Bipentatonic scale - Revision history

Lhearne: /* Bipentatonic scales in regular temperaments */ updated di to bi

2026-01-05T03:40:50Z

Bipentatonic scales in regular temperaments: updated di to bi

← Older revision		Revision as of 03:40, 5 January 2026
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	Just a couple examples linked for now:		Just a couple examples linked for now:

	[[Marvel ~~dipentatonic~~]]		[[Marvel bipentatonic]]

	[[Prodigy ~~dipentatonic~~]]		[[Prodigy bipentatonic]]

	[[Category:10-tone scales]]		[[Category:10-tone scales]]
	[[Category:Terms]]		[[Category:Terms]]

Inthar: /* Bipentatonic scales in regular temperaments */

2024-01-25T00:14:43Z

Bipentatonic scales in regular temperaments

← Older revision		Revision as of 00:14, 25 January 2024
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	Just a couple examples linked for now:		Just a couple examples linked for now:

	[[Marvel ~~bipentatonic~~]]		[[Marvel dipentatonic]]

	[[Prodigy ~~bipentatonic~~]]		[[Prodigy dipentatonic]]

	[[Category:10-tone scales]]		[[Category:10-tone scales]]
	[[Category:Terms]]		[[Category:Terms]]

Inthar at 00:14, 25 January 2024

2024-01-25T00:14:32Z

← Older revision		Revision as of 00:14, 25 January 2024
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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''' 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.		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.

Inthar: /* Bipentic scales using the Pythagorean pentic and arbitrary offsets */

2024-01-25T00:13:56Z

Bipentic scales using the Pythagorean pentic and arbitrary offsets

← Older revision		Revision as of 00:13, 25 January 2024
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	== Bipentic scales using the Pythagorean pentic and arbitrary offsets ==		== 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		* 9/8 ≤ δ ≤ 32/27
	* 81/64 ≤ δ ≤ 4/3		* 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), we obtain the ternary scale 2L3m5s (smsmsLsmsL).
	* If δ = sqrt(9/8) = 101.955c, then we have a permutation of 2L8s (sssssLsssL).		* If δ = sqrt(9/8) = 101.955c, then we have a permutation of 2L8s (sssssLsssL).

Inthar: /* Dipentic scales using the Pythagorean pentic and arbitrary offsets */

2024-01-25T00:13:35Z

Dipentic scales using the Pythagorean pentic and arbitrary offsets

Inthar at 00:11, 25 January 2024

2024-01-25T00:11:46Z

← Older revision		Revision as of 00:11, 25 January 2024
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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]]. ~~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]]. 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 ==
	=== ~~Dipentatonic~~ mosses ===		=== 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)

	=== ~~Dipentatonic~~ ternary scale patterns ===		=== 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. ~~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. 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

	== ~~Dipentatonic~~ scales in regular temperaments ==		== Bipentatonic scales in regular temperaments ==
	Just a couple examples linked for now:		Just a couple examples linked for now:

Inthar at 00:11, 25 January 2024

2024-01-25T00:11:34Z

← Older revision		Revision as of 00:11, 25 January 2024
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	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).		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]]. 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.

Inthar: Inthar moved page Dipentatonic scale to Bipentatonic scale

2024-01-25T00:11:24Z

Inthar moved page Dipentatonic scale to Bipentatonic scale

← Older revision	Revision as of 00:11, 25 January 2024
(No difference)

Inthar: Latin bi- is better to separate it from the Greek penta- in "pentatonic".

2024-01-25T00:11:16Z

Latin bi- is better to separate it from the Greek penta- in "pentatonic".

← Older revision		Revision as of 00:11, 25 January 2024
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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 ===		=== 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)		* 1L 9s: Lsssssssss (pentatonic 1L 4s)
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	=== Dipentatonic ternary scale patterns ===		=== 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		* 1L 4M 5s, LsMsMsMsMs
	* 2L 3M 5s, MsLsMsLsMs		* 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.		* 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.		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:		Just a couple examples linked for now:

	[[Marvel ~~dipentatonic~~]]		[[Marvel bipentatonic]]

	[[Prodigy ~~dipentatonic~~]]		[[Prodigy bipentatonic]]

	[[Category:10-tone scales]]		[[Category:10-tone scales]]
	[[Category:Terms]]		[[Category:Terms]]

Inthar: /* Dipentatonic ternary scale patterns */

2024-01-05T17:19:49Z

Dipentatonic ternary scale patterns

← Older revision		Revision as of 17:19, 5 January 2024
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	* 4L 5M 1s, LMLMLMLMsM		* 4L 5M 1s, LMLMLMLMsM
	* 5L 1M 4s, LsLsLsLsLM		* 5L 1M 4s, LsLsLsLsLM
	* [[Blackdye]] (5L 2M 3s), LsLMLsLMLs ~~<!-- Named by ground -->~~		* [[Blackdye]] (5L 2M 3s), LsLMLsLMLs
	* ~~[[Onyxwood]] (~~5L 3M 2s), LMLsLMLsLM ~~<!-- Named by inthar -->~~		* 5L 3M 2s, LMLsLMLsLM
	* 5L 4M 1s, LMLMLMLMLs		* 5L 4M 1s, LMLMLMLMLs

← Older revision		Revision as of 00:13, 25 January 2024
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	* 5L 4M 1s, LMLMLMLMLs		* 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.		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.		* 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.		* 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.		* 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.		* 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.		* 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.		* The offset δ = sqrt(243/128) = 554.888c implies d > b = c > a > 0 and yields the ternary scale smsmmmsmsL.