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'''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.
{{Interwiki
| en = Lemba
| de = Slendrisch #Lemba
| es =
| ja =
}}
{{Infobox regtemp
| Title = Lemba
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[50/49]], [[525/512]] (7-limit);<br>[[45/44]], [[50/49]], [[385/384]] (11-limit);<br>[[45/44]], [[50/49]], [[65/64]], [[78/77]]<br>(13-limit)
| Edo join 1 = 10 | Edo join 2 = 16
| Mapping = 2; 3 -1 -1 5 1
| Generators = 8/7 | Generators tuning = 231.2 | Optimization method = CWE
| MOS scales = [[4L 2s]], [[6L 4s]], [[10L 6s]]
| Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 16
| Odd limit 2 = 13 | Mistuning 2 = 21.5 | Complexity 2 = 16
}}
'''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 scale]]s 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.  
Related temperaments include [[Gamelismic clan #Baladic|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 ==
== Interval chain ==
In the following table, prime harmonics are in '''bold'''.  
In the following table, odd harmonics 1–13 and their inverses are in '''bold'''.  


{| class="wikitable center-1 right-2 right-4"
{| class="wikitable center-1 right-2 right-4"
! rowspan="2" | # gens
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 1
|-
|-
! Cents*
! Cents*
! Approximate Ratios
! Approximate ratios
! Cents
! Cents
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| '''1/1'''
| 600.000
| 600.0
| 7/5, 10/7
| 7/5, 10/7
|-
|-
| 1
| 1
| 230.966
| 231.2
| '''8/7'''
| '''8/7'''
| 830.966
| 831.2
| '''8/5''', '''13/8'''
| '''8/5''', '''13/8'''
|-
|-
| 2
| 2
| 461.932
| 462.3
| 21/16, 13/10
| 13/10, 21/16
| 1061.932
| 1062.3
| 11/6, 15/8, 24/13
| 11/6, 13/7, 15/8, 24/13
|-
|-
| 3
| 3
| 692.898
| 693.5
| '''3/2'''
| '''3/2'''
| 92.898
| 93.5
| 15/14
| 15/14
|-
|-
| 4
| 4
| 923.864
| 924.6
| 12/7, 22/13
| 12/7, 22/13
| 323.864
| 324.6
| 6/5, 39/32
| 6/5, 39/32
|-
|-
| 5
| 5
| 1154.830
| 1155.8
| 48/25, 63/32, 96/48
| 39/20, 48/25, 63/32, 96/48
| 554.830
| 555.8
| '''11/8'''
| '''11/8'''
|-
|-
| 6
| 6
| 185.796
| 187.0
| 9/8
| '''9/8''', 11/10
| 785.796
| 787.0
| 11/7
| 11/7
|-
|-
| 7
| 7
| 416.762
| 418.1
| 9/7
| 9/7
| 1016.762
| 1018.1
| 9/5
| 9/5
|-
|-
| 8
| 8
| 647.728
| 649.3
| 36/25
| 36/25
| 47.728
| 49.3
| 33/32, 36/35
| 33/32, 36/35
|}
|}
<nowiki>*</nowiki> In 13-limit POTE tuning. This tuning is very close to [[26edo]].
<nowiki>*</nowiki> In 13-limit CWE tuning, octave reduced


== Tuning spectrum ==
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~8/7 = 232.9268{{c}}
| CWE: ~8/7 = 232.2655{{c}}
| POTE: ~8/7 = 232.0888{{c}}
|}


Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77]
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~8/7 = 231.9971{{c}}
| CWE: ~8/7 = 231.1781{{c}}
| POTE: ~8/7 = 230.9742{{c}}
|}


Gencom mapping: [{{val| 2 2 5 6 5 7 }},[{{val| 0 3 -1 -1 5 1 }}]
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~8/7 = 232.0999{{c}}
| CWE: ~8/7 = 231.1617{{c}}
| POTE: ~8/7 = 230.9665{{c}}
|}


