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Mothra: Difference between revisions

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Line 8: Line 8:
| Mapping = 1; 3 12 -1
| Mapping = 1; 3 12 -1
| 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''' 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.
'''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.


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.


{| class="wikitable sortable center-all right-2"
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" | 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 41:
|-
|-
| 1
| 1
| SM2
| 232.3
| 232.3
| '''8/7'''
| '''8/7'''
Line 44: Line 48:
|-
|-
| 2
| 2
| s4
| 464.5
| 464.5
| '''21/16''', 35/27, 64/49
| '''21/16''', 35/27, 64/49
Line 50: Line 55:
|-
|-
| 3
| 3
| P5
| 696.8
| 696.8
| '''3/2'''
| '''3/2'''
Line 56: Line 62:
|-
|-
| 4
| 4
| SM6
| 929.0
| 929.0
| 12/7
| 12/7
Line 62: Line 69:
|-
|-
| 5
| 5
| s8
| 1161.3
| 1161.3
| 35/18, 63/32, 96/49
| 35/18, 63/32, 96/49
Line 68: Line 76:
|-
|-
| 6
| 6
| M2
| 193.5
| 193.5
| '''9/8''', 10/9
| '''9/8''', 10/9
Line 74: Line 83:
|-
|-
| 7
| 7
| SM3
| 425.8
| 425.8
| 9/7
| 9/7
Line 80: Line 90:
|-
|-
| 8
| 8
| s5
| 658.0
| 658.0
| 35/24, 72/49
| 35/24, 72/49
Line 86: Line 97:
|-
|-
| 9
| 9
| M6
| 890.3
| 890.3
| 5/3, 27/16
| 5/3, 27/16
Line 92: Line 104:
|-
|-
| 10
| 10
| SM7
| 1122.5
| 1122.5
| 40/21, 27/14
| 40/21, 27/14
Line 98: Line 111:
|-
|-
| 11
| 11
| sM2
| 154.8
| 154.8
| 35/32, 54/49
| 35/32, 54/49
Line 104: Line 118:
|-
|-
| 12
| 12
| M3
| 387.0
| 387.0
| '''5/4'''
| '''5/4'''
Line 110: Line 125:
|-
|-
| 13
| 13
| SA4
| 619.3
| 619.3
| 10/7
| 10/7
Line 116: Line 132:
|-
|-
| 14
| 14
| sM6
| 851.5
| 851.5
| 80/49
| 80/49
Line 122: Line 139:
|-
|-
| 15
| 15
| M7
| 1083.8
| 1083.8
| '''15/8''', 50/27
| '''15/8''', 50/27
Line 128: Line 146:
|-
|-
| 16
| 16
| SA1
| 116.0
| 116.0
| 15/14
| 15/14
Line 134: Line 153:
|-
|-
| 17
| 17
| sM3
| 348.3
| 348.3
| 60/49
| 60/49
Line 140: Line 160:
|-
|-
| 18
| 18
| A4
| 580.5
| 580.5
| 25/18, 45/32
| 25/18, 45/32
Line 146: Line 167:
|-
|-
| 19
| 19
| SA5
| 812.8
| 812.8
| 45/28, 100/63
| 45/28, 100/63
Line 152: Line 174:
|-
|-
| 20
| 20
| sM7
| 1045.0
| 1045.0
| 90/49
| 90/49
Line 158: Line 181:
|-
|-
| 21
| 21
| A1
| 77.3
| 77.3
| 25/24
| 25/24
Line 164: Line 188:
|-
|-
| 22
| 22
| SA2
| 309.5
| 309.5
| 25/21
| 25/21
Line 170: Line 195:
|-
|-
| 23
| 23
| sA4
| 541.8
| 541.8
|  
|  
Line 176: Line 202:
|-
|-
| 24
| 24
| A5
| 774.0
| 774.0
| 25/16
| 25/16
Line 182: Line 209:
|-
|-
| 25
| 25
| SA6
| 1006.3
| 1006.3
| 25/14
| 25/14
Line 188: Line 216:
|-
|-
| 26
| 26
| sA1
| 38.5
| 38.5
| 50/49
| 50/49
Line 193: Line 222:
| 33/32, 55/54
| 33/32, 55/54
|}
|}
<nowiki/>* In 7-limit [[CWE tuning]]
<nowiki/>* In 7-limit [[CWE tuning]], octave reduced


== Tuning spectrum ==
== Tuning spectrum ==
{{see also|Slendric #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.


Line 294: Line 325:
| 232.123
| 232.123
|  
|  
| As sP5
| As s5
|-
|-
|  
|  
Line 300: Line 331:
| 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 337:
| 232.214
| 232.214
|  
|  
| As sP4
| As s4
|-
|-
| [[31edo|6\31]]
| [[31edo|6\31]]
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