Nicetone: Difference between revisions

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== 5-limit Zarlino scale ==

'''[[Ptolemy's intense diatonic]]''', '''5-limit Zarlino''' or sometimes '''Zarlino''' for short is a [[heptatonic]] [[5-limit]] [[JI]] [[scale]] with the [[nicetone]] step pattern. It consists of the intervals [[1/1]] ~~- 9~~/~~8 -~~ [[5/4]] - [[4/3]] - [[3/2]] - [[5/3]] - [[15/8]] - [[2/1]]. It corresponds to the case where L represents [[9/8]], M represents [[10/9]], and s represents [[16/15]]. See the [[Ptolemy's intense diatonic|dedicated page]] for Scala files and Fokker blocks information.

'''[[Ptolemy's intense diatonic]]''', '''5-limit Zarlino''' or sometimes '''Zarlino''' for short is a [[heptatonic]] [[5-limit]] [[JI]] [[scale]] with the [[nicetone]] step pattern. It consists of the intervals [[1/1]]–9/8–[[5/4]]–[[4/3]]–[[3/2]]–[[5/3]]–[[15/8]]–[[2/1]]. It corresponds to the case where L represents [[9/8]], M represents [[10/9]], and s represents [[16/15]]. See the [[Ptolemy's intense diatonic|dedicated page]] for Scala files and Fokker blocks information.

== Intervals ==

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|-

! Outer generator ({{nowrap|''G''1 {{=}} 2L + M + s}})

| <math>\displaystyle \frac{4}{7} < G_\text{1} < \frac{2}{3}</math>

| <math>\displaystyle \frac{4}{7} \lt G_\text{1} \lt \frac{2}{3}</math>

|-

! RH inner generator ({{nowrap|''G''2R {{=}} L + M}})

| <math>\displaystyle \frac{1}{2} G_\text{1} < G_\text{2R} < 4 G_\text{1} - 2 \,\text{ for }\, \frac{4}{7} < G_\text{1} ≤ \frac{3}{5}</math> <math>\displaystyle \frac{1}{2} G_\text{1} < G_\text{2R} < 1 - G_\text{1} \,\text{ for }\,\frac{3}{5} ≤ G_\text{1} < \frac{2}{3}</math>

| <math>\displaystyle \frac{1}{2} G_\text{1} \lt G_\text{2R} \lt 4 G_\text{1} - 2 \,\text{ for }\, \frac{4}{7} \lt G_\text{1} \le \frac{3}{5}</math> <math>\displaystyle \frac{1}{2} G_\text{1} \lt G_\text{2R} \lt 1 - G_\text{1} \,\text{ for }\,\frac{3}{5} \le G_\text{1} \lt \frac{2}{3}</math>

|-

! LH inner generator ({{nowrap|''G''2L {{=}} L + s}})

| <math>\displaystyle 2 - 3 G_\text{1} < G_\text{2L} < \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{4}{7} < G_\text{1} ≤ \frac{3}{5}</math> <math>\displaystyle 2 G_\text{1} - 1 < G_\text{2L} < \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{3}{5} ≤ G_\text{1} < \frac{2}{3}</math>

| <math>\displaystyle 2 - 3 G_\text{1} \lt G_\text{2L} \lt \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{4}{7} \lt G_\text{1} \le \frac{3}{5}</math> <math>\displaystyle 2 G_\text{1} - 1 \lt G_\text{2L} \lt \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{3}{5} \le G_\text{1} \lt \frac{2}{3}</math>

|-

! Large step ({{nowrap|L {{=}} 2''G''1 − 1}})

| <math>\displaystyle \frac{1}{7} < L < \frac{1}{3}</math>

| <math>\displaystyle \frac{1}{7} \lt L \lt \frac{1}{3}</math>

|-

! Middle step ({{nowrap|M {{=}} 1 − ''G''1 − ''G''2L}})

| <math>\displaystyle \frac{1}{4} (1 - 3 L) < M < L \,\text{ for }\, \frac{1}{7} < L ≤ \frac{1}{5}</math> <math>\displaystyle \frac{1}{4} (1 - 3 L) < M < \frac{1}{2} (1 - 3 L) \,\text{ for }\, \frac{1}{5} ≤ L < \frac{1}{3}</math>

| <math>\displaystyle \frac{1}{4} (1 - 3 L) \lt M \lt L \,\text{ for }\, \frac{1}{7} \lt L \le \frac{1}{5}</math> <math>\displaystyle \frac{1}{4} (1 - 3 L) \lt M \lt \frac{1}{2} (1 - 3 L) \,\text{ for }\, \frac{1}{5} \le L \lt \frac{1}{3}</math>

|-

! Small step ({{nowrap|s {{=}} 1 − ''G''1 − ''G''2R}})

| <math>\displaystyle \frac{1}{2} (1 - 5 L) < s < \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{7} < L ≤ \frac{1}{5}</math> <math>\displaystyle 0 < s < \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{5} ≤ L < \frac{1}{3}</math>

| <math>\displaystyle \frac{1}{2} (1 - 5 L) \lt s \lt \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{7} \lt L \le \frac{1}{5}</math> <math>\displaystyle 0 \lt s \lt \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{5} \le L \lt \frac{1}{3}</math>

