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{{Infobox Regtemp
{{Interwiki
| Title = Slendric
| en = Mothra
| de = Slendrisch #Mothra
}}
{{Infobox regtemp
| Title = Mothra
| Subgroups = 2.3.5.7
| Subgroups = 2.3.5.7
| Comma basis = [[81/80]], [[1029/1024]]
| Comma basis = [[81/80]], [[1029/1024]]
| Edo join 1 = 26 | Edo join 2 = 31
| Edo join 1 = 26 | Edo join 2 = 31
| Generator = 8/7 | Generator tuning = 232.4 | Optimization method = CTE
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], ... [[5L 21s]]
| Mapping = 1; 3 12 -1
| 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)
| Pergen = (P8, P5/3)
| Odd limit 1 = 7 | Mistuning 1 = ? | Complexity 1 = ?
| Odd limit 1 = 7 | Mistuning 1 = 5.4 | Complexity 1 = 31
| Odd limit 2 = (2.3.5.7) 27 | Mistuning 2 = ? | Complexity 2 = ?
| 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.


'''Mothra''' is a temperament in the [[7-limit]] that is a strong extension to [[slendric]], which is defined by splitting the interval of [[3/2]] into three [[8/7]]s and 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]] is larger here. [[EDOs]] that support mothra include [[26edo]], [[31edo]], and [[36edo]], and 31 is a particularly good tuning.
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 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]].
For technical data, see [[Gamelismic clan #Mothra]].


== Interval chains ==
== Intervals ==
In the following tables, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.  
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-all right-2"
{| class="wikitable sortable center-1 center-2 right-3"
|-
|-
! rowspan="3" | # !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
! 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 undecimal extensions
! rowspan="2" | 7-limit intervals !! colspan="2" | Intervals of 11-limit extensions
|-
|-
! Undecimal mothra !! Mosura
! Undecimal mothra !! Mosura
|-
|-
| 0
| 0
| P1
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
Line 36: Line 44:
|-
|-
| 1
| 1
| SM2
| 232.3
| 232.3
| '''8/7'''
| '''8/7'''
Line 42: Line 51:
|-
|-
| 2
| 2
| s4
| 464.5
| 464.5
| '''21/16''', 35/27, 64/49
| '''21/16''', 35/27, 64/49
Line 48: Line 58:
|-
|-
| 3
| 3
| P5
| 696.8
| 696.8
| '''3/2'''
| '''3/2'''
Line 54: Line 65:
|-
|-
| 4
| 4
| SM6
| 929.0
| 929.0
| 12/7
| 12/7
Line 60: Line 72:
|-
|-
| 5
| 5
| s8
| 1161.3
| 1161.3
| 35/18, 63/32, 96/49
| 35/18, 63/32, 96/49
Line 66: Line 79:
|-
|-
| 6
| 6
| M2
| 193.5
| 193.5
| '''9/8''', 10/9
| '''9/8''', 10/9
Line 72: Line 86:
|-
|-
| 7
| 7
| SM3
| 425.8
| 425.8
| 9/7
| 9/7
Line 78: Line 93:
|-
|-
| 8
| 8
| s5
| 658.0
| 658.0
| 35/24, 72/49
| 35/24, 72/49
Line 84: Line 100:
|-
|-
| 9
| 9
| M6
| 890.3
| 890.3
| 5/3, 27/16
| 5/3, 27/16
Line 90: Line 107:
|-
|-
| 10
| 10
| SM7
| 1122.5
| 1122.5
| 40/21, 27/14
| 40/21, 27/14
Line 96: Line 114:
|-
|-
| 11
| 11
| sM2
| 154.8
| 154.8
| 35/32, 54/49
| 35/32, 54/49
Line 102: Line 121:
|-
|-
| 12
| 12
| M3
| 387.0
| 387.0
| '''5/4'''
| '''5/4'''
Line 108: Line 128:
|-
|-
| 13
| 13
| SA4
| 619.3
| 619.3
| 10/7
| 10/7
Line 114: Line 135:
|-
|-
| 14
| 14
| sM6
| 851.5
| 851.5
| 80/49
| 80/49
Line 120: Line 142:
|-
|-
| 15
| 15
| M7
| 1083.8
| 1083.8
| '''15/8''', 50/27
| '''15/8''', 50/27
Line 126: Line 149:
|-
|-
| 16
| 16
| SA1
| 116.0
| 116.0
| 15/14
| 15/14
Line 132: Line 156:
|-
|-
| 17
| 17
| sM3
| 348.3
| 348.3
| 60/49
| 60/49
Line 138: Line 163:
|-
|-
| 18
| 18
| A4
| 580.5
| 580.5
| 25/18, 45/32
| 25/18, 45/32
Line 144: Line 170:
|-
|-
| 19
| 19
| SA5
| 812.8
| 812.8
| 45/28, 100/63
| 45/28, 100/63
Line 150: Line 177:
|-
|-
| 20
| 20
| sM7
| 1045.0
| 1045.0
| 90/49
| 90/49
Line 156: Line 184:
|-
|-
| 21
| 21
| A1
| 77.3
| 77.3
| 25/24
| 25/24
Line 162: Line 191:
|-
|-
| 22
| 22
| SA2
| 309.5
| 309.5
| 25/21
| 25/21
Line 168: Line 198:
|-
|-
| 23
| 23
| sA4
| 541.8
| 541.8
|  
|  
Line 174: Line 205:
|-
|-
| 24
| 24
| A5
| 774.0
| 774.0
| 25/16
| 25/16
Line 180: Line 212:
|-
|-
| 25
| 25
| SA6
| 1006.3
| 1006.3
| 25/14
| 25/14
Line 186: Line 219:
|-
|-
| 26
| 26
| sA1
| 38.5
| 38.5
| 50/49
| 50/49
Line 191: Line 225:
| 33/32, 55/54
| 33/32, 55/54
|}
|}
<nowiki/>* In 7-limit [[CWE tuning]]
<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 }}


