Aberschismic
| Aberschismic |
Aberschismic (formerly hemifamity) is a rank-3 temperament that equates the syntonic comma of 81/80 with the septimal comma of 64/63, bisecting their sum, the septimal quartertone of 36/35, and tempering out their difference, the aberschisma. This reduces the two formal commas required for 7-limit JI to one generic comma step, and classical and septimal intervals are found on the same chain of fifths inflected by this step to the opposite sides. In addition, 10/7 is identified with the diatonic augmented fourth (C–F♯), and 50/49 is identified with the Pythagorean comma (C–B♯′). Because of this, aberschismic favors a slightly sharp perfect fifth, in contrast to simpler temperaments like marvel or starling.
Aberschismic is the head of the aberschismic family; notable 11-limit extensions include pele (41 & 46 & 58), laka (41 & 53 & 58), akea (41 & 46 & 53), and lono (46 & 53 & 58).
This article concerns the basic 7-limit temperament, aberschismic itself.
The temperament's name was derived from Tristan Bay's name for the corresponding comma, aberschisma, proposed in 2024 and officialized in 2026. The former name hemifamity was a contraction of hemififths and amity, two of the supporting rank-2 temperaments.
See Aberschismic family #Aberschismic for technical data.
Notation
Aberschismic is easily notated with chain-of-fifths notation with an extra pair of accidental for the generic comma step. As an example, we can use up and down arrows (^/v) for the comma step. Note that this comma step does not represent the Pythagorean comma, unless it is further tempered to cassandra. Aberschismic can therefore be seen cassandra with the Pythagorean comma decoupled from the generic comma step.
| Ratio | Name | Example on C |
|---|---|---|
| 3/2 | Perfect fifth | C–G |
| 5/4 | Down major third | C–vE |
| 7/4 | Down minor seventh | C–vB♭ |
Tunings
Edo tunings where the comma step is one step wide and all 9-odd-limit intervals are distinct include 41, 46, 53, and 58. Sums of these edos tend to give more accurate tunings.
| Target | Minimax | |
|---|---|---|
| Generators | Eigenmonzo basis | |
| 7-odd-limit | ~3/2 = 702.7775 ¢, ~5/4 = 386.3137 ¢ | 2.5.7/3 |
| 9-odd-limit | ~3/2 = 702.6747 ¢, ~5/4 = 386.3137 ¢ | 2.5.9/7 |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 702.7918 ¢, ~5/4 = 386.0144 ¢ | CWE: ~3/2 = 700.8166 ¢, ~5/4 = 386.5465 ¢ | POTE: ~3/2 = 702.8292 ¢, ~5/4 = 386.8177 ¢ |