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Move counterpyth to subgroup extensions section as it skips multiple primes
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{{Interwiki
| en =
| de = Hemifamity
| es =
| ja =
| ro =
| ko = 헤미패미티 (음률)
}}
{{Technical data page}}
{{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.  
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]].  
 
Hemifamity can be further tempered to [[garibaldi]], which expands 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).  
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).  
Line 10: Line 20:


{{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}
{{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}
: mapping generators: ~2, ~3, ~5
: mapping generators: ~2, ~3, ~5


Line 19: Line 28:
: Angle (3/2, 10/9) = 82.112 degrees
: Angle (3/2, 10/9) = 82.112 degrees


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.7918, ~5/4 = 386.0144
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}, ~5/4 = 386.7266{{c}}
: [[error map]]: {{val| -0.283 +0.426 -0.153 +0.222 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}, ~5/4 = 386.5465{{c}}
: error map: {{val| 0.000 +0.862 +0.233 +0.821 }}


[[Minimax tuning]]: c = 5120/5103
[[Minimax tuning]]: c = 5120/5103
* [[7-odd-limit]]: 3 and 7 1/7c sharp, 5 just
* [[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 }}
: {{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
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
* [[9-odd-limit]]: 3 1/8c sharp, 5 just, 7 1/4c sharp
* [[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 }}
: {{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
: [[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 }}
{{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}


[[Badness]] (Smith): 0.153 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.675


[[Projection pair]]s: 7 5120/729
[[Projection pair]]s: 7 5120/729
Line 42: Line 55:
==== 11- and 13-limit extensions ====
==== 11- and 13-limit extensions ====
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–v<sup>3</sup>F#). 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]].  
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–v<sup>3</sup>F#). 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]].  
Temperaments discussed elsewhere include:
* ''[[Kahoupokane]]'' (+121/120) → [[Biyatismic clan #Kahoupokane|Biyatismic clan]]


==== Subgroup extensions ====
==== Subgroup extensions ====
A notable 2.3.5.7.19 subgroup extension, counterpyth, is given right below.
A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]].  
 
=== Counterpyth ===
{{Main| Counterpyth }}
 
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#).
 
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
 
Comma list: 400/399, 1216/1215
 
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<sup>-3</sup>


== Pele ==
== Pele ==
{{Main| Pele }}
{{Main| Pele }}
{{See also| Pentacircle clan }}
{{See also| Pentacircle clan }}
Pele tempers out [[441/440]] as well as [[896/891]] and may be described as the {{nowrap| 41 & 46 & 58 }} temperament, finding the interval class of 11 at the down diminished fifth (C–vGb). It also extends [[parapyth]]. [[145edo]] makes for an excellent tuning.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 81: Line 80:
: Angle(3/2, 56/55) = 90.4578 degrees
: Angle(3/2, 56/55) = 90.4578 degrees


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


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 17/10 0 1/10 0 -1/10 }}, {{monzo| 17/5 -2 6/5 0 -1/5 }}, {{monzo| 16/5 -2 3/5 0 2/5 }}, {{monzo| 17/5 -2 1/5 0 4/5 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 17/10 0 1/10 0 -1/10 }}, {{monzo| 17/5 -2 6/5 0 -1/5 }}, {{monzo| 16/5 -2 3/5 0 2/5 }}, {{monzo| 17/5 -2 1/5 0 4/5 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


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


[[Badness]] (Smith): 0.648 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.779


[[Projection pair]]s: 7 5120/729 11 655360/59049
[[Projection pair]]s: 7 5120/729 11 655360/59049
Line 101: Line 104:
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 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}


Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.4398, ~5/4 = 386.8933
Optimal tunings:
* WE: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}, ~5/4 = 388.0342{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}, ~5/4 = 387.7761{{c}}


Minimax tuning:  
Minimax tuning:  
Line 109: Line 114:
{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 145, 232 }}
{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 145, 232 }}