== Tuning spectrum ==
Vals refer to the appropriate undecimal extension in the edo's range.
Vals refer to the appropriate undecimal extension in the EDO's range.


{| class="wikitable center-all left-4 left-5"
{| class="wikitable center-all left-4 left-5"
Line 275: Line 329:
|  
|  
| 150be val
| 150be val
|-
|
| [[19/17]]
| 232.093
|
| As M2
|-
|-
|  
|  
Line 281: Line 341:
|  
|  
|  
|  
|-
|
| [[19/13]]
| 232.123
|
| As s5
|-
|-
|  
|  
Line 286: Line 352:
| 232.193
| 232.193
|  
|  
| 1/4-comma meantone fifth
| 1/4-comma meantone fifth, (7-limit) 5- through 21-odd-limit minimax
|-
|
| [[17/13]]
| 232.214
|
| As s4
|-
|-
| '''[[31edo|6\31]]'''
| [[31edo|6\31]]
|  
|  
| '''232.258'''
| 232.258
| ↑ Undecimal mothra (99/98) <br /> ↓ Mosura (176/175)
| ↑ Undecimal mothra (99/98) <br /> ↓ Mosura (176/175)
| '''Lower bound of (7-limit) 15- and 21-odd-limit diamond monotone'''
|  
|-
|-
|  
|  
Line 323: Line 395:
|  
|  
|  
|  
|-
| [[165edo|32\165]]
|
| 232.727
|
| 165bc val
|-
|-
| [[67edo|13\67]]
| [[67edo|13\67]]
Line 370: Line 436:
| '''240.000'''
| '''240.000'''
|  
|  
| 5e val, '''upper bound of (7-limit) 5- to 21-odd-limit diamond monotone'''
| 5e val, '''upper bound of 5- to 9-odd-limit diamond monotone'''
|}
|}
<nowiki/>* Besides the octave
<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:Mothra| ]] <!-- main article -->
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