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The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]].  
{{Technical data page}}
The '''hemimean family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] 3136/3125, the [[hemimean comma]].  


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 is the difference 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


[[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]]: {{mapping| 0 0 2 5 | 0 1 0 0 }}
 
[[Mapping to lattice]]: [{{val| 0 0 2 5 }}, {{val| 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]]s:
* [[WE]]: ~2 = 1199.8194{{c}}, ~3/2 = 702.1353{{c}}, ~28/25 = 193.7425{{c}}
: [[error map]]: {{val| -0.181 -0.000 +0.810 -0.474 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.1118{{c}}, ~28/25 = 193.7167{{c}}
: error map: {{val| 0.000 +0.157 +1.120 -0.243 }}


[[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]] (Sintel): 0.706


[[Complexity spectrum]]: 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7
[[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: <code>5 3136/625, 7 68841472/9765625</code> to 2.3.25/7


== Belobog ==
== Belobog ==
 
[[Subgroup]]: 2.3.5.7.11
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: {{mapping| 0 -2 2 5 4 | 0 -1 0 0 -2 }}
 
Mapping to lattice: [{{val| 0 -2 2 5 4 }}, {{val| 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]]s:
* [[WE]]: ~2 = 1200.0098{{c}}, ~3/2 = 701.7170{{c}}, ~28/25 = 193.5520{{c}}
: [[error map]]: {{val| +0.010 -0.228 +0.810 -1.046 +0.542 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7144{{c}}, ~28/25 = 193.5518{{c}}
: error map: {{val| 0.000 -0.241 +0.790 -1.067 +0.525 }}


[[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]] (Sintel): 0.732


[[Projection pair]]s: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7
[[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 ===
 
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 tunings:
* WE: ~2 = 1199.9154{{c}}, ~3/2 = 701.7875{{c}}, ~28/25 = 193.5853{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8346{{c}}, ~28/25 = 193.5962{{c}}


Badness: 1.11 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}


=== Bellowblog  ===
Badness (Sintel): 1.03


=== 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 tunings:
* WE: ~2 = 1200.000{{c}}, ~3/2 = 702.5857{{c}}, ~28/25 = 193.2930{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.6342{{c}}, ~28/25 = 193.2932{{c}}


Badness: 1.26 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}


== Siebog  ==
Badness (Sintel): 1.18


Subgroup: 2.3.5.7.11
== Siebog ==
[[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]]s:
* [[WE]]: ~2 = 1199.5790{{c}}, ~3/2 = 701.8397{{c}}, ~28/25 = 194.0111{{c}}
: [[error map]]: {{val| -0.421 -0.536 +0.867 +0.388 +0.849 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7235{{c}}, ~28/25 = 193.9948{{c}}
: error map: {{val| 0.000 -0.232 +1.676 +1.148 +1.894 }}


[[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
 
{{Optimal ET sequence|legend=1| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}
 
[[Badness]] (Sintel): 1.05
 
== Triglav ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 3136/3125
 
{{Mapping|legend=1| 1 0 -2 -8 0 | 0 1 2 5 2 | 0 0 4 10 1 }}
: Mapping generators: ~2, ~3, ~11/9
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8764{{c}}, ~3/2 = 702.4302{{c}}, ~11/9 = 345.5856{{c}}
: [[error map]]: {{val| -0.124 +0.352 +0.889 -0.448 -1.119 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.4067{{c}}, ~11/9 = 345.6505{{c}}
: error map: {{val| 0.000 +0.452 +1.102 -0.288 -0.854 }}
 
{{Optimal ET sequence|legend=1| 24d, 31, 80, 87, 111, 118, 198, 316, 514c, 545c }}
 
[[Badness]] (Sintel): 0.984
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 1001/1000, 3025/3024
 
Mapping: {{Mapping| 1 0 -2 -8 0 5 | 0 1 2 5 2 -1 | 0 0 4 10 1 1 }}
 
Optimal tunings:
* WE: ~2 = 1199.6554{{c}}, ~3/2 = 702.8049{{c}}, ~11/9 = 345.3412{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9366{{c}}, ~11/9 = 345.4458{{c}}
 
{{Optimal ET sequence|legend=0| 24d, 31, 56, 80, 87, 111, 118, 167, 198 }}
 
Badness (Sintel): 1.16
 
== Semihemimean ==
Named by [[Xenllium]] in 2026, semihemimean tempers out [[9801/9800]] and splits the octave in halves.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3136/3125, 9801/9800
 
{{Mapping|legend=1| 2 0 0 -6 -3 | 0 1 0 0 -2 | 0 0 2 5 7 }}
: Mapping generators: ~99/70, ~3, ~56/25
 
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 599.9102{{c}}, ~3/2 = 702.1314{{c}}, ~28/25 = 193.7431{{c}}
: [[error map]]: {{val| -0.180 -0.003 +0.813 -0.470 -0.008 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.1353{{c}}, ~28/25 = 193.7122{{c}}
: error map: {{val| 0.000 +0.180 +1.111 -0.265 +0.397 }}
 
