Hemimean family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''hemimean family''' of [[temperament]]s are [[rank-3 temperament]]s which [[temper out]] [[3136/3125]].  
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 [[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.  
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>.  
Other equivalences characteristic to hemimean are [[128/125]]~[[50/49]] and [[49/45]]~([[25/24]])<sup>2</sup>.  
Line 9: Line 9:
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 0 0 -3 | 0 1 0 0 | 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]]: {{mapping| 0 0 2 5 | 0 1 0 0 }}
Line 21: Line 20:
: 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
[[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-odd-limit|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|Eigenmonzo (unchanged-interval) basis]]: 2.3.7
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7


{{Optimal ET sequence|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 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 ==
[[Subgroup]]: 2.3.5.7.11


=== Hemimean orion ===
[[Comma list]]: 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]])}.
{{Mapping|legend=1| 1 0 0 -3 -9 | 0 1 0 0 2 | 0 0 2 5 8 }}
: Mapping generators: ~2, ~3, ~56/25


Subgroup: 2.3.5.7.17
Mapping to lattice: {{mapping| 0 -2 2 5 4 | 0 -1 0 0 -2 }}


Comma list: 1701/1700, 3136/3125
Lattice basis:  
: 28/25 length = 0.3829, 16/15 length = 1.1705
: Angle (28/25, 16/15) = 93.2696


Sval mapping: {{mapping| 1 0 0 -3 -5 | 0 1 0 0 5 | 0 0 2 5 1 }}
[[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 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548
[[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, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg }}
{{Optimal ET sequence|legend=1| 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248 }}


Badness: 0.573
[[Badness]] (Sintel): 0.732


==== 2.3.5.7.17.19 subgroup ====
[[Projection pair]]s: 5 3136/625, 7 68841472/9765625, 11 1700108992512/152587890625 to 2.3.25/7
Subgroup: 2.3.5.7.17.19


Comma list: 476/475, 1216/1215, 1445/1444
Scales: [[belobog31]]


Sval mapping: {{mapping| 1 0 0 -3 -5 -6 | 0 1 0 0 5 5 | 0 0 2 5 1 2 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647
Comma list: 441/440, 1001/1000, 3136/3125


{{Optimal ET sequence|legend=1| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
Mapping: {{mapping| 1 0 0 -3 -9 15 | 0 1 0 0 2 -2 | 0 0 2 5 8 -7 }}


Badness: 0.456
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}}


=== Semiorion ===
{{Optimal ET sequence|legend=0| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
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
Badness (Sintel): 1.03


Comma list: 289/288, 3136/3125
=== Bellowblog ===
Subgroup: 2.3.5.7.11.13


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


: sval mapping generators: ~17/12, ~3, ~56/25
Mapping: {{mapping| 1 0 0 -3 -9 -4 | 0 1 0 0 2 -1 | 0 0 2 5 8 8 }}


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


{{Optimal ET sequence|legend=1| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}
{{Optimal ET sequence|legend=0| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}


Badness: 1.095
Badness (Sintel): 1.18


==== 2.3.5.7.17.19 subgroup ====
== Siebog ==
Subgroup: 2.3.5.7.17.19
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 3136/3125


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: {{mapping| 2 0 0 -6 5 3 | 0 1 0 0 1 1 | 0 0 2 5 0 1 }}
[[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 }}


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 ET sequence|legend=1| 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): 1.05


== 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 -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 }}


{{Mapping|legend=1| 1 0 0 -3 -9 | 0 1 0 0 2 | 0 0 2 5 8 }}
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}}


: mapping generators: ~2, ~3, ~56/25
{{Optimal ET sequence|legend=0| 24d, 31, 56, 80, 87, 111, 118, 167, 198 }}


Mapping to lattice: {{mapping| 0 -2 2 5 4 | 0 -1 0 0 -2 }}
Badness (Sintel): 1.16


Lattice basis:
== Semihemimean ==
: 28/25 length = 0.3829, 16/15 length = 1.1705
Named by [[Xenllium]] in 2026, semihemimean tempers out [[9801/9800]] and splits the octave in halves.  
: Angle (28/25, 16/15) = 93.2696


