Slendric
| Slendric |
((2.3.7) 27-odd limit) 2.81 ¢
((2.3.7) 27-odd limit) 21 notes
Slendric, alternatively and originally named wonder by Margo Schulter[1], or systematically gamelic, is a regular temperament generated by 8/7, so that three of them stack to 3/2. Thus the gamelisma, 1029/1024, is tempered out, which defines the gamelismic clan. Since 1029/1024 is a relatively small comma (8.4¢), and the error is distributed over a few intervals, slendric is quite an accurate temperament (approximating many intervals within 1 or 2 cents in optimal tunings).
The disadvantage, if you want to think of it that way, is that approximations to the 5th harmonic do not occur until you go a large number of generators away from the unison. In other words, the 5th harmonic must have a large complexity. Possible extensions of slendric to the full 7-limit include mothra, rodan, and guiron, where mothra tempers out 81/80, placing 5/1 at 12 generators (4 fifths) up; rodan tempers out 245/243, placing 10/1 at 17 generators up; and guiron tempers out the schisma, 32805/32768, placing the 5th harmonic 24 generators (8 fifths) down. From there, it is easy to extend these temperaments to the 11-limit since 1029/1024 factorizes in this limit into (385/384) × (441/440), and so the logical extension of slendric is to temper out both commas; this places the interval of 55/32 at four generators up.
This article concerns the basic 2.3.7 subgroup temperament, slendric itself.
For technical data, see Gamelismic clan #Slendric.
Intervals
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: "super"/"sub", which can be abbreviated with the prefixes S and s, respectively. We define the slendric generator to be the supermajor second, and therefore the 2-generator interval is a subfourth (as a major second and a perfect fourth together reach a perfect fifth) as well as a supersupermajor third. Between a major third and perfect fourth is a minor second, which is therefore equivalent to three repetitions of "super" (implying that "super" is rigorously an inflection by the "quark" of 49/48~64/63); because of this equivalence, it is never necessary to attach more than one "super"/"sub" to a diatonic interval.
Interval chains
In the following tables, odd harmonics and subharmonics 1–27 are labeled in bold.
| # | Extended diatonic category |
Cents* | Approximate ratios |
|---|---|---|---|
| 0 | P1 | 0.0 | 1/1 |
| 1 | SM2 | 233.7 | 8/7 |
| 2 | s4 | 467.5 | 21/16, 64/49 |
| 3 | P5 | 701.2 | 3/2 |
| 4 | SM6 | 935.0 | 12/7 |
| 5 | s8 | 1168.7 | 63/32, 96/49 |
| 6 | M2 | 202.5 | 9/8 |
| 7 | SM3 | 436.2 | 9/7 |
| 8 | s5 | 670.0 | 72/49 |
| 9 | M6 | 903.7 | 27/16 |
| 10 | SM7 | 1137.5 | 27/14 |
| 11 | sM2 | 171.2 | 54/49 |
| # | Extended diatonic category |
Cents* | Approximate ratios |
|---|---|---|---|
| 0 | P1 | 0.0 | 1/1 |
| −1 | sm7 | 966.3 | 7/4 |
| −2 | S5 | 732.5 | 32/21, 49/32 |
| −3 | P4 | 498.8 | 4/3 |
| −4 | sm3 | 265.0 | 7/6 |
| −5 | S1 | 31.3 | 49/48, 64/63 |
| −6 | m7 | 997.5 | 16/9 |
| −7 | sm6 | 763.8 | 14/9 |
| −8 | S4 | 530.0 | 49/36 |
| −9 | m3 | 296.3 | 32/27 |
| −10 | sm2 | 62.5 | 28/27 |
| −11 | Sm7 | 1028.8 | 49/27 |
* In 2.3.7-subgroup CWE tuning
Alternate way of organizing intervals
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).
| 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 |
Chords
Scales
5-note and 6-note (both proper)
The 5-note MOS of slendric is Lssss, in which L is 7/6 and s is 8/7; this serves as an approximation to 5edo. This expands to the 6-note MOS, LLLLLs, in which L is 8/7 and s is the characteristic small interval of slendric (sometimes known as the quark) representing both 64/63 and 49/48.
