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{{Infobox Regtemp
{{Interwiki
| en = Mothra
| de = Slendrisch #Mothra
}}
{{Infobox regtemp
| Title = Mothra
| 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) 21 | 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]] (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.


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-approximated. 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]].
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 38: Line 44:
|-
|-
| 1
| 1
| SM2
| 232.3
| 232.3
| '''8/7'''
| '''8/7'''
Line 44: Line 51:
|-
|-
| 2
| 2
| s4
| 464.5
| 464.5
| '''21/16''', 35/27, 64/49
| '''21/16''', 35/27, 64/49
Line 50: Line 58:
|-
|-
| 3
| 3
| P5
| 696.8
| 696.8
| '''3/2'''
| '''3/2'''
Line 56: Line 65:
|-
|-
| 4
| 4
| SM6
| 929.0
| 929.0
| 12/7
| 12/7
Line 62: Line 72:
|-
|-
| 5
| 5
| s8
| 1161.3
| 1161.3
| 35/18, 63/32, 96/49
| 35/18, 63/32, 96/49
Line 68: Line 79:
|-
|-
| 6
| 6
| M2
| 193.5
| 193.5
| '''9/8''', 10/9
| '''9/8''', 10/9
Line 74: Line 86:
|-
|-
| 7
| 7
| SM3
| 425.8
| 425.8
| 9/7
| 9/7
Line 80: Line 93:
|-
|-
| 8
| 8
| s5
| 658.0
| 658.0
| 35/24, 72/49
| 35/24, 72/49
Line 86: Line 100:
|-
|-
| 9
| 9
| M6
| 890.3
| 890.3
| 5/3, 27/16
| 5/3, 27/16
Line 92: Line 107:
|-
|-
| 10
| 10
| SM7
| 1122.5
| 1122.5
| 40/21, 27/14
| 40/21, 27/14
Line 98: Line 114:
|-
|-
| 11
| 11
| sM2
| 154.8
| 154.8
| 35/32, 54/49
| 35/32, 54/49
Line 104: Line 121:
|-
|-
| 12
| 12
| M3
| 387.0
| 387.0
| '''5/4'''
| '''5/4'''
Line 110: Line 128:
|-
|-
| 13
| 13
| SA4
| 619.3
| 619.3
| 10/7
| 10/7
Line 116: Line 135:
|-
|-
| 14
| 14
| sM6
| 851.5
| 851.5
| 80/49
| 80/49
Line 122: Line 142:
|-
|-
| 15
| 15
| M7
| 1083.8
| 1083.8
| '''15/8''', 50/27
| '''15/8''', 50/27
Line 128: Line 149:
|-
|-
| 16
| 16
| SA1
| 116.0
| 116.0
| 15/14
| 15/14
Line 134: Line 156:
|-
|-
| 17
| 17
| sM3
| 348.3
| 348.3
| 60/49
| 60/49
Line 140: Line 163:
|-
|-
| 18
| 18
| A4
| 580.5
| 580.5
| 25/18, 45/32
| 25/18, 45/32
Line 146: Line 170:
|-
|-
| 19
| 19
| SA5
| 812.8
| 812.8
| 45/28, 100/63
| 45/28, 100/63
Line 152: Line 177:
|-
|-
| 20
| 20
| sM7
| 1045.0
| 1045.0
| 90/49
| 90/49
Line 158: Line 184:
|-
|-
| 21
| 21
| A1
| 77.3
| 77.3
| 25/24
| 25/24
Line 164: Line 191:
|-
|-
| 22
| 22
| SA2
| 309.5
| 309.5
| 25/21
| 25/21
Line 170: Line 198:
|-
|-
| 23
| 23
| sA4
| 541.8
| 541.8
|  
|  
Line 176: Line 205:
|-
|-
| 24
| 24
| A5
| 774.0
| 774.0
| 25/16
| 25/16
Line 182: Line 212:
|-
|-
| 25
| 25
| SA6
| 1006.3
| 1006.3
| 25/14
| 25/14
Line 188: Line 219:
|-
|-
| 26
| 26
| sA1
| 38.5
| 38.5
| 50/49
| 50/49
Line 193: 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 294: Line 346:
| 232.123
| 232.123
|  
|  
| As sP5
| As s5
|-
|-
|  
|  
Line 300: Line 352:
| 232.193
| 232.193
|  
|  
| 1/4-comma meantone fifth
| 1/4-comma meantone fifth, (7-limit) 5- through 21-odd-limit minimax
|-
|-
|  
|  
Line 306: Line 358:
| 232.214
| 232.214
|  
|  
| As sP4
| As s4
|-
|-
| [[31edo|6\31]]
| [[31edo|6\31]]
Line 343: Line 395:
|  
|  
|  
|  
|-
| [[165edo|32\165]]
|
| 232.727
|
| 165bc val
|-
|-
| [[67edo|13\67]]
| [[67edo|13\67]]
Line 395: Line 441:


== Music ==
== Music ==
[http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] by [http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ Chris Vaisvil]
; [[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 -->

Latest revision as of 07:32, 20 March 2026

Mothra
Subgroups 2.3.5.7
Comma basis 81/80, 1029/1024
Reduced mapping ⟨1; 3 12 -1]
ET join 26 & 31
Generators (CWE) ~8/7 = 232.3 ¢
MOS scales 1L 4s, 5L 1s, 5L 6s, …, 5L 21s
Ploidacot tricot
Pergen (P8, P5/3)
Minimax error 7-odd-limit: 5.4 ¢;
(2.3.5.7) 21-odd-limit: 10.8 ¢
Target scale size 7-odd-limit: 31 notes;
(2.3.5.7) 21-odd-limit: 36 notes

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 scales 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.

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 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.

# Extended
diatonic
interval		 Cents* Approximate ratios
7-limit intervals 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

* In 7-limit CWE tuning, octave reduced

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~8/7 = 232.3996 ¢ CWE: ~8/7 = 232.2514 ¢ POTE: ~8/7 = 232.1933 ¢

Tuning spectrum

See also: Slendric § Tuning spectrum

Vals refer to the appropriate undecimal extension in the edo's range.

Edo
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) Extension Comments
4\21 228.571 21c val, lower bound of 5-odd-limit diamond monotone
10/9 230.401 1/2-comma meantone fifth
5\26 230.769 Lower bound of 7- and 9-odd-limit diamond monotone
8/7 231.174 Untempered tuning
16\83 231.325 83bc val
40/21 231.553
11\57 231.579
5/3 231.595 1/3-comma meantone fifth
17\88 231.818
23\119 231.933 119be val
25/24 231.937 2/7-comma meantone fifth
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
6\31 232.258 ↑ Undecimal mothra (99/98)
↓ Mosura (176/175)
15/14 232.465
31\160 232.500 160be val
15/8 232.551 1/5-comma meantone fifth
25\129 232.558
19\98 232.653
13\67 232.836
96/49 232.861 1/5-comma slendric
20\103 233.010 103ce val
12/7 233.282 1/4-comma slendric
7\36 233.333
3/2 233.985 1/3-comma slendric
1\5 240.000 5e val, upper bound of 5- to 9-odd-limit diamond monotone

* Besides the octave

Music

Chris Vaisvil
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