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 of these is taken to represent 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. Slendric is also supported by edos with 5edo's 3/2.
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
The intervals of Slendric can be organized according to how many steps of 5edo, or equivalently the 5-note MOS, they correspond to, since the MOS scales of Slendric up to at least 26 notes have 5 large steps and many small steps, each the size of a quark. The "major" interval of a class here is the one 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. 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¢, very 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¢, very 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
As Slendric is a structure present in very many EDOs of note and an obvious simplification of the 2.3.7 subgroup aside from that, it behooves us to consider how this structure interacts with other harmonies within the 17-limit. Fortunately, there are a variety of choices for how each of the primes 5, 11, 13, and 17 fits into the Slendric framework.
Prime 5
There are two most important strong extensions to reach prime 5 and complete the 7-limit, these being Mothra and Rodan. Other mappings of 5 can be employed, such as the Schismic one (known as Guiron), but these are significantly more complex and give prime 5 less structural presence.
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.
As regards weak extensions, notable ones include Miracle, which splits 8/7 in two, and Valentine, which splits it in three (therefore dividing the perfect fifth into 6 and 9 parts, respectively). They also include Superkleismic (15 & 26), which splits 7/4 into three intervals of 6/5; this is supported by 26, 41, and 56edo.
Miracle's generator (known as the "secor") represents 16/15 and 15/14 simultaneously (tempering out 225/224, the marvel comma), so that 8/5 is placed at 7 secors. As a consequence of splitting 8/7 in half, Miracle also includes an exact neutral third, interpretable in the 7-limit as 49/40. Miracle is 10 & 21, and 31, 41, and 72edo support it.
Valentine places 6/5 and 5/4 at 4 and 5 steps respectively; the generator thus represents (5/4)/(6/5) = 25/24 and (6/5)/(8/7) = 21/20, and their ratio (126/125, the starling comma) is tempered out. Valentine is 15 & 16, and 31, 46, and 77edo support it.
Prime 11
As mentioned before, extensions to 11 can be created off of these by tempering out 385/384 and 441/440. This works almost perfectly in the 41 & 46 Rodan range, and the diatonic major third is identified with 14/11. This also applies to the 26 & 31 Mothra range, yet the case with Mothra is slightly more complicated, as the interval formed from (sharpened) 7/6 stacked twice can reasonably represent either 11/8 or 15/11, depending on the tuning (note that in 31edo, it represents both).
The former is supported by 26 & 31, and the latter by 31 & 36; the resulting extensions are called "undecimal Mothra" and "Mosura" respectively. Undecimal Mothra equates 14/11 to 9/7 (tempering out 99/98), and Mosura equates 14/11 to 32/25 (tempering out 176/175). Of the two, the former is taken to be canonical primarily as 11/8 itself is reached by far fewer generators (-8, compared to 23).
Miracle, Valentine, and Superkleismic all receive extensions to 11 in this manner as well. In Miracle's case, the neutral third is mapped to 11/9~27/22, while in Valentine's, the neutral second formed by two steps represents 12/11~11/10. Superkleismic, in fact, tempers out 100/99, whereby 16/11 is reached at only two of its 6/5 generators, which produces the notable 2.7.11 subgroup structure known as Orgone.
Primes 13 and 17
While 36edo's representation of the 2.3.7 subgroup fails to provide comparably accurate harmonies of 5 and 11, it does somewhat better with the next higher primes: 13, 17, and 19 (though the latter two descend from 12edo). Looking at 36edo's mapping of 13, we see that it divides 7/6 into halves that can each be taken as 14/13~13/12, and further that two quarks represent 28/27 and 27/26 simultaneously.
The former leads us to a weak extension, known as Baladic, that tempers out 169/168 and splits the octave in two; equating 17/13 to the downfourth, we see 9/8 is also split into 18/17~17/16, and therefore that 17/12 is a semioctave.
The latter leads us to a strong extension, called Euslendric (36 & 77), that reaches 13/8 after 19 generators, as the up-augmented fifth, and 17/16 after 21 generators, as the augmented unison. Euslendric is notable as its harmonies can extend to even higher limits, reaching 19/16 as the minor third (-9 generators), 23/16 as the up-diminished fifth (-23 generators), and 29/16 as the upminor seventh (-11 generators), all within the optimal tuning band for 2.3.7 accuracy.
Revisiting the 11-limit extensions mentioned above, Rodan naturally obtains 13/11 as the minor third to find 13 at 22 generators down. Meanwhile, Mothra's 9/8 is flat enough that it is very close to 143/128 = (11/8)/(16/13), so 144/143 can be tempered out as a way to extend each 11-limit extension of Mothra further to the 13-limit. All of these take on the obvious mapping to reach the full 17-limit, though in the case of Rodan, 17 receives greater damage than any lower prime in the most accurate Rodan tunings (such as 87edo).
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged interval)* |
Generator (¢) | Mapping of 5 | Comments |
|---|---|---|---|---|
| 2\11 | 218.182 | Lower bound of {1, 3, 7, 9} diamond monotone | ||
| 3\16 | 225.000 | ↓ +7 gens "Gorgo" {36/35} |
||
| 7\37 | 227.027 | 37b val | ||
| 4\21 | 228.571 | ↑ Gorgo ↓ -14 gens "Archaeotherium" {405/392} |
||
| 9\47 | 229.787 | |||
| 5\26 | 230.769 | ↑ Archaeotherium ↓ +12 gens "Mothra" {81/80} |
||
| 8/7 | 231.174 | Untempered tuning | ||
| 11\57 | 231.579 | |||
| 17/13 | 232.214 | As s4, approx. 1/8-comma | ||
| 6\31 | 232.258 | |||
| 13\67 | 232.836 | |||
| 96/49 | 232.861 | 1/5-comma | ||
| φ/√2 | 233.090 | As generator | ||
| 12/7 | 233.282 | 1/4-comma; (2.3.7) 7-odd-limit minimax tuning | ||
| 7\36 | 233.333 | ↑ Mothra ↓ -24 gens "Guiron" {10976/10935} |
||
| 9/7 | 233.583 | 2/7-comma; (2.3.7) 9-odd-limit minimax tuning | ||
| 22\113 | 233.628 | 113c val (guiron) | ||
| 27/14 | 233.704 | 3/10-comma; 2.3.7 CEE tuning | ||
| 15\77 | 233.766 | |||
| 23\118 | 233.898 | |||
| 3/2 | 233.985 | 1/3-comma; (2.3.7) 21- and 27-odd-limit minimax tuning | ||
| 8\41 | 234.146 | ↑ Guiron ↓ +17 gens "Rodan" {245/243} |
||
| 55/32 | 234.408 | As SM6, approx. 3/8-comma | ||
| 17\87 | 234.483 | |||
| 63/32 | 234.547 | 2/5-comma | ||
| 9\46 | 234.783 | ↑ Rodan | ||
| 19\97 | 235.052 | |||
| 10\51 | 235.294 | |||
| 21/16 | 235.390 | 1/2-comma | ||
| 11\56 | 235.714 | |||
| 12\61 | 236.066 | |||
| 13\66 | 236.364 | |||
| 14\71 | 236.620 | |||
| 1\5 | 240.000 | Upper bound of {1, 3, 7, 9} diamond monotone |
* Besides the octave
