Slendric
Slendric (also known as "Wonder" or "Gamelic") is the basic harmonic interpretation of the structure where the perfect fifth (~3/2) is split into three equal parts, each representing the interval 8/7. Since the 7th harmonic is less than 3 cents from just when 3/2 is pure, Slendric constitutes an exceptionally good rank-2 traversal of the 2.3.7 tuning space for its simplicity. Its corresponding comma is the difference between 3/2 and (8/7)3, which is 1029/1024.
Melodically, the Slendric generator stack forms a 5-note scale (1L 4s) that is nearly equipentatonic. MOSes further down the hierarchy (6, 11, 16, ... notes) can be thought of as the notes of a basic pentatonic form, inflected by multiples of a characteristic small interval known as the quark (representing a third of a diatonic semitone, and the commas 49/48 and 64/63 tempered together). As a result, these MOS scales tend to be extremely hard.
Slendric can exhibit a wide range of tunings, with fifths between those of 26edo (692c) and 56edo (707c), or generators roughly between 231 and 236c, while maintaining the recognizability of the 2.3.7 structure. Notable EDO tunings are in between these, and include EDOs that end in "1" or "6", i.e. 31edo, 36edo, 41edo, and 46edo.
Structural theory
General theory
Interval categories
It is possible to define the intervals of Slendric in terms of diatonic categories, for at three steps is the perfect fifth, and at every three steps further are all of the standard fifth-generated intervals. For the remaining steps, a single pair of inflections suffices: "up"/"down", which can be abbreviated with the prefixes S and s, respectively (standing in for "super" and "sub", which can be used synonymously). An "up" is rigorously defined to be an inflection by the "quark" of 49/48~64/63. The slendric generator is then the upmajor second, and therefore the 2-generator interval is a downfourth (as a major second and a perfect fourth together reach a perfect fifth) as well as a double-upmajor third. Between a major third and perfect fourth is a minor second, which is therefore equivalent to three repetitions of "up"; because of this equivalence, it is never necessary to attach more than one "up"/"down" to a diatonic interval.
The pentatonic framework
Instead of organizing the intervals according to larger and larger MOSes (none of which are proper until at least 26 notes), the intervals of Slendric can be organized according to how many steps of 5edo, or equivalently the 5-note MOS, they correspond to. The "major" interval of a class is the one that's just larger than the corresponding 5edo interval, and the "minor" interval is just smaller. Below are the intervals of the symmetric mode of Slendric[21] (5L 16s). The generator tuning here is 3/10-comma, where the quark is exactly sqrt(28/27), or about 31.5 cents.
| Steps of 5edo | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| "Augmented" interval | 63.12 | 296.81 | 530.50 | 764.19 | 997.88 | |
| JI intervals represented | 28/27 | 32/27 | 49/36 | 14/9 | 16/9 | |
| "Major" interval | 31.56 | 265.25 | 498.94 | 732.63 | 966.31 | 1200.00 |
| JI intervals represented | 49/48, 64/63 | 7/6 | 4/3 | 32/21, 49/32 | 7/4 | 2/1 |
| "Minor" interval | 0.00 | 233.69 | 467.37 | 701.06 | 934.75 | 1168.44 |
| JI intervals represented | 1/1 | 8/7 | 21/16, 64/49 | 3/2 | 12/7 | 63/32, 96/49 |
| "Diminished" interval | 202.12 | 435.81 | 669.50 | 903.19 | 1136.88 | |
| JI intervals represented | 9/8 | 9/7 | 72/49 | 27/16 | 27/14 |
Interval chains
In the following tables, odd harmonics and subharmonics 1–27 are labeled in bold. Cent values reflect 3/10-comma tuning.
| # | Extended diatonic category |
Cents | Approximate ratios |
|---|---|---|---|
| 0 | P1 | 0 | 1/1 |
| 1 | SM2 | 234 | 8/7 |
| 2 | s4 | 467 | 21/16, 64/49 |
| 3 | P5 | 701 | 3/2 |
| 4 | SM6 | 935 | 12/7 |
| 5 | s8 | 1169 | 63/32, 96/49 |
| 6 | M2 | 202 | 9/8 |
| 7 | SM3 | 436 | 9/7 |
| 8 | s5 | 670 | 72/49 |
| 9 | M6 | 903 | 27/16 |
| 10 | SM7 | 1137 | 27/14 |
| 11 | sM2 | 171 | 54/49 |
| # | Extended diatonic category |
Cents | Approximate ratios |
|---|---|---|---|
| 0 | P1 | 0 | 1/1 |
| −1 | sm7 | 966 | 7/4 |
| −2 | S5 | 733 | 32/21, 49/32 |
| −3 | P4 | 499 | 4/3 |
| −4 | sm3 | 265 | 7/6 |
| −5 | S1 | 31 | 49/48, 64/63 |
| −6 | m7 | 998 | 16/9 |
| −7 | sm6 | 764 | 14/9 |
| −8 | S4 | 530 | 49/36 |
| −9 | m3 | 297 | 32/27 |
| −10 | sm2 | 63 | 28/27 |
| −11 | Sm7 | 1029 | 49/27 |
Tunings and extensions
While, for pure 2.3.7 subgroup accuracy, 36edo is a practically optimal tuning, it is essentially straddle-5 and straddle-11, being between two full 11-limit interpretations (the 36p and 36ce vals). Thus, other extensions of Slendric should be sought to improve the accuracy of 5-limit and 11-limit harmony.
There are two most important strong extensions to prime 5, these being Mothra and Rodan.
Mothra uses a meantone fifth in order to find 5/4 at the diatonic major third (12 generators up) and temper out 81/80. As the fifth is flattened, Mothra tunings have a more melodically salient quark (serving as an aberrisma), which now represents 36/35 in addition to 49/48 and 64/63, and bring the 7th harmonic closer to purity. The most important Mothra tunings are 31edo, at the optimum for this temperament with a close-to-just 5/4, and 26edo, which approximates the tuning formed by stacking a purely tuned 8/7. 36edo using the 12edo major third of 400¢ as 5/4 also qualifies as Mothra.
Rodan, meanwhile, slightly sharpens the fifth and can be constructed by equating 81/80 to the quark. This thereby tempers out the aberschisma (5120/5103), and furthermore implies that 9/7 forms half of 5/3, the Sensamagic (245/243) equivalence. From this, it can be seen that 5/4 is found at a perfect fifth (3 generators) above twice 9/7 (7 generators each), or 17 generators in all, the downmajor third in the system described earlier. 41edo and 46edo bound the main Rodan tuning range, but their sum, 87edo, is essentially optimal with a nearly just 5/4. 36edo using the flat major third of 367¢ as 5/4 also qualifies as Rodan.
