Hemifamity

Hemifamity or Aberschismic, 41 & 46 & 53, is a 7-limit rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:

1. equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63
2. equates the limma 256/243 with 21/20
3. equates the apotome 2187/2048 with 15/14
4. equates the Pythagorean augmented fourth 729/512 with 10/7
5. splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63

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

As of early 2026, the temperament has no single agreed-upon name.

The older name Hemifamity is a portmanteau of Hemififths and Amity, rank-2 temperaments which both support this rank-3 temperament.

The newer 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.

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

• 2.3.5.7[41 & 46]: Rodan (Aberschismic + Slendric)
• 2.3.5.7[41 & 48]: Monkey (Aberschismic + Tetracot)
• 2.3.5.7[41 & 53]: Garibaldi (Aberschismic + Schismic, Aberschismic + Marvel)
• 2.3.5.7[41 & 58]: Hemififths (Aberschismic + 2401/2400)
• 2.3.5.7[46 & 53]: Amity (Aberschismic + 4375/4374)
• 2.3.5.7[46 & 58]: Septimal Diaschismic (Aberschismic + Diaschismic)
• 2.3.5.7[53 & 58]: Buzzard (Aberschismic + 2.3.7 Buzzard)
• 2.3.5.7[87 & 99]: Misty (Aberschismic + Didacus)

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. This is the most efficient extension that adds prime 13 and is supported by edos as big as 140edo.

List of patent vals

Main article: Hemifamity/Patent vals