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'''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. 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.
'''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. 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.  
The name is from [[Herman Miller]]'s conlang name for the temperament.  
Line 6: Line 6:


== 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 ==
=== Prime-optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized 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}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized 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: [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;" | 13-limit prime-optimized 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}}
|}


Gencom mapping: [{{val| 2 2 5 6 5 7 }}, {{val| 0 3 -1 -1 5 1 }}]
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings
|-
! Target
! Generator
! Eigenmonzo*
|-
| 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"
{| class="wikitable center-all left-4"
|-
|-
! EDO<br>generator
! Edo<br>generator
! [[eigenmonzo|eigenmonzo<br>(unchanged interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]
! generator (¢)
! Generator (¢)
! comments
! Comments
|-
|-
|
|
Line 98: Line 204:
|-
|-
|
|
| 12/11
| 11/6
| 224.681
| 224.681
|  
|  
Line 105: Line 211:
|  
|  
| 225.000
| 225.000
|  
| Lower bound of 7-odd-limit diamond monotone
|-
|-
|
|
Line 125: Line 231:
|  
|  
| 228.571
| 228.571
|  
| 42bc val
|-
|-
|
|
| 6/5
| 5/3
| 228.910
| 228.910
| 5-odd-limit minimax
| 5-odd-limit minimax
Line 148: Line 254:
|-
|-
|
|
| 14/11
| 11/7
| 230.415
| 230.415
| 15-odd-limit minimax
| 15-odd-limit minimax
Line 160: Line 266:
|  
|  
| 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 200: Line 286:
|  
|  
| 232.258
| 232.258
|  
| 62c val
|-
|-
|
|
| 129140163/1500625
| 13/9
| 232.418
| 9-odd-limit least squares
|-
|
| 18/13
| 232.676
| 232.676
|  
|  
Line 225: Line 306:
|  
|  
| 233.333
| 233.333
|  
| 36c val
|-
|-
|
|
Line 233: Line 314:
|-
|-
|
|
| 4/3
| 3/2
| 233.985
| 233.985
|  
|  
Line 245: Line 326:
|  
|  
| 234.783
| 234.783
|  
| 46ce val
|-
|-
|
|
Line 260: Line 341:
|  
|  
| 235.714
| 235.714
|  
| 56ccee val
|-
|-
|
|
| 14/13
| 13/7
| 235.851
| 235.851
|  
|  
Line 271: Line 352:
| 239.814
| 239.814
|  
|  
|-
| 2\10
|
| 240.000
| Upper bound of 7- and 9-odd-limit diamond monotone
|-
|-
|
|
| 16/13
| 13/8
| 240.528
| 240.528
|  
|  
Line 290: Line 376:
== Music ==
== Music ==
; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=2ziAZx03KF8 ''Lemba Suite, for Two Organs''] (Prelude, Aria &amp; Fugue) in 8/7 eigenmonzo tuning
* [https://www.youtube.com/watch?v=2ziAZx03KF8 ''Lemba Suite, for Two Organs''] (Prelude, Aria & Fugue) in 8/7 eigenmonzo tuning


; [[Herman Miller]]
; [[Herman Miller]]
* [https://sites.google.com/site/teamouse/LembaGalatsia.mp3 ''Lemba Galatsia'']
* [https://sites.google.com/site/teamouse/LembaGalatsia.mp3 ''Lemba Galatsia'']
* [https://sites.google.com/site/teamouse/lemba-gpo-test.mp3 ''GPO Lemb''a]
* [https://sites.google.com/site/teamouse/lemba-gpo-test.mp3 ''GPO Lemb''a]
[[Billy Stiltner]]
* [https://billystiltner.bandcamp.com/track/26lembaforxmt ''26lembaforxmt'']


== Notes ==
== Notes ==

Latest revision as of 09:03, 17 August 2025

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

Prime-optimized 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 ¢

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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