2L 11s (3/1-equivalent): Difference between revisions

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This MOS scale pattern is most notable for corresponding to the 13-note [[Trithagorean]]/[[3.5 subgroup]] scale generated by a just [[5/3]], which has an L:s ratio of approximately 1.64. In this case, the large step represents [[48828125/43046721]] and the small step represents [[27/25]]. It is also the 13-note MOS of several tritave-equivalent temperaments generated by ~5/3 (similar to how the [[5L 2s]] diatonic scale is the 7-note MOS of chain-of-fifth temperaments). The [[3.5.11 subgroup]] [[microtemperament]] [[~~Delta~~ centauri]] (which is generated by a near-just ~5/3 such that 9 generators stacked represent ~[[11/9]]) has this as its 13-note MOS scale, although the 15-note {{MOS scalesig|13L 2s<3/1>|link=1}} ~~scale~~ may be preferred ~~as it has a much greater abundance~~ of 9~~:11:15 triads for only 2 more notes~~. [[Arcturus]] temperament tunings with ~5/3 sharper than 6 steps of [[13edt]] will generate this MOS, but tunings with ~5/3 flatter than 6 steps of [[13edt]] will instead generate the sister MOS {{MOS scalesig|11L 2s<3/1>|link=1}}.

This MOS scale pattern is most notable for corresponding to the 13-note [[Trithagorean]]/[[3.5 subgroup]] scale generated by a just [[5/3]], which has an L:s ratio of approximately 1.64, being the smallest non-trivial [[proper]] MOS scale generated by the 3.5 subgroup. In this case, the large step represents [[48828125/43046721]] and the small step represents [[27/25]]. It is also the 13-note MOS of several tritave-equivalent temperaments generated by ~5/3 (similar to how the [[5L 2s]] diatonic scale is the 7-note MOS of chain-of-fifth temperaments). The [[3.5.11 subgroup]] [[microtemperament]] [[delta centauri]] (which is generated by a near-just ~5/3 such that 9 generators stacked represent ~[[11/9]]) has this as its 13-note MOS scale, although the 15-note daughter scale {{MOS scalesig|13L 2s<3/1>|link=1}} may be preferred for that temperament due to the complexity of 11/9. [[Arcturus]] temperament tunings with ~5/3 sharper than 6 steps of [[13edt]] will generate this MOS, but tunings with ~5/3 flatter than 6 steps of [[13edt]] will instead generate the sister MOS {{MOS scalesig|11L 2s<3/1>|link=1}}.

In the moderately hard-of-basic region is the 3.5.11 subgroup [[Polaris]] temperament which, like Delta centauri, constructs an ~11/9 from stacking ~5/3 generators, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 [[9/5]]s stacked) tritave-reduced is ~11/9. If the L:s ratio is very high (starting at around 5:1), the generator becomes too sharp to approximate 5/3, and everything gets closer and closer to collapsing into [[2edt]].

In the region very near basic ([[15edt]]) tuning is the 3.5.13 subgroup [[Aldebaran]] temperament, which is generated by ~5/3 such that 5 generators stacked represent ~[[13/9]]. In the moderately hard-of-basic region is the 3.5.11 subgroup [[Polaris]] temperament which, like Delta centauri, constructs an ~11/9 from stacking ~5/3 generators, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 [[9/5]]s stacked) tritave-reduced is ~11/9. If the L:s ratio is very high (starting at around 5:1), the generator becomes too sharp to approximate 5/3, and everything gets closer and closer to collapsing into [[2edt]].

It can be considered as a kind of detempering of [[13edt]], but it would probably make more sense to consider the daughter MOSes of [[Lambda]] instead.

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== Scale tree ==

{{MOS tuning spectrum

|1/1=[[Arcturus]]

|1/1=[[CWE]]-optimized [[Arcturus]] (878.291{{c}})

|8/5=[[Delta centauri]]

|8/5=[[CWE]]-optimized [[Delta centauri]] (883.901{{c}})

|13/8=[[Trithagorean]] (884.359{{c}})

|13/8=Just-[[5/3]] a.k.a. [[Trithagorean]] tuning (884.359{{c}})

|13/5=[[Polaris]]

|2/1=[[CWE]]-optimized [[Aldebaran]] (888.125{{c}})

|13/5=[[CWE]]-optimized [[Polaris]] (892.050{{c}})

