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#redirect [[Meantone family#Mothra]]
{{Interwiki
| en = Mothra
| de = Slendrisch #Mothra
}}
{{Infobox regtemp
| Title = Mothra
| Subgroups = 2.3.5.7
| Comma basis = [[81/80]], [[1029/1024]]
| Edo join 1 = 26 | Edo join 2 = 31
| Mapping = 1; 3 12 -1
| Generators = 8/7 | Generators tuning = 232.3 | Optimization method = CWE
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], …, [[5L 21s]]
| Pergen = (P8, P5/3)
| Odd limit 1 = 7 | Mistuning 1 = 5.4 | Complexity 1 = 31
| Odd limit 2 = (2.3.5.7) 21 | Mistuning 2 = 10.8 | Complexity 2 = 36
}}
'''Mothra''', also known as '''cynder''', is a temperament of the [[7-limit]] that is a strong extension to [[slendric]], which is defined by splitting a perfect fifth representing [[3/2]] into three intervals of [[8/7]], tempering out [[1029/1024]]. The fifth of mothra is flattened to a [[meantone]] fifth, so that it reaches [[5/4]] when stacked four times and [[81/80]] is tempered out, unlike that of the other slendric extension [[rodan]], which is sharpened from just. This has the effect of bringing the generator 8/7 considerably closer to just, and also allowing [[MOS scale]]s of mothra to be more melodically usable than those of other forms of slendric, as the structurally-pervasive small step known as the [[quark]] (the residue between the octave and 5 generators, representing [[49/48]], [[64/63]], and in mothra also [[36/35]]) is larger here. [[EDOs]] that support mothra include [[26edo]], [[31edo]], and [[36edo]], and 31 is a particularly good tuning.


[[Category:Temperaments]]
In the [[11-limit]], two extensions are of note: undecimal mothra (26 & 31), which tempers out [[99/98]], [[385/384]] and [[441/440]] to find the 11th harmonic at 8 generators down, and mosura (31 & 36), which tempers out [[176/175]] to find the 11th harmonic at 23 generators up. These two mappings merge at 31edo, which is therefore a uniquely suitable tuning for 11-limit mothra.
 
In higher limits, one may note that the two-generator interval closely approximates [[17/13]], and that the six-generator interval - the meantone whole tone of [[9/8]][[~]][[10/9]], approximates [[19/17]] - so that the 13:17:19 chord is well-represented; it is worth noting also that this chord is entirely included within the subtemperament obtained from taking every other generator of mothra, which is [[A-team]] (the crawma, [[83521/83486]], is the relevant comma tempered out here). This can be combined with the canonical mapping of 13 for each undecimal extension, which tempers out [[144/143]], to provide a natural route to the [[19-limit]].
 
For technical data, see [[Gamelismic clan #Mothra]].
 
== Intervals ==
As a strong extension of slendric, mothra's intervals can be expressed using the same system of extended diatonic interval naming [[Slendric #Interval categories|used for slendric]]. It is particularly convenient to use diatonic conventions for mothra, because its chain of fifths is meantone, and therefore 5/4 is simply read as a major third.
 
In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.
 
{| class="wikitable sortable center-1 center-2 right-3"
|-
! rowspan="3" | # !! rowspan="3" | Extended <br> diatonic <br> interval !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
|-
! rowspan="2" | 7-limit intervals !! colspan="2" | Intervals of 11-limit extensions
|-
! Undecimal mothra !! Mosura
|-
| 0
| P1
| 0.0
| '''1/1'''
|
|
|-
| 1
| SM2
| 232.3
| '''8/7'''
| 55/48, 63/55
| 25/22
|-
| 2
| s4
| 464.5
| '''21/16''', 35/27, 64/49
| 55/42, 72/55
| 33/25
|-
| 3
| P5
| 696.8
| '''3/2'''
| 49/33
|
|-
| 4
| SM6
| 929.0
| 12/7
| 55/32, 56/33
|
|-
| 5
| s8
| 1161.3
| 35/18, 63/32, 96/49
| 55/28, 64/33, 108/55
| 88/45
|-
| 6
| M2
| 193.5
| '''9/8''', 10/9
| 49/44, 55/49
|
|-
| 7
| SM3
| 425.8
| 9/7
| 14/11
|
|-
| 8
| s5
| 658.0
| 35/24, 72/49
| '''16/11'''
| 22/15
|-
| 9
| M6
| 890.3
| 5/3, 27/16
|
|
|-
| 10
| SM7
| 1122.5
| 40/21, 27/14
| 21/11
|
|-
| 11
| sM2
| 154.8
| 35/32, 54/49
| 12/11
| 11/10
|-
| 12
| M3
| 387.0
| '''5/4'''
|
| 44/35
|-
| 13
| SA4
| 619.3
| 10/7
| 63/44
|
|-
| 14
| sM6
| 851.5
| 80/49
| 18/11
| 44/27, 33/20
|-
| 15
| M7
| 1083.8
| '''15/8''', 50/27
|
| 66/35
|-
| 16
| SA1
| 116.0
| 15/14
| 35/33
|
|-
| 17
| sM3
| 348.3
| 60/49
| 27/22, 40/33
| 11/9
|-
| 18
| A4
| 580.5
| 25/18, 45/32
|
| 88/63
|-
| 19
| SA5
| 812.8
| 45/28, 100/63
| 35/22
|
|-
| 20
| sM7
| 1045.0
| 90/49
| 20/11
| 11/6
|-
| 21
| A1
| 77.3
| 25/24
|
| 22/21
|-
| 22
| SA2
| 309.5
| 25/21
|
|
|-
| 23
| sA4
| 541.8
|
| 15/11
| '''11/8'''
|-
| 24
| A5
| 774.0
| 25/16
|
| 11/7
|-
| 25
| SA6
| 1006.3
| 25/14
|
| 88/49
|-
| 26
| sA1
| 38.5
| 50/49
| 45/44
| 33/32, 55/54
|}
<nowiki/>* In 7-limit [[CWE tuning]], octave reduced
 
