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Slendric: Difference between revisions

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'''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).
'''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 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. 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.
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. 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.


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]].


== Interval chains ==
== Theory ==
=== 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.
 
=== Interval chains ===
In the following tables, odd harmonics and subharmonics 1–27 are labeled in '''bold'''.  
In the following tables, odd harmonics and subharmonics 1–27 are labeled in '''bold'''.  


<div><div style="display: inline-grid; margin-right: 25px;">
<div><div style="display: inline-grid; margin-right: 25px;">


{| class="wikitable sortable center-1 right-2"
{| class="wikitable sortable center-1 center-2 right-3"
|-
|-
! &#35;
! #
! class="unsortable" | Extended <br> diatonic <br> category
! Cents*
! Cents*
! class="unsortable" | Approximate ratios
! class="unsortable" | Approximate ratios
|-
|-
| 0
| 0
| P1
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|-
|-
| 1
| 1
| SM2
| 233.7
| 233.7
| '''8/7'''
| '''8/7'''
|-
|-
| 2
| 2
| s4
| 467.5
| 467.5
| '''21/16''', 64/49
| '''21/16''', 64/49
|-
|-
| 3
| 3
| P5
| 701.2
| 701.2
| '''3/2'''
| '''3/2'''
|-
|-
| 4
| 4
| SM6
| 935.0
| 935.0
| 12/7
| 12/7
|-
|-
| 5
| 5
| s8
| 1168.7
| 1168.7
| 63/32, 96/49
| 63/32, 96/49
|-
|-
| 6
| 6
| M2
| 202.5
| 202.5
| '''9/8'''
| '''9/8'''
|-
|-
| 7
| 7
| SM3
| 436.2
| 436.2
| 9/7
| 9/7
|-
|-
| 8
| 8
| s5
| 670.0
| 670.0
| 72/49
| 72/49
|-
|-
| 9
| 9
| M6
| 903.7
| 903.7
| '''27/16'''
| '''27/16'''
|-
|-
| 10
| 10
| SM7
| 1137.5
| 1137.5
| 27/14
| 27/14
|-
|-
| 11
| 11
| sM2
| 171.2
| 171.2
| 54/49
| 54/49
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</div>
</div>
<div style="display: inline-grid; margin-right: 25px;">
<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable sortable center-1 right-2"
{| class="wikitable sortable center-1 center-2 right-3"
|-
|-
! &#35;
! #
! class="unsortable" | Extended <br> diatonic <br> category
! Cents*
! Cents*
! class="unsortable" | Approximate ratios
! class="unsortable" | Approximate ratios
|-
|-
| 0
| 0
| P1
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|-
|-
| −1
| −1
| sm7
| 966.3
| 966.3
| '''7/4'''
| '''7/4'''
|-
|-
| −2
| −2
| S5
| 732.5
| 732.5
| '''32/21''', 49/32
| '''32/21''', 49/32
|-
|-
| −3
| −3
| P4
| 498.8
| 498.8
| '''4/3'''
| '''4/3'''
|-
|-
| −4
| −4
| sm3
| 265.0
| 265.0
| 7/6
| 7/6
|-
|-
| −5
| −5
| S1
| 31.3
| 31.3
| 49/48, 64/63
| 49/48, 64/63
|-
|-
| −6
| −6
| m7
| 997.5
| 997.5
| '''16/9'''
| '''16/9'''
|-
|-
| −7
| −7
| sm6
| 763.8
| 763.8
| 14/9
| 14/9
|-
|-
| −8
| −8
| S4
| 530.0
| 530.0
| 49/36
| 49/36
|-
|-
| −9
| −9
| m3
| 296.3
| 296.3
| '''32/27'''
| '''32/27'''
|-
|-
| −10
| −10
| sm2
| 62.5
| 62.5
| 28/27
| 28/27
|-
|-
| −11
| −11
| Sm7
| 1028.8
| 1028.8
| 49/27
| 49/27
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</div>
</div>
<nowiki/>* In 2.3.7-subgroup [[CWE tuning]]
<nowiki/>* 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 [[1L 4s|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, [[5L 1s|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, [[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.
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, [[5L 6s|LsLsLsLsLss]], has [[9/8]] "whole tones" in alternation with ~32 cent "sixth tones", with the exception of one pair of adjacent "sixth tones".
{| class="wikitable"
|-
! 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 ===
=== Alternate way of organizing intervals ===
Line 210: Line 175:
|-
|-
! Steps of 5edo
! Steps of 5edo
| 0
| '''0'''
| 1
| '''1'''
| 2
| '''2'''
| 3
| '''3'''
| 4
| '''4'''
| 5
| '''5'''
|- style="background-color: #DFDFDF;"
|- style="background-color: #DFDFDF;"
! "Augmented" interval
! "Augmented" interval
Line 280: Line 245:
| 27/16
| 27/16
| 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 ==
* [[Slendric pentad]]
== Scales ==
=== 5-note and 6-note (both proper) ===
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.
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, [[5L 6s|LsLsLsLsLss]], has [[9/8]] "whole tones" in alternation with ~32 cent "sixth tones", with the exception of one pair of adjacent "sixth tones".
{| class="wikitable"
|-
! 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
|}
|}


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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)
| ↑ 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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|  
|  
|
|
|-
|
| [[17/13]]
| 232.214
|
| As s4
|-
|-
| [[31edo|6\31]]
| [[31edo|6\31]]
Line 386: Line 427:
|  
|  
| 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)
|  
|  
|-
|-
Line 450: Line 497:
|  
|  
| 234.146
| 234.146
| ↑ Guiron <br> ↓ [[Rodan]] (245/243)
| ↑ Guiron <br> ↓ [[Rodan#Tuning spectrum|Rodan]] (245/243)
|  
|  
|-
|-
Line 458: Line 505:
|  
|  
|  
|  
|-
|
| [[55/32]]
| 234.408
|
| As SM6
|-
|-
| [[87edo|17\87]]
| [[87edo|17\87]]
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| '''Upper bound of {1, 3, 7, 9} diamond monotone'''
| '''Upper bound of {1, 3, 7, 9} diamond monotone'''
|}
|}
<nowiki/>* Besides the octave
<nowiki>*</nowiki> Besides the octave
 
== 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
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 ==
Retrieved from "https://en.xen.wiki/w/Slendric"