Hemifamity family: Difference between revisions

{{Technical data page}}

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).  

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

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

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

[[Badness]] (Smith): 0.153 × 10^-3

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]].  

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#).  

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

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

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

[[Badness]] (Smith): 0.648 × 10^-3

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.4398, ~5/4 = 386.8933

* 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 (Smith): 0.703 × 10^-3

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.5544, ~5/4 = 387.9654

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

Badness (Smith): 0.930 × 10^-3

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

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

[[Badness]] (Smith): 0.825 × 10^-3

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4078, ~5/4 = 385.5405

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

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

Badness (Smith): 0.822 × 10^-3

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4062, ~5/4 = 385.5254

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

Badness (Smith): 0.661 × 10^-3

: 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

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

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

[[Badness]] (Smith): 0.998 × 10^-3

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

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

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

Badness (Smith): 0.822 × 10^-3

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8941, ~5/4 = 388.5932

[[Badness]] (Smith): 1.18 × 10^-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

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

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

[[Badness]] (Smith): 0.994 × 10^-3

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

[[Badness]] (Smith): 1.74 × 10^-3

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