Slendric: Difference between revisions
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'''Slendric''' 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 [[cent]]s from just when 3/2 is pure, Slendric constitutes an exceptionally good [[ | '''Slendric''' 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 [[cent]]s from just when 3/2 is pure, Slendric constitutes an exceptionally good [[Rank-2 temperament|rank-2]] traversal of the [[2.3.7 subgroup|2.3.7]] tuning space for its simplicity. Its corresponding [[comma]], the '''slendrisma''' or '''gamelisma,''' is the difference between 3/2 and (8/7)<sup>3</sup>, which is 1029/1024. | ||
Melodically, the Slendric generator stack forms a 5-note scale (1L 4s) that is nearly [[equipentatonic]]. [[MOS]]es 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]]. | Melodically, the Slendric generator stack forms a 5-note scale (1L 4s) that is nearly [[equipentatonic]]. [[MOS]]es 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]]. | ||
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* 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 [[7/4|the 7th harmonic]] closer to purity. | * 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 [[7/4|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. | * 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 [[ | * 36edo is excellent when restricted to 2.3.7, but JI interpretations outside this subgroup are less clear (in particular, it's [[Straddle primes|straddle-5]], being between two full 7-limit interpretations). | ||
== Structural theory == | == Structural theory == | ||
Revision as of 15:54, 21 January 2026
Slendric 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, the slendrisma or gamelisma, 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
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 |
