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]] with the augmented fourth (C–F#) and [[50/49]] with the [[Pythagorean comma]].  

 

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).  

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}, ~5/4 = 386.7266{{c}}

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7

[[Badness]] (Sintel): 0.675

Strong extensions of hemifamity 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]] at the down diminished fifth (C–vGb); 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]].  

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.5424{{c}}, ~3/2 = 703.0109{{c}}, ~5/4 = 387.6427{{c}}

: [[error map]]: {{val| -0.458 +0.598 +0.414 -1.995 +2.097 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2804{{c}}, ~5/4 = 387.3911{{c}}

: error map: {{val| 0.000 +1.325 +1.077 -1.117 +3.269 }}

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9

[[Badness]] (Sintel): 0.779

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}, ~5/4 = 387.7761{{c}}

* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9

* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9

{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 145, 232 }}

Badness (Sintel): 0.658

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4518{{c}}, ~5/4 = 388.4909{{c}}

{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 99ef, 145 }}

Badness (Sintel): 0.884

[[Optimal tuning]]s:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6175{{c}}, ~5/4 = 386.4170{{c}}

: error map: {{val| 0.000 +0.663 +0.103 +1.886 +1.528 }}

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7

[[Badness]] (Sintel): 0.992

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

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5780{{c}}, ~5/4 = 386.7718{{c}}

: unchanged-interval (eigenmonzo) basis: 2.11.13/7

{{Optimal ET sequence|legend=0| 41, 53, 58, 94, 111, 152f, 415dff }} *

Badness (Sintel): 0.769

Comma list: 352/351, 400/399, 456/455, 495/494

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5613{{c}}, ~5/4 = 386.6230{{c}}

{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }} *

Badness (Sintel): 0.647

== Akea ==

[[File:Lattice Akea.png|thumb|Lattice for 13-limit akea.]]

[[File:Lattice Akea-commatic.png|thumb|Ditto, but rearranged to basis {~2, ~3, ~81/80}.]]

Akea tempers out [[385/384]] and may be described as the {{nowrap| 41 & 46 & 53 }} temperament, finding the interval class of 11 at the double-up fourth (C–^^F). [[140edo]], [[181edo]] and especially [[321edo]] can be used as tunings. Note that [[94edo]] is a notable tuning not appearing on the optimal ET sequence.

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.1396{{c}}, ~3/2 = 702.9241{{c}}, ~5/4 = 385.1817{{c}}

: [[error map]]: {{val| +0.140 +1.109 -0.853 -0.351 -1.213 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8511{{c}}, ~5/4 = 385.1712{{c}}

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5

[[Badness]] (Sintel): 1.20

Optimal tunings:  

* WE: ~2 = 1200.0943{{c}}, ~3/2 = 702.9377{{c}}, ~5/4 = 385.4278{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.8853{{c}}, ~5/4 = 385.4002{{c}}

: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

{{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}

Badness (Sintel): 0.769

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.3368{{c}}, ~3/2 = 702.5643{{c}}, ~5/4 = 389.5319{{c}}

: [[error map]]: {{val| -0.663 -0.054 +1.892 +1.341 -2.088 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.9356{{c}}, ~5/4 = 389.4076{{c}}

: error map: {{val| 0.000 +0.981 +3.094 +2.968 -0.708 }}

[[Badness]] (Sintel): 1.41

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9205{{c}}, ~5/4 = 389.4341{{c}}

{{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}

Badness (Sintel): 0.850

[[Optimal tuning]]s:

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

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

{{Optimal ET sequence|legend=1| 41, 65d, 87, 111, 152, 239, 391, 980bcde, 1132bcdde, 1371bbcddee }}

[[Badness]] (Sintel): 1.19

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.7179{{c}}, ~400/231 = 951.2909{{c}}, ~5/4 = 387.4982{{c}}

: [[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

Optimal tunings:

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

 

 

Badness (Sintel): 0.731

 

 

 

 

 

 

 

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