Omnidiatonic: Difference between revisions

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'''Omnidiatonic''' (also known as '''interdia''' and '''archylino''') is a 7-note [[Maximum variety|maximum-variety-3]] scale with the [[step signature]] 2L 3M 2s. Omnidiatonic is a [[chiral]] scale with LMsMLsM and LMsLMsM variants. [[14edo]] is the first equal division that supports omnidiatonic. The name "omnidiatonic" was given by [[User:CompactStar|CompactStar]] and the name "interdia" was given by [[User:Xenllium|Xenllium]], both of which refer to this scale being intermediate between the [[5L 2s]] diatonic scale and the [[2L 5s]] antidiatonic scale. The name "archylino" was given by [[User:AthiTrydhen|Praveen Venkataramana]], which refers to intervals separated by 64/63, the Archytas comma, being mapped to the same number of scale steps of 2.3.7 JI archylino 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1 (MsLMsML).

'''Omnidiatonic''' (also known as '''interdia''' and '''archylino''') is a 7-note [[Maximum variety|maximum-variety-3]] scale with the [[step signature]] 2L 3M 2s. Omnidiatonic is a [[chiral]] scale with LMsMLsM and LMsLMsM variants. [[14edo]] is the first equal division that supports omnidiatonic. The name "omnidiatonic" was given by [[User:CompactStar|CompactStar]] and the name "interdia" was given by [[User:Xenllium|Xenllium]], both of which refer to this scale being intermediate between the [[5L 2s]] diatonic scale and the [[2L 5s]] antidiatonic scale. The name "archylino" was given by [[User:AthiTrydhen|Praveen Venkataramana]], which refers to intervals separated by 64/63, the Archytas comma, being mapped to the same number of scale steps of 2.3.7 JI archylino 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1 (MsLMsML). In terms of Greek scales, this can be seen as Archytas' diatonic.

Omnidiatonic can be tuned as a 7-limit JI scale or a tempered version thereof, where L represents 8/7, M represents 9/8, and s represents 28/27.

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|-

! Outer generator (''G''1 = L + 2M + s)

| <math>\displaystyle \frac{1}{2} < G_\text{1} < \frac{3}{5}</math>

| <math>\displaystyle \frac{1}{2} \lt G_\text{1} \lt \frac{3}{5}</math>

|-

! RH inner generator (''G''2R = M + s)

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

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

|-

! LH inner generator (''G''2L = L + M)

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

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

|-

! Large step (L = 1 - ''G''1 - ''G''2R)

| <math>\displaystyle \frac{1}{7} < L < \frac{1}{2}</math>

| <math>\displaystyle \frac{1}{7} \lt L \lt \frac{1}{2}</math>

|-

! Middle step (M = 2''G''1 - 1)

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

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

|-

! Small step (s = 1 - ''G''1 - ''G''2L)

