Aberrisma - Revision history

Inthar: /* Extension to higher ranks */

2026-08-26T06:50:10Z

Extension to higher ranks

← Older revision		Revision as of 06:50, 26 August 2026
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	== Extension to higher ranks ==		== Extension to higher ranks ==
	Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have a GS with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.		Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have a [[GS]] with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.

	{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''.}}		{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''.}}

Inthar: /* Extension to higher ranks */

2026-08-26T06:49:59Z

Extension to higher ranks

← Older revision		Revision as of 06:49, 26 August 2026
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	== Extension to higher ranks ==		== Extension to higher ranks ==
	Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have ~~an AGS~~ with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.		Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have a GS with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.

	{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''.}}		{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''.}}

Ground at 03:50, 26 August 2026

2026-08-26T03:50:17Z

← Older revision		Revision as of 03:50, 26 August 2026
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	Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have an AGS with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.		Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have an AGS with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.

	{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''}}		{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''.}}

	However, this is often not enough to accurately represent a larger JI subgroup. Portent penslen (5L5m6s) is a somewhat unique counterexample, being bicatenary yet microtempered in the 11-limit. The solution is to add more step sizes while maintaining as much lower-rank structure as possible. For example, while penslen is found in edos like 4 and 46, the equivalent in the similarly LCJI-accurate 53 and 58 is 5L2m3s4t, a quaternary subset of Buzzard[28]. [[Ground's composition theory#Interval logic deviation theory\|ILD]] is a simple way to access notes outside of a scale with a higher rank.		However, this is often not enough to accurately represent a larger JI subgroup. Portent penslen (5L5m6s) is a somewhat unique counterexample, being bicatenary yet microtempered in the 11-limit. The solution is to add more step sizes while maintaining as much lower-rank structure as possible. For example, while penslen is found in edos like 41 and 46, the equivalent in the similarly LCJI-accurate 53 and 58 is 5L2m3s4t, a quaternary subset of Buzzard[28] with a JI tuning of 35/32 : 21/20 : 36/35 : 64/63. [[Ground's composition theory#Interval logic deviation theory\|ILD]] is a simple way to access notes outside of a scale with a higher rank.

	A common tempering of blackdye involves a sharper 3 to significantly inflate 81/80, both for melodic utility and to equate 27/20 to a superfourth like 15/11 or 11/8. It also becomes natural to temper out 64/63. ~~If less tempering~~ is ~~desired~~, (81/80)^2 = 43¢ ~~becomes a desirable~~ aberrisma ~~instead~~. This does not exist in blackdye, but becomes accessible in certain variants of 5L2m8s as well as numerous quaternary scales with even more options for available intervals.		A common tempering of blackdye involves a sharper 3 to significantly inflate 81/80, both for melodic utility and to equate 27/20 to a superfourth like 15/11 or 11/8. It also becomes natural to temper out 64/63. Possibly the most common ratio representation of an aberrisma is 36/35, but few ternary scales use it without tempering out 81/80 or 64/63.

			(81/80)^2 = 43¢ is notably in the middle of the usual aberrisma size range as well. This does not exist in blackdye, but becomes accessible in certain variants of 5L2m8s as well as numerous quaternary scales with even more options for available intervals.

	== Compositional examples ==		== Compositional examples ==

Ground at 03:40, 26 August 2026

2026-08-26T03:40:30Z

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	* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.		* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
	* '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.		* '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.

			== Extension to higher ranks ==
			Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have an AGS with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.

			{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''}}

			However, this is often not enough to accurately represent a larger JI subgroup. Portent penslen (5L5m6s) is a somewhat unique counterexample, being bicatenary yet microtempered in the 11-limit. The solution is to add more step sizes while maintaining as much lower-rank structure as possible. For example, while penslen is found in edos like 4 and 46, the equivalent in the similarly LCJI-accurate 53 and 58 is 5L2m3s4t, a quaternary subset of Buzzard[28]. [[Ground's composition theory#Interval logic deviation theory\|ILD]] is a simple way to access notes outside of a scale with a higher rank.

			A common tempering of blackdye involves a sharper 3 to significantly inflate 81/80, both for melodic utility and to equate 27/20 to a superfourth like 15/11 or 11/8. It also becomes natural to temper out 64/63. If less tempering is desired, (81/80)^2 = 43¢ becomes a desirable aberrisma instead. This does not exist in blackdye, but becomes accessible in certain variants of 5L2m8s as well as numerous quaternary scales with even more options for available intervals.