Badness (Smith): 0.703 × 10<sup>-3</sup>
Badness (Sintel): 0.658


=== 17-limit ===
=== 17-limit ===
Line 118: Line 123:
Mapping: {{mapping| 1 0 0 10 17 22 8 | 0 1 0 -6 -10 -13 -1 | 0 0 1 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 }}


Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.5544, ~5/4 = 387.9654
Optimal tunings:
* WE: ~2 = 1199.3960{{c}}, ~3/2 = 703.0725{{c}}, ~5/4 = 388.4246{{c}}
* 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 }}
{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 99ef, 145 }}


Badness (Smith): 0.930 × 10<sup>-3</sup>
Badness (Sintel): 0.884


== Laka ==
== Laka ==
{{Main| Laka }}
{{Main| Laka }}


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.
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.  
 
<blockquote>
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.  
</blockquote>
—[[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.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 142: Line 142:
{{Mapping|legend=1| 1 0 0 10 -18 | 0 1 0 -6 15 | 0 0 1 1 -1 }}
{{Mapping|legend=1| 1 0 0 10 -18 | 0 1 0 -6 15 | 0 0 1 1 -1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.5133, ~5/4 = 385.5563
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6201{{c}}, ~3/2 = 702.4416{{c}}, ~5/4 = 386.6781{{c}}
: [[error map]]: {{val| -0.380 +0.107 -0.395 +0.924 +0.527 }}
* [[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 }}


[[Minimax tuning]]
[[Minimax tuning]]
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 0 2/21 -1/21 1/21 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 2 0 3/7 2/7 -2/7 }}, {{monzo| 2 0 3/7 -5/7 5/7 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 0 2/21 -1/21 1/21 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 2 0 3/7 2/7 -2/7 }}, {{monzo| 2 0 3/7 -5/7 5/7 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7


{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}
{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}


[[Badness]] (Smith): 0.825 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.992


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


=== 13-limit ===
=== 13-limit ===
Line 162: Line 166:
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 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }}


Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4078, ~5/4 = 385.5405
Optimal tunings:
* WE: ~2 = 1199.4742{{c}}, ~3/2 = 702.3385{{c}}, ~5/4 = 387.0965{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5780{{c}}, ~5/4 = 386.7718{{c}}


Minimax tuning:  
Minimax tuning:  
Line 169: Line 175:
: unchanged-interval (eigenmonzo) basis: 2.11.13/7
: unchanged-interval (eigenmonzo) basis: 2.11.13/7


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


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]


Badness (Smith): 0.822 × 10<sup>-3</sup>
Badness (Sintel): 0.769


=== 2.3.5.7.11.13.19 subgroup ===
=== 2.3.5.7.11.13.19 subgroup ===
Line 182: Line 188:
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 -18 -13 -6 | 0 1 0 -6 15 12 5 | 0 0 1 1 -1 -1 1 }}


Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4062, ~5/4 = 385.5254
Optimal tunings:
* WE: ~2 = 1199.4881{{c}}, ~3/2 = 702.3224{{c}}, ~5/4 = 386.8881{{c}}
* 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 }}*
{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }} *


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]


Badness (Smith): 0.661 × 10<sup>-3</sup>
Badness (Sintel): 0.647
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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:
* 17-odd-limit
: [{{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<sup>-3</sup>


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


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 216: Line 210:
{{Mapping|legend=1| 1 0 0 10 -3 | 0 1 0 -6 7 | 0 0 1 1 -2 }}
{{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
[[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}}
: error map: {{val| 0.000 +0.896 -1.143 -0.761 -1.703 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5


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


[[Badness]] (Smith): 0.998 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.20


=== 13-limit ===
=== 13-limit ===
Line 240: Line 238:
Mapping to lattice: [{{val| 0 1 3 -3 1 -2 }}, {{val| 0 0 -1 -1 2 2 }}]
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
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}}


