Ternary scale - Revision history

Inthar: /* Twisted MV3 scales */

2026-03-27T01:53:48Z

Twisted MV3 scales

← Older revision		Revision as of 01:53, 27 March 2026
Line 34:		Line 34:
	Now do the letterwise substitution procedure, as in [[MOS substitution]], with X as the slot letter, with filling MOS alternating: LmLsmsLms = XXXsXsXXs(LmLmLm).		Now do the letterwise substitution procedure, as in [[MOS substitution]], with X as the slot letter, with filling MOS alternating: LmLsmsLms = XXXsXsXXs(LmLmLm).

	Twisted MV3 scales always have step signature ka'''x'''ka'''y'''kb'''z'''.		Twisted MV3 scales always have step signature ka'''x'''ka'''y'''kb'''z''' with k > 1.

	== Interleaving ==		== Interleaving ==

Inthar: /* Even-regular scales */

2026-03-27T01:50:23Z

Even-regular scales

← Older revision		Revision as of 01:50, 27 March 2026
Line 21:		Line 21:

	=== Even-regular scales ===		=== Even-regular scales ===
	'''Even-regular scales''' have two parallel generator chains offset by a len(scale)/2-step interval, with "bad" generators only occurring on ~~2 notes~~. This generator comes from detempering a generator of ax2bX (a 2-period MOS) for an even-regular scale of MOS substitution type ax(bybz). For example, achiral diachrome (LsLsLmsLsLsm, 2m(5L5s)) has a 5-step interval 2L + m + 2s (interpreted as 4/3) as a generator, and m + 4X (period-complement of X) is a generator of the MOS 2m10X. Even-regular scales are achiral.		'''Even-regular scales''' have two parallel generator chains offset by a len(scale)/2-step interval, with "bad" generators only occurring on 1 note per chain. This generator comes from detempering a generator of ax2bX (a 2-period MOS) for an even-regular scale of MOS substitution type ax(bybz). For example, achiral diachrome (LsLsLmsLsLsm, 2m(5L5s)) has a 5-step interval 2L + m + 2s (interpreted as 4/3) as a generator, and m + 4X (period-complement of X) is a generator of the MOS 2m10X. Even-regular scales are achiral.

	Even-regular scales have an interleaving structure: if n > 4, an n-note even-regular scale is an [[interleaving]] of two copies of an n/2-note even-regular scale if n is a multiple of 4, and is an interleaving of the two opposite chiralities of an n/2-note odd-regular scale otherwise. Examples:		Even-regular scales have an interleaving structure: if n > 4, an n-note even-regular scale is an [[interleaving]] of two copies of an n/2-note even-regular scale if n is a multiple of 4, and is an interleaving of the two opposite chiralities of an n/2-note odd-regular scale otherwise. Examples:

Inthar: /* Relationship with rank */

2026-03-15T00:12:00Z

Relationship with rank

← Older revision		Revision as of 00:12, 15 March 2026
Line 2:		Line 2:

	== Relationship with rank ==		== Relationship with rank ==
	Ternary scales with [[independent]] step sizes are rank-3. However, rank-3 scales ~~may~~ not be ternary.		Ternary scales with [[independent]] step sizes are rank-3. However, rank-3 scales need not be ternary. Duodene (16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1) is an example of a rank-3 (2.3.5) quaternary scale.

	== MV3 ternary scales ==		== MV3 ternary scales ==

Inthar at 15:50, 6 March 2026

2026-03-06T15:50:39Z

← Older revision		Revision as of 15:50, 6 March 2026
Line 1:		Line 1:
	A '''ternary scale''' is a scale with exactly 3 step sizes.		A '''ternary scale''' is a scale with exactly 3 step sizes. This article is an overview of the theory of ternary scale patterns.

	== Relationship with rank ==		== Relationship with rank ==

Inthar: /* See also */

2026-03-06T15:48:59Z

Inthar: /* Twisted MV3 scales */

2026-03-06T15:34:42Z

Twisted MV3 scales

← Older revision		Revision as of 15:34, 6 March 2026
Line 30:		Line 30:
	'''Twisted MV3 scales''' are much like MOS substitution scales, but the "template MOS" is not actually a MOS, but a "twisted" multiMOS, which is constructed as follows:		'''Twisted MV3 scales''' are much like MOS substitution scales, but the "template MOS" is not actually a MOS, but a "twisted" multiMOS, which is constructed as follows:
	# Take the ''brightest'' (or ''darkest'') mode of a multiMOS, where the period (one MOS unit) has exactly one step count X even (e.g. 2X 1s, XXs -> 6X 3s, XXsXXsXXs)		# Take the ''brightest'' (or ''darkest'') mode of a multiMOS, where the period (one MOS unit) has exactly one step count X even (e.g. 2X 1s, XXs -> 6X 3s, XXsXXsXXs)
	# ~~Put~~ borders between units: XXs\|XXs\|XXs		# Mark borders between units: XXs\|XXs\|XXs
	# Change s\|X -> X\|s (or vice versa) at some of the borders: XXX\|sXs\|XXs		# Change s\|X -> X\|s (or vice versa) at some of the borders: XXX\|sXs\|XXs
	Now do the letterwise substitution procedure, as in [[MOS substitution]], with X as the slot letter, with filling MOS alternating: LmLsmsLms = XXXsXsXXs(LmLmLm).		Now do the letterwise substitution procedure, as in [[MOS substitution]], with X as the slot letter, with filling MOS alternating: LmLsmsLms = XXXsXsXXs(LmLmLm).