Both of these scales are somewhat lacking in harmonic resources relative to similar-sized scales of other temperaments. Even within the 2.3.7 subgroup, archy and Semaphore have pentatonic scales with more consonant intervals and chords; or if more accuracy is desired a 2.3.7 JI scale could be used.
Slendric really shines when used with larger scales than these. The 5-note MOS, however, has a special role in organizing the intervals of slendric because it is so close to 5edo - that is, slendric is very suitable for a pentatonic framework of categorization, rather than a heptatonic/diatonic one.
11-note (LsLsLsLsLss, improper)
The 11-note MOS, LsLsLsLsLss, has 9/8 "whole tones" in alternation with ~32 cent "sixth tones", with the exception of one pair of adjacent "sixth tones".
| Small ("minor") interval | 31.56 | 63.13 | 265.25 | 296.81 | 498.94 | 530.50 | 732.63 | 764.19 | 966.31 | 997.88 |
|---|---|---|---|---|---|---|---|---|---|---|
| JI intervals represented | 49/48, 64/63 | 28/27 | 7/6 | 32/27 | 4/3 | 49/36 | 32/21, 49/32 | 14/9 | 7/4 | 16/9 |
| Large ("major") interval | 202.12 | 233.69 | 435.81 | 467.37 | 669.50 | 701.06 | 903.19 | 934.75 | 1136.87 | 1168.44 |
| JI intervals represented | 9/8 | 8/7 | 9/7 | 21/16, 64/49 | 72/49 | 3/2 | 27/16 | 12/7 | 27/14 | 63/32, 96/49 |
Scala files
Tunings
Notable edos that support slendric include 31, 36, 41, 46, and 77. Constrained Tenney–Euclidean slendric is extremely well-approximated by 2160edo.
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged interval)* |
Generator (¢) | Extension | Comments |
|---|---|---|---|---|
| 2\11 | 218.182 | Lower bound of {1, 3, 7, 9} diamond monotone | ||
| 3\16 | 225.000 | ↓ Gorgo (36/35) | ||
| 7\37 | 227.027 | 37b val | ||
| 4\21 | 228.571 | ↑ Gorgo ↓ Archaeotherium (405/392) |
||
| 9\47 | 229.787 | |||
| 5\26 | 230.769 | ↑ Archaeotherium ↓ Mothra (81/80) |
||
| 8/7 | 231.174 | Untempered tuning | ||
| 11\57 | 231.579 | |||
| 17\88 | 231.818 | |||
| 17/13 | 232.214 | As s4 | ||
| 6\31 | 232.258 | |||
| 19\98 | 232.653 | |||
| 13\67 | 232.836 | |||
| 96/49 | 232.861 | 1/5-comma | ||
| 20\103 | 233.010 | 103c val (mothra) | ||
| 12/7 | 233.282 | 1/4-comma; (2.3.7) 7-odd-limit minimax tuning | ||
| 7\36 | 233.333 | ↑ Mothra ↓ Guiron (10976/10935) |
||
| 29\149 | 233.557 | 149cc val (guiron) | ||
| 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 | |||
| 31\159 | 233.962 | |||
| 3/2 | 233.985 | 1/3-comma; (2.3.7) 21- and 27-odd-limit minimax tuning | ||
| 8\41 | 234.146 | ↑ Guiron ↓ Rodan (245/243) |
||
| 25\128 | 234.375 | |||
| 55/32 | 234.408 | As SM6 | ||
| 17\87 | 234.483 | |||
| 63/32 | 234.547 | 2/5-comma | ||
| 26\133 | 234.586 | |||
| 9\46 | 234.783 | ↑ Rodan | ||
| 19\97 | 235.052 | |||
| 10\51 | 235.294 | |||
| 21/16 | 235.390 | 1/2-comma | ||
| 21\107 | 235.514 | |||
| 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
Etymology
Slendric is so named because the basic slendric scale is a near-equalized form of 1L 4s, and thus an equipentatonic scale, similar to (but not exactly) the slendro scale used in Indonesian music. This relation is rough and tenuous at best, hence the alternative names such as wonder (especially given how many other 2.3.7 structures such as archy and buzzard also have equipentatonic scales).
Music
- Slendric[11] wailing (2012)