	== Compositional examples ==		== Compositional examples ==

Ground at 18:31, 31 July 2026

2026-07-31T18:31:30Z

← Older revision		Revision as of 18:31, 31 July 2026
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	{{Proposed}}		{{Proposed}}

	An '''aberrisma''' is an interval representing some comma as an additional ~~smaller~~ type of melodic step. Its name originates from the comparison of its sound to a chromatic aberration, with the -isma suffix often used for commas. ~~Though~~ they are typically between 20 and 55 cents (around 81/80 to half of 16/15) and often inflated commas, Larger and smaller sizes can function as aberrismas depending on the scale tuning. ~~The aberrisma~~ is ~~used as one of~~ the ~~parameters in constructing an aberrismic scale, a type~~ of ~~ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but~~ with ~~a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals~~.		An '''aberrisma''' is an interval representing some comma as an additional type of melodic step smaller than a semitone. Its name originates from the comparison of its sound to a chromatic aberration, with the -isma suffix often used for commas. Although they are typically between 20 and 55 cents (around 81/80 to half of 16/15) and often inflated commas about the size of a diesis, the aberrisma is a melodic function and not an interval size. Larger and smaller sizes can function as aberrismas depending on the scale tuning. '''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground\|Ground]] and [[User:Inthar\|Inthar]] that deals with aberrismas.

	Aberrismas may also appear in MOS scales, such as ~~garibaldi~~[17].		The aberrisma is used as one of the parameters in constructing an aberrismic scale, a type of ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but with a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals. Aberrismas may also appear in MOS scales, such as Garibaldi[17]. Superpyth[12] is a good example of the distinction between interval function and size. The diatonic semitone, a deflated 256/243, is equated to the inflated 81/80. Despite being the same size, the former represents a narrow semitone, the latter is an aberrisma, and a stack of the two is a classic semitone.

	~~'''Aberrismic theory'''~~ is the ~~subset~~ of ~~microtonal theory pioneered by [[User:Ground\|Ground]] and [[User:Inthar\|Inthar]] that deals with aberrismas~~.

	== Example: The emergence of blackdye ==		== Example: The emergence of blackdye ==
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	\|\|Two interleaved diatonics		\|\|Two interleaved diatonics
	\|}		\|}
			See [[Ground's composition theory#Quasi-diatonic scales]] for more tunings of diasem and blackdye.

	=== Other aberrismic scales ===		=== Other aberrismic scales ===

Inthar: /* Glossary */

2026-07-27T01:14:40Z

Glossary

← Older revision		Revision as of 01:14, 27 July 2026
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	== Glossary ==		== Glossary ==
	* '''Aberration scale''': A scale made by interleaving aberrismas into a binary scale and stretching or compressing, usually a [[MOS substitution]] scale of type {{nowrap\|[a+b+1]s(aLbm)}} (compression, called '''tractaberrated''') or {{nowrap\|[a+b-1]s(aLbm)}} (stretching, called '''tensaberrated'''). For example, sLsmsLsLsLsmsLs is an aberration scale made from diatonic (a MOS substitution scale of type 8s(5L2m)).		* '''Aberration scale''': A scale made by interleaving aberrismas into a binary scale and stretching or compressing, usually a [[MOS substitution]] scale of type {{nowrap\|[a+b+1]s(aLbm)}} (compression, called '''tractaberrated''') or {{nowrap\|[a+b-1]s(aLbm)}} (stretching, called '''tensaberrated'''). For example, sLsmsLsLsLsmsLs is an aberration scale made from diatonic (a MOS substitution scale of type 8s(5L2m)), also called trackdye.
	* '''Aberrisma''': The smallest interval region that melodically sounds like a step.		* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
	* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.		* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.

Ground at 16:37, 24 July 2026

2026-07-24T16:37:40Z

← Older revision		Revision as of 16:37, 24 July 2026
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	{{Proposed}}		{{Proposed}}

	An '''aberrisma''' is an interval representing some comma as an additional smaller type of melodic step. Its name originates from the comparison of its sound to a chromatic aberration, with the -isma suffix often used for commas. Though they are ~~often diesis-sized inflated commas (~~between 25 and 55 cents), Larger and smaller sizes can function as aberrismas depending on the scale tuning. The aberrisma is used as one of the parameters in constructing an aberrismic scale, a type of ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but with a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals.		An '''aberrisma''' is an interval representing some comma as an additional smaller type of melodic step. Its name originates from the comparison of its sound to a chromatic aberration, with the -isma suffix often used for commas. Though they are typically between 20 and 55 cents (around 81/80 to half of 16/15) and often inflated commas, Larger and smaller sizes can function as aberrismas depending on the scale tuning. The aberrisma is used as one of the parameters in constructing an aberrismic scale, a type of ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but with a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals.

	Aberrismas may also appear in MOS scales, such as garibaldi[17].		Aberrismas may also appear in MOS scales, such as garibaldi[17].