|}

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 == 5-limit Zarlino scale ==
-'''[[Ptolemy's intense diatonic]]''', '''5-limit Zarlino''' or sometimes '''Zarlino''' for short is a [[heptatonic]] [[5-limit]] [[JI]] [[scale]] with the [[nicetone]] step pattern. It consists of the intervals [[1/1]] - 9/8 - [[5/4]] - [[4/3]] - [[3/2]] - [[5/3]] - [[15/8]] - [[2/1]]. It corresponds to the case where L represents [[9/8]], M represents [[10/9]], and s represents [[16/15]]. See the [[Ptolemy's intense diatonic|dedicated page]] for Scala files and Fokker blocks information.
+'''[[Ptolemy's intense diatonic]]''', '''5-limit Zarlino''' or sometimes '''Zarlino''' for short is a [[heptatonic]] [[5-limit]] [[JI]] [[scale]] with the [[nicetone]] step pattern. It consists of the intervals [[1/1]]–9/8–[[5/4]]–[[4/3]]–[[3/2]]–[[5/3]]–[[15/8]]–[[2/1]]. It corresponds to the case where L represents [[9/8]], M represents [[10/9]], and s represents [[16/15]]. See the [[Ptolemy's intense diatonic|dedicated page]] for Scala files and Fokker blocks information.
 == Intervals ==
@@ Line 190: / Line 190: @@
 |-
 ! Outer generator<br />({{nowrap|''G''<sub>1</sub> {{=}} 2L + M + s}})
-| <math>\displaystyle \frac{4}{7} &lt; G_\text{1} &lt; \frac{2}{3}</math>
+| <math>\displaystyle \frac{4}{7} \lt G_\text{1} \lt \frac{2}{3}</math>
 |-
 ! RH inner generator<br />({{nowrap|''G''<sub>2R</sub> {{=}} L + M}})
-| <math>\displaystyle \frac{1}{2} G_\text{1} &lt; G_\text{2R} &lt; 4 G_\text{1} - 2 \,\text{ for }\, \frac{4}{7} &lt; G_\text{1} &le; \frac{3}{5}</math> <br><math>\displaystyle \frac{1}{2} G_\text{1} &lt; G_\text{2R} &lt; 1 - G_\text{1} \,\text{ for }\,\frac{3}{5} &le; G_\text{1} &lt; \frac{2}{3}</math>
+| <math>\displaystyle \frac{1}{2} G_\text{1} \lt G_\text{2R} \lt 4 G_\text{1} - 2 \,\text{ for }\, \frac{4}{7} \lt G_\text{1} \le \frac{3}{5}</math> <br><math>\displaystyle \frac{1}{2} G_\text{1} \lt G_\text{2R} \lt 1 - G_\text{1} \,\text{ for }\,\frac{3}{5} \le G_\text{1} \lt \frac{2}{3}</math>
 |-
 ! LH inner generator<br />({{nowrap|''G''<sub>2L</sub> {{=}} L + s}})
-| <math>\displaystyle 2 - 3 G_\text{1} &lt; G_\text{2L} &lt; \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{4}{7} &lt; G_\text{1} &le; \frac{3}{5}</math> <br><math>\displaystyle 2 G_\text{1} - 1 &lt; G_\text{2L} &lt; \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{3}{5} &le; G_\text{1} &lt; \frac{2}{3}</math>
+| <math>\displaystyle 2 - 3 G_\text{1} \lt G_\text{2L} \lt \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{4}{7} \lt G_\text{1} \le \frac{3}{5}</math> <br><math>\displaystyle 2 G_\text{1} - 1 \lt G_\text{2L} \lt \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{3}{5} \le G_\text{1} \lt \frac{2}{3}</math>
 |-
 ! Large step<br />({{nowrap|L {{=}} 2''G''<sub>1</sub> &minus; 1}})
-| <math>\displaystyle \frac{1}{7} &lt; L &lt; \frac{1}{3}</math>
+| <math>\displaystyle \frac{1}{7} \lt L \lt \frac{1}{3}</math>
 |-
 ! Middle step<br />({{nowrap|M {{=}} 1 &minus; ''G''<sub>1</sub> &minus; ''G''<sub>2L</sub>}})
-| <math>\displaystyle \frac{1}{4} (1 - 3 L) &lt; M &lt; L \,\text{ for }\, \frac{1}{7} &lt; L &le; \frac{1}{5}</math> <br><math>\displaystyle \frac{1}{4} (1 - 3 L) &lt; M &lt; \frac{1}{2} (1 - 3 L) \,\text{ for }\, \frac{1}{5} &le; L &lt; \frac{1}{3}</math>
+| <math>\displaystyle \frac{1}{4} (1 - 3 L) \lt M \lt L \,\text{ for }\, \frac{1}{7} \lt L \le \frac{1}{5}</math> <br><math>\displaystyle \frac{1}{4} (1 - 3 L) \lt M \lt \frac{1}{2} (1 - 3 L) \,\text{ for }\, \frac{1}{5} \le L \lt \frac{1}{3}</math>
 |-
 ! Small step<br />({{nowrap|s {{=}} 1 &minus; ''G''<sub>1</sub> &minus; ''G''<sub>2R</sub>}})
-| <math>\displaystyle \frac{1}{2} (1 - 5 L) &lt; s &lt; \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{7} &lt; L &le; \frac{1}{5}</math> <br><math>\displaystyle 0 &lt; s &lt; \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{5} &le; L &lt; \frac{1}{3}</math>
+| <math>\displaystyle \frac{1}{2} (1 - 5 L) \lt s \lt \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{7} \lt L \le \frac{1}{5}</math> <br><math>\displaystyle 0 \lt s \lt \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{5} \le L \lt \frac{1}{3}</math>
 |}