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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.
 
Other equivalences characteristic to hemimean are [[128/125]]~[[50/49]] and [[49/45]]~([[25/24]])<sup>2</sup>.  


== Hemimean ==
== Hemimean ==
[[Subgroup]]: 2.3.5.7
[[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 generators: ~2, ~3, ~56/25
{{Mapping|legend=1| 1 0 0 -3 | 0 1 0 0 | 0 0 2 5 }}
: 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]]s:
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~28/25 = 193.650{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.112{{c}}, ~28/25 = 193.717{{c}}


[[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 basis]]: 2.3.7
: [[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


=== Hemimean orion ===
== Belobog ==
As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[5/4]], 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 [[square superparticular|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. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.
[[Subgroup]]: 2.3.5.7.11


Subgroup: 2.3.5.7.17
[[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


Comma list: 1701/1700, 3136/3125
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.720{{c}}, ~28/25 = 193.554{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.714{{c}}, ~28/25 = 193.552{{c}}


Sval mapping: [{{val| 1 0 0 -3 -5 }}, {{val| 0 1 0 0 5 }}, {{val| 0 0 2 5 1 }}]
[[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 tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548
{{Optimal ET sequence|legend=1| 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248 }}


Optimal GPV sequence: {{Val list| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg }}
[[Badness]] (Sintel): 0.732


Badness: 0.573
[[Projection pair]]s: 5 3136/625, 7 68841472/9765625, 11 1700108992512/152587890625 to 2.3.25/7


==== 2.3.5.7.17.19 subgroup ====
Scales: [[belobog31]]
Subgroup: 2.3.5.7.17.19


Comma list: 476/475, 1216/1215, 1445/1444
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Sval mapping: [{{val| 1 0 0 -3 -5 -6 }}, {{val| 0 1 0 0 5 5 }}, {{val| 0 0 2 5 1 2 }}]
Comma list: 441/440, 1001/1000, 3136/3125


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647
Mapping: {{mapping| 1 0 0 -3 -9 15 | 0 1 0 0 2 -2 | 0 0 2 5 8 -7 }}


Optimal GPV sequence: {{Val list| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.822{{c}}, ~28/25 = 193.582{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.835{{c}}, ~28/25 = 193.596{{c}}


Badness: 0.456
{{Optimal ET sequence|legend=0| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}


=== Semiorion ===
Badness (Sintel): 1.034
Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two.  


Subgroup: 2.3.5.7.17
=== Bellowblog ===
Subgroup: 2.3.5.7.11.13


Comma list: 289/288, 3136/3125
Comma list: 196/195, 352/351, 625/624


Sval mapping: [{{val| 2 0 0 -6 5 }}, {{val| 0 1 0 0 1 }}, {{val| 0 0 2 5 0 }}]
Mapping: {{mapping| 1 0 0 -3 -9 -4 | 0 1 0 0 2 -1 | 0 0 2 5 8 8 }}


Sval mapping generators: ~17/12, ~3, ~56/25
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.567{{c}}, ~28/25 = 193.249{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.634{{c}}, ~28/25 = 193.293{{c}}


Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3471, ~28/25 = 193.6499
{{Optimal ET sequence|legend=0| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}


Optimal GPV sequence: {{Val list| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}
Badness (Sintel): 1.183


Badness: 1.095
== Siebog ==
[[Subgroup]]: 2.3.5.7.11


==== 2.3.5.7.17.19 subgroup ====
[[Comma list]]: 540/539, 3136/3125
Subgroup: 2.3.5.7.17.19


Comma list: 289/288, 361/360, 476/475
{{Mapping|legend=1| 1 0 0 -3 8 | 0 1 0 0 3 | 0 0 2 5 -8 }}
: Mapping generators: ~2, ~3, ~56/25


Mapping: [{{val| 2 0 0 -6 5 3 }}, {{val| 0 1 0 0 1 1 }}, {{val| 0 0 2 5 0 1 }}]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.164{{c}}, ~28/25 = 193.865{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.723{{c}}, ~28/25 = 193.995{{c}}


Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669
[[Minimax tuning]]:
* [[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


Optimal GPV sequence: {{Val list| 12, …, 50, 68, 106d, 118, 248g, 316g }}
{{Optimal ET sequence|legend=1| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}


Badness: 0.569
[[Badness]] (Sintel): 1.045


== Belobog ==
== Triglav ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 441/440, 3136/3125
[[Comma list]]: 3025/3024, 3136/3125
 
{{Mapping|legend=1| 1 0 2 2 1 | 0 1 2 5 2 | 0 0 -4 -10 -1 }}
: Mapping generators: ~2, ~3, ~18/11
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.288{{c}}, ~18/11 = 854.313{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.407{{c}}, ~18/11 = 854.350{{c}}
 
{{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 2 1 6 | 0 1 2 5 2 -1 | 0 0 -4 -10 -1 -1 }}


