Mothra
| Mothra |
((2.3.5.7) 21-odd limit) ? ¢
((2.3.5.7) 21-odd limit) ? notes
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 (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-approximated; 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. 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.
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 | |||
| 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 sP5 | ||
| 5/4 | 232.193 | 1/4-comma meantone fifth | ||
| 17/13 | 232.214 | As sP4 | ||
| 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 | |||
| 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 5- to 9-odd-limit diamond monotone |
* Besides the octave
Music
Prelude for solo piano in mothra16, brat 4 tuning by Chris Vaisvil