Mothra: Difference between revisions

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#redirect [[Gamelismic clan #Mothra]]
{{Infobox Regtemp
| Title = Slendric
| Subgroups = 2.3.5.7
| Comma basis = [[81/80]], [[1029/1024]]
| 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
| Pergen = (P8, P5/3)
| Odd limit 1 = 7 | Mistuning 1 = ? | Complexity 1 = ?
| Odd limit 2 = (2.3.5.7) 27 | Mistuning 2 = ? | Complexity 2 = ?
}}


[[Category:Temperaments]]
'''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.
 
For technical data, see [[Gamelismic clan #Mothra]].
 
== Interval chains ==
In the following tables, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.
 
{| class="wikitable sortable center-all right-2"
|-
! rowspan="3" | # !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
|-
! rowspan="2" | 7-limit intervals !! colspan="2" | Intervals of undecimal extensions
|-
! Undecimal mothra !! Mosura
|-
| 0
| 0.0
| '''1/1'''
|
|
|-
| 1
| 232.3
| '''8/7'''
| 55/48, 63/55
| 25/22
|-
| 2
| 464.5
| '''21/16''', 35/27, 64/49
| 55/42, 72/55
| 33/25
|-
| 3
| 696.8
| '''3/2'''
| 49/33
|
|-
| 4
| 929.0
| 12/7
| 55/32, 56/33
|
|-
| 5
| 1161.3
| 35/18, 63/32, 96/49
| 55/28, 64/33, 108/55
| 88/45
|-
| 6
| 193.5
| '''9/8''', 10/9
| 49/44, 55/49
|
|-
| 7
| 425.8
| 9/7
| 14/11
|
|-
| 8
| 658.0
| 35/24, 72/49
| '''16/11'''
| 22/15
|-
| 9
| 890.3
| 5/3, 27/16
|
|
|-
| 10
| 1122.5
| 40/21, 27/14
| 21/11
|
|-
| 11
| 154.8
| 35/32, 54/49
| 12/11
| 11/10
|-
| 12
| 387.0
| '''5/4'''
|
| 44/35
|-
| 13
| 619.3
| 10/7
| 63/44
|
|-
| 14
| 851.5
| 80/49
| 18/11
| 44/27, 33/20
|-
| 15
| 1083.8
| '''15/8''', 50/27
|
| 66/35
|-
| 16
| 116.0
| 15/14
| 35/33
|
|-
| 17
| 348.3
| 60/49
| 27/22, 40/33
| 11/9
|-
| 18
| 580.5
| 25/18, 45/32
|
| 88/63
|-
| 19
| 812.8
| 45/28, 100/63
| 35/22
|
|-
| 20
| 1045.0
| 90/49
| 20/11
| 11/6
|-
| 21
| 77.3
| 25/24
|
| 22/21
|-
| 22
| 309.5
| 25/21
|
|
|-
| 23
| 541.8
|
| 15/11
| '''11/8'''
|-
| 24
| 774.0
| 25/16
|
| 11/7
|-
| 25
| 1006.3
| 25/14
|
| 88/49
|-
| 26
| 38.5
| 50/49
| 45/44
| 33/32, 55/54
|}
<nowiki/>* In 7-limit [[CWE tuning]]
 
== 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
|-
|
| [[10/7]]
| 232.114
|
|
|-
|
| [[5/4]]
| 232.193
|
| 1/4-comma meantone fifth
|-
| '''[[31edo|6\31]]'''
|
| '''232.258'''
| ↑ Undecimal mothra (99/98) <br /> ↓ Mosura (176/175)
| '''Lower bound of (7-limit) 15- and 21-odd-limit diamond monotone'''
|-
|
| [[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
|
|
|-
| [[165edo|32\165]]
|
| 232.727
|
| 165bc val
|-
| [[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 (7-limit) 5- to 21-odd-limit diamond monotone'''
|}
<nowiki/>* Besides the octave
 
[[Category:Mothra| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Meantone family]]
[[Category:Meantone family]]
[[Category:Gamelismic clan]]
[[Category:Gamelismic clan]]
[[Category:Orwellismic temperaments]]
[[Category:Orwellismic temperaments]]

Revision as of 20:22, 29 April 2025

Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value).

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 8/7s 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 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 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.

For technical data, see Gamelismic clan #Mothra.

Interval chains

In the following tables, odd harmonics and subharmonics 1–21 are labeled in bold.

# Cents* Approximate ratios
7-limit intervals Intervals of undecimal extensions
Undecimal mothra Mosura
0 0.0 1/1
1 232.3 8/7 55/48, 63/55 25/22
2 464.5 21/16, 35/27, 64/49 55/42, 72/55 33/25
3 696.8 3/2 49/33
4 929.0 12/7 55/32, 56/33
5 1161.3 35/18, 63/32, 96/49 55/28, 64/33, 108/55 88/45
6 193.5 9/8, 10/9 49/44, 55/49
7 425.8 9/7 14/11
8 658.0 35/24, 72/49 16/11 22/15
9 890.3 5/3, 27/16
10 1122.5 40/21, 27/14 21/11
11 154.8 35/32, 54/49 12/11 11/10
12 387.0 5/4 44/35
13 619.3 10/7 63/44
14 851.5 80/49 18/11 44/27, 33/20
15 1083.8 15/8, 50/27 66/35
16 116.0 15/14 35/33
17 348.3 60/49 27/22, 40/33 11/9
18 580.5 25/18, 45/32 88/63
19 812.8 45/28, 100/63 35/22
20 1045.0 90/49 20/11 11/6
21 77.3 25/24 22/21
22 309.5 25/21
23 541.8 15/11 11/8
24 774.0 25/16 11/7
25 1006.3 25/14 88/49
26 38.5 50/49 45/44 33/32, 55/54

* In 7-limit CWE tuning

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
[[119edo|23\119] 231.933 119be val
25/24 231.937 2/7-comma meantone fifth
[[150edo|29\150] 232.000 150be val
10/7 232.114
5/4 232.193 1/4-comma meantone fifth
6\31 232.258 ↑ Undecimal mothra (99/98)
↓ Mosura (176/175) 		Lower bound of (7-limit) 15- and 21-odd-limit diamond monotone
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
32\165 232.727 165bc val
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 (7-limit) 5- to 21-odd-limit diamond monotone

* Besides the octave

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