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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.

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.

== Example: The emergence of blackdye ==
The Zarlino diatonic is chiral - there are two different, equally valid second degrees of the Ionian mode. Both are useful, as the sharp one forms a perfect fifth with the fifth degree but a wolf fifth with the sixth degree, and the flat one forms a perfect fifth with the sixth degree but a wolf fifth with the fifth degree.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
One way to make it achiral is to temper out 81/80, the difference between these two steps, resulting in [[Meantone]] diatonic; intuitively this requires flattening the fifth and sharpening the sixth somewhat. However, an alternative way, if you wish to observe 81/80 or to use just intonation, is to include both varieties of whole tone over the unison, treating 81/80 as a melodic step between them. This can be thought of as dividing up a 9/8 into a 10/9 and an 81/80. It is then reasonable to extend this action to all instances of 9/8 in the scale (as, for instance, the Didymic diatonic has 27/16 as opposed to 5/3). The result is a 10-note ternary scale called '''blackdye'''.

== The "why" of aberrismic theory ==
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.
* A style of music that is melodic and heavy in modulations benefits from
** Multiple step sizes for melodic interest, for example diesis-sized steps that are below conventional semitones, specifically ones large enough to be melodically distinct but small enough to represent intonational changes.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation. 
* It is widely agreed that lower primes are more robust to detuning. Hence for approximating JI with edos, we use lower prime temperaments, and which also represent either 81/80 or 64/63 steps for greater accuracy.
The above suggests temperaments, in particular edos, that use tempered lower primes, and edos large enough to have small diesis-sized steps. In the context of fifth-based modulation, scales also benefit from having offset arcs of fifths. One simple way to have this is to detemper MOS scales into ternary scales with an additional smaller melodic step size, which have a generator arc with fifths or a generator arc that stacks to fifths via a detempered generator chain.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{adv|"Almost" a cross-set means that one or two notes may be missing from the full cross-set and one note may have been added. Exact cross-sets are italicized.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of...<br/>(interpreted)}}
!|Notes
|-
!|pinedye / dia1s<br/>(5L2m1s)<br/>{{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm<br/>1sR: LLmLLmLs<br/>1sL: LLmLLsLm
||2.3.5<br/>[L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s<br/>(5L2m2s)<br/>{{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL<br/>2sL: LsLmLsLmL
||2.3.7<br/>[L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s<br/>(5L2m3s)<br/>{{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5<br/>[L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s<br/>(5L2m4s)<br/>{{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss<br/>4sR: LsLmLsLsLms<br/>4sL: LsLsLmLsLsm
||2.3.7<br/>[L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]].<br/>{{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s<br/>(5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}}<br/>5sR: LmsLsLsLmsLs<br/>5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7<br/>[L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s<br/>(5L2m7s)<br/>{{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7<br/>[L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}
See [[Ground's composition theory#Quasi-diatonic scales]] for more tunings of diasem and blackdye.

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of...<br/>(interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m<br/>(5L5m6s)<br/>{{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46]<br/>[L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|prejubilic / citro3s<br/>(4L2m3s)<br/>{{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms<br/>R: LmLsLmsLs<br/>L: LmLsLsmLs
||2.5.7<br/>[L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s<br/>(7L2m5s)<br/>{{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s<br/>(3L4m3s)<br/>{{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s<br/>(4L3m2s)<br/>{{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm<br/>R: LmLmsLmLs<br/>L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

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

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.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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).
* [[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 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.

There are two choices involved in interpreting a given ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.

=== Example: Blackdye ===
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.
* [[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"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s<br/>m<br/>L 
| 81/80<br/>16/15<br/>10/9
| 143/140<br/>22/21<br/>160/143
|-
! 2-step
| L + s<br/>L + m 
| 9/8<br/>32/27
| 8/7, 9/8<br/>7/6
|- 
! 3-step
| L + 2s<br/>L + m + s<br/>2L + s<br/>2L + m
| 729/640<br/>6/5<br/>5/4<br/>320/243
| 7/6<br/>13/11<br/>14/11<br/>13/10
|- 
! 4-step
| 2L + 2s<br/>2L + m + s
| 81/64<br/>4/3
| 13/10<br/>4/3
|-
! 5-step
| 2L + m + 2s<br/>2L + 2m + s<br/>3L + 2s<br/>3L + m + s
| 27/20<br/>64/45<br/>45/32<br/>40/27
| 66/49<br/>11/8<br/>16/11<br/>49/33
|-
! 6-step
| 3L + m + 2s<br/>3L + 2m + s
| 3/2<br/>128/81
| 3/2<br/>20/13
|- 
! 7-step
| 3L + m + 3s<br/>3L + 2m + 2s<br/>4L + m + 2s<br/>4L + 2m + s
| 243/160<br/>8/5<br/>5/3<br/>1280/729
| 20/13<br/>11/7<br/>22/13<br/>12/7
|- 
! 8-step
| 4L + m + 3s<br/>4L + 2m + 2s
| 27/16<br/>16/9
| 12/7<br/>7/4, 16/9
|-
! 9-step
| 5L + 2m + s<br/>5L + m + 2s<br/>4L + 2m + 2s
| 9/5<br/>15/8<br/>160/81
| 143/80<br/>21/11<br/>280/143
|}

== 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)), also called trackdye.
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}