Xenharmonic Wiki

Lemba: Difference between revisions

← Older edit
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,170 edits
→Spectrum of lemba tunings by eigenmonzos: improve and standardize tuning spectra tables
+ mos
 
(26 intermediate revisions by 9 users not shown)
Line 1: Line 1:
'''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, 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}}
|}
 
{| 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}}
|}
 
=== 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
|}


Gencom map: [{{val| 2 2 5 6 5 7 }},[{{val| 0 3 -1 -1 5 1 }}]
{| 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 }}
|}


{| class="wikitable center-all"
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
|-
! ET<br>generator
! Edo<br>generator
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]
! supermajor<br>second (¢)
! Generator (¢)
! comments
! 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]]
* [http://soonlabel.com/xenharmonic/archives/1232 Lemba Suite] (Prelude, Aria &amp; Fugue) {{dead link}}
* [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: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]]
{{IoT}}
Retrieved from "https://en.xen.wiki/w/Lemba"