== 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 = 232.3996{{c}}
| CWE: ~8/7 = 232.2514{{c}}
| POTE: ~8/7 = 232.1933{{c}}
|}
 
=== Tuning spectrum ===
{{See also| Slendric #Tuning spectrum }}
 
Vals refer to the appropriate undecimal extension in the edo's range.
 
{| class="wikitable center-all left-4 left-5"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Extension
! Comments
|-
| '''[[21edo|4\21]]'''
|
| '''228.571'''
|
| 21c val, '''lower bound of 5-odd-limit diamond monotone'''
|-
|
| [[10/9]]
| 230.401
|
| 1/2-comma meantone fifth
|-
| '''[[26edo|5\26]]'''
|
| '''230.769'''
|
| '''Lower bound of 7- and 9-odd-limit diamond monotone'''
|-
|
| [[8/7]]
| 231.174
|
| Untempered tuning
|-
| [[83edo|16\83]]
|
| 231.325
|
| 83bc val
|-
|
| [[40/21]]
| 231.553
|
|
|-
| [[57edo|11\57]]
|
| 231.579
|
|
|-
|
| [[5/3]]
| 231.595
|
| 1/3-comma meantone fifth
|-
| [[88edo|17\88]]
|
| 231.818
|
|
|-
| [[119edo|23\119]]
|
| 231.933
|
| 119be val
|-
|
| [[25/24]]
| 231.937
|
| 2/7-comma meantone fifth
|-
| [[150edo|29\150]]
|
| 232.000
|
| 150be val
|-
|
| [[19/17]]
| 232.093
|
| As M2
|-
|
| [[10/7]]
| 232.114
|
|
|-
|
| [[19/13]]
| 232.123
|
| As s5
|-
|
| [[5/4]]
| 232.193
|
| 1/4-comma meantone fifth, (7-limit) 5- through 21-odd-limit minimax
|-
|
| [[17/13]]
| 232.214
|
| As s4
|-
| [[31edo|6\31]]
|
| 232.258
| ↑ Undecimal mothra (99/98) <br /> ↓ Mosura (176/175)
|
|-
|
| [[15/14]]
| 232.465
|
|
|-
| [[160edo|31\160]]
|
| 232.500
|
| 160be val
|-
|
| [[15/8]]
| 232.551
|
| 1/5-comma meantone fifth
|-
| [[129edo|25\129]]
|
| 232.558
|
|
|-
| [[98edo|19\98]]
|
| 232.653
|
|
|-
| [[67edo|13\67]]
|
| 232.836
|
|
|-
|
| [[96/49]]
| 232.861
|
| 1/5-comma slendric
|-
| [[103edo|20\103]]
|
| 233.010
|
| 103ce val
|-
|
| [[12/7]]
| 233.282
|
| 1/4-comma slendric
|-
| [[36edo|7\36]]
|
| 233.333
|
|
|-
|
| [[3/2]]
| 233.985
|
| 1/3-comma slendric
|-
| '''[[5edo|1\5]]'''
|
| '''240.000'''
|
| 5e val, '''upper bound of 5- to 9-odd-limit diamond monotone'''
|}
<nowiki/>* Besides the octave
 
== Music ==
; [[Chris Vaisvil]]
* ''Prelude for solo piano'' (2014) by [[Chris Vaisvil]] – [https://web.archive.org/web/20201127013310/http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 play] | [https://www.chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ blog] – in Mothra[16], brat 4 tuning
 
[[Category:Mothra| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Meantone family]]
[[Category:Meantone family]]
[[Category:Gamelismic clan]]
[[Category:Orwellismic temperaments]]
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