Φ, and MOS scales are always precisely 2Φ
| Golden Lemba<ref>L/s ratios are always precisely Φ, and MOS scales are always precisely 2Φ</ref>
|-
|-
|
| 21/13
| 21/13
| 230.253
| 230.253
|  
|  
|-
|-
|
| 11/8
| 11/8
| 230.264
| 230.264
|  
|  
|-
|-
|
| 14/11
| 14/11
| 230.415
| 230.415
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|-
|
| 13/12
| 13/12
| 230.714
| 230.714
|  
|  
|-
|-
| (5\26)
| 5\26
|
| 230.769
| 230.769
|  
|  
|-
|-
|
| 10/9
| 10/9
| 231.085
| 231.085
| 9-, 11- and 13-odd-limit minimax
| 9-, 11- and 13-odd-limit minimax
|-
|-
|
| 8/7
| 8/7
| 231.174
| 231.174
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|
| {{monzo| 0 63 -20 -20 22 -6 }}
| {{monzo| 0 63 -20 -20 22 -6 }}
| 231.250
| 231.250
| 13-odd-limit least squares
| 13-odd-limit least squares
|-
|-
|
| {{monzo| 0 17 -6 -6 6 }}
| {{monzo| 0 17 -6 -6 6 }}
| 231.294
| 231.294
| 11-odd-limit least squares
| 11-odd-limit least squares
|-
|-
|
| 52521875/177147
| 52521875/177147
| 231.298
| 231.298
| 7-odd-limit least squares
| 7-odd-limit least squares
|-
|-
|
| {{monzo| 0 66 -17 -23 25 -7 }}
| {{monzo| 0 66 -17 -23 25 -7 }}
| 231.399
| 231.399
| 15-odd-limit least squares
| 15-odd-limit least squares
|-
|-
|
| 17/13
| 17/13
| 232.213
| 232.213
|  
|  
|-
|-
| (12\62)
| 12\62
|
| 232.258
| 232.258
|  
|  
|-
|-
|
| 129140163/1500625
| 129140163/1500625
| 232.418
| 232.418
| 9-odd-limit least squares
| 9-odd-limit least squares
|-
|-
|
| 18/13
| 18/13
| 232.676
| 232.676
|  
|  
|-
|-
|
| Φ
| Φ
| 233.090
| 233.090
|  
|  
|-
|-
|
| 7/6
| 7/6
| 233.282
| 233.282
|  
|  
|-
|-
| (7\36)
| 7\36
|
| 233.333
| 233.333
|  
|  
|-
|-
|
| 9/7
| 9/7
| 233.583
| 233.583
|  
|  
|-
|-
|
| 4/3
| 4/3
| 233.985
| 233.985
|  
|  
|-
|-
|
| 21/17
| 21/17
| 234.274
| 234.274
|  
|  
|-
|-
|
|  
|  
| 234.485
| 234.485
| 2.3.7.17 subgroup least squares {{clarify}}
| 2.3.7.17 subgroup least squares {{clarify}}
|-
|-
| (9\46)
| 9\46
|
| 234.783
| 234.783
|  
|  
|-
|-
|
| 17/16
| 17/16
| 234.985
| 234.985
|  
|  
|-
|-
|
| 21/16
| 21/16
| 235.390
| 235.390
|  
|  
|-
|-
| (11\56)
| 11\56
|
| 235.714
| 235.714
|  
|  
|-
|-
|
| 14/13
| 14/13
| 235.851
| 235.851
|  
|  
|-
|-
|
| 11/9
| 11/9
| 236.851
| 236.851
|  
|  
|-
|-
|
| 16/15
| 16/15
| 237.243
| 237.243
|  
|  
|-
|-
|
| 15/14
| 15/14
| 239.814
| 239.814
|  
|  
|-
|-
|
| 16/13
| 16/13
| 240.528
| 240.528
|  
|  
|-
|-
|
| 15/13
| 15/13
| 247.741
| 247.741

Revision as of 00:24, 30 September 2021

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 map: [⟨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

in 8/7 eigenmonzo tuning

By Herman Miller

Template:IoT

  1. ↑ L/s ratios are always precisely Φ, and MOS scales are always precisely 2Φ
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