Slendric: Difference between revisions
No edit summary |
m added mention of 18 as a Lucas number |
||
| (8 intermediate revisions by 3 users not shown) | |||
| Line 21: | Line 21: | ||
For technical data, see [[Gamelismic clan #Slendric]]. | For technical data, see [[Gamelismic clan #Slendric]]. | ||
== | == Theory == | ||
=== Interval categories === | === 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. | 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. | ||
| Line 32: | Line 32: | ||
{| class="wikitable sortable center-1 center-2 right-3" | {| class="wikitable sortable center-1 center-2 right-3" | ||
|- | |- | ||
! | ! # | ||
! class="unsortable" | Extended <br | ! class="unsortable" | Extended <br> diatonic <br> category | ||
! Cents* | ! Cents* | ||
! class="unsortable" | Approximate ratios | ! class="unsortable" | Approximate ratios | ||
| Line 101: | Line 101: | ||
{| class="wikitable sortable center-1 center-2 right-3" | {| class="wikitable sortable center-1 center-2 right-3" | ||
|- | |- | ||
! | ! # | ||
! class="unsortable" | Extended <br | ! class="unsortable" | Extended <br> diatonic <br> category | ||
! Cents* | ! Cents* | ||
! class="unsortable" | Approximate ratios | ! class="unsortable" | Approximate ratios | ||
| Line 246: | Line 246: | ||
| 27/14 | | 27/14 | ||
|} | |} | ||
=== Relationship with acoustic phi === | |||
One may note that the two-generator interval is tempered somewhat flat of 21/16 in most tunings of slendric. This allows it to be interpretable as [[17/13]], tempering out [[273/272]] and [[833/832]], into which 1029/1024 factors. Combining that with the mapping of 55/32 at four generators, we obtain the representations 21/16, 17/13, [[55/42]], and [[72/55]] for this interval. 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 slendric 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 φ<sup>6</sup>; approximating 18<sup>1/6</sup> by φ gives us φ/√2 as an approximation of (3/2)<sup>1/3</sup>. This interval's precise value is about 233.0903{{c}}, 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, quite close to the 27\139 (233.0935{{c}}) tuning in [[139edo]]. | |||
== Chords == | == Chords == | ||
| Line 252: | Line 257: | ||
== Scales == | == Scales == | ||
=== 5-note and 6-note (both proper) === | === 5-note and 6-note (both proper) === | ||
The 5-note [[MOS]] of slendric is [[1L 4s|Lssss]], in which L | The 5-note [[MOS]] of slendric is [[1L 4s|Lssss]], in which L represents [[7/6]] and s [[8/7]]; this serves as an approximation to [[5edo]]. This expands to the 6-note MOS, [[5L 1s|LLLLLs]], in which L represents 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, [[Superpyth|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. | 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, [[Superpyth|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. | ||
| Line 342: | Line 347: | ||
| | | | ||
| 225.000 | | 225.000 | ||
| ↓ [[Gorgo]] (36/35) | | ↓ ''[[Gorgo]]'' (36/35) | ||
| | | | ||
|- | |- | ||
| Line 354: | Line 359: | ||
| | | | ||
| 228.571 | | 228.571 | ||
| ↑ | | ↑ Gorgo <br> ↓ ''[[Gamelismic clan#Archaeotherium|Archaeotherium]]'' (405/392) | ||
| | | | ||
|- | |- | ||
| Line 366: | Line 371: | ||
| | | | ||
| 230.769 | | 230.769 | ||
| ↑ Archaeotherium <br> ↓ [[Mothra]] (81/80) | | ↑ Archaeotherium <br> ↓ [[Mothra#Tuning spectrum|Mothra]] (81/80) | ||
| | | | ||
|- | |- | ||
| Line 422: | Line 427: | ||
| | | | ||
| 103c val (mothra) | | 103c val (mothra) | ||
|- | |||
| | |||
| φ/√2 | |||
| 233.090 | |||
| | |||
| As generator | |||
|- | |- | ||
| | | | ||
| Line 432: | Line 443: | ||
| | | | ||
| 233.333 | | 233.333 | ||
| ↑ Mothra <br> ↓ [[Guiron]] (10976/10935) | | ↑ Mothra <br> ↓ ''[[Guiron]]'' (10976/10935) | ||
| | | | ||
|- | |- | ||
| Line 486: | Line 497: | ||
| | | | ||
| 234.146 | | 234.146 | ||
| ↑ Guiron <br> ↓ [[Rodan]] (245/243) | | ↑ Guiron <br> ↓ [[Rodan#Tuning spectrum|Rodan]] (245/243) | ||
| | | | ||
|- | |- | ||
| Line 582: | Line 593: | ||
== Etymology == | == 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 | 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 == | == Music == | ||
; [[Keenan Pepper]] | ; [[Keenan Pepper]] | ||
| Line 604: | Line 608: | ||
[[Category:Slendric| ]] <!-- main article --> | [[Category:Slendric| ]] <!-- main article --> | ||
[[Category:Rank-2 temperaments]] | |||
[[Category:Subgroup temperaments]] | |||
[[Category:Gamelismic clan]] | |||
[[Category:Listen]] | |||