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'''Hemifamity''' or '''Aberschismic''', 2.3.5.7[[[41edo|41]] & [[46edo|46]] & [[53edo|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:
'''Aberschismic''' (formerly known as '''Hemifamity'''), [[41edo|41]] & [[46edo|46]] & [[53edo|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 3-spine prime commas for 5 and 7, 81/80 and 64/63
# equates the limma 256/243 with 21/20
# equates the Pythagorean minor second with 21/20
# equates the apotome 2187/2048 with 15/14
# equates the Pythagorean chroma with 15/14
# equates the Pythagorean augmented fourth 729/512 with 10/7
# equates the Pythagorean augmented fourth with 10/7
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63
# 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 rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].
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 ==
As of early 2026, the temperament has no single agreed-upon name.
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 older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.
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 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 older name ''Hemifamity'' is a portmanteau of ''[[Hemififths]]'' and ''[[Amity]]'', both rank-2 temperaments which support this rank-3 temperament.
 
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.


== Structural theory ==
== Structural theory ==
=== Notation ===
=== Notation ===
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.
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.)
{| class="wikitable"
!# commas
! colspan="2" |-2
! colspan="2" |-1
! colspan="2" |0
! colspan="2" |+1
! colspan="2" |+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
|class="thl"|'''839.0'''
|class="thl"|'''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
|class="thl"|'''968.8'''
|class="thl"|'''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
|class="thl"|'''547.8'''
|class="thl"|'''11/8''', 48/35
|-
! |0
|1149.3
|35/36, 32/33
| |1174.7
| |80/81, 63/64
|class="thl"|'''0'''
|class="thl"|'''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
|class="thl"|'''702.9'''
|class="thl"|'''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
|class="thl"|'''386.3'''
|class="thl"|'''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 ==
== Supporting edos ==
=== Comma resolution ===
=== Comma-level resolution ===
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]], [[70edo]].
 
=== Half-comma resolution ===
=== Half-comma resolution ===
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].
Line 29: Line 232:


== Supporting rank-2 temperaments ==
== Supporting rank-2 temperaments ==
* 2.3.5.7[41 & 46]: [[Rodan]]
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:
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]
* 41 & 46, [[Rodan]] (Aberschismic + [[Slendric]], Aberschismic + [[Sensamagic]])
* 2.3.5.7[41 & 53]: [[Garibaldi]]
* 41 & 48, [[Tetracot (temperament)#Extensions|Monkey]] (Aberschismic + [[Tetracot (temperament)|Tetracot]])
* 2.3.5.7[41 & 58]: [[Hemififths]]
* 41 & 53, [[Garibaldi]] (Aberschismic + Schismic, Aberschismic + [[Marvel]])
* 2.3.5.7[46 & 53]: [[Amity]]
* 41 & 58, [[Hemififths]] (Aberschismic + 2401/2400)
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]
* 46 & 53, [[Amity]] (Aberschismic + 4375/4374)
* 2.3.5.7[53 & 58]: [[Buzzard]]
* 46 & 58, Septimal [[Diaschismic]] (Aberschismic + Diaschismic)
* 2.3.5.7[87 & 99]: [[Misty]]
* 53 & 58, [[Buzzard]] (Aberschismic + 2.3.7 Buzzard)
* 87 & 99, [[Misty]] (Aberschismic + [[Didacus]])
* 7 & 29, Aberschismic + [[Porcupine]]


== Extensions ==
== 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 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 is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].
* 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 ==
== List of patent vals ==
:''Main article: [[Hemifamity/Patent vals]]''
:''Main article: [[Aberschismic/Patent vals]]''
{{navbox regtemp}}
{{navbox regtemp}}
{{Cat|Temperaments}}
{{Cat|Temperaments}}

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:

  1. equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63
  2. equates the Pythagorean minor second with 21/20
  3. equates the Pythagorean chroma with 15/14
  4. equates the Pythagorean augmented fourth with 10/7
  5. 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")
  6. splits 7/5 into two equal parts each representing 32/27
  7. 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:

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


View • Talk • EditRegular temperaments
Rank-2
Acot Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave)
Monocot Meantone • Schismic • Gentle-fifth temperaments • Archy
Complexity 2 Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot)
Complexity 3 Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot)
Complexity 4 Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot)
Complexity 5-6 Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot)
Higher complexity Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (delta-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot)
Straddle-3 A-Team (alter-tricot) • Machine (alter-monocot)
No-3 Mabilic (alpha-triseph) • Orgone (trimech) • Didacus (diseph) • Sidewalk (epsilon-heptaseph)
No-octaves Sensamagic (monogem)
Exotemperaments Dicot • Mavila • Father
Higher-rank
Rank-3 Aberschismic • Marvel • Parapyth • Pentagoth
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