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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.
{{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.  
 
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.  


See [[Jubilismic clan #Lemba]] for more technical data.  
See [[Jubilismic clan #Lemba]] for more technical data.


== Interval chain ==
== Interval chain ==
Line 56: Line 75:
| 6
| 6
| 187.0
| 187.0
| '''9/8'''
| '''9/8''', 11/10
| 787.0
| 787.0
| 11/7
| 11/7
Line 75: Line 94:


== Tunings ==
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
Line 92: Line 112:


{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
Line 108: Line 128:


{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
Line 121: Line 141:
| CWE: ~8/7 = 231.1617{{c}}
| CWE: ~8/7 = 231.1617{{c}}
| POTE: ~8/7 = 230.9665{{c}}
| POTE: ~8/7 = 230.9665{{c}}
|}
=== 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 }}
|}
|}


Line 142: Line 223:
|-
|-
|
|
| 12/11
| 11/6
| 224.681
| 224.681
|  
|  
Line 149: Line 230:
|  
|  
| 225.000
| 225.000
|  
| Lower bound of 7-odd-limit diamond monotone
|-
|-
|
|
Line 169: Line 250:
|  
|  
| 228.571
| 228.571
|  
| 42bc val
|-
|-
|
|
| 6/5
| 5/3
| 228.910
| 228.910
| 5-odd-limit minimax
| 5-odd-limit minimax
Line 192: Line 273:
|-
|-
|
|
| 14/11
| 11/7
| 230.415
| 230.415
| 15-odd-limit minimax
| 15-odd-limit minimax
Line 204: 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 244: Line 305:
|  
|  
| 232.258
| 232.258
|  
| 62c val
|-
|
| 129140163/1500625
| 232.418
| 9-odd-limit least squares
|-
|-
|
|
| 18/13
| 13/9
| 232.676
| 232.676
|  
|  
Line 269: Line 325:
|  
|  
| 233.333
| 233.333
|  
| 36c val
|-
|-
|
|
Line 277: Line 333:
|-
|-
|
|
| 4/3
| 3/2
| 233.985
| 233.985
|  
|  
Line 289: Line 345:
|  
|  
| 234.783
| 234.783
|  
| 46ce val
|-
|-
|
|
Line 304: Line 360:
|  
|  
| 235.714
| 235.714
|  
| 56ccee val
|-
|-
|
|
| 14/13
| 13/7
| 235.851
| 235.851
|  
|  
Line 315: Line 371:
| 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 339: Line 400:
* [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 ==
Retrieved from "https://en.xen.wiki/w/Lemba"