Bipentatonic scale: Difference between revisions

(3 intermediate revisions by one other user not shown)

Line 1:

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

Line 31:

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

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.

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 ''bipythpentic'' 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.

Assuming octave equivalence, the offsets δ and 1200 - δ behave the same. Taking that fact into account, the following offset ranges do not yield bipythpentic 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.

Offsets between 1/1 and 9/8 yield the three ternary bipythpentic scale patterns of the form ababacabac, where c > a. Outside of this range, ternary bipythpentic 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).

@@ Line 1: / Line 1: @@
-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''' or '''dipentatonic 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]]. Bipentatonic scales also exist in [[regular temperaments]] of course, but exploration of such scales is not included in the article for now at least.
@@ Line 31: / Line 31: @@
 == Bipentic scales using the Pythagorean pentic and arbitrary offsets ==
-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.
+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 ''bipythpentic'' 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.
+Assuming octave equivalence, the offsets δ and 1200 - δ behave the same. Taking that fact into account, the following offset ranges do not yield bipythpentic 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.
+Offsets between 1/1 and 9/8 yield the three ternary bipythpentic scale patterns of the form ababacabac, where c > a. Outside of this range, ternary bipythpentic 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).