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.
Note that "up" intervals and "down" intervals can be represented as fractions with a single factor of 7 in the denominator and numerator (compactly, "/7" or "ru", and "7/" or "zo" intervals), respectively, with uninflected diatonic intervals representing the 3-limit. Considering extensions to prime 5, Rodan maps 7/5 onto the chain of fifths so that "up" and "down" also comprise the /5 and 5/ classes of intervals, while Mothra maps 5 directly onto the chain of fifths. Each of these provides a very intuitive way to notate the full 7-limit.
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 | 63.0 | 296.7 | 530.4 | 764.1 | 997.8 | |
| JI intervals represented | 28/27 | 32/27 | 49/36 | 14/9 | 16/9 | |
| "Major" interval | 31.5 | 265.2 | 498.9 | 732.6 | 966.3 | 1200.0 |
| JI intervals represented | 49/48, 64/63 | 7/6 | 4/3 | 32/21, 49/32 | 7/4 | 2/1 |
| "Minor" interval | 0.0 | 233.7 | 467.4 | 701.1 | 934.8 | 1168.5 |
| JI intervals represented | 1/1 | 8/7 | 21/16, 64/49 | 3/2 | 12/7 | 63/32, 96/49 |
| "Diminished" interval | 202.2 | 435.9 | 669.6 | 903.3 | 1137.0 | |
| JI intervals represented | 9/8 | 9/7 | 72/49 | 27/16 | 27/14 |
Notable features and related structures
A distinctive feature of slendric tuning systems is the subfourth of two generators, which represents 21/16. Additionally, it serves as (8/7)2 = 64/49, and thus is tempered a few cents flat of 21/16 in most tunings, making it an interseptimal naiadic. Another interpretation then is 17/13, tempering out 273/272 and 833/832, into which 1029/1024 factors. (31edo's tuning comes particularly close to 17/13.)
Taking every other step of Slendric results in a subtemperament generated by this subfourth, which is known as A-team. It is one of the main regular temperaments representing the oneirotonic (5L 3s) scale - specifically the hard tunings thereof such as in 18, 23, and 31edo. The core subgroup interpretation of A-team is 2.9.21.55: note that two A-team generators, representing 12/7, come close to 55/32 and therefore 385/384 and 441/440, which again multiply to 1029/1024, can be tempered out. Different A-team tunings can pick up other harmonic approximations; an interesting one is the 13:17:19 chord found in Mothra (and especially 31edo)'s version of A-team.
As a result of the ease of finding 55/32 and 17/13 along the Slendric chain, any extension to the full 7-limit can also find prime 11, and any extension to 2.3.7.13 can also find prime 17. This applies both to strong and weak extensions.
Relationship with acoustic phi
The A-team generator acquires the representations 21/16, 17/13, 55/42, and 72/55. But if we look one octave higher, a pattern becomes clear: 21/8, 34/13, 55/21, and 144/55 are all ratios of two-apart Fibonacci numbers, which therefore closely approximate the square of acoustic phi, entailing that acoustic phi squared over 2 is close to two Slendric generators. A single generator, therefore, is approximable by acoustic phi divided by sqrt(2). This can also be explained by 18 being the 6th Lucas number, and therefore a close approximation to φ6; approximating 181/6 by φ gives us φ/√2 as an approximation of (3/2)1/3. This interval's precise value is about 233.09¢, and using it as a generator produces a form of Slendric too sharp to be Mothra but flat of 36edo, with a fifth about 2.7 cents flat.
Interval chain
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 2.3.7 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 2.3.7 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
Tuning considerations
The error induced by the comma 1029/1024, about 8.4¢, has to be distributed between three factors of 7 and one factor of 3, and ideally both 3 and 7 should be flattened; we can define tunings of Slendric by the fraction of this comma by which 8/7 is sharpened. As representations of 2.3.7 intervals generally stack more factors of 3 than factors of 7, it can be argued 3 should be flattened less than 7. This occurs between 1/3-comma tuning (234.0¢, just flat of 41edo) which sets 3/2, and thus the entire Pythagorean chain, just while 8/7 is sharpened by 2.8¢; and 1/4-comma tuning (233.3¢, extremely close to 36edo) which sets them equally flat, so that 7/6 is just. A notable EDO tuning in this range is 77edo.
But, especially if 6:7:8 is considered the fundamental 2.3.7 harmony, it is reasonable to want a tuning where the error of 4/3 is split between that of 7/6 and 8/7. Furthermore, sharpward error is often considered more acceptable than flatward error on the interval 7/6, and these flatter tunings of slendric are those which happen to tune 7/6 sharp. 1/5-comma tuning (232.9¢, near 67edo) sets 7/6 and 8/7 equally sharp, by about 1.7¢ each.
Based on the above, 36edo can be considered a practically optimal tuning, as it is an EDO of reasonable size in the best range for pure 2.3.7 subgroup accuracy; however, it is essentially straddle-5 and straddle-11, being between two full 11-limit interpretations (the 36p and 36ce vals). Thus, other tunings of Slendric should be sought to improve the accuracy of 5-limit and 11-limit harmony.
Additional particularities of Slendric to consider include the tuning of the subfourth, and the size of the quark. The subfourth varies between nearly 13/10 in the flattest tunings, which has a potentially tertian function (e.g. in 10:13:15 triads), and near-just 21/16 in the sharpest tunings, which much more closely resembles a fourth; of the intervals of Slendric, this is the one with the least clear independent role and the most variability in function between the different tunings. As for the quark, its size can vary between that of a comma and that of a quartertone. Tunings where the fifth is flattened significantly (specifiable as Mothra in the full 7-limit) have a more melodically salient quark that serves as an aberrisma, and bring the 7th harmonic closer to purity.
Extensions
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. The exaggerated quark now represents 36/35 in addition to 49/48 and 64/63. 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 pure 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 the Sensamagic (245/243) equivalence, that 9/7 forms half of 5/3. 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: this is 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.