{| class="wikitable center-all"
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings
|-
|-
! ET<br>generator
! Target
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! Generator
! supermajor<br>second (¢)
! Eigenmonzo*
! comments
|-
| 5-odd-limit
| ~8/7 = 228.910{{c}}
| 5/3
|-
| 7-odd-limit
| ~8/7 = 231.174{{c}}
| 7/4
|-
| 9-odd-limit
| ~8/7 = 231.085{{c}}
| 9/5
|-
| 11-odd-limit
| ~8/7 = 231.085{{c}}
| 9/5
|-
| 13-odd-limit
| ~8/7 = 231.085{{c}}
| 9/5
|-
| 15-odd-limit
| ~8/7 = 230.415{{c}}
| 11/7
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings
|-
! Target
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~8/7 = 231.298{{c}}
| {{Monzo| 0 -11 5 5 }}
|-
| 9-odd-limit
| ~8/7 = 232.418{{c}}
| {{Monzo| 0 17 -4 -4 }}
|-
| 11-odd-limit
| ~8/7 = 231.294{{c}}
| {{Monzo| 0 17 -6 -6 6 }}
|-
| 13-odd-limit
| ~8/7 = 231.250{{c}}
| {{Monzo| 0 63 -20 -20 22 -6 }}
|-
| 15-odd-limit
| ~8/7 = 231.399{{c}}
| {{Monzo| 0 66 -17 -23 25 -7 }}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]
! Generator (¢)
! Comments
|-
|-
|
|
Line 96: Line 223:
|-
|-
|
|
| 12/11
| 11/6
| 224.681
| 224.681
|  
|  
Line 103: Line 230:
|  
|  
| 225.000
| 225.000
|  
| Lower bound of 7-odd-limit diamond monotone
|-
|-
|
|
Line 123: Line 250:
|  
|  
| 228.571
| 228.571
|  
| 42bc val
|-
|-
|
|
| 6/5
| 5/3
| 228.910
| 228.910
|  
| 5-odd-limit minimax
|-
|-
| (2 - Φ)\2
| (2 - Φ)\2
|  
|  
| 229.179
| 229.179
| Golden Lemba<ref>L/s ratios are always precisely Φ, and MOS scales are always precisely 2Φ</ref>
| Golden Lemba<ref>L/s ratios are always precisely Φ, and mos scales are always precisely 2Φ</ref>
|-
|-
|
|
Line 146: Line 273:
|-
|-
|
|
| 14/11
| 11/7
| 230.415
| 230.415
| 15-odd-limit minimax
| 15-odd-limit minimax
Line 158: Line 285:
|  
|  
| 230.769
| 230.769
|  
| Lower bound of 9-odd-limit diamond monotone<br>11- and 13-odd-limit diamond monotone (singleton)
|-
|-
|
|
| 10/9
| 9/5
| 231.085
| 231.085
| 9-, 11- and 13-odd-limit minimax
| 9-, 11- and 13-odd-limit minimax
|-
|-
|
|
| 8/7
| 7/4
| 231.174
| 231.174
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|
| {{monzo| 0 63 -20 -20 22 -6 }}
| 231.250
| 13-odd-limit least squares
|-
|
| {{monzo| 0 17 -6 -6 6 }}
| 231.294
| 11-odd-limit least squares
|-
|
| 52521875/177147
| 231.298
| 7-odd-limit least squares
|-
|
| {{monzo| 0 66 -17 -23 25 -7 }}
| 231.399
| 15-odd-limit least squares
|-
|-
|
|
Line 198: Line 305:
|  
|  
| 232.258
| 232.258
|  
| 62c val
|-
|-
|
|
| 129140163/1500625
| 13/9
| 232.418
| 9-odd-limit least squares
|-
|
| 18/13
| 232.676
| 232.676
|  
|  
Line 223: Line 325:
|  
|  
| 233.333
| 233.333
|  
| 36c val
|-
|-
|
|
Line 231: Line 333:
|-
|-
|
|
| 4/3
| 3/2
| 233.985
| 233.985
|  
|  
Line 238: Line 340:
| 21/17
| 21/17
| 234.274
| 234.274
|
|-
|
|
|
| 234.485
| 2.3.7.17 subgroup least squares {{clarify}}
|-
|-
| 9\46
| 9\46
|  
|  
| 234.783
| 234.783
|  
| 46ce val
|-
|-
|
|
Line 263: Line 360:
|  
|  
| 235.714
| 235.714
|  
| 56ccee val
|-
|-
|
|
| 14/13
| 13/7
| 235.851
| 235.851
|  
|  
|-
|-
|
|
| 11/9
| 15/14
| 236.851
| 239.814
|  
|  
|-
|-
|
| 2\10
| 16/15
| 237.243
|  
|  
| 240.000
| Upper bound of 7- and 9-odd-limit diamond monotone
|-
|-
|
|
| 15/14
| 13/8
| 239.814
| 240.528
|  
|  
|-
|-
|
|
| 16/13
| 15/13
| 240.528
| 247.741
|  
|  
|-
|-
|
|
| 15/13
| 11/9
| 247.741
| 252.592
|  
|  
|}
|}


== Music ==
== Music ==
By Claudi Meneghin
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=2ziAZx03KF8 Lemba Suite, for Two Organs] (Prelude, Aria &amp; Fugue)
* [https://www.youtube.com/watch?v=2ziAZx03KF8 ''Lemba Suite, for Two Organs''] (Prelude, Aria & Fugue) in 8/7 eigenmonzo tuning
: in 8/7 eigenmonzo tuning
 
