Slendric: Difference between revisions
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{{Infobox | {{Interwiki | ||
| en = Slendric | |||
| de = Slendrisch | |||
}} | |||
{{Infobox regtemp | |||
| Title = Slendric | | Title = Slendric | ||
| Subgroups = 2.3.7 | | Subgroups = 2.3.7 | ||
| Comma basis = [[1029/1024]] | | Comma basis = [[1029/1024]] | ||
| Edo join 1 = 5 | Edo join 2 = 21 | | Edo join 1 = 5 | Edo join 2 = 21 | ||
| Mapping = 1; 3 -1 | | Mapping = 1; 3 -1 | ||
| Generators = 8/7 | Generators tuning = 233.7 | Optimization method = CWE | |||
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], … | |||
| Pergen = (P8, P5/3) | | Pergen = (P8, P5/3) | ||
| Color name = Latrizoti | | Color name = Latrizoti | ||
| Odd limit 1 = 7 | Mistuning 1 = 2.11 | Complexity 1 = 11 | | Odd limit 1 = 7 | Mistuning 1 = 2.11 | Complexity 1 = 11 | ||
| Odd limit 2 = | | Odd limit 2 = 2.3.7 27 | Mistuning 2 = 2.81 | Complexity 2 = 21 | ||
}} | }} | ||
'''Slendric''', alternatively and originally named '''wonder''' by [[Margo Schulter]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_76975.html#77043 Yahoo! Tuning Group | ''Music Theory (was Re: How to keep discussions on-topic)''], and [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_87455.html#88377 Yahoo! Tuning Group | ''The "best" scale.'']</ref>, 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 [[5/1|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 [[extension]]s 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 5/1 at 17 generators up; and guiron tempers out the schisma, [[32805/32768]], placing 5/1 at 24 generators (8 fifths) down. [[Weak extension]]s such as [[miracle]] and [[valentine]] are somewhat more efficient, but they change the generator chain. | |||
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. Alternatively, giving 5/1 an independent generator and then tempering out 385/384 and 441/440 results in the rank-3 temperament [[portent]], the natural 11-limit [[expansion]] of slendric. Portent is [[support]]ed by the slendric extensions listed above. | |||
This article concerns the basic [[2.3.7 subgroup]] temperament, slendric itself. | This article concerns the basic [[2.3.7 subgroup]] temperament, slendric itself. | ||
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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. | ||
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| 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 == | ||
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== 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. | ||
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== Tunings == | == Tunings == | ||
Notable edos that support slendric include {{EDOs| 31, 36, 41, 46, and 77}}. | Notable edos that support slendric include {{EDOs| 31, 36, 41, 46, and 77}}. | ||
=== Norm-based tunings === | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7-subgroup norm-based tunings (tempered-octave) | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Free | |||
! Free & skewed | |||
|- | |||
! Tenney | |||
| TE: ~2 = 1200.4862{{c}}, ~8/7 = 233.7822{{c}} | |||
| WE: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}} | |||
|} | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
* [[DKW theory|DKW]]: ~2 = 1200.000, ~8/7 = 233.042 | |+ style="font-size: 105%; white-space: nowrap;" | 2.3.7-subgroup norm-based tunings (pure-octave) | ||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~8/7 = 233.8889{{c}} | |||
| CWE: ~8/7 = 233.7474{{c}} | |||
| POTE: ~8/7 = 233.6875{{c}} | |||
|} | |||
=== Other tunings === | |||
* [[DKW theory|DKW]]: ~2 = 1200.000{{c}}, ~8/7 = 233.042{{c}} | |||
=== Tuning spectrum === | === Tuning spectrum === | ||
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| | | | ||
| 225.000 | | 225.000 | ||
| ↓ [[Gorgo]] (36/35) | | ↓ ''[[Gorgo]]'' (36/35) | ||
| | | | ||
|- | |- | ||
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| | | | ||
| 228.571 | | 228.571 | ||
| ↑ | | ↑ Gorgo <br> ↓ ''[[Gamelismic clan#Archaeotherium|Archaeotherium]]'' (405/392) | ||
| | | | ||
|- | |- | ||
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| | | | ||
| 230.769 | | 230.769 | ||
| ↑ Archaeotherium <br> ↓ [[Mothra]] (81/80) | | ↑ Archaeotherium <br> ↓ [[Mothra#Tuning spectrum|Mothra]] (81/80) | ||
| | | | ||
|- | |- | ||
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| | | | ||
| 103c val (mothra) | | 103c val (mothra) | ||
|- | |||
| | |||
| φ/√2 | |||
| 233.090 | |||
| | |||
| As generator | |||
|- | |- | ||
| | | | ||
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| | | | ||
| 233.333 | | 233.333 | ||
| ↑ Mothra <br> ↓ [[Guiron]] (10976/10935) | | ↑ Mothra <br> ↓ ''[[Guiron]]'' (10976/10935) | ||
| | | | ||
|- | |- | ||
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| | | | ||
| 234.146 | | 234.146 | ||
| ↑ Guiron <br> ↓ [[Rodan]] (245/243) | | ↑ Guiron <br> ↓ [[Rodan#Tuning spectrum|Rodan]] (245/243) | ||
| | | | ||
|- | |- | ||