[[Mapping]]: [{{val| 1 0 0 -3 -9 }}, {{val| 0 1 0 0 2 }}, {{val| 0 0 2 5 8 }}]
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.707{{c}}, ~18/11 = 854.537{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.937{{c}}, ~18/11 = 854.554{{c}}


Mapping generators: ~2, ~3, ~56/25
{{Optimal ET sequence|legend=0| 24d, 31, 80, 87, 111, 118, 198 }}


Mapping to lattice: [{{val| 0 -2 2 5 4 }}, {{val| 0 -1 0 0 -2 }}]
Badness (Sintel): 1.159


Lattice basis:
== Semihemimean ==
: 28/25 length = 0.3829, 16/15 length = 1.1705
[[Subgroup]]: 2.3.5.7.11
: Angle (28/25, 16/15) = 93.2696


[[Minimax tuning]]:  
[[Comma list]]: 3136/3125, 9801/9800
* [[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]]: 2.9/7.11/5


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


[[Badness]]: 0.609 × 10<sup>-3</sup>
[[Optimal tuning]]s:  
* [[CTE]]: ~99/70 = 600.000{{c}}, ~3/2 = 702.002{{c}}, ~28/25 = 193.633{{c}}
* [[CWE]]: ~99/70 = 600.000{{c}}, ~3/2 = 702.135{{c}}, ~28/25 = 193.712{{c}}


[[Projection pair]]s: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7
{{Optimal ET sequence|legend=1| 12, 50, 68, 80, 118, 130, 198 }}


Scales: [[belobog31]]
[[Badness]] (Sintel): 1.787


=== 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: 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:
* CTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.838{{c}}, ~28/25 = 193.671{{c}}
* CWE: ~99/70 = 600.000{{c}}, ~3/2 = 702.174{{c}}, ~28/25 = 193.787{{c}}
 
{{Optimal ET sequence|legend=0| 12, 50, 68, 80, 118, 130, 198 }}
 
[[Badness]] (Sintel): 1.550
 
=== 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:
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.108{{c}}, ~28/25 = 193.723{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.269{{c}}, ~28/25 = 193.776{{c}}
 
{{Optimal ET sequence|legend=0| 12, 50, 68, 80, 118, 130, 198 }}
 
[[Badness]] (Sintel): 1.743
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 289/288, 361/360, 456/455, 476/475, 561/560
 
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 }}
 
Optimal tunings:
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.252{{c}}, ~19/17 = 193.758{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.355{{c}}, ~19/17 = 193.792{{c}}
 
{{Optimal ET sequence|legend=0| 12, 50, 68, 80, 118, 130, 198 }}
 
[[Badness]] (Sintel): 1.318
 
== 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 ([[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
 
Subgroup-val mapping: {{mapping| 1 0 0 -3 -5 | 0 1 0 0 5 | 0 0 2 5 1 }}


Mapping: [{{val| 1 0 0 -3 -9 15 }}, {{val| 0 1 0 0 2 -2 }}, {{val| 0 0 2 5 8 -7 }}]
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.196{{c}}, ~28/25 = 193.655{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.304{{c}}, ~28/25 = 193.737{{c}}


Optimal GPV sequence: {{Val list| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
{{Optimal ET sequence|legend=1| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg }}


Badness: 1.11 × 10<sup>-3</sup>
Badness (Sintel): 0.884


=== Bellowblog ===
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.17.19
 
Comma list: 476/475, 1216/1215, 1445/1444


Comma list: 196/195, 352/351, 625/624
Subgroup-val mapping: {{mapping| 1 0 0 -3 -5 -6 | 0 1 0 0 5 5 | 0 0 2 5 1 2 }}


Mapping: [{{val| 0 0 -3 -9 -4 }}, {{val| 0 1 0 0 2 -1 }}, {{val| 0 0 2 5 8 8 }}]
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.132{{c}}, ~19/17 = 193.647{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.213{{c}}, ~19/17 = 193.716{{c}}


Optimal GPV sequence: {{Val list| 12f, 19e, 31, 56, 68e, 87, 118, 205d, 263f, 304f, 391df, 509df }}
{{Optimal ET sequence|legend=0| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}


Badness: 1.26 × 10<sup>-3</sup>
Badness (Sintel): 0.578


== Siebog ==
=== Semiorion (2.3.5.7.17) ===
[[Subgroup]]: 2.3.5.7.11
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]])}.


