Aberschismic family: Difference between revisions

The '''aberschismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: [[5120/5103]]).  

 

 

It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have [[5/4]] at the down major third (C–vE) and [[7/4]] at the down minor seventh (C–vB♭).  

Strong extensions of aberschismic are [[#Pele|pele]], [[#Laka|laka]], [[#Akea|akea]], and [[#Lono|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] as a down diminished fifth (C–vG♭); laka, up augmented third (C–^E♯); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v³F♯). All these extensions follow the trend of tuning the fifth a little sharp, so a mapping for 13 arises from identifying [[13/11]] with the minor third (C–E♭), which is further facilitated by the fact that the minor third in aberschismic is the exact mean of [[6/5]] and [[7/6]], thus tempering out the [[semiparticular]] [[847/845]] ([[S-expression|S11/S13]]) and by extension [[352/351]] and [[2080/2079]].  

 

A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]].

{{Mapping|legend=0| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}

{{Mapping|legend=0| 1 0 0 10 17 22 8 | 0 1 0 -6 -10 -13 -1 | 0 0 1 1 1 1 -1 }}

Laka can be described as the {{nowrap| 41 & 53 & 58 }} temperament, tempering out [[540/539]], and finds the interval class of 11 at the up augmented third (C–^E♯). [[Gene Ward Smith]] considered it a [[17-limit]] temperament, assigning the vanishing of [[442/441]] ({{nowrap| 41g & 53 & 58 }}) as the main extension, but {{nowrap| 41 & 53g & 58 }} also makes for a competitive extension.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group | ''Laka 17-limit minimax planar temperament'']</ref> Indeed, laka makes most sense as a 2.3.5.7.11.13.19-[[subgroup]] temperament, skipping prime 17, as the 19 is accurate and easily available in a 24-tone scale. [[152edo]] makes for an excellent tuning, using the 152f val for prime 13.  

[[Projection pair]]s: <code>7 5120/729 11 14348907/1310720</code>

{{Mapping|legend=0| 1 0 0 10 -18 -13 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }}

{{Mapping|legend=0| 1 0 0 10 -18 -13 -6 | 0 1 0 -6 15 12 5 | 0 0 1 1 -1 -1 1 }}

{{Mapping|legend=0| 1 0 0 10 -3 2 | 0 1 0 -6 7 4 | 0 0 1 1 -2 -2 }}

{{Mapping|legend=0| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.7125{{c}}, ~3/2 = 702.6631{{c}}, ~128/99 = 441.8973{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8413{{c}}, ~128/99 = 441.9493{{c}}

: error map: {{val| 0.000 +0.886 +0.426 +0.866 +1.050 }}

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/9

 

 

 

 

{{Mapping|legend=0| 1 0 0 10 7 12 | 0 1 1 -5 -2 -5 | 0 0 2 2 -1 -1 }}

 

 

Badness (Sintel): 0.938

[[Optimal tuning]]s:

: [[error map]]: {{val| -0.282 +0.627 +0.620 -0.203 -1.074 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.5081{{c}}, ~5/4 = 387.3182{{c}}

: error map: {{val| 0.000 +1.061 +1.004 +0.395 -0.426 }}

[[Badness]] (Sintel): 2.09

{{Mapping|legend=0| 1 0 0 10 -6 -1 | 0 2 0 -12 9 3 | 0 0 1 1 1 1 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}, ~5/4 = 387.2360{{c}}

{{Optimal ET sequence|legend=0| 29, 53, 58, 87, 111, 140, 198, 536f }}

Badness (Sintel): 0.731

== Subgroup extensions ==

=== Counterpyth (2.3.5.7.19) ===

Inspired by [[Margo Schulter]]'s [[parapyth]], counterpyth was named and first explored by [[Flora Canou]] in 2024. It is an extension of aberschismic with an even milder fifth, as it finds [[19/15]] at the major third (C–E) and [[19/10]] at the major seventh (C–B), taking advantage of the factorization {{nowrap| 5120/5103 {{=}} ([[400/399]])⋅([[1216/1215]]) }}. Other important ratios are [[21/19]] at the diminished third (C–E𝄫) and [[19/14]] at the augmented third (C–E♯).

It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths.

Subgroup: 2.3.5.7.19

Comma list: 400/399, 1216/1215

{{Mapping|legend=0| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}

Optimal tunings:  

* WE: ~2 = 1199.6953{{c}}, ~3/2 = 702.5169{{c}}, ~5/4 = 386.2648{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6771{{c}}, ~5/4 = 386.0544{{c}}

{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}

Badness (Sintel): 0.347

[[Category:Catalogs of rank-3 temperaments]]