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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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. | 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)]], | 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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== 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" | |||
|+ 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, ~8/7 = 233.042 | * [[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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| | | | ||
| 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) | ||
| | | | ||
|- | |- | ||