; [[Herman Miller]]
* [https://sites.google.com/site/teamouse/LembaGalatsia.mp3 ''Lemba Galatsia'']
* [https://sites.google.com/site/teamouse/lemba-gpo-test.mp3 ''GPO Lemb''a]
 
; [[Billy Stiltner]]
* [https://billystiltner.bandcamp.com/track/26lembaforxmt ''26lembaforxmt'']


By [[Herman Miller]]
== Notes ==
* [https://sites.google.com/site/teamouse/LembaGalatsia.mp3 Lemba Galatsia]
<references/>
* [https://sites.google.com/site/teamouse/lemba-gpo-test.mp3 GPO Lemba]


[[Category:Temperaments]]
[[Category:Lemba| ]] <!-- main article -->
[[Category:Lemba| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Jubilismic clan]]
[[Category:Jubilismic clan]]
[[Category:Avicennmic temperaments]]
[[Category:Avicennmic temperaments]]
[[Category:Gamelismic clan]]
[[Category:Gamelismic clan]]
[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 05:59, 25 February 2026

Lemba
Subgroups 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis 50/49, 525/512 (7-limit);
45/44, 50/49, 385/384 (11-limit);
45/44, 50/49, 65/64, 78/77
(13-limit)
Reduced mapping ⟨2; 3 -1 -1 5 1]
ET join 10 & 16
Generators (CWE) ~8/7 = 231.2 ¢
MOS scales 4L 2s, 6L 4s, 10L 6s
Ploidacot diploid tricot
Minimax error 9-odd-limit: 17.5 ¢;
13-odd-limit: 21.5 ¢
Target scale size 9-odd-limit: 16 notes;
13-odd-limit: 16 notes

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, 11/10 787.0 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

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~8/7 = 232.9268 ¢ CWE: ~8/7 = 232.2655 ¢ POTE: ~8/7 = 232.0888 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~8/7 = 231.9971 ¢ CWE: ~8/7 = 231.1781 ¢ POTE: ~8/7 = 230.9742 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~8/7 = 232.0999 ¢ CWE: ~8/7 = 231.1617 ¢ POTE: ~8/7 = 230.9665 ¢

Target tunings

Minimax tunings
Target Generator Eigenmonzo*
5-odd-limit ~8/7 = 228.910 ¢ 5/3
7-odd-limit ~8/7 = 231.174 ¢ 7/4
9-odd-limit ~8/7 = 231.085 ¢ 9/5
11-odd-limit ~8/7 = 231.085 ¢ 9/5
13-odd-limit ~8/7 = 231.085 ¢ 9/5
15-odd-limit ~8/7 = 230.415 ¢ 11/7
Least squares tunings
Target Generator Eigenmonzo*
7-odd-limit ~8/7 = 231.298 ¢ [0 -11 5 5⟩
9-odd-limit ~8/7 = 232.418 ¢ [0 17 -4 -4⟩
11-odd-limit ~8/7 = 231.294 ¢ [0 17 -6 -6 6⟩
13-odd-limit ~8/7 = 231.250 ¢ [0 63 -20 -20 22 -6⟩
15-odd-limit ~8/7 = 231.399 ¢ [0 66 -17 -23 25 -7⟩

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged interval) 		Generator (¢) Comments
5/4 213.686
15/11 221.016
11/6 224.681
3\16 225.000 Lower bound of 7-odd-limit diamond monotone
13/10 227.107
11/10 227.501
13/11 227.698
8\42 228.571 42bc val
5/3 228.910 5-odd-limit minimax
(2 - Φ)\2 229.179 Golden Lemba[1]
21/13 230.253
11/8 230.264
11/7 230.415 15-odd-limit minimax
13/12 230.714
5\26 230.769 Lower bound of 9-odd-limit diamond monotone
11- and 13-odd-limit diamond monotone (singleton)
9/5 231.085 9-, 11- and 13-odd-limit minimax
7/4 231.174 7-odd-limit minimax
17/13 232.213
12\62 232.258 62c val
13/9 232.676
Φ 233.090
7/6 233.282
7\36 233.333 36c val
9/7 233.583
3/2 233.985
21/17 234.274
9\46 234.783 46ce val
17/16 234.985
21/16 235.390
11\56 235.714 56ccee val
13/7 235.851
15/14 239.814
2\10 240.000 Upper bound of 7- and 9-odd-limit diamond monotone
13/8 240.528
15/13 247.741
11/9 252.592

Music

Claudi Meneghin
Herman Miller
Billy Stiltner

Notes

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