Bipentatonic scale: Difference between revisions

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A ~~'''dipentatonic''' or~~ '''bipentatonic ~~scale~~''' is a 10-note octave-equivalent scale where every other note gives a fixed choice of pentatonic scale; hence a ~~dipentatonic~~ scale is a type of [[flought scale]]. Following from this, ~~dipentatonic~~ scales based on a [[MOS]] pentatonic scale have a maximum of two sizes for intervals that are an even number of steps. Many ~~dipentatonic~~ scales are generated by a diatonic-sized fifth (between the 7edo fifth and the 5edo fifth) of a fixed size. Some ~~dipentatonic~~ scales are MOSes, such as the Erlich decatonic in [[22edo]]. Modulating by fifths is easy in ~~dipentatonic~~ scales where the interleaved pentatonics are generated by a fifth (i.e. [[2L 3s]] and [[3L 2s]], depending on tuning).

A '''bipentatonic''' 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 ~~dipentatonic~~ scales by the pentatonics they interleave and by their rank (whether they are mosses or rank-3). The second part of this article surveys ~~dipentatonic~~ scales in [[JI]]. Dipentatonic scales also exist in [[regular temperaments]] of course, but exploration of such scales is not included in the article for now at least.

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]]. Dipentatonic scales also exist in [[regular temperaments]] of course, but exploration of such scales is not included in the article for now at least.

== Abstract ~~dipentatonic~~ scales ==

== Abstract bipentatonic scales ==

=== Dipentatonic mosses ===

Every 10 note mos except 5L 5s is ~~dipentatonic~~. The degenerate case of a ~~dipentatonic~~ 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:

* 1L 9s: Lsssssssss (pentatonic 1L 4s)

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=== Dipentatonic ternary scale patterns ===

A ternary ~~dipentatonic~~ 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

* 2L 3M 5s, MsLsMsLsMs

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

== Just ~~dipentatonic~~ scales ==

== Just bipentatonic scales ==

Every other note of a just ~~dipentatonic~~ scale gives a Pythagorean pentatonic. Dipentatonic 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.

Every other note of a just bipentatonic scale gives a Pythagorean pentatonic. Dipentatonic 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.

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Just a couple examples linked for now:

[[Marvel ~~dipentatonic~~]]

[[Marvel bipentatonic]]

[[Prodigy ~~dipentatonic~~]]

[[Prodigy bipentatonic]]

[[Category:10-tone scales]]

[[Category:Terms]]

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-A '''dipentatonic''' or '''bipentatonic scale''' is a 10-note octave-equivalent scale where every other note gives a fixed choice of pentatonic scale; hence a dipentatonic scale is a type of [[flought scale]]. Following from this, dipentatonic scales based on a [[MOS]] pentatonic scale have a maximum of two sizes for intervals that are an even number of steps. Many dipentatonic scales are generated by a diatonic-sized fifth (between the 7edo fifth and the 5edo fifth) of a fixed size. Some dipentatonic scales are MOSes, such as the Erlich decatonic in [[22edo]]. Modulating by fifths is easy in dipentatonic scales where the interleaved pentatonics are generated by a fifth (i.e. [[2L 3s]] and [[3L 2s]], depending on tuning).
+A '''bipentatonic''' 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 dipentatonic scales by the pentatonics they interleave and by their rank (whether they are mosses or rank-3). The second part of this article surveys dipentatonic scales in [[JI]]. Dipentatonic scales also exist in [[regular temperaments]] of course, but exploration of such scales is not included in the article for now at least.
+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]]. Dipentatonic scales also exist in [[regular temperaments]] of course, but exploration of such scales is not included in the article for now at least.
-== Abstract dipentatonic scales ==
+== Abstract bipentatonic scales ==
 === Dipentatonic mosses ===
-Every 10 note mos except 5L 5s is dipentatonic. The degenerate case of a dipentatonic 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:
 * 1L 9s: Lsssssssss (pentatonic 1L 4s)
@@ Line 16: / Line 16: @@
 === Dipentatonic ternary scale patterns ===
-A ternary dipentatonic 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
 * 2L 3M 5s, MsLsMsLsMs
@@ Line 63: / Line 63: @@
 * If δ = 1\\2, then d = c > b = a > 0, and we get the 2L8s MOS ssssLssssL.
-== Just dipentatonic scales ==
+== Just bipentatonic scales ==
-Every other note of a just dipentatonic scale gives a Pythagorean pentatonic. Dipentatonic 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.
+Every other note of a just bipentatonic scale gives a Pythagorean pentatonic. Dipentatonic 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.
@@ Line 178: / Line 178: @@
 Just a couple examples linked for now:
-[[Marvel dipentatonic]]
+[[Marvel bipentatonic]]
-[[Prodigy dipentatonic]]
+[[Prodigy bipentatonic]]
 [[Category:10-tone scales]]
 [[Category:Terms]]