Inthar: /* MV3 ternary scales */

2026-03-06T15:21:57Z

MV3 ternary scales

← Older revision		Revision as of 15:21, 6 March 2026
Line 10:		Line 10:
	# '''pairwise-MOS''': scales such that identifying any two step sizes always results in a MOS		# '''pairwise-MOS''': scales such that identifying any two step sizes always results in a MOS
	#* '''abacaba''' scales		#* '''abacaba''' scales
	#* '''odd-regular''' scales: ~~an infinite family of~~ [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is odd and gcd(a, b) = 1 (example: [[diasem]])		#* '''odd-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is odd and gcd(a, b) = 1 (example: [[diasem]])
	#* '''even-regular''' scales: ~~an infinite family of~~ [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is even and gcd(a, b) = 1 (example: [[penslen]])		#* '''even-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is even and gcd(a, b) = 1 (example: [[penslen]])
	# '''abcba''' scales: the unique sporadic non-pairwise-MOS MV3 ternary pattern		# '''abcba''' scales: the unique sporadic non-pairwise-MOS MV3 ternary pattern
	# '''twisted MV3''' scales (see below): an infinite family of non-pairwise-MOS MV3 ternary scales		# '''twisted MV3''' scales (see below): an infinite family of non-pairwise-MOS MV3 ternary scales

Inthar at 15:15, 6 March 2026

2026-03-06T15:15:09Z

← Older revision		Revision as of 15:15, 6 March 2026
Line 1:		Line 1:
	A '''ternary scale''' is a scale with exactly 3 step sizes.		A '''ternary scale''' is a scale with exactly 3 step sizes.

			== Relationship with rank ==
	Ternary scales with [[independent]] step sizes are rank-3. However, rank-3 scales may not be ternary.		Ternary scales with [[independent]] step sizes are rank-3. However, rank-3 scales may not be ternary.

Line 43:		Line 44:
	: ''Main article: [[MOS substitution]]''		: ''Main article: [[MOS substitution]]''
	MOS substitution generalizes odd-regular and even-regular scales (as well as the above interleaving construction).		MOS substitution generalizes odd-regular and even-regular scales (as well as the above interleaving construction).

			== See also ==
			* [[Aberrisma]]
	{{cat\|Scales		{{cat\|Scales
	Ternary scales{{!}}*}}		Ternary scales{{!}}*}}

Inthar at 15:13, 6 March 2026

2026-03-06T15:13:11Z

← Older revision		Revision as of 15:13, 6 March 2026
Line 43:		Line 43:
	: ''Main article: [[MOS substitution]]''		: ''Main article: [[MOS substitution]]''
	MOS substitution generalizes odd-regular and even-regular scales (as well as the above interleaving construction).		MOS substitution generalizes odd-regular and even-regular scales (as well as the above interleaving construction).
	{{cat\|Scales}}		{{cat\|Scales
			Ternary scales{{!}}*}}

Inthar: /* Twisted MV3 scales */

2026-03-06T15:03:36Z

Twisted MV3 scales

← Older revision		Revision as of 15:03, 6 March 2026
Line 27:		Line 27:

	=== Twisted MV3 scales ===		=== Twisted MV3 scales ===
	Twisted MV3 scales are much like MOS substitution scales, but the "template MOS" is not actually a MOS, but a "twisted" multiMOS, which is constructed as follows:		'''Twisted MV3 scales''' are much like MOS substitution scales, but the "template MOS" is not actually a MOS, but a "twisted" multiMOS, which is constructed as follows:
	# Take the ''brightest'' (or ''darkest'') mode of a multiMOS, where the period (one MOS unit) has exactly one step count X even (e.g. 2X 1s, XXs -> 6X 3s, XXsXXsXXs)		# Take the ''brightest'' (or ''darkest'') mode of a multiMOS, where the period (one MOS unit) has exactly one step count X even (e.g. 2X 1s, XXs -> 6X 3s, XXsXXsXXs)
	# Put borders between units: XXs\|XXs\|XXs		# Put borders between units: XXs\|XXs\|XXs

← Older revision		Revision as of 15:48, 6 March 2026
Line 46:		Line 46:

	== See also ==		== See also ==
	* [[Aberrisma]]		* [[Aberrisma\|Aberrismic theory]]: A compositional theory using ternary scales
	{{cat\|Scales		{{cat\|Scales
	Ternary scales{{!}}*}}		Ternary scales{{!}}*}}