Hemifamity 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]]. Hemifamity can be compared to [[garibaldi]], with garibaldi expanding the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.

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).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: [[5120/5103]]

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

: mapping generators: ~2, ~3, ~5

[[Mapping to lattice]]: [{{val| 0 1 2 -4 }}, {{val| 0 0 1 1 }}]

Lattice basis:

: 3/2 length = 0.5670, 10/9 length = 1.8063

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.7918, ~5/4 = 386.0144

[[Minimax tuning]]: c = 5120/5103

* [[7-odd-limit]]: 3 and 7 1/7c sharp, 5 just

: {{monzo list| 1 0 0 0 | 10/7 1/7 1/7 -1/7 | 0 0 1 0 | 10/7 -6/7 1/7 6/7 }}

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

* [[9-odd-limit]]: 3 1/8c sharp, 5 just, 7 1/4c sharp

: {{monzo list| 1 0 0 0 | 5/4 1/4 1/8 -1/8 | 0 0 1 0 | 5/2 -3/2 1/4 3/4 }}

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

{{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}

[[Badness]] (Smith): 0.153 × 10^-3

[[Projection pair]]s: 7 5120/729

* [http://www.archive.org/details/Choraled ''Choraled''] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]

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]].

==== Subgroup extensions ====

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

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

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.6411, ~5/4 = 385.4452

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

Badness (Smith): 0.212 × 10^-3

{{Main| Pele }}

{{See also| Pentacircle clan }}

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 441/440, 896/891

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

[[Mapping to lattice]]: [{{val| 0 1 4 -2 -6 }}, {{val| 0 0 -1 -1 -1 }}]

Lattice basis:  

: Angle(3/2, 56/55) = 90.4578 degrees

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 703.2829, ~5/4 = 386.5647

* [[11-odd-limit]]

{{Optimal ET sequence|legend=1| 29, 41, 58, 87, 99e, 145, 186e }}

[[Badness]] (Smith): 0.648 × 10^-3

[[Projection pair]]s: 7 5120/729 11 655360/59049

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363

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

 

 

 

 

 

 

 

 

 

 

== Laka ==

 

 

 

 

 

[[Comma list]]: 540/539, 5120/5103

 

 

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.5133, ~5/4 = 385.5563

Comma list: 352/351, 442/441, 540/539, 561/560

Mapping: {{mapping| 1 0 0 10 -18 -13 32 | 0 1 0 -6 15 12 -22 | 0 0 1 1 -1 -1 3 }}

Minimax tuning:  

: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 13/12 0 0 1/12 1/6 -1/12 0 }}, {{monzo| -7/4 0 0 5/4 3/2 -5/4 0 }}, {{monzo| 7/4 0 0 3/4 1/2 -3/4 0 }}, {{monzo| 0 0 0 0 1 0 0 }}, {{monzo| 7/4 0 0 -1/4 1/2 1/4 0 }}, {{monzo| 35/12 0 0 23/12 5/6 -23/12 0 }}]

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

{{Optimal ET sequence|legend=0| 58, 94, 111, 152f, 205, 263df }}

Badness (Smith): 1.19 × 10^-3

== 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}.]]

[[Comma list]]: 385/384, 2200/2187

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8909, ~5/4 = 385.3273

* [[11-odd-limit]]

{{Optimal ET sequence|legend=1| 34, 41, 53, 87, 140, 181, 321 }}

[[Badness]] (Smith): 0.998 × 10^-3

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384

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

Lattice basis:  

: 3/2 length = 0.5354, 27/20 length = 1.0463

: Angle (3/2, 27/20) = 80.5628 degrees

Mapping to lattice: [{{val| 0 1 3 -3 1 -2 }}, {{val| 0 0 -1 -1 2 2 }}]

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.9018, ~5/4 = 385.4158

: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 0 }}, {{monzo| 26/9 0 -7/9 1/9 2/3 0 }}]

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

Badness (Smith): 0.822 × 10^-3

Scales: [[akea46_13]]

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 176/175, 5120/5103

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

{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 111, 268cd }}

[[Badness]] (Smith): 1.18 × 10^-3

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845

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

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.8670, ~5/4 = 388.6277

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

Badness (Smith): 0.908 × 10^-3

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 3025/3024, 5120/5103

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

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

[[Minimax tuning]]:

* [[11-odd-limit]]:  

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/5 2/5 0 -1/15 -2/15 }}, {{monzo| 14/5 6/5 0 7/15 -16/15 }}, {{monzo| 16/5 -6/5 0 13/15 -4/15 }}, {{monzo| 16/5 -6/5 0 -2/15 11/15 }}]

: [[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

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 3388/3375, 5120/5103

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

: mapping generators: ~2, ~400/231, ~5

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

{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}

[[Badness]] (Smith): 1.74 × 10^-3

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845

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

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

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

Badness (Smith): 0.781 × 10^-3

[[Category:Pages with mostly numerical content]]

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

[[Category:Hemifamity| ]] <!-- key article -->

[[Category:Rank 3]]

[[Category:Listen]]