Bipentatonic scale: Difference between revisions
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A '''bipentatonic scale''' is a 10-note octave-equivalent scale where every other note gives a fixed choice of pentatonic scale; hence a bipentatonic scale is a type of [[flought scale]]. Following from this, bipentatonic scales based on a [[MOS]] pentatonic scale have a maximum of two sizes for intervals that are an even number of steps. Many bipentatonic scales are generated by a diatonic-sized fifth (between the 7edo fifth and the 5edo fifth) of a fixed size. Some bipentatonic scales are MOSes, such as the Erlich decatonic in [[22edo]]. Modulating by fifths is easy in bipentatonic scales where the interleaved pentatonics are generated by a fifth (i.e. [[2L 3s]] and [[3L 2s]], depending on tuning). | A '''bipentatonic scale''' is a 10-note octave-equivalent scale where every other note gives a fixed choice of pentatonic scale; hence a bipentatonic scale is a type of [[flought scale]]. Following from this, bipentatonic scales based on a [[MOS]] pentatonic scale have a maximum of two sizes for intervals that are an even number of steps. Many bipentatonic scales are generated by a diatonic-sized fifth (between the 7edo fifth and the 5edo fifth) of a fixed size. Some bipentatonic scales are MOSes, such as the Erlich decatonic in [[22edo]]. Modulating by fifths is easy in bipentatonic scales where the interleaved pentatonics are generated by a fifth (i.e. [[2L 3s]] and [[3L 2s]], depending on tuning). | ||
The first part of this article classifies abstract bipentatonic scales by the pentatonics they interleave and by their rank (whether they are mosses or rank-3). The second part of this article surveys bipentatonic scales in [[JI]]. | The first part of this article classifies abstract bipentatonic scales by the pentatonics they interleave and by their rank (whether they are mosses or rank-3). The second part of this article surveys bipentatonic scales in [[JI]]. Bipentatonic scales also exist in [[regular temperaments]] of course, but exploration of such scales is not included in the article for now at least. | ||
== Abstract bipentatonic scales == | == Abstract bipentatonic scales == | ||
=== | === Bipentatonic mosses === | ||
Every 10 note mos except 5L 5s is bipentatonic. The degenerate case of a bipentatonic mos is 5L 5s, where the interleaved pentatonic is 5edo. The nondegenerate cases are: | Every 10 note mos except 5L 5s is bipentatonic. The degenerate case of a bipentatonic mos is 5L 5s, where the interleaved pentatonic is 5edo. The nondegenerate cases are: | ||
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* 9L 1s: LLLLLLLLLs (pentatonic 4L 1s) | * 9L 1s: LLLLLLLLLs (pentatonic 4L 1s) | ||
=== | === Bipentatonic ternary scale patterns === | ||
A ternary bipentatonic scale with step sizes 5x ay (5-a)z has the form xYxYxYxYxY where the Y's are replaced with y and z steps arranged in the [[MOS]] pattern ay (5-a)z. The interlocking pentatonics are copies of the mos aL (5-a)s if y > z and (5-a)L as if z > y. The following is a complete list of such abstract patterns, assuming octave equivalence: | A ternary bipentatonic scale with step sizes 5x ay (5-a)z has the form xYxYxYxYxY where the Y's are replaced with y and z steps arranged in the [[MOS]] pattern ay (5-a)z. The interlocking pentatonics are copies of the mos aL (5-a)s if y > z and (5-a)L as if z > y. The following is a complete list of such abstract patterns, assuming octave equivalence: | ||
* 1L 4M 5s, LsMsMsMsMs | * 1L 4M 5s, LsMsMsMsMs | ||
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== Just bipentatonic scales == | == Just bipentatonic scales == | ||
Every other note of a just bipentatonic scale gives a Pythagorean pentatonic. | Every other note of a just bipentatonic scale gives a Pythagorean pentatonic. Bipentatonic scales that are also [[SN scales|3-SN scales]] can be constructed by placing the same interval above or below each step of a pentatonic scale. This interval defines the scale, and the logic behind the listing below. | ||
These scales can be considered the minimum complexity rank-3 decatonic scales that are supersets of Pythagorean[5]. The can be thought of as Blackwood decatonics, but without 256/243 tempered out. Instead of 5 240c intervals in an octave as one generator and a 5/4 as the other, these scales have (one incstance of) a third generator of a prime > 3 along with Pythagorean[5], or two parallel Pythagorean[5]s, seperated by a prime or a prime to some power of three. They have form ABACABABAC (or CABABACABA, inverted, or beginning after ABA) where AB=9/8 and AC=32/27. | These scales can be considered the minimum complexity rank-3 decatonic scales that are supersets of Pythagorean[5]. The can be thought of as Blackwood decatonics, but without 256/243 tempered out. Instead of 5 240c intervals in an octave as one generator and a 5/4 as the other, these scales have (one incstance of) a third generator of a prime > 3 along with Pythagorean[5], or two parallel Pythagorean[5]s, seperated by a prime or a prime to some power of three. They have form ABACABABAC (or CABABACABA, inverted, or beginning after ABA) where AB=9/8 and AC=32/27. | ||
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112/99 (32/27) 14/11 4/3 (63/44) 3/2 56/33 (16/9) 21/11 2/1 | 112/99 (32/27) 14/11 4/3 (63/44) 3/2 56/33 (16/9) 21/11 2/1 | ||
== | == Bipentatonic scales in regular temperaments == | ||
Just a couple examples linked for now: | Just a couple examples linked for now: | ||