}}

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 {{MOS intro}}
-This MOS scale pattern is most notable for corresponding to the 13-note [[Trithagorean]]/[[3.5 subgroup]] scale generated by a just [[5/3]], which has an L:s ratio of approximately 1.64. In this case, the large step represents [[48828125/43046721]] and the small step represents [[27/25]]. It is also the 13-note MOS of several tritave-equivalent temperaments generated by ~5/3 (similar to how the [[5L 2s]] diatonic scale is the 7-note MOS of chain-of-fifth temperaments).  The [[3.5.11 subgroup]] [[microtemperament]] [[Delta centauri]] (which is generated by a near-just ~5/3 such that 9 generators stacked represent ~[[11/9]]) has this as its 13-note MOS scale, although the 15-note {{MOS scalesig|13L 2s<3/1>|link=1}} scale may be preferred as it has a much greater abundance of 9:11:15 triads for only 2 more notes. [[Arcturus]] temperament tunings with ~5/3 sharper than 6 steps of [[13edt]] will generate this MOS, but tunings with ~5/3 flatter than 6 steps of [[13edt]] will instead generate the sister MOS {{MOS scalesig|11L 2s<3/1>|link=1}}.
+This MOS scale pattern is most notable for corresponding to the 13-note [[Trithagorean]]/[[3.5 subgroup]] scale generated by a just [[5/3]], which has an L:s ratio of approximately 1.64, being the smallest non-trivial [[proper]] MOS scale generated by the 3.5 subgroup. In this case, the large step represents [[48828125/43046721]] and the small step represents [[27/25]]. It is also the 13-note MOS of several tritave-equivalent temperaments generated by ~5/3 (similar to how the [[5L 2s]] diatonic scale is the 7-note MOS of chain-of-fifth temperaments).  The [[3.5.11 subgroup]] [[microtemperament]] [[delta centauri]] (which is generated by a near-just ~5/3 such that 9 generators stacked represent ~[[11/9]]) has this as its 13-note MOS scale, although the 15-note daughter scale {{MOS scalesig|13L 2s<3/1>|link=1}} may be preferred for that temperament due to the complexity of 11/9. [[Arcturus]] temperament tunings with ~5/3 sharper than 6 steps of [[13edt]] will generate this MOS, but tunings with ~5/3 flatter than 6 steps of [[13edt]] will instead generate the sister MOS {{MOS scalesig|11L 2s<3/1>|link=1}}.
-In the moderately hard-of-basic region is the 3.5.11 subgroup [[Polaris]] temperament which, like Delta centauri, constructs an ~11/9 from stacking ~5/3 generators, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 [[9/5]]s stacked) tritave-reduced is ~11/9. If the L:s ratio is very high (starting at around 5:1), the generator becomes too sharp to approximate 5/3, and everything gets closer and closer to collapsing into [[2edt]].
+In the region very near basic ([[15edt]]) tuning is the 3.5.13 subgroup [[Aldebaran]] temperament, which is generated by ~5/3 such that 5 generators stacked represent ~[[13/9]]. In the moderately hard-of-basic region is the 3.5.11 subgroup [[Polaris]] temperament which, like Delta centauri, constructs an ~11/9 from stacking ~5/3 generators, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 [[9/5]]s stacked) tritave-reduced is ~11/9. If the L:s ratio is very high (starting at around 5:1), the generator becomes too sharp to approximate 5/3, and everything gets closer and closer to collapsing into [[2edt]].
 It can be considered as a kind of detempering of [[13edt]], but it would probably make more sense to consider the daughter MOSes of [[Lambda]] instead.
@@ Line 33: / Line 33: @@
 == Scale tree ==
 {{MOS tuning spectrum
-|1/1=[[Arcturus]]
+|1/1=[[CWE]]-optimized [[Arcturus]] (878.291{{c}})
-|8/5=[[Delta centauri]]
+|8/5=[[CWE]]-optimized [[Delta centauri]] (883.901{{c}})
-|13/8=[[Trithagorean]] (884.359{{c}})
+|13/8=Just-[[5/3]] a.k.a. [[Trithagorean]] tuning (884.359{{c}})
-|13/5=[[Polaris]]
+|2/1=[[CWE]]-optimized [[Aldebaran]] (888.125{{c}})
+|13/5=[[CWE]]-optimized [[Polaris]] (892.050{{c}})
 }}
 {{todo|inline=1| expand}}

2L 11s (3/1-equivalent): Difference between revisions

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