Slendric

Slendric, alternatively and originally named wonder by Margo Schulter, 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.

This article concerns the basic 2.3.7 subgroup temperament, slendric itself.

For technical data, see Gamelismic clan #Slendric.

Interval chains

In the following tables, odd harmonics and subharmonics 1–27 are labeled in bold.

#	Cents*	Approximate ratios
0	0.0	1/1
1	233.7	8/7
2	467.5	21/16, 64/49
3	701.2	3/2
4	935.0	12/7
5	1168.7	63/32, 96/49
6	202.5	9/8
7	436.2	9/7
8	670.0	72/49
9	903.7	27/16
10	1137.5	27/14
11	171.2	54/49

#	Cents*	Approximate ratios
0	0.0	1/1
−1	966.3	7/4
−2	732.5	32/21, 49/32
−3	498.8	4/3
−4	265.0	7/6
−5	31.3	49/48, 64/63
−6	997.5	16/9
−7	763.8	14/9
−8	530.0	49/36
−9	296.3	32/27
−10	62.5	28/27
−11	1028.8	49/27

* In 2.3.7-subgroup CWE tuning

Chords

• Slendric pentad

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

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

Scala files

• Slendric5
• Slendric6
• Slendric11
• Slendric16

Tunings

Notable edos that support slendric include 31, 36, 41, 46, and 77. Constrained Tenney–Euclidean slendric is extremely well-approximated by 2160edo.

• TE: ~2 = 1200.486, ~8/7 = 233.782
• DKW: ~2 = 1200.000, ~8/7 = 233.042

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)
	7/4	231.174		Untempered tuning
16\83		231.325		83bc val (mothra)
11\57		231.579
17\88		231.818
6\31		232.258
19\98		232.653
13\67		232.836
	49/48	232.861		1/5-comma
20\103		233.010		103c val (mothra)
	7/6	233.282		1/4-comma
7\36		233.333	↑ Mothra ↓ Guiron (10976/10935)
	9/7	233.583		2/7-comma
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
8\41		234.146	↑ Guiron ↓ Rodan (245/243)
25\128		234.375
17\87		234.483
	64/63	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

Music

Keenan Pepper

• Slendric[11] wailing (2012)

Chris Vaisvil

• The 11th Slendric Fanfare
• 16 Slendric Virgins

References