Aberrisma

An aberrisma is an interval between roughly 20 and 55 cents representing some comma as an additional smaller type of melodic step. 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.

Aberrismic/split-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 split primes, like 3+ and 3- (in a dual-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 split 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).

Aberrismic theory is the subset of microtonal theory pioneered by Ground and Inthar that deals with aberrismas.

• pinedye (5L2m1s or 1s)
• diasem (5L2m2s or 2s)
• blackdye (5L2m3s or 3s)
• diaslen (5L2m4s or 4s)
• diachrome (5L2m5s or 5s)

For more information on the math notation on this page, see User:Inthar/Math style guide.

Edos with quasi-diatonic aberrismic scales

All edos 20 and above have an aberrismic scale of the form 5L2mks. If n = 5p + 2q where p > q > 1 (n-edo is a diatonic edo with step ratio p:q), then (n + k)edo has a 5L2mks scale with step ratio p:q:1 for 1 ≤ k < q, and (n + rk)edo has a 5L2mks scale with step ratio p:q:r if 1 ≤ r ≤ rk < q.

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 temperaments. 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 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, or that the temperament before applying the tuning map has the same rank as the JI subgroup. This also implies that there is only one JI ratio for each interval under such interpretations, unlike in temperaments that temper out commas.
• 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, since Flutterpyth temperament is restricted to (maximally) 2.3.7.11.13.19 and does not include 5-limit thirds.

Blackdye intervals in two temperaments

Interval class	Sizes	Untempered 2.3.5	2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
1-steps	s m L	81/80 16/15 10/9	143/140 22/21 160/143
2-steps	L + s L + m	9/8 32/27	8/7, 9/8 7/6
3-steps	L + 2s L + m + s 2L + s 2L + m	729/640 6/5 5/4 320/243	7/6 13/11 14/11 13/10
4-steps	2L + 2s 2L + m + s	81/64 4/3	13/10 4/3
5-steps	2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s	27/20 64/45 45/32 40/27	66/49 11/8 16/11 49/33
6-steps	3L + m + 2s 3L + 2m + s	3/2 128/81	3/2 20/13
7-steps	3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s	243/160 8/5 5/3 1280/729	20/13 11/7 22/13 12/7
8-steps	4L + m + 3s 4L + 2m + 2s	27/16 16/9	12/7 7/4, 16/9
9-steps	5L + 2m + s 5L + m + 2s 4L + 2m + 2s	9/5 15/8 160/81	143/80 21/11 280/143

External links

• Aberrismic DR Dual server