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

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

|}

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-'''Omnidiatonic''' (also known as '''interdia''' and '''archylino''') is a 7-note [[Maximum variety|maximum-variety-3]] scale with the [[step signature]] 2L 3M 2s. Omnidiatonic is a [[chiral]] scale with LMsMLsM and LMsLMsM variants. [[14edo]] is the first equal division that supports omnidiatonic. The name "omnidiatonic" was given by [[User:CompactStar|CompactStar]] and the name "interdia" was given by [[User:Xenllium|Xenllium]], both of which refer to this scale being intermediate between the [[5L 2s]] diatonic scale and the [[2L 5s]] antidiatonic scale. The name "archylino" was given by [[User:AthiTrydhen|Praveen Venkataramana]], which refers to intervals separated by 64/63, the Archytas comma, being mapped to the same number of scale steps of 2.3.7 JI archylino 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1 (MsLMsML).
+'''Omnidiatonic''' (also known as '''interdia''' and '''archylino''') is a 7-note [[Maximum variety|maximum-variety-3]] scale with the [[step signature]] 2L 3M 2s. Omnidiatonic is a [[chiral]] scale with LMsMLsM and LMsLMsM variants. [[14edo]] is the first equal division that supports omnidiatonic. The name "omnidiatonic" was given by [[User:CompactStar|CompactStar]] and the name "interdia" was given by [[User:Xenllium|Xenllium]], both of which refer to this scale being intermediate between the [[5L 2s]] diatonic scale and the [[2L 5s]] antidiatonic scale. The name "archylino" was given by [[User:AthiTrydhen|Praveen Venkataramana]], which refers to intervals separated by 64/63, the Archytas comma, being mapped to the same number of scale steps of 2.3.7 JI archylino 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1 (MsLMsML). In terms of Greek scales, this can be seen as Archytas' diatonic.
 Omnidiatonic can be tuned as a 7-limit JI scale or a tempered version thereof, where L represents 8/7, M represents 9/8, and s represents 28/27.
@@ Line 40: / Line 40: @@
 |-
 ! Outer generator <br>(''G''<sub>1</sub> = L + 2M + s)
-| <math>\displaystyle \frac{1}{2} &lt; G_\text{1} &lt; \frac{3}{5}</math>
+| <math>\displaystyle \frac{1}{2} \lt G_\text{1} \lt \frac{3}{5}</math>
 |-
 ! RH inner generator <br>(''G''<sub>2R</sub> = M + s)
-| <math>\displaystyle 2 G_\text{1} - 1 &lt; G_\text{2R} &lt; 4 G_\text{1} - 2 \text{ for }\frac{1}{2} &lt; G_\text{1} &le; \frac{4}{7}</math> <br><math>\displaystyle 2 G_\text{1} - 1 &lt; G_\text{2R} &lt; 2 - 3 G_\text{1} \text{ for }\frac{4}{7} &le; G_\text{1} &lt; \frac{3}{5}</math>
+| <math>\displaystyle 2 G_\text{1} - 1 \lt G_\text{2R} \lt 4 G_\text{1} - 2 \text{ for }\frac{1}{2} \lt G_\text{1} \le \frac{4}{7}</math> <br><math>\displaystyle 2 G_\text{1} - 1 \lt G_\text{2R} \lt 2 - 3 G_\text{1} \text{ for }\frac{4}{7} \le G_\text{1} \lt \frac{3}{5}</math>
 |-
 ! LH inner generator <br>(''G''<sub>2L</sub> = L + M)
-| <math>\displaystyle 2 - 3 G_\text{1} &lt; G_\text{2L} &lt; 1 - G_\text{1} \text{ for } \frac{1}{2} &lt; G_\text{1} &le; \frac{4}{7}</math> <br><math>\displaystyle 4 G_\text{1} - 2 &lt; G_\text{2L} &lt; 1 - G_\text{1} \text{ for }\frac{4}{7} &le; G_\text{1} &lt; \frac{3}{5}</math>
+| <math>\displaystyle 2 - 3 G_\text{1} \lt G_\text{2L} \lt 1 - G_\text{1} \text{ for } \frac{1}{2} \lt G_\text{1} \le \frac{4}{7}</math> <br><math>\displaystyle 4 G_\text{1} - 2 \lt G_\text{2L} \lt 1 - G_\text{1} \text{ for }\frac{4}{7} \le G_\text{1} \lt \frac{3}{5}</math>
 |-
 ! Large step <br>(L = 1 - ''G''<sub>1</sub> - ''G''<sub>2R</sub>)
-| <math>\displaystyle \frac{1}{7} &lt; L &lt; \frac{1}{2}</math>
+| <math>\displaystyle \frac{1}{7} \lt L \lt \frac{1}{2}</math>
 |-
 ! Middle step <br>(M = 2''G''<sub>1</sub> - 1)
-| <math>\displaystyle \frac{1}{5} (1 - 2 L) &lt; M &lt; L \text{ for } \frac{1}{7} &lt; L &le; \frac{1}{5}</math> <br><math>\displaystyle \frac{1}{5} (1 - 2 L) &lt; M &lt; \frac{1}{3} (1 - 2 L) \text{ for } \frac{1}{5} &le; L &lt; \frac{1}{2}</math>
+| <math>\displaystyle \frac{1}{5} (1 - 2 L) \lt M \lt L \text{ for } \frac{1}{7} \lt L \le \frac{1}{5}</math> <br><math>\displaystyle \frac{1}{5} (1 - 2 L) \lt M \lt \frac{1}{3} (1 - 2 L) \text{ for } \frac{1}{5} \le L \lt \frac{1}{2}</math>
 |-
 ! Small step <br>(s = 1 - ''G''<sub>1</sub> - ''G''<sub>2L</sub>)
-| <math>\displaystyle \frac{1}{2} (1 - 5 L) &lt; s &lt; \frac{1}{5} (1 - 2 L) \text{ for } \frac{1}{7} &lt; L &le; \frac{1}{5}</math> <br><math>\displaystyle 0 &lt; s &lt; \frac{1}{5} (1 - 2 L) \text{ for } \frac{1}{5} &le; L &lt; \frac{1}{2}</math>
+| <math>\displaystyle \frac{1}{2} (1 - 5 L) \lt s \lt \frac{1}{5} (1 - 2 L) \text{ for } \frac{1}{7} \lt L \le \frac{1}{5}</math> <br><math>\displaystyle 0 \lt s \lt \frac{1}{5} (1 - 2 L) \text{ for } \frac{1}{5} \le L \lt \frac{1}{2}</math>
 |}