[[Comma list]]: 540/539, 3136/3125
Subgroup: 2.3.5.7.17


[[Mapping]]: [{{val| 1 0 0 -3 8 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 2 5 -8 }}]
Comma list: 289/288, 3136/3125


Mapping generators: ~2, ~3, ~768/343
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


[[Minimax tuning]]:  
Optimal tunings:  
* [[11-odd-limit]]
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.347{{c}}, ~28/25 = 193.650{{c}}
: [{{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 }}]
* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.218{{c}}, ~28/25 = 193.604{{c}}
: [[Eigenmonzo basis]]: 2.3.11/10


{{Val list|legend=1| 12e, 19, 31, 68e, 80, 99e, 130, 241, 340ce, 371ce, 470cde, 711cde }}
{{Optimal ET sequence|legend=1| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}


[[Badness]]: 0.870 × 10<sup>-3</sup>
Badness (Sintel): 1.690


== Triglav ==
==== 2.3.5.7.17.19 subgroup ====
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.17.19


[[Comma list]]: 3025/3024, 3136/3125
Comma list: 289/288, 361/360, 476/475


[[Mapping]]: [{{val| 1 0 2 2 1 }}, {{val| 0 1 2 5 2 }}, {{val| 0 0 -4 -10 -1 }}]
Mapping: {{mapping| 2 0 0 -6 5 3 | 0 1 0 0 1 1 | 0 0 2 5 0 1 }}


Mapping generators: ~2, ~3, ~18/11
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.509{{c}}, ~19/17 = 193.669{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.279{{c}}, ~19/17 = 193.592{{c}}


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


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


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Hemimean family| ]] <!-- main article -->
[[Category:Hemimean family| ]] <!-- main article -->
[[Category:Hemimean]]
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 12:25, 30 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The hemimean family of rank-3 temperaments tempers out 3136/3125, the hemimean comma.

The hemimean comma is the difference between the septimal semicomma (126/125) and the 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 syntonic comma (81/80) into two equal parts, each representing 126/125~225/224.

Other equivalences characteristic to hemimean are 128/125~50/49 and 49/45~(25/24)2.

Hemimean

Subgroup: 2.3.5.7

Comma list: 3136/3125

Mapping: [⟨1 0 0 -3], ⟨0 1 0 0], ⟨0 0 2 5]]

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

Mapping to lattice: [⟨0 0 2 5], ⟨0 1 0 0]]

Lattice basis:

28/25 length = 0.5055, 3/2 length = 1.5849
Angle (28/25, 3/2) = 90 degrees

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.955 ¢, ~28/25 = 193.650 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.112 ¢, ~28/25 = 193.717 ¢

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
Unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 675c

Badness (Sintel): 0.706

Complexity spectrum: 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7

Projection pairs: 5 3136/625, 7 68841472/9765625 to 2.3.25/7

Belobog

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125

Mapping: [⟨1 0 0 -3 -9], ⟨0 1 0 0 2], ⟨0 0 2 5 8]]

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

Mapping to lattice: [⟨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 tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.720 ¢, ~28/25 = 193.554 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.714 ¢, ~28/25 = 193.552 ¢

Minimax tuning:

[[1 0 0 0 0⟩, [27/22 6/11 -5/22 -3/11 5/22⟩, [24/11 -4/11 -2/11 2/11 2/11⟩, [27/11 -10/11 -5/11 5/11 5/11⟩, [24/11 -4/11 -13/11 2/11 13/11⟩]
Unchanged-interval (eigenmonzo) basis: 2.9/7.11/5

Optimal ET sequence: 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248

Badness (Sintel): 0.732

Projection pairs: 5 3136/625, 7 68841472/9765625, 11 1700108992512/152587890625 to 2.3.25/7

Scales: belobog31

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -3 -9 15], ⟨0 1 0 0 2 -2], ⟨0 0 2 5 8 -7]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.822 ¢, ~28/25 = 193.582 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.835 ¢, ~28/25 = 193.596 ¢

Optimal ET sequence: 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e

Badness (Sintel): 1.034

Bellowblog

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -3 -9 -4], ⟨0 1 0 0 2 -1], ⟨0 0 2 5 8 8]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.567 ¢, ~28/25 = 193.249 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.634 ¢, ~28/25 = 193.293 ¢

Optimal ET sequence: 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d

Badness (Sintel): 1.183

Siebog

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125

Mapping: [⟨1 0 0 -3 8], ⟨0 1 0 0 3], ⟨0 0 2 5 -8]]

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.164 ¢, ~28/25 = 193.865 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.723 ¢, ~28/25 = 193.995 ¢

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [8/5 3/5 1/5 0 -1/5⟩, [1 3/2 1/2 0 -1/2⟩, [8/5 3/5 -4/5 0 4/5⟩]
Unchanged-interval (eigenmonzo) basis: 2.3.11/5

Optimal ET sequence: 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee

Badness (Sintel): 1.045

Triglav

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125

Mapping: [⟨1 0 2 2 1], ⟨0 1 2 5 2], ⟨0 0 -4 -10 -1]]

Mapping generators: ~2, ~3, ~18/11

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.288 ¢, ~18/11 = 854.313 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.407 ¢, ~18/11 = 854.350 ¢

