Mothra
| Mothra |
((2.3.5.7) 21-odd limit) 10.8 ¢
((2.3.5.7) 21-odd limit) 36 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.
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
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 | |||
| 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