Ground at 16:18, 24 July 2026

2026-07-24T16:18:26Z

Inthar: /* Other aberrismic scales */

2026-07-17T02:56:04Z

Other aberrismic scales

← Older revision		Revision as of 02:56, 17 July 2026
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	\|\| Has two aberrisma sizes, s and m.		\|\| Has two aberrisma sizes, s and m.
	\|-		\|-
	!\|~~prelembic~~ / citro3s<br/>(4L2m3s)<br/>{{adv\|{{nowrap\|subst 2m(4L3s)}}}}		!\|prejubilic / citro3s<br/>(4L2m3s)<br/>{{adv\|{{nowrap\|subst 2m(4L3s)}}}}
	\|\|C: LLsmLsLms<br/>R: LmLsLmsLs<br/>L: LmLsLsmLs		\|\|C: LLsmLsLms<br/>R: LmLsLmsLs<br/>L: LmLsLsmLs
	\|\|2.5.7<br/>[L, m, s] = [28/25, 35/32, 50/49]		\|\|2.5.7<br/>[L, m, s] = [28/25, 35/32, 50/49]

Inthar: /* Systematic naming */

2026-07-16T20:04:45Z

Systematic naming

← Older revision		Revision as of 20:04, 16 July 2026
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	Basic systematic names for aberrismic scales are of the form		Basic systematic names for aberrismic scales are of the form

	[mos_prefix]n[added_step_size] (e.g. dia2s for diasem),		[mos_prefix]n[added_step_size] (e.g. dia2s for diasem).

	~~where~~ the ~~MOS prefix (a TAMNAMS prefix if one is available) is chosen based on~~ the ~~aberrismic~~-~~theoretical generator (as opposed to the offset), rather than from any particular mathematical construction. For example, penslen has~~ MOS ~~substitution type 6s(5L5m), but~~ the ~~systematic name is slen5m, not penwd6s, since the generator is conceived as a generator of 5L6s (Slendric[11])~~.		Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.

	This is subject to change as aberrismic theory notation is updated in the future.		This is subject to change as aberrismic theory notation is updated in the future.

	~~Proposal: Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.~~

	== Notation ==		== Notation ==

← Older revision		Revision as of 02:56, 17 July 2026
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	\|\| Has two aberrisma sizes, s and m.		\|\| Has two aberrisma sizes, s and m.
	\|-		\|-
	!\|~~prelembic~~ / citro3s<br/>(4L2m3s)<br/>{{adv\|{{nowrap\|subst 2m(4L3s)}}}}		!\|prejubilic / citro3s<br/>(4L2m3s)<br/>{{adv\|{{nowrap\|subst 2m(4L3s)}}}}
	\|\|C: LLsmLsLms<br/>R: LmLsLmsLs<br/>L: LmLsLsmLs		\|\|C: LLsmLsLms<br/>R: LmLsLmsLs<br/>L: LmLsLsmLs
	\|\|2.5.7<br/>[L, m, s] = [28/25, 35/32, 50/49]		\|\|2.5.7<br/>[L, m, s] = [28/25, 35/32, 50/49]

← Older revision		Revision as of 16:18, 24 July 2026
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	{{Proposed}}		{{Proposed}}

	An '''aberrisma''' is an interval representing some comma as an additional smaller type of melodic step. Though they are often between 20 and 55 cents, Larger and smaller sizes can function as aberrismas depending on the scale tuning. The aberrisma is used as one of the parameters in constructing an aberrismic scale, a type of ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but with a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals.		An '''aberrisma''' is an interval representing some comma as an additional smaller type of melodic step. Its name originates from the comparison of its sound to a chromatic aberration, with the -isma suffix often used for commas. Though they are often diesis-sized inflated commas (between 25 and 55 cents), Larger and smaller sizes can function as aberrismas depending on the scale tuning. The aberrisma is used as one of the parameters in constructing an aberrismic scale, a type of ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but with a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals.

