Aberschismic: Difference between revisions

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#REDIRECT [[Aberschismic family #Aberschismic]]
{{Interwiki
| en = Aberschismic
| de = Aberrschismisch
| es =
| ja =
| ko = 애버스키스믹 (음률)
}}
{{Infobox regtemp
| Title = Aberschismic
| Subgroups = 2.3.5.7
| Comma basis = [[5120/5103]]
| Edo join 1 = 12 | Edo join 2 = 41 | Edo join 3 = 46
| Mapping = 1; 1 0 -6; 0 1 1
| Generators = 3/2; 5/4 | Generators tuning = 702.8166; 386.5465
| Optimization method = CWE
| Odd limit 1 = 9 | Mistuning 1 = 1.44 | Complexity 1 = ?
}}
'''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|tempering out]] their difference, the [[aberschisma]]. This reduces the two [[formal comma]]s 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 [[3/2|perfect fifth]], in contrast to simpler temperaments like [[marvel]] or [[starling]].
 
Aberschismic is the head of the [[aberschismic family]]; notable [[11-limit]] [[extension]]s 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 [[support]]ing 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.
 
{| class="wikitable center-1 center-3"
|+Nomenclature of selected intervals
! Ratio
! Name
! Example<br>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 [[41edo|41]], [[46edo|46]], [[53edo|53]], and [[58edo|58]]. Sums of these edos tend to give more accurate tunings.
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generators
! Eigenmonzo basis
|-
| 7-odd-limit
| ~3/2 = 702.7775{{c}}, ~5/4 = 386.3137{{c}}
| 2.5.7/3
|-
| 9-odd-limit
| ~3/2 = 702.6747{{c}}, ~5/4 = 386.3137{{c}}
| 2.5.9/7
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.7918{{c}}, ~5/4 = 386.0144{{c}}
| CWE: ~3/2 = 700.8166{{c}}, ~5/4 = 386.5465{{c}}
| POTE: ~3/2 = 702.8292{{c}}, ~5/4 = 386.8177{{c}}
|}
 
== Music ==
; [[Gene Ward Smith]]
* ''Choraled'' (archived 2010) – [http://www.archive.org/details/Choraled details] | [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play]
 
; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127015604/http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 ''Hemifamity27'']


[[Category:Aberschismic| ]] <!-- main article -->
[[Category:Aberschismic| ]] <!-- main article -->
[[Category:Rank-3 temperaments]]
[[Category:Rank-3 temperaments]]
[[Category:Aberschismic family]]
[[Category:Aberschismic family]]

Revision as of 11:08, 8 June 2026

Aberschismic
Subgroups 2.3.5.7
Comma basis 5120/5103
Reduced mapping ⟨1; 1 0 -6; 0 1 1]
ET join 12 & 41 & 46
Generators (CWE) ~3/2 = 702.8166 ¢, ~5/4 = 386.5465 ¢
MOS scales n/a
Ploidacot n/a
Minimax error 9-odd-limit: 1.44 ¢
Target scale size 9-odd-limit: ? notes

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.

Nomenclature of selected intervals
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 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
Norm-based tunings
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 ¢

Music

Gene Ward Smith
Chris Vaisvil
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