Aberschismic family: Difference between revisions

The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[5120/5103]] ({{monzo|legend=1| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]].  

A notable 2.3.5.7.19 subgroup extension, counterpyth, is given right below.

 

 

Developed analogous to [[parapyth]], counterpyth is an extension of hemifamity 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#).

 

 

 

 

 

 

 

Badness (Sintel): 0.347

Laka can be described as the {{nowrap| 41 & 53 & 58 }} temperament, tempering out [[540/539]]. [[Gene Ward Smith]] considered it to be a [[17-limit]] temperament, assigning †442/441 ({{nowrap| 41g & 53 & 58 }}) as the main extension. It should be noted that {{nowrap| 41 & 53g & 58 }} also makes for a possible extension.

 

It's the way the numbers fall. The Laka geometry happens to work reasonably well in the 13-limit but not so well in the 17-limit. There isn't one obvious 17-limit extension and none of them are competitive with other 17-limit temperaments.  

—[[Graham Breed]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group | ''Laka 17-limit minimax planar temperament'']</ref>

 

It makes most sense as a 2.3.5.7.11.13.19-[[subgroup]] temperament, omitting harmonic 17, as the 19 is accurate and easily available in a 24-tone scale.  

[[Projection pair]]s: 5120/729 11 14348907/1310720

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.8776{{c}}, ~128/99 = 441.7516{{c}}

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

{{Optimal ET sequence|legend=1| 41, 87, 111, 152, 239, 391 }}

[[Badness]] (Smith): 0.994 × 10^-3

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~400/231 = 951.4956{{c}}, ~5/4 = 386.7868{{c}}

[[Badness]] (Smith): 1.74 × 10^-3

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 951.4871{{c}}, ~5/4 = 386.6606{{c}}

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

Badness (Smith): 0.781 × 10^-3

== Kahoupokane ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 121/120, 5120/5103

: mapping generators: ~2, ~3, ~11/10

[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000{{c}}, ~3 = 1903.042{{c}}, ~11/10 = 157.992{{c}}

Badness (Sintel): 2.734

=== 13-limit ===

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 121/120, 169/168, 352/351

{{Mapping|legend=1| 1 0 1 11 2 7 | 0 1 1 -5 1 -2 | 0 0 -2 -2 -1 -1 }}

 

Badness (Sintel): 1.269