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Hemimean family: Difference between revisions

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The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]].  
{{Technical data page}}
The '''hemimean family''' of [[temperament]]s are [[rank-3 temperament]]s which [[temper out]] [[3136/3125]].  


The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). This is a no-threes comma, with [[Spectrum of a temperament|spectrum]] 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7.
The hemimean comma, 3136/3125, is the ratio between the [[126/125|septimal semicomma (126/125)]] and the [[225/224|septimal kleisma (225/224)]]. This fact alone makes hemimean a very notable rank-3 temperament, as any non-meantone tuning of hemimean will split the [[81/80|syntonic comma (81/80)]] into two equal parts, each representing 126/125~225/224.  


= Hemimean =
Other equivalences characteristic to hemimean are [[128/125]]~[[50/49]] and [[49/45]]~([[25/24]])<sup>2</sup>.
[[Comma]]: 3136/3125, hemimean


7 and 9 limit minimax
== Hemimean ==
[[Subgroup]]: 2.3.5.7


[|1 0 0 0&gt;, |0 1 0 0&gt;, |6/5 0 0 2/5&gt;, |0 0 0 1&gt;]
[[Comma list]]: 3136/3125 (hemimean)


[[Eigenmonzo]]s: 2, 7/6, 4/3
{{Mapping|legend=1| 1 0 0 -3 | 0 1 0 0 | 0 0 2 5 }}


Lattice basis: 28/25 length 0.5055 3/2 length 1.5849
: mapping generators: ~2, ~3, ~56/25


Angle(28/25, 3/2) = 90 degrees
[[Mapping to lattice]]: {{mapping| 0 0 2 5 | 0 1 0 0 }}


Map to lattice: [&lt;0 0 2 5|, &lt;0 1 0 0|]
Lattice basis:  
: 28/25 length = 0.5055, 3/2 length = 1.5849
: Angle (28/25, 3/2) = 90 degrees


Map: [&lt;1 0 0 -3|, &lt;0 1 0 0|, &lt;0 0 2 5|]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9550, ~28/25 = 193.6499


[[Generator]]s: 2, 3, 56/25
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}]
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7


[[EDO|Edos]]: {{EDOs|68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c}}
{{Optimal ET sequence|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 675c }}


Badness: 0.000160
[[Badness]]: 0.160 × 10<sup>-3</sup>
 
[[Complexity spectrum]]: 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7


[[Projection pair]]s: 5 3136/625 7 68841472/9765625 to 2.3.25/7
[[Projection pair]]s: 5 3136/625 7 68841472/9765625 to 2.3.25/7


= Belobog =
=== Hemimean orion ===
[[Comma]]s: 441/440, 3136/3125
As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[10/9]] and [[9/8]], it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] (5/4)/(19/17)<sup>2</sup> = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. An interesting tuning not in the optimal ET sequence is [[111edo]]. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.
 
The [[S-expression]]-based comma list for the 2.3.5.7.17.19 subgroup extension is {[[1216/1215|S16/S18]], [[1445/1444|S17/S19]], [[1701/1700|S18/S20]](, ([[136/135|S16*S17]])/([[190/189|S19*S20]]) = [[476/475|S16/S18 * S17/S19 * S18/S20]])}.
 
Subgroup: 2.3.5.7.17
 
Comma list: 1701/1700, 3136/3125
 
Sval mapping: {{mapping| 1 0 0 -3 -5 | 0 1 0 0 5 | 0 0 2 5 1 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548
 
{{Optimal ET sequence|legend=1| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg }}
 
Badness: 0.573
 
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19
 
Comma list: 476/475, 1216/1215, 1445/1444


11 limit minimax
Sval mapping: {{mapping| 1 0 0 -3 -5 -6 | 0 1 0 0 5 5 | 0 0 2 5 1 2 }}


[|1 0 0 0 0&gt;, |27/22 6/11 -5/22 -3/11 5/22&gt;,
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647
|24/11 -4/11 -2/11 2/11 2/11&gt;,
|27/11 -10/11 -5/11 5/11 5/11&gt;,  
|24/11 -4/11 -13/11 2/11 13/11&gt;]


[[Eigenmonzo]]s: 2, 11/10, 9/7
{{Optimal ET sequence|legend=1| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}


Lattice basis: 28/25 0.3829 16/15 1.1705
Badness: 0.456


Angle(28/25, 16/15) = 93.2696
=== Semiorion ===
Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two. The [[S-expression]]-based comma list for the 2.3.5.7.17.19 subgroup extension is {[[289/288|S17]], [[361/360|S19]], [[1216/1215|S16/S18]](, [[1701/1700|S18/S20]], [[476/475]] = [[2128/2125|S16/S20]] * [[1445/1444|S17/S19]])}.  


