Aberschismic family: Difference between revisions

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–vBb).  

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. Thus a successful mapping of 13 can be found by fixing the [[13/11]] at the minor third (C–Eb), tempering out [[352/351]], [[847/845]], and [[2080/2079]].  

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

Mapping: {{mapping| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}

Mapping: {{mapping| 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.  

Mapping: {{mapping| 1 0 0 10 -18 -13 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }}

Mapping: {{mapping| 1 0 0 10 -18 -13 -6 | 0 1 0 -6 15 12 5 | 0 0 1 1 -1 -1 1 }}

Mapping: {{mapping| 1 0 0 10 -3 2 | 0 1 0 -6 7 4 | 0 0 1 1 -2 -2 }}

Lono tempers out [[176/175]] and may be described as the {{nowrap| 46 & 53 & 58 }} temperament, finding the interval class of 11 at the triple-down augmented fourth (C–v³F#). It notably also tempers out [[8019/8000]], thus setting 11/10, 10/9, 9/8, and 8/7 a comma apart from each other. [[111edo]] is a great tuning for it. [[157edo]] is a viable alternative, which is almost as good.  

Mapping: {{mapping| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }}

Mapping: {{mapping| 1 0 0 10 -6 -1 | 0 2 0 -12 9 3 | 0 0 1 1 1 1 }}

{{Main| Counterpyth }}

Developed analogous to [[parapyth]], counterpyth 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). Notice the factorization {{nowrap| 5120/5103 {{=}} ([[400/399]])⋅([[1216/1215]]) }}. Other important ratios are [[21/19]] at the diminished third (C–Ebb) and [[19/14]] at the augmented third (C–E#).  

Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}

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

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