Aberschismic: Difference between revisions
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Restricted to the subgroup 2.3.7/5, it becomes the fifth-generated rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103] = 2.3.7/5[12 & 29]. | Restricted to the subgroup 2.3.7/5, it becomes the fifth-generated rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103] = 2.3.7/5[12 & 29]. | ||
== Names == | == Names == | ||
The | The name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63. | ||
The | The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated (slightly hard of just) [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic. | ||
The | The older name ''Hemifamity'' is a portmanteau of ''[[Hemififths]]'' and ''[[Amity]]'', both rank-2 temperaments which support this rank-3 temperament. | ||
== Structural theory == | == Structural theory == | ||
Latest revision as of 16:56, 6 June 2026
Aberschismic (formerly known as Hemifamity), 41 & 46 & 53, is a 7-limit rank-3 temperament that tempers out 5120/5103 (known as the aberschisma), which manifests in several different ways:
- equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63
- equates the Pythagorean minor second with 21/20
- equates the Pythagorean chroma with 15/14
- equates the Pythagorean augmented fourth with 10/7
- splits 36/35 into two equal parts each representing the "common comma" 81/80 ~ 64/63 (hence 36/35 might humorously be referred to as the "famity")
- splits 7/5 into two equal parts each representing 32/27
- makes 28/27 - 21/20 - 16/15 - 27/25 equidistant and 35/32 - 10/9 - 9/8 - 8/7 equidistant
Restricted to the subgroup 2.3.7/5, it becomes the fifth-generated rank-2 temperament Argent or Argentic, 2.3.7/5[5120/5103] = 2.3.7/5[12 & 29].
Names
The name Aberschismic, coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in aberrismic theory where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.
The names Argent and Argentic come from the relationship to the silver ratio and the associated (slightly hard of just) argent tuning of MOS diatonic.
The older name Hemifamity is a portmanteau of Hemififths and Amity, both rank-2 temperaments which support this rank-3 temperament.
Structural theory
Notation
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.
Intervals
(* Cent values in pure-2/1, pure-7/4 tuning. JI interpretations are in the 13-limit strong extension Akea, 41 & 46 & 53. Prime harmonics are in bold.)
| # commas | -2 | -1 | 0 | +1 | +2 | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| # fifths | Cents* | JI | Cents* | JI | Cents* | JI | Cents* | JI | Cents* | JI |
| -7 | 1028.904 | 1054.2 | 1079.6 | 28/15 | 1104.9 | 128/135 | 1130.3 | 26/27 | ||
| -6 | 531.82 | 49/36 | 557.162 | 112/81 | 582.5 | 7/5 | 607.8 | 64/45 | 633.2 | 36/25, 13/9 |
| -5 | 34.736 | 49/48 | 60.1 | 28/27 | 85.4 | 21/20 | 110.8 | 16/15 | 136.1 | 13/12 |
| -4 | 737.652 | 49/32 | 763.0 | 14/9 | 788.3 | 128/81 | 813.7 | 8/5 | 839.0 | 13/8 |
| -3 | 240.6 | 147/128 | 265.9 | 7/6 | 291.252 | 32/27, 13/11 | 316.6 | 6/5 | 341.9 | 11/9, 39/32, 128/105 |
| -2 | 943.5 | 140/81 | 968.8 | 7/4 | 994.2 | 16/9 | 1019.5 | 9/5 | 1044.9 | 11/6, 64/35 |
| -1 | 446.4 | 35/27 | 471.7 | 21/16 | 497.1 | 4/3 | 522.4 | 27/20 | 547.8 | 11/8, 48/35 |
| 0 | 1149.3 | 35/36, 32/33 | 1174.7 | 80/81, 63/64 | 0 | 1/1 | 25.3 | 81/80, 64/63 | 50.7 | 36/35, 33/32 |
| 1 | 652.2 | 16/11, 35/24 | 677.6 | 40/27 | 702.9 | 3/2 | 728.3 | 32/21 | 753.6 | 54/35 |
| 2 | 155.1 | 12/11, 35/32 | 180.5 | 10/9 | 205.8 | 9/8 | 231.2 | 8/7 | 256.5 | 81/70 |
| 3 | 858.1 | 18/11, 64/39, 105/64 | 883.4 | 5/3 | 908.7 | 27/16, 22/13 | 934.1 | 12/7 | 959.4 | 256/147 |
| 4 | 361.0 | 16/13 | 386.3 | 5/4 | 411.7 | 81/64 | 437.0 | 9/7 | 462.3 | 64/49 |
| 5 | 1063.9 | 24/13 | 1089.2 | 15/8 | 1114.6 | 20/21 | 1139.9 | 27/14 | 1165.3 | 48/49 |
| 6 | 566.8 | 25/18, 18/13 | 592.2 | 45/32 | 617.5 | 10/7 | 642.8 | 81/56 | 668.2 | 72/49 |
| 7 | 69.7 | 27/26 | 95.1 | 135/128 | 120.4 | 15/14 | 145.8 | 171.1 | ||
Supporting edos
Comma-level resolution
The following edos map both 81/80 and 64/63 to one edostep: 41edo, 46edo, 48edo, 53edo, 58edo, 70edo.
Half-comma resolution
The following edos map both 81/80 and 64/63 to two edosteps: 87edo, 94edo, 99edo, 111edo.
Third-comma resolution
The following edos map both 81/80 and 64/63 to three edosteps: 128edo, 133edo, 135edo, 140edo, 145edo, 147edo, 152edo, 157edo.
Supporting rank-2 temperaments
Given any rank-2 temperament on a rank-3 subgroup of the 7-limit that is not 2.3.7/5 (e.g. 2.3.5, 2.3.7, 2.5.7), Aberschismic tempering may be added to extend the temperament to the full 7-limit. Examples:
- 41 & 46, Rodan (Aberschismic + Slendric, Aberschismic + Sensamagic)
- 41 & 48, Monkey (Aberschismic + Tetracot)
- 41 & 53, Garibaldi (Aberschismic + Schismic, Aberschismic + Marvel)
- 41 & 58, Hemififths (Aberschismic + 2401/2400)
- 46 & 53, Amity (Aberschismic + 4375/4374)
- 46 & 58, Septimal Diaschismic (Aberschismic + Diaschismic)
- 53 & 58, Buzzard (Aberschismic + 2.3.7 Buzzard)
- 87 & 99, Misty (Aberschismic + Didacus)
- 7 & 29, Aberschismic + Porcupine
Extensions
- Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).
- Prime 13 may be added by tempering out 4096/4095 (which equates 128/105 to 39/32); this is what 13-limit Akea (41 & 46 & 53) adds. An alternate extension adds Parapyth tempering and is called Pele (41 & 46 & 58).
Note that Akea and Pele together imply Rodan (41 & 46).
List of patent vals
- Main article: Aberschismic/Patent vals