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545
[[Subgroup]]: 2.3.5.7.11


[[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|Eigenmonzo (unchanged-interval) basis]]: 2.9/7.11/5


{{Optimal ET sequence|legend=1| 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248 }}
{{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:  
* [[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 }}


[[Projection pair]]s: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7
{{Optimal ET sequence|legend=1| 12, …, 38d, 50, 68, 80, 118, 130, 198, 248, 328, 446, 774c }}


Scales: [[belobog31]]
[[Badness]] (Sintel): 1.79


=== 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:
* 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
 
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:
* 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}}
 
{{Optimal ET sequence|legend=0| 12, …, 50, 68, 80, 118, 130, 198 }}
 
Badness (Sintel): 1.32
 
== 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.
 
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)]])}.


Mapping: {{mapping| 1 0 0 -3 -9 15 | 0 1 0 0 2 -2 | 0 0 2 5 8 -7 }}
Subgroup: 2.3.5.7.17


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.8219, ~28/25 = 193.5816
Comma list: 1701/1700, 3136/3125


{{Optimal ET sequence|legend=1| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
Subgroup-val mapping: {{mapping| 1 0 0 -3 -5 | 0 1 0 0 5 | 0 0 2 5 1 }}


Badness: 1.11 × 10<sup>-3</sup>
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}}


=== Bellowblog ===
{{Optimal ET sequence|legend=0| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328, 446, 545c, 873cg }}
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 625/624
Badness (Sintel): 0.884


Mapping: {{mapping| 1 0 0 -3 -9 -4 | 0 1 0 0 2 -1 | 0 0 2 5 8 8 }}
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493
Comma list: 476/475, 1216/1215, 1445/1444


{{Optimal ET sequence|legend=1| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}
Subgroup-val mapping: {{mapping| 1 0 0 -3 -5 -6 | 0 1 0 0 5 5 | 0 0 2 5 1 2 }}


Badness: 1.26 × 10<sup>-3</sup>
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}}


== Siebog ==
{{Optimal ET sequence|legend=0| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 3136/3125
Badness (Sintel): 0.578


{{Mapping|legend=1| 1 0 0 -3 8 | 0 1 0 0 3 | 0 0 2 5 -8 }}
=== 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]])}.


: mapping generators: ~2, ~3, ~56/25
Subgroup: 2.3.5.7.17


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645
Comma list: 289/288, 3136/3125


[[Minimax tuning]]:
Subgroup-val mapping: {{mapping| 2 0 0 -6 5 | 0 1 0 0 1 | 0 0 2 5 0 }}
* [[11-odd-limit]]
: mapping generators: ~17/12, ~3, ~56/25
: [{{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|Eigenmonzo (unchanged-interval) 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 }}
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}}


[[Badness]]: 0.870 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}


== Triglav ==
Badness (Sintel): 1.69
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 3136/3125
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19


{{Mapping|legend=1| 1 0 2 2 1 | 0 1 2 5 2 | 0 0 -4 -10 -1 }}
Comma list: 289/288, 361/360, 476/475


: mapping generators: ~2, ~3, ~18/11
Mapping: {{mapping| 2 0 0 -6 5 3 | 0 1 0 0 1 1 | 0 0 2 5 0 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2875, ~18/11 = 854.3132
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}}


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

Latest revision as of 12:00, 14 August 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:

  • WE: ~2 = 1199.8194 ¢, ~3/2 = 702.1353 ¢, ~28/25 = 193.7425 ¢
error map: ⟨-0.181 -0.000 +0.810 -0.474]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.1118 ¢, ~28/25 = 193.7167 ¢
error map: ⟨0.000 +0.157 +1.120 -0.243]

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:

  • WE: ~2 = 1200.0098 ¢, ~3/2 = 701.7170 ¢, ~28/25 = 193.5520 ¢
error map: ⟨+0.010 -0.228 +0.810 -1.046 +0.542]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7144 ¢, ~28/25 = 193.5518 ¢
error map: ⟨0.000 -0.241 +0.790 -1.067 +0.525]

