Bipentatonic scale: Difference between revisions

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* [[Onyxwood]] (5L 3M 2s), LMLsLMLsLM

* 5L 4M 1s, LMLMLMLMLs

=== Dipentic 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.

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.

* 9/8 ≤ δ ≤ 32/27

* 81/64 ≤ δ ≤ 4/3

Offsets between 1/1 and 9/8 yield the three ternary dipythpentic scale patterns of the form ababacabac, where c > a. Outside of this range, ternary dipythpentic scales only occur with 3 values for offsets.

* If δ < sqrt(9/8), we obtain the ternary scale 2L3m5s (smsmsLsmsL).

* If δ = sqrt(9/8) = 101.955c, then we have a permutation of 2L8s (sssssLsssL).

* If sqrt(9/8) < δ < sqrt(32/27), we have 2L5m3s (msmsmLmsmL).

* If δ = sqrt(32/27) = 147.067c, we have a permutation of 7L3s (LsLsLLLsLL).

* If sqrt(32/27) < δ < 9/8, we have 5L2m3s (LsLsLmLsLm).

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

Offsets between 32/27 and 81/64 yield dipentic 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.

* 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.

* If sqrt(729/512) < δ < sqrt(3/2), then b > c > d > a > 0.

* The offset δ = sqrt(3/2) = 350.978c implies b = c > d = a > 0 and yields the 3L7s MOS sLLsLLsLLL.

* If sqrt(3/2) < δ < sqrt(128/81), then c > b > a > d > 0.

* The offset δ = sqrt(128/81) = 396.090c implies c > b = a > d > 0 and yields the ternary scale mmLsLmmmLm.

* If sqrt(128/81) < δ < 81/64, then c > a > b > d > 0.

* 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.

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

* If sqrt(243/128) < δ < 1\\2, then d > c > b > a > 0.

* If δ = 1\\2, then d = c > b = a > 0, and we get the 2L8s MOS ssssLssssL.

== Just dipentatonic scales ==

@@ Line 29: / Line 29: @@
 * [[Onyxwood]] (5L 3M 2s), LMLsLMLsLM <!-- Named by inthar -->
 * 5L 4M 1s, LMLMLMLMLs
+=== Dipentic 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.
+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.
+* 9/8 ≤ δ ≤ 32/27
+* 81/64 ≤ δ ≤ 4/3
+Offsets between 1/1 and 9/8 yield the three ternary dipythpentic scale patterns of the form ababacabac, where c > a. Outside of this range, ternary dipythpentic scales only occur with 3 values for offsets.
+* If δ < sqrt(9/8), we obtain the ternary scale 2L3m5s (smsmsLsmsL).
+* If δ = sqrt(9/8) = 101.955c, then we have a permutation of 2L8s (sssssLsssL).
+* If sqrt(9/8) < δ < sqrt(32/27), we have 2L5m3s (msmsmLmsmL).
+* If δ = sqrt(32/27) = 147.067c, we have a permutation of 7L3s (LsLsLLLsLL).
+* If sqrt(32/27) < δ < 9/8, we have 5L2m3s (LsLsLmLsLm).
+* In the degenerate case δ = 9/8, we have the 5L2s diatonic MOS LLLsLLs.
+Offsets between 32/27 and 81/64 yield dipentic 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.
+* 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.
+* If sqrt(729/512) < δ < sqrt(3/2), then b > c > d > a > 0.
+* The offset δ = sqrt(3/2) = 350.978c implies b = c > d = a > 0 and yields the 3L7s MOS sLLsLLsLLL.
+* If sqrt(3/2) < δ < sqrt(128/81), then c > b > a > d > 0.
+* The offset δ = sqrt(128/81) = 396.090c implies c > b = a > d > 0 and yields the ternary scale mmLsLmmmLm.
+* If sqrt(128/81) < δ < 81/64, then c > a > b > d > 0.
+* 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.
+* In the degenerate case δ = 4/3, d > b > c > a = 0, and we obtain a non-dipentic 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.
+* If sqrt(243/128) < δ < 1\\2, then d > c > b > a > 0.
+* If δ = 1\\2, then d = c > b = a > 0, and we get the 2L8s MOS ssssLssssL.
 == Just dipentatonic scales ==