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, 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": 31edo, 36edo, 41edo, and 46edo.

• Flatter tunings of Slendric (which can be interpreted as the full 7-limit temperament Mothra) have a more melodically salient quark (serving as an aberrisma) and bring the 7th harmonic closer to purity.
• Somewhat sharper tunings of Slendric such as 41 and 46 (supporting the full 7-limit temperament Rodan) temper out the Aberschisma (5120/5103), an important structural feature equating 81/80 and 64/63. The combination of Slendric and Aberschismic also implies tempering out 245/243.
• 36edo is excellent when restricted to 2.3.7, but JI interpretations outside this subgroup are less clear (in particular, it's straddle-5, being between two full 7-limit interpretations).

Structural theory

General theory

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.

Interval chains

In the following tables, odd harmonics and subharmonics 1–27 are labeled in bold. The generator tuning here is 3/10-comma, where the quark is exactly sqrt(28/27), or about 31.5 cents.

#	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