Optimal ET sequence: 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: [⟨1 0 2 2 1 6], ⟨0 1 2 5 2 -1], ⟨0 0 -4 -10 -1 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.707 ¢, ~18/11 = 854.537 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.937 ¢, ~18/11 = 854.554 ¢

Optimal ET sequence: 24d, 31, 80, 87, 111, 118, 198

Badness (Sintel): 1.159

Semihemimean

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 9801/9800

Mapping: [⟨2 0 0 -6 -3], ⟨0 1 0 0 -2], ⟨0 0 2 5 7]]

Mapping generators: ~99/70, ~3, ~56/25

Optimal tunings:

  • CTE: ~99/70 = 600.000 ¢, ~3/2 = 702.002 ¢, ~28/25 = 193.633 ¢
  • CWE: ~99/70 = 600.000 ¢, ~3/2 = 702.135 ¢, ~28/25 = 193.712 ¢

Optimal ET sequence: 12, 50, 68, 80, 118, 130, 198

Badness (Sintel): 1.787

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 3136/3125, 4459/4455

Mapping: [⟨2 0 0 -6 -3 15], ⟨0 1 0 0 -2 2], ⟨0 0 2 5 7 -6]]

Optimal tunings:

  • CTE: ~99/70 = 600.000 ¢, ~3/2 = 701.838 ¢, ~28/25 = 193.671 ¢
  • CWE: ~99/70 = 600.000 ¢, ~3/2 = 702.174 ¢, ~28/25 = 193.787 ¢

Optimal ET sequence: 12, 50, 68, 80, 118, 130, 198

Badness (Sintel): 1.550

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 561/560, 1001/1000, 1632/1625

Mapping: [⟨2 0 0 -6 -3 15 5], ⟨0 1 0 0 -2 2 1], ⟨0 0 2 5 7 -6 0]]

Optimal tunings:

  • CTE: ~17/12 = 600.000 ¢, ~3/2 = 702.108 ¢, ~28/25 = 193.723 ¢
  • CWE: ~17/12 = 600.000 ¢, ~3/2 = 702.269 ¢, ~28/25 = 193.776 ¢

Optimal ET sequence: 12, 50, 68, 80, 118, 130, 198

Badness (Sintel): 1.743

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 361/360, 456/455, 476/475, 561/560

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]]

Optimal tunings:

  • CTE: ~17/12 = 600.000 ¢, ~3/2 = 702.252 ¢, ~19/17 = 193.758 ¢
  • CWE: ~17/12 = 600.000 ¢, ~3/2 = 702.355 ¢, ~19/17 = 193.792 ¢

Optimal ET sequence: 12, 50, 68, 80, 118, 130, 198

Badness (Sintel): 1.318

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 (28/25)/(19/17) = 476/475, or equivalently stated, the semiparticular (5/4)/(19/17)2 = 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 {S16/S18, S17/S19, S18/S20(, (S16*S17)/(S19*S20) = S16/S18 * S17/S19 * S18/S20)}.

Subgroup: 2.3.5.7.17

Comma list: 1701/1700, 3136/3125

Subgroup-val mapping: [⟨1 0 0 -3 -5], ⟨0 1 0 0 5], ⟨0 0 2 5 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.196 ¢, ~28/25 = 193.655 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.304 ¢, ~28/25 = 193.737 ¢

Optimal ET sequence: 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 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: [⟨1 0 0 -3 -5 -6], ⟨0 1 0 0 5 5], ⟨0 0 2 5 1 2]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.132 ¢, ~19/17 = 193.647 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.213 ¢, ~19/17 = 193.716 ¢

Optimal ET sequence: 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 {S17, S19, S16/S18(, S18/S20, 476/475 = S16/S20 * S17/S19)}.

Subgroup: 2.3.5.7.17

Comma list: 289/288, 3136/3125

Subgroup-val 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:

  • CTE: ~17/12 = 600.000 ¢, ~3/2 = 702.347 ¢, ~28/25 = 193.650 ¢
  • CWE: ~17/12 = 600.000 ¢, ~3/2 = 702.218 ¢, ~28/25 = 193.604 ¢

Optimal ET sequence: 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g

Badness (Sintel): 1.690

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 289/288, 361/360, 476/475

Mapping: [⟨2 0 0 -6 5 3], ⟨0 1 0 0 1 1], ⟨0 0 2 5 0 1]]

Optimal tunings:

  • CTE: ~17/12 = 600.000 ¢, ~3/2 = 702.509 ¢, ~19/17 = 193.669 ¢
  • CWE: ~17/12 = 600.000 ¢, ~3/2 = 702.279 ¢, ~19/17 = 193.592 ¢

Optimal ET sequence: 12, …, 50, 68, 106d, 118, 248g, 316g

Badness (Sintel): 0.722

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