Map to lattice: [&lt;0 -2 2 5 4|, &lt;0 -1 0 0 -2|]
Subgroup: 2.3.5.7.17


Map: [&lt;1 0 0 -3 -9|, &lt;0 1 0 0 2|, [0 0 2 5 8|]
Comma list: 289/288, 3136/3125


[[Generator]]s: 2, 3, 56/25
Sval mapping: {{mapping| 2 0 0 -6 5 | 0 1 0 0 1 | 0 0 2 5 0 }}


[[EDO|Edos]]: 12, 31, 87, 118, 130, 217, [[248edo|248]], [[378edo|378]], [[626edo|626]], [[961edo|961cd]]
: sval mapping generators: ~17/12, ~3, ~56/25


Badness: 0.000609
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3471, ~28/25 = 193.6499


Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7
{{Optimal ET sequence|legend=1| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}
 
Badness: 1.095
 
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19
 
Comma list: 289/288, 361/360, 476/475
 
Mapping: {{mapping| 2 0 0 -6 5 3 | 0 1 0 0 1 1 | 0 0 2 5 0 1 }}
 
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669
 
{{Optimal ET sequence|legend=1| 12, …, 50, 68, 106d, 118, 248g, 316g }}
 
Badness: 0.569
 
== Belobog ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 441/440, 3136/3125
 
{{Mapping|legend=1| 1 0 0 -3 -9 | 0 1 0 0 2 | 0 0 2 5 8 }}
 
: mapping generators: ~2, ~3, ~56/25
 
Mapping to lattice: {{mapping| 0 -2 2 5 4 | 0 -1 0 0 -2 }}
 
Lattice basis:
: 28/25 length = 0.3829, 16/15 length = 1.1705
: Angle (28/25, 16/15) = 93.2696
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 27/22 6/11 -5/22 -3/11 5/22 }}, {{monzo| 24/11 -4/11 -2/11 2/11 2/11 }}, {{monzo| 27/11 -10/11 -5/11 5/11 5/11 }}, {{monzo| 24/11 -4/11 -13/11 2/11 13/11 }}]
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/5
 
{{Optimal ET sequence|legend=1| 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248 }}
 
[[Badness]]: 0.609 × 10<sup>-3</sup>
 
[[Projection pair]]s: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7


Scales: [[belobog31]]
Scales: [[belobog31]]


== 13-limit ==
=== 13-limit ===
Commas: 441/440, 1001/1000, 3136/3125
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 1001/1000, 3136/3125
 
Mapping: {{mapping| 1 0 0 -3 -9 15 | 0 1 0 0 2 -2 | 0 0 2 5 8 -7 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.8219, ~28/25 = 193.5816
 
{{Optimal ET sequence|legend=1| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
 
Badness: 1.11 × 10<sup>-3</sup>
 
=== Bellowblog ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 625/624


Map: [&lt;1 0 0 -3 -9 15|, &lt;0 1 0 0 2 -2|, &lt;0 0 2 5 8 -7|]
Mapping: {{mapping| 1 0 0 -3 -9 -4 | 0 1 0 0 2 -1 | 0 0 2 5 8 8 }}


EDOs: 12, 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493


Badness: 0.00111
{{Optimal ET sequence|legend=1| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}


== Bellowblog ==
Badness: 1.26 × 10<sup>-3</sup>
Commas: 196/195, 352/351, 625/624


Map: [&lt;1 0 0 -3 -9 -4|, &lt;0 1 0 0 2 -1|, &lt;0 0 2 5 8 8|]
== Siebog ==
[[Subgroup]]: 2.3.5.7.11


EDOs: 31, 56, 87, 118, 205d, 263f, 304f, 391df, 509df
[[Comma list]]: 540/539, 3136/3125


Badness: 0.00126
{{Mapping|legend=1| 1 0 0 -3 8 | 0 1 0 0 3 | 0 0 2 5 -8 }}


= Siebog =
: mapping generators: ~2, ~3, ~56/25
[[Comma]]s: 540/539, 3136/3125


11 limit minimax
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645


[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |8/5 3/5 1/5 0 -1/5&gt;,  
[[Minimax tuning]]:
|1 3/2 1/2 0 -1/2&gt;, |8/5 3/5 -4/5 0 4/5&gt;]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 8/5 3/5 1/5 0 -1/5 }}, {{monzo| 1 3/2 1/2 0 -1/2 }}, {{monzo| 8/5 3/5 -4/5 0 4/5 }}]
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/5


[[Eigenmonzo]]s: 2, 11/10, 4/3
{{Optimal ET sequence|legend=1| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}


Map: [&lt;1 0 0 -3 8|, &lt;0 1 0 0 3|, &lt;0 0 2 5 -8|]
[[Badness]]: 0.870 × 10<sup>-3</sup>


[[generator|Generators]]: 2, 3, 768/343
== Triglav ==
[[Subgroup]]: 2.3.5.7.11


[[EDO|Edos]]: 19, 31, 80, 99d, 111, [[130edo|130]], [[241edo|241]], 340ce, 371ce, 470cde, 711cde
[[Comma list]]: 3025/3024, 3136/3125


Badness: 0.000870
{{Mapping|legend=1| 1 0 2 2 1 | 0 1 2 5 2 | 0 0 -4 -10 -1 }}


= Triglav =
: mapping generators: ~2, ~3, ~18/11
Commas: 3025/3024, 3136/3125


Map: [&lt;1 0 2 2 1|, &lt;0 1 2 5 2|, &lt;0 0 -4 -10 -1|]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2875, ~18/11 = 854.3132


EDOs: 31, 80, 87, 111, 118, 198, 316, 545c, 861ce
{{Optimal ET sequence|legend=1| 24d, 31, 80, 87, 111, 118, 198, 316, 514c, 545c }}


Badness: 0.000819
[[Badness]]: 0.819 × 10<sup>-3</sup>


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