	Aberrismas may also appear in MOS scales, such as garibaldi[17].		Aberrismas may also appear in MOS scales, such as garibaldi[17].
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	* Plus / Minus +/- : These tune a note sharp or flat by a small aberrisma. They reference Johnston notation because 81/80 is a common aberrisma, but they can also represent 64/63 or any other interval of similar function. <!--They're also used to denote straddle primes, like 3+ and 3- (in a straddle-3 subgroup, these can be abbreviated as 3±). This results in some pretty heavy overloading, but +/- are only used on notes when they represent an aberrisma and only used on ratios when they represent straddle primes. -->		* Plus / Minus +/- : These tune a note sharp or flat by a small aberrisma. They reference Johnston notation because 81/80 is a common aberrisma, but they can also represent 64/63 or any other interval of similar function. <!--They're also used to denote straddle primes, like 3+ and 3- (in a straddle-3 subgroup, these can be abbreviated as 3±). This results in some pretty heavy overloading, but +/- are only used on notes when they represent an aberrisma and only used on ratios when they represent straddle primes. -->
	* Duplus / Duminus ≠/= : Short for ++/--, most often representing 36/35~33/32~1053/1024, which is the large aberrisma in scales like penslen, or two small aberrismas in Akea temperament. Of all options, this set of characters is the easiest to type, looks the best in various fonts, and is least likely to be confused for the similar semisharp accidental (although they happen to represent the same size of interval).		* Duplus / Duminus ≠/= : Short for ++/--, most often representing 36/35~33/32~1053/1024, which is the large aberrisma in scales like penslen, or two small aberrismas in Akea temperament. Of all options, this set of characters is the easiest to type, looks the best in various fonts, and is least likely to be confused for the similar semisharp accidental (although they happen to represent the same size of interval).
			* [[Erac\|Eracs]] may be used to represent the inflating of commas typically found in aberrismic scales, although they represent a specific unit of error.

	== Aberrismic theory and RTT ==		== Aberrismic theory and RTT ==
	Aberrismic theory often applies RTT to ternary LCJI scales with comma steps. Certain scales with aberrismas may thus be endowed with JI interpretations via [[RTT]] temperaments, which may be used in suitable [[equal temperament]]s. Under ~~groundfault~~'s use of edos (usually patent vals) as RTT temperaments, the aberrisma tends to become a [[81/80]] in a 2.3.5 context and a [[64/63]] in a 2.3.7 context. Some scales such as 5L2m5s and 5L2m7s admit a more accurate 2.3.5.7 interpretation that tempers out neither 81/80 nor 64/63 but identifies the two commas, tempering out [[5120/5103]]. Tempering is important in aberrismic theory as a way to <!--simultaneously achieve sufficient accuracy to LCJI and -->improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.		Aberrismic theory often applies RTT to ternary LCJI scales with comma steps. Certain scales with aberrismas may thus be endowed with JI interpretations via [[RTT]] temperaments, which may be used in suitable [[equal temperament]]s. Under Ground's use of edos (usually patent vals) as RTT temperaments, the aberrisma tends to become a [[81/80]] in a 2.3.5 context and a [[64/63]] in a 2.3.7 context. Some scales such as 5L2m5s and 5L2m7s admit a more accurate 2.3.5.7 interpretation that tempers out neither 81/80 nor 64/63 but identifies the two commas, tempering out [[5120/5103]]. Tempering is important in aberrismic theory as a way to <!--simultaneously achieve sufficient accuracy to LCJI and -->improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

	At times, a scale pattern has varying temperaments according to the tuning. For example, 5L2m3s may be given the temperament structure of either untempered 2.3.5 or [[Ultrapyth]] temperament.		At times, a scale pattern has varying temperaments according to the tuning. For example, 5L2m3s may be given the temperament structure of either untempered 2.3.5 or [[Ultrapyth]] temperament.
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	The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.		The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
	* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.		* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
	* [[Ultrapyth]], 2.3.5.7.11.13[32 & 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. ~~groundfault terms~~ the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.		* [[Ultrapyth]], 2.3.5.7.11.13[32 & 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. Ground once termed the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.

	{\| class="wikitable right-2 right-3 right-4 right-5"		{\| class="wikitable right-2 right-3 right-4 right-5"
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	* '''Monotone-MOS''': A ternary scale is ''monotone-MOS'' if it becomes a MOS under all three of the identifications L = M, M = s, and s = 0. If ''any'' (not necessarily all) of the identifications make the scale a MOS, the scale is said to ''satisfy a monotone-MOS subcondition''. For example, diasem (LmLsLmLsL) satisfies all three monotone-MOS subconditions, but blackdye (sLmLsLmLsL) satisfies only the m = s and s = 0 monotone-MOS subconditions. An aberrismic scale is required to satisfy the s = 0 monotone-MOS subcondition.		* '''Monotone-MOS''': A ternary scale is ''monotone-MOS'' if it becomes a MOS under all three of the identifications L = M, M = s, and s = 0. If ''any'' (not necessarily all) of the identifications make the scale a MOS, the scale is said to ''satisfy a monotone-MOS subcondition''. For example, diasem (LmLsLmLsL) satisfies all three monotone-MOS subconditions, but blackdye (sLmLsLmLsL) satisfies only the m = s and s = 0 monotone-MOS subconditions. An aberrismic scale is required to satisfy the s = 0 monotone-MOS subcondition.
	* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.		* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
	* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.		* '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.

	== Compositional examples ==		== Compositional examples ==