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 cents~~), and the error is distributed over several 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 several 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.

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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~~|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.

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.

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=== Alternate way of organizing intervals ===

Instead of organizing the intervals according to larger and larger MOSes (none of which are proper until at least 26 notes), the intervals of slendric can be organized according to how many steps of [[~~5edo|~~5edo]], or equivalently the 5-note MOS, they correspond to. The "major" interval of a class is the one that's just larger than the corresponding 5edo interval, and the "minor" interval is just smaller. Below are the intervals of the symmetric mode of Slendric[21] ([[5L 16s]]).

Instead of organizing the intervals according to larger and larger MOSes (none of which are proper until at least 26 notes), the intervals of slendric can be organized according to how many steps of [[5edo]], or equivalently the 5-note MOS, they correspond to. The "major" interval of a class is the one that's just larger than the corresponding 5edo interval, and the "minor" interval is just smaller. Below are the intervals of the symmetric mode of Slendric[21] ([[5L 16s]]).

{| class="wikitable"

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== Tunings ==

Notable edos that support slendric include ~~[[31edo~~|31]], ~~[[36edo|~~36]], ~~[[41edo|~~41]], and ~~[[77edo|~~77]]. [[Constrained tuning|~~CTE~~]] slendric is extremely well-approximated by [[~~2160edo|~~2160edo]].

Notable edos that support slendric include {{EDOs| 31, 36, 41, 46, and 77}}. [[Constrained tuning|Constrained Tenney–Euclidean]] slendric is extremely well-approximated by [[2160edo]].

* [[TE]]: ~2 = 1200.486, ~8/7 = 233.782

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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 cents), and the error is distributed over several 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 several 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.
@@ Line 132: / Line 132: @@
 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|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.
 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.
@@ Line 191: / Line 191: @@
 === Alternate way of organizing intervals ===
-Instead of organizing the intervals according to larger and larger MOSes (none of which are proper until at least 26 notes), the intervals of slendric can be organized according to how many steps of [[5edo|5edo]], or equivalently the 5-note MOS, they correspond to. The "major" interval of a class is the one that's just larger than the corresponding 5edo interval, and the "minor" interval is just smaller. Below are the intervals of the symmetric mode of Slendric[21] ([[5L 16s]]).
+Instead of organizing the intervals according to larger and larger MOSes (none of which are proper until at least 26 notes), the intervals of slendric can be organized according to how many steps of [[5edo]], or equivalently the 5-note MOS, they correspond to. The "major" interval of a class is the one that's just larger than the corresponding 5edo interval, and the "minor" interval is just smaller. Below are the intervals of the symmetric mode of Slendric[21] ([[5L&nbsp;16s]]).
 {| class="wikitable"
@@ Line 275: / Line 275: @@
 == Tunings ==
-Notable edos that support slendric include [[31edo|31]], [[36edo|36]], [[41edo|41]], and [[77edo|77]]. [[Constrained tuning|CTE]] slendric is extremely well-approximated by [[2160edo|2160edo]].
+Notable edos that support slendric include {{EDOs| 31, 36, 41, 46, and 77}}. [[Constrained tuning|Constrained Tenney–Euclidean]] slendric is extremely well-approximated by [[2160edo]].
 * [[TE]]: ~2 = 1200.486, ~8/7 = 233.782