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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).  
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>.


Subgroup: 2.3.5.7
== Hemimean ==
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125 (hemimean)
[[Comma list]]: 3136/3125 (hemimean)


[[Mapping]]: [{{val| 1 0 0 -3 }}, {{val| 0 1 0 0 }}, {{val| 0 0 2 5 }}]
{{Mapping|legend=1| 1 0 0 -3 | 0 1 0 0 | 0 0 2 5 }}


Mapping generators: ~2, ~3, ~56/25
: mapping generators: ~2, ~3, ~56/25


[[Mapping to lattice]]: [{{val| 0 0 2 5 }}, {{val| 0 1 0 0 }}]
[[Mapping to lattice]]: {{mapping| 0 0 2 5 | 0 1 0 0 }}


Lattice basis:  
Lattice basis:  
: 28/25 length = 0.5055, 3/2 length = 1.5849
: 28/25 length = 0.5055, 3/2 length = 1.5849
: Angle (28/25, 3/2) = 90 degrees
: Angle (28/25, 3/2) = 90 degrees
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9550, ~28/25 = 193.6499


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
* [[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 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6, 4/3
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7


{{Val list|legend=1| 12, 19, 31, 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]]: 0.160 × 10<sup>-3</sup>
[[Badness]]: 0.160 × 10<sup>-3</sup>
Line 32: Line 36:
[[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 ===
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
 
Sval mapping: {{mapping| 1 0 0 -3 -5 -6 | 0 1 0 0 5 5 | 0 0 2 5 1 2 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647
 
{{Optimal ET sequence|legend=1| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
 
Badness: 0.456
 
=== 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]])}.
 
Subgroup: 2.3.5.7.17
 
Comma list: 289/288, 3136/3125
 
Sval mapping: {{mapping| 2 0 0 -6 5 | 0 1 0 0 1 | 0 0 2 5 0 }}
 
: sval mapping generators: ~17/12, ~3, ~56/25
 
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3471, ~28/25 = 193.6499
 
{{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 }}


Subgroup: 2.3.5.7.11
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
[[Comma list]]: 441/440, 3136/3125


[[Mapping]]: [{{val| 1 0 0 -3 -9 }}, {{val| 0 1 0 0 2 }}, {{val| 0 0 2 5 8 }}]
{{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 generators: ~2, ~3, ~56/25


Mapping to lattice: [{{val| 0 -2 2 5 4 }}, {{val| 0 -1 0 0 -2 }}]
Mapping to lattice: {{mapping| 0 -2 2 5 4 | 0 -1 0 0 -2 }}


Lattice basis:  
Lattice basis:  
: 28/25 length = 0.3829, 16/15 length = 1.1705
: 28/25 length = 0.3829, 16/15 length = 1.1705
: Angle (28/25, 16/15) = 93.2696
: Angle (28/25, 16/15) = 93.2696
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[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 }}]
: [{{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]]s (unchanged intervals): 2, 11/10, 9/7
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/5


{{Val list|legend=1| 12, 19e, 31, 87, 99e, 118, 130, 217, 248, 378, 626, 961cd }}
{{Optimal ET sequence|legend=1| 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248 }}


[[Badness]]: 0.609 × 10<sup>-3</sup>
[[Badness]]: 0.609 × 10<sup>-3</sup>
Line 61: Line 126:
Scales: [[belobog31]]
Scales: [[belobog31]]


=== 13-limit ===
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 441/440, 1001/1000, 3136/3125
Comma list: 441/440, 1001/1000, 3136/3125


Mapping: [{{val| 1 0 0 -3 -9 15 }}, {{val| 0 1 0 0 2 -2 }}, {{val| 0 0 2 5 8 -7 }}]
Mapping: {{mapping| 1 0 0 -3 -9 15 | 0 1 0 0 2 -2 | 0 0 2 5 8 -7 }}


{{Val list|legend=1| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
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>
Badness: 1.11 × 10<sup>-3</sup>


=== Bellowblog ===
=== Bellowblog ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 625/624
Comma list: 196/195, 352/351, 625/624


Mapping: [{{val| 0 0 -3 -9 -4 }}, {{val| 0 1 0 0 2 -1 }}, {{val| 0 0 2 5 8 8 }}]
Mapping: {{mapping| 1 0 0 -3 -9 -4 | 0 1 0 0 2 -1 | 0 0 2 5 8 8 }}


{{Val list|legend=1| 12f, 19e, 31, 56, 68e, 87, 118, 205d, 263f, 304f, 391df, 509df }}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493
 
{{Optimal ET sequence|legend=1| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}


Badness: 1.26 × 10<sup>-3</sup>
Badness: 1.26 × 10<sup>-3</sup>


== Siebog ==
== Siebog ==
[[Subgroup]]: 2.3.5.7.11


Subgroup: 2.3.5.7.11
[[Comma list]]: 540/539, 3136/3125


[[Comma list]]: 540/539, 3136/3125
{{Mapping|legend=1| 1 0 0 -3 8 | 0 1 0 0 3 | 0 0 2 5 -8 }}
 
: mapping generators: ~2, ~3, ~56/25
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[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 }}]
: [{{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]]s (unchanged intervals): 2, 11/10, 4/3
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/5


[[Mapping]]: [{{val| 1 0 0 -3 8 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 2 5 -8 }}]
{{Optimal ET sequence|legend=1| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}


Mapping generators: ~2, ~3, ~768/343
[[Badness]]: 0.870 × 10<sup>-3</sup>


{{Val list|legend=1| 12e, 19, 31, 68e, 80, 99e, 130, 241, 340ce, 371ce, 470cde, 711cde }}
== Triglav ==
[[Subgroup]]: 2.3.5.7.11


[[Badness]]: 0.870 × 10<sup>-3</sup>
[[Comma list]]: 3025/3024, 3136/3125


== Triglav  ==
{{Mapping|legend=1| 1 0 2 2 1 | 0 1 2 5 2 | 0 0 -4 -10 -1 }}


Subgroup: 2.3.5.7.11
: mapping generators: ~2, ~3, ~18/11
 
[[Comma list]]: 3025/3024, 3136/3125


[[Mapping]]: [{{val| 1 0 2 2 1 }}, {{val| 0 1 2 5 2 }}, {{val| 0 0 -4 -10 -1 }}]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2875, ~18/11 = 854.3132


{{Val list|legend=1| 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]]: 0.819 × 10<sup>-3</sup>
[[Badness]]: 0.819 × 10<sup>-3</sup>


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