Minimax tuning:  
Minimax tuning:  
Line 249: Line 249:
{{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}
{{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}


Badness (Smith): 0.822 × 10<sup>-3</sup>
Badness (Sintel): 0.769


Scales: [[akea46_13]]
Scales: [[akea46_13]]


== Lono ==
== Lono ==
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–v<sup>3</sup>F#). 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.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 260: Line 262:
{{Mapping|legend=1| 1 0 0 10 6 | 0 1 0 -6 -6 | 0 0 1 1 3 }}
{{Mapping|legend=1| 1 0 0 10 6 | 0 1 0 -6 -6 | 0 0 1 1 3 }}


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


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


[[Badness]] (Smith): 1.18 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.41


=== 13-limit ===
=== 13-limit ===
Line 273: Line 279:
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 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 tunings:
* WE: ~2 = 1199.3329{{c}}, ~3/2 = 702.5519{{c}}, ~5/4 = 389.5508{{c}}
* 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 }}
{{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}


Badness (Smith): 0.908 × 10<sup>-3</sup>
Badness (Sintel): 0.850


== Kapo ==
== Kapo ==
Line 285: Line 293:


{{Mapping|legend=1| 1 0 0 10 7 | 0 1 1 -5 -2 | 0 0 2 2 -1 }}
{{Mapping|legend=1| 1 0 0 10 7 | 0 1 1 -5 -2 | 0 0 2 2 -1 }}
: mapping generators: ~2, ~3, ~128/99
: mapping generators: ~2, ~3, ~128/99


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8776, ~128/99 = 441.7516
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7125{{c}}, ~3/2 = 702.6631{{c}}, ~128/99 = 441.8973{{c}}
: [[error map]]: {{val| -0.287 +0.421 -0.143 +0.216 +0.021 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8413{{c}}, ~128/99 = 441.9493{{c}}
: error map: {{val| 0.000 +0.886 +0.426 +0.866 +1.050 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]:  
* [[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 }}]
: [{{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
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/9


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


[[Badness]] (Smith): 0.994 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.19


== Namaka ==
== Namaka ==
Line 305: Line 316:


{{Mapping|legend=1| 1 0 0 10 -6 | 0 2 0 -12 9 | 0 0 1 1 1 }}
{{Mapping|legend=1| 1 0 0 10 -6 | 0 2 0 -12 9 | 0 0 1 1 1 }}
: mapping generators: ~2, ~400/231, ~5
: mapping generators: ~2, ~400/231, ~5


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


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


[[Badness]] (Smith): 1.74 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
Line 321: Line 335:
Mapping: {{mapping| 1 0 0 10 -6 -1 | 0 2 0 -12 9 3 | 0 0 1 1 1 1 }}
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 tunings:
* WE: ~2 = 1199.7072{{c}}, ~26/15 = 951.2767{{c}}, ~5/4 = 387.4314{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}, ~5/4 = 387.2360{{c}}
 
{{Optimal ET sequence|legend=0| 29, 53, 58, 87, 111, 140, 198, 536f }}
 
Badness (Sintel): 0.731
 
== Subgroup extensions ==
=== Counterpyth (2.3.5.7.19) ===
{{Main| Counterpyth }}
 
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 ET sequence|legend=0| 29, 53, 58, 87, 111, 140, 198 }}
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
 
Comma list: 400/399, 1216/1215
 
Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}
 
Optimal tunings:
* WE: ~2 = 1199.6953{{c}}, ~3/2 = 702.5169{{c}}, ~5/4 = 386.2648{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6771{{c}}, ~5/4 = 386.0544{{c}}
 
{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}


Badness (Smith): 0.781 × 10<sup>-3</sup>
Badness (Sintel): 0.347


== Notes ==
== References ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemifamity family| ]] <!-- main article -->
[[Category:Hemifamity family| ]] <!-- main article -->
[[Category:Hemifamity| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Listen]]
[[Category:Listen]]
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