{{Optimal ET sequence|legend=1| 12, …, 38d, 50, 68, 80, 118, 130, 198, 248, 328, 446, 774c }}
 
[[Badness]] (Sintel): 1.79
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 3136/3125, 4459/4455
 
Mapping: {{Mapping| 2 0 0 -6 -3 15 | 0 1 0 0 -2 2 | 0 0 2 5 7 -6 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8791{{c}}, ~3/2 = 702.1476{{c}}, ~28/25 = 193.7877{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.1740{{c}}, ~28/25 = 193.7869{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 50, 68, 80, 118, 130, 198, 328, 576cf, 774cf, 904cef }}
 
Badness (Sintel): 1.55
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 561/560, 1001/1000, 1632/1625
 
Mapping: {{Mapping| 2 0 0 -6 -3 15 5 | 0 1 0 0 -2 2 1 | 0 0 2 5 7 -6 0 }}
 
Optimal tunings:
* WE: ~17/12 = 599.9481{{c}}, ~3/2 = 702.2594{{c}}, ~28/25 = 193.7763{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 702.2692{{c}}, ~28/25 = 193.7762{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 50, 68, 80, 118, 130, 198 }}
 
Badness (Sintel): 1.74
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[Mapping]]: [{{val| 1 0 0 -3 8 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 2 5 -8 }}]
Comma list: 289/288, 361/360, 456/455, 476/475, 561/560


Mapping generators: ~2, ~3, ~768/343
Mapping: {{Mapping| 2 0 0 -6 -3 15 5 3 | 0 1 0 0 -2 2 1 1 | 0 0 2 5 7 -6 0 1 }}


{{Val list|legend=1| 12e, 19, 31, 68e, 80, 99e, 130, 241, 340ce, 371ce, 470cde, 711cde }}
Optimal tunings:
* WE: ~17/12 = 599.9687{{c}}, ~3/2 = 702.3501{{c}}, ~19/17 = 193.7922{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 702.3549{{c}}, ~19/17 = 193.7919{{c}}


[[Badness]]: 0.870 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 12, …, 50, 68, 80, 118, 130, 198 }}


== Triglav  ==
Badness (Sintel): 1.32


Subgroup: 2.3.5.7.11
== Subgroup extensions ==
=== Hemimean orion (2.3.5.7.17) ===
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 their difference [[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.


[[Comma list]]: 3025/3024, 3136/3125
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
 
Subgroup-val mapping: {{mapping| 1 0 0 -3 -5 | 0 1 0 0 5 | 0 0 2 5 1 }}
 
Optimal tunings:
* WE: ~2 = 1199.7919{{c}}, ~3/2 = 702.2561{{c}}, ~28/25 = 193.7586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3042{{c}}, ~28/25 = 193.7365{{c}}
 
{{Optimal ET sequence|legend=0| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328, 446, 545c, 873cg }}
 
Badness (Sintel): 0.884
 
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19
 
Comma list: 476/475, 1216/1215, 1445/1444
 
Subgroup-val mapping: {{mapping| 1 0 0 -3 -5 -6 | 0 1 0 0 5 5 | 0 0 2 5 1 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.8239{{c}}, ~3/2 = 702.1623{{c}}, ~19/17 = 193.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2129{{c}}, ~19/17 = 193.7161{{c}}
 
{{Optimal ET sequence|legend=0| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
 
Badness (Sintel): 0.578
 
=== Semiorion (2.3.5.7.17) ===
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
 
Subgroup-val mapping: {{mapping| 2 0 0 -6 5 | 0 1 0 0 1 | 0 0 2 5 0 }}
: mapping generators: ~17/12, ~3, ~56/25
 
Optimal tunings:
* WE: ~17/12 = 600.0551{{c}}, ~3/2 = 702.1998{{c}}, ~28/25 = 193.5929{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 702.2183{{c}}, ~28/25 = 193.6044{{c}}
 
{{Optimal ET sequence|legend=0| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}
 
Badness (Sintel): 1.69
 
==== 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 }}


[[Mapping]]: [{{val| 1 0 2 2 1 }}, {{val| 0 1 2 5 2 }}, {{val| 0 0 -4 -10 -1 }}]
Optimal tunings:  
* WE: ~17/12 = 600.0873{{c}}, ~3/2 = 702.2450{{c}}, ~19/17 = 193.5751{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 702.2791{{c}}, ~19/17 = 193.5923{{c}}


{{Val list|legend=1| 31, 80, 87, 111, 118, 198, 316, 545c, 861ce }}
{{Optimal ET sequence|legend=0| 12, 30dh, 38d, 50, 68, 106d, 118, 248g, 316g }}


[[Badness]]: 0.819 × 10<sup>-3</sup>
Badness (Sintel): 0.722


[[Category:Regular temperament theory]]
[[Category:Hemimean family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Hemimean family| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Hemimean]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Hemimean_family"