Bipentatonic scale: Difference between revisions

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* 5L 4M 1s, LMLMLMLMLs

== ~~Dipentic~~ scales using the Pythagorean pentic and arbitrary offsets ==

== Bipentic scales using the Pythagorean pentic and arbitrary offsets ==

A ''~~dipentic~~'' scale is a 10-note scale such that the even degrees form one copy of the [[pentic]] MOS 2L3s and the odd degrees form a second, shifted copy of 2L3s with the same generator tuning as the first copy. ~~Dipentic~~ scales are classified by the generator used by the pentic and the ''offset'' between the two copies of pentic. Here we classify ''dipythpentic'' scales, ~~dipentic~~ scales that use the 3/2-generated Pythagorean tuning for the two copies of pentic.

A ''bipentic'' scale is a 10-note scale such that the even degrees form one copy of the [[pentic]] MOS 2L3s and the odd degrees form a second, shifted copy of 2L3s with the same generator tuning as the first copy. Bipentic scales are classified by the generator used by the pentic and the ''offset'' between the two copies of pentic. Here we classify ''dipythpentic'' scales, bipentic scales that use the 3/2-generated Pythagorean tuning for the two copies of pentic.

Assuming octave equivalence, the offsets δ and 1200 - δ behave the same. Taking that fact into account, the following offset ranges do not yield dipythpentic scales. With these offsets, the two pentic scales do not interleave because there is an s-step of one copy of pentic that is contained entirely within a pentic L-step of the second.

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* In the degenerate case δ = 9/8, we have the ternary scale 5L2s diatonic MOS LLLsLLs.

Offsets between 32/27 and 81/64 yield ~~dipentic~~ scales of the pattern abcdcbabcb.

Offsets between 32/27 and 81/64 yield bipentic scales of the pattern abcdcbabcb.

* In the degenerate case δ = 32/27, then b > d > c > a = 0 and we have the non-~~dipentic~~ ternary scale LsmsLLsL.

* In the degenerate case δ = 32/27, then b > d > c > a = 0 and we have the non-bipentic ternary scale LsmsLLsL.

* If 32/27 < δ < sqrt(729/512), then b > d > c > a > 0.

* The offset δ = sqrt(729/512) = 305.865c implies b > d = c > a and we have the ternary scale sLmmmLsLmL.

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* In the degenerate case δ = 81/64, c > a > b > d = 0 and we have msLLsmsLs.

Offsets between 4/3 and 1\\2 yield ~~dipentic~~ scales of the pattern ababcbabad.

Offsets between 4/3 and 1\\2 yield bipentic scales of the pattern ababcbabad.

* In the degenerate case δ = 4/3, d > b > c > a = 0, and we obtain a non-~~dipentic~~ ternary scale mmsmmL.

* In the degenerate case δ = 4/3, d > b > c > a = 0, and we obtain a non-bipentic ternary scale mmsmmL.

* If 4/3 < δ < sqrt(243/128), then d > b > c > a > 0.

* The offset δ = sqrt(243/128) = 554.888c implies d > b = c > a > 0 and yields the ternary scale smsmmmsmsL.

@@ Line 30: / Line 30: @@
 * 5L 4M 1s, LMLMLMLMLs
-== Dipentic scales using the Pythagorean pentic and arbitrary offsets ==
+== Bipentic scales using the Pythagorean pentic and arbitrary offsets ==
-A ''dipentic'' scale is a 10-note scale such that the even degrees form one copy of the [[pentic]] MOS 2L3s and the odd degrees form a second, shifted copy of 2L3s with the same generator tuning as the first copy. Dipentic scales are classified by the generator used by the pentic and the ''offset'' between the two copies of pentic. Here we classify ''dipythpentic'' scales, dipentic scales that use the 3/2-generated Pythagorean tuning for the two copies of pentic.
+A ''bipentic'' scale is a 10-note scale such that the even degrees form one copy of the [[pentic]] MOS 2L3s and the odd degrees form a second, shifted copy of 2L3s with the same generator tuning as the first copy. Bipentic scales are classified by the generator used by the pentic and the ''offset'' between the two copies of pentic. Here we classify ''dipythpentic'' scales, bipentic scales that use the 3/2-generated Pythagorean tuning for the two copies of pentic.
 Assuming octave equivalence, the offsets δ and 1200 - δ behave the same. Taking that fact into account, the following offset ranges do not yield dipythpentic scales. With these offsets, the two pentic scales do not interleave because there is an s-step of one copy of pentic that is contained entirely within a pentic L-step of the second.
@@ Line 45: / Line 45: @@
 * In the degenerate case δ = 9/8, we have the ternary scale 5L2s diatonic MOS LLLsLLs.
-Offsets between 32/27 and 81/64 yield dipentic scales of the pattern abcdcbabcb.
+Offsets between 32/27 and 81/64 yield bipentic scales of the pattern abcdcbabcb.
-* In the degenerate case δ = 32/27, then b > d > c > a = 0 and we have the non-dipentic ternary scale LsmsLLsL.
+* In the degenerate case δ = 32/27, then b > d > c > a = 0 and we have the non-bipentic ternary scale LsmsLLsL.
 * If 32/27 < δ < sqrt(729/512), then b > d > c > a > 0.
 * The offset δ = sqrt(729/512) = 305.865c implies b > d = c > a and we have the ternary scale sLmmmLsLmL.
@@ Line 56: / Line 56: @@
 * In the degenerate case δ = 81/64, c > a > b > d = 0 and we have msLLsmsLs.
-Offsets between 4/3 and 1\\2 yield dipentic scales of the pattern ababcbabad.
+Offsets between 4/3 and 1\\2 yield bipentic scales of the pattern ababcbabad.
-* In the degenerate case δ = 4/3, d > b > c > a = 0, and we obtain a non-dipentic ternary scale mmsmmL.
+* In the degenerate case δ = 4/3, d > b > c > a = 0, and we obtain a non-bipentic ternary scale mmsmmL.
 * If 4/3 < δ < sqrt(243/128), then d > b > c > a > 0.
 * The offset δ = sqrt(243/128) = 554.888c implies d > b = c > a > 0 and yields the ternary scale smsmmmsmsL.