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:

  • WE: ~2 = 1199.9154 ¢, ~3/2 = 701.7875 ¢, ~28/25 = 193.5853 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8346 ¢, ~28/25 = 193.5962 ¢

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

Badness (Sintel): 1.03

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:

  • WE: ~2 = 1200.000 ¢, ~3/2 = 702.5857 ¢, ~28/25 = 193.2930 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.6342 ¢, ~28/25 = 193.2932 ¢

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

Badness (Sintel): 1.18

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:

  • WE: ~2 = 1199.5790 ¢, ~3/2 = 701.8397 ¢, ~28/25 = 194.0111 ¢
error map: ⟨-0.421 -0.536 +0.867 +0.388 +0.849]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7235 ¢, ~28/25 = 193.9948 ¢
error map: ⟨0.000 -0.232 +1.676 +1.148 +1.894]

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): 1.05

Triglav

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125

Mapping: [⟨1 0 -2 -8 0], ⟨0 1 2 5 2], ⟨0 0 4 10 1]]

Mapping generators: ~2, ~3, ~11/9

Optimal tunings:

  • WE: ~2 = 1199.8764 ¢, ~3/2 = 702.4302 ¢, ~11/9 = 345.5856 ¢
error map: ⟨-0.124 +0.352 +0.889 -0.448 -1.119]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.4067 ¢, ~11/9 = 345.6505 ¢
error map: ⟨0.000 +0.452 +1.102 -0.288 -0.854]

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 -8 0 5], ⟨0 1 2 5 2 -1], ⟨0 0 4 10 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.6554 ¢, ~3/2 = 702.8049 ¢, ~11/9 = 345.3412 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9366 ¢, ~11/9 = 345.4458 ¢

Optimal ET sequence: 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: [⟨2 0 0 -6 -3], ⟨0 1 0 0 -2], ⟨0 0 2 5 7]]

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

Optimal tunings:

  • WE: ~99/70 = 599.9102 ¢, ~3/2 = 702.1314 ¢, ~28/25 = 193.7431 ¢
error map: ⟨-0.180 -0.003 +0.813 -0.470 -0.008]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.1353 ¢, ~28/25 = 193.7122 ¢
error map: ⟨0.000 +0.180 +1.111 -0.265 +0.397]

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 702.1476 ¢, ~28/25 = 193.7877 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.1740 ¢, ~28/25 = 193.7869 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 702.2594 ¢, ~28/25 = 193.7763 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.2692 ¢, ~28/25 = 193.7762 ¢

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

Badness (Sintel): 1.74

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:

  • WE: ~17/12 = 599.9687 ¢, ~3/2 = 702.3501 ¢, ~19/17 = 193.7922 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.3549 ¢, ~19/17 = 193.7919 ¢

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

Badness (Sintel): 1.32

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)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:

  • WE: ~2 = 1199.7919 ¢, ~3/2 = 702.2561 ¢, ~28/25 = 193.7586 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3042 ¢, ~28/25 = 193.7365 ¢

Optimal ET sequence: 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: [⟨1 0 0 -3 -5 -6], ⟨0 1 0 0 5 5], ⟨0 0 2 5 1 2]]

Optimal tunings:

  • WE: ~2 = 1199.8239 ¢, ~3/2 = 702.1623 ¢, ~19/17 = 193.7325 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2129 ¢, ~19/17 = 193.7161 ¢

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:

  • WE: ~17/12 = 600.0551 ¢, ~3/2 = 702.1998 ¢, ~28/25 = 193.5929 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.2183 ¢, ~28/25 = 193.6044 ¢

Optimal ET sequence: 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: [⟨2 0 0 -6 5 3], ⟨0 1 0 0 1 1], ⟨0 0 2 5 0 1]]

Optimal tunings:

  • WE: ~17/12 = 600.0873 ¢, ~3/2 = 702.2450 ¢, ~19/17 = 193.5751 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.2791 ¢, ~19/17 = 193.5923 ¢

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

